Thermal Damage and Rate Processes in Biologic Tissues

Thomsen, Sharon; Pearce, John A.

doi:10.1007/978-90-481-8831-4_13

Sharon Thomsen³ &
John A. Pearce⁴

5803 Accesses
19 Citations

Abstract

Heat is generated in laser irradiated tissues by absorption and transformation of the light energy into heat. Once generated within tissues, heat is heat no matter what original energy source is used to produce it. Heating of cells and biological tissues can produce reversible injury and dysfunction that can be repaired by innate cellular and host mechanisms. However, more severe, irreversible damage leads to death immediately (primary thermal effects) or after (delayed secondary effects) the heating event. Sometimes, the dividing line between reversible (repairable) and irreversible (lethal) damage in living, surviving tissues is not easily observed at the time of heating. Therefore, to determine accurately the extent of effective (killing) thermal treatment, the observer has to wait for all the moribund, injured cells to die and undergo post-mortem necrosis (two-four days) [1–6].

Access provided by Autonomous University of Puebla. Download chapter PDF

Variable heat shock response model for medical laser procedures

Article Open access 03 January 2019

An analytical solution of the bioheat model in a spherical tissue due to laser irradiation

Article 14 September 2019

Energy-Based Devices: Radiofrequency and High-Intensity Focused Ultrasound

Keywords

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction

Heat is generated in laser irradiated tissues by absorption and transformation of the light energy into heat. Once generated within tissues, heat is heat no matter what original energy source is used to produce it. Heating of cells and biological tissues can produce reversible injury and dysfunction that can be repaired by innate cellular and host mechanisms. However, more severe, irreversible damage leads to death immediately (primary thermal effects) or after (delayed secondary effects) the heating event. Sometimes, the dividing line between reversible (repairable) and irreversible (lethal) damage in living, surviving tissues is not easily observed at the time of heating. Therefore, to determine accurately the extent of effective (killing) thermal treatment, the observer has to wait for all the moribund, injured cells to die and undergo post-mortem necrosis (two-four days) [1–6]. In addition, low temperature heat can directly trigger cell death by apoptosis, a sequential series of complicated processes that requires energy, signal transduction, enzyme activation and time (hours to days) to develop [6–9].

The effects of cellular and tissue heating can be studied using both qualitative and quantitative gross and microscopic pathologic methods. Some of these effects are characteristic of direct thermal damage, but others are delayed non-specific responses and thus are attributable to thermal injury only when a history of thermal damage is provided.

In this chapter, the qualitative and quantitative pathologic markers of thermal damage will be identified, described and illustrated. In addition, experimental and computational methods will be described for estimating kinetic coefficients for those particular markers that develop as a result of physical and chemical reactions that can be described by first-order kinetics. The experimental results can be obtained under constant temperature or transient thermal history situations simulating continuous and pulsed laser irradiations. To be useful for the prediction of these temperature/time treatment dosimetries, the kinetic model must be coupled to quantitative pathological analyses of well-defined pathologic end points showing thermal effects. For many heat treatment clinical goals that involve cell killing, the more useful endpoints are the spatial boundaries of (1) cell and tissue death (effective treatment volume) and (2) full extent of the lethal and non-lethal thermal effects (total treatment volume). In the case of in vivo experiments with transient thermal histories, the experimental data yield a noisy kinetic plot; however, good estimates of the appropriate rate coefficients can be made using a relatively new method described in Section 13.3.

2 Pathophysiology and Pathogenesis of Photothermal Lesions

2.1 General Principles

2.1.1 Primary Thermal Injury

Pathologic mechanisms of primary thermal injury are those that produce acute structural and functional abnormalities in cells and tissues due to the direct physical interaction of the heat with cellular and tissue proteins, lipoproteins and water. These pathologic effects can be detected immediately, during, or just after the heating episode in either living tissues (in vivo), living cells in culture (in vitro), or non-living tissues or organs recently removed from animals (ex vivo). The most characteristic and reliable diagnostic histopathophysiologic markers (indicators) of thermal injury are those produced by primary mechanisms of pathologic effects (see Table 13.1).

Table 13.1 Primary thermal effects in non-viable and viable tissue targets: low to high temperature sequence

Full size table

2.1.2 Secondary (Delayed) Thermal Injury

Pathologic mechanisms of secondary (delayed) thermal injuries are based on delayed pathophysiologic cellular, tissue or host responses triggered by cellular and tissue injuries and/or death. These secondary pathologic responses can occur soon (several seconds, minutes or hours) or long (days, weeks or months) after the heating event. Therefore, the full story of the development, progression and resolution (healing) of photothermal lesions can only be studied in living and surviving cells, tissues and organisms (see Table 13.2). These delayed responses are not unique to thermal lesions but are characteristic of the cellular and host reactions to any chemical, mechanical or physical insult that results in reversible and irreversible (lethal) cellular and tissue injury [6].

Table 13.2 Secondary (delayed) thermal effects: pathologic responses to cell and tissue injury and death

Full size table

2.1.3 Identification of Thermal Lesions

The determination that a lesion is due to thermal injury requires (1) the observation of characteristic (diagnostic) pathologic markers of thermal damage (mainly primary effects) in the tissues, (2) having the history of a heating episode and/or (3) the documentation of the heating parameters (tissue temperature and time at temperature) to show that significant heat injury could have been produced in that tissue or organ. Satisfaction of at least two or all three of these criteria is necessary to establish the role of heat in the creation of the lesion.

2.1.4 Factors Associated with Thermal Lesion Size, Development and Progression

The extent and severity of the pathologic effects produced by the interactions of heat with biologic tissue are determined by (1) the geometry of the heat source volume and thermal gradients created in the tissue over time, (2) tissue optical and thermal properties before heating and their dynamic changes during and after heating, (3) tissue temperatures and time at temperature (thermal history) at any point along the thermal gradients, (4) target tissue composition and structure including protein, lipid and water content and their structural anatomy, (5) target tissue viability, (6) mechanisms of primary (direct) and secondary (delayed) thermal injury, and (7) the various cellular, tissue and host responses to the injury over time in surviving organisms. The signals for these responses involve the release of several soluble tissue cytokines, cytotactic factors and growth factors from (1) injured, dying and/or necrotic cells, (2) invading inflammatory cells, and (3) surrounding viable parenchymal and connective tissue cells. Usually, these factors are various polypeptides that can trigger (1) local vascular and cellular inflammatory events, (2) immune responses, and (3) control regional cell proliferation and protein synthesis over time [6, 10–13].

2.1.5 Cells: Basic Units of Life and Death

In general, when considering lethal thermal injury of tissues, it is good to remember that individual cells are the basic unit of life. All tissues and organs are a mixture of living cells residing in a more heat resistant, non-living extracellular matrix. The extracellular matrix is a structurally and functionally complex tangle of formed and unformed proteins such as collagens, elastins, glycoproteins, globular proteins and polypeptides suspended in a solution of water, sugars and salts. The more formed matrix elements such as the collagens, elastin and basement membranes are arranged into a scaffolding of support platforms and compartments to provide the functional organization for parenchymal cells and tissues such as glands, skin, nerves, muscle and bone. The matrix components are produced and maintained by living cells within the tissue including parenchymal cells (epithelial, muscle and nerve cells) and connective tissue cells such as fibroblasts, lipocytes and myofibrocytes [14–18].

At low temperatures and heating times, individual tissue cells can die, but the extracellular matrix remains intact leaving a preformed scaffolding to guide regenerating parenchymal cells into reconstituted, functioning structures such as glandular ducts and liver lobules. However, when higher temperatures and longer heating times are applied, then the scaffolding collapses and the whole necrotic tissue is replaced by scar tissue. Also, if the rate of formation of scar tissue is greater than that of parenchymal cell regeneration, the volume of the necrotic tissue will be filled by scar rather than regenerating parenchyma. These rates vary greatly among different species and among different tissues within the same species [19].

2.2 Zones of Thermal Damage: Gross and Histologic Pathologic Features

2.2.1 Heat Source Volumes and Thermal Gradients

The geometry of the heat source volume in laser irradiation is determined by the volume distribution of the absorbed light and the transformation of the absorbed light energy to heat [20, 21]. Initially, heat energy is confined to the light absorbing tissue volume. But, over time, temperature gradients between the heated source volume and the cooler peripheral tissues will be produced by the radial and axial transport of the heat energy away from the original heat source volume. Heat produces many pathological lesions within biologic tissues depending on the tissue temperature T(r, t) and time at temperature t(r, T) at any point along the gradients. These pathologic lesions will form concentric zones of thermal damage around the single volume heat source with the more severe effects found closer to the hot center and less severe in the cooler periphery (Fig. 13.1). Since many of these zones are separated by distinct boundaries, the pathologic effects of heating can not only be recognized and described (qualitative pathology), but also their boundaries can be measured and mapped (quantitative pathology). These data can be used to generate mathematical models of thermal effects in tissues as discussed in Section 13.3 [3, 10, 19, 22, 23].

2.2.2 Ex Vivo and In Vivo Heating: No Survival

Beginning at the hot center and progressing to the cooler periphery, the thermal damage zones include (1) tissue defects due to tissue ablation, (2) carbonization, (3) water vaporization, (4) thermal tissue and cellular protein coagulation or denaturation, and (5) cellular and intracellular organelle lipoprotein membrane rupture and collapse. All of these primary pathological effects reflect heating conditions sufficient to cause lethal thermal injury, that is, injury leading to cell death in living tissues and organs. All these pathologic zones can be created in recently extracted, fresh (ex vivo) and living (in vivo) tissues and organs.

2.2.3 In Vivo Heating with Short Term Survival

However, other pathologic lesions resulting from heating require that the host animals are alive, have an intact blood supply and blood pressure and survive at least for several seconds to a few hours after the heating event (Fig. 13.2). Thermal protein denaturation, cellular membrane damage and endothelial cell death by apoptosis produces blood vessel damage that results in a red thermal zone of vascular damage at the interface of the thermal lesion and the surrounding normal tissue.

Depending on the irradiation/heating conditions, the outer boundary of this red damage zone usually marks the outer boundary of cell/tissue death [2, 3, 19]. However, the repair mechanisms of some viable, yet heat-injured cells located at the cooler, outer edges of the red zone may remain intact. These individual cells can repair themselves and survive. Therefore, for heat treatments of invasive cancers, standard practice involves creating an additional therapeutic margin of 5–10 mm around the tumor to make sure that the thermal treatment volume encompasses as many invading cancer cells as possible.

2.2.4 In Vivo Heating with Prolonged Survival

Upon prolonged survival (hours to months), the host response to lethal thermal injury will involve a series of pathophysiological events that are triggered by any injurious insult to living cells and tissues, not just heat. Initially, these non-specific, general responses include post-mortem cell and tissue necrosis (Fig. 13.3). Then, resolution and repair of the tissue structural and functional void left by the necrotic process involves (1) organization, that is, clean up and removal of necrotic debris soon followed by (2) tissue repair that includes tissue regeneration and wound healing (scar formation) (see Figs. 13.6, 13.7, 13.8, and 13.9).

2.3 Mechanisms of Thermal Injury

2.3.1 Primary (Direct) Thermal Injury

The mechanisms of primary direct thermal injury of cells and tissues involve the numerous processes of tissue ablation, carbonization, water vaporization, thermal denaturation of proteins and cell membrane rupture (see Table 13.3). The first three mechanisms are very complicated physiochemical reactions that are not explained by simple first-order kinetics. On the other hand, heat-induced intracellular and extracellular protein molecular denaturation and cellular lipoprotein membrane dissociation “melting” and disruption are primary processes that have been shown to be first-order kinetic processes that can be described by the Arrhenius equation as will be discussed in Section 13.3.

Table 13.3 Treatment effects produced along heat gradients extending from hot center of single volume heat source to cooler periphery in vivo and in vitro heating

Full size table

2.3.2 Tissue Mass Ablation

For the purposes of this chapter, tissue ablation is defined as removal of tissue mass to create a space within the tissue. Functional ablation is the removal or modification of a tissue function due to actual removal of mass or lethal injury of tissue. “Ablation” as used in the cardiac clinical literature usually means functional ablation. For example, creation of a thermal coagulation lesion in heart muscle is done to “remove” a functional source of cardiac arrhythmias. The treatment endpoint is the “ablation” or disappearance of the arrhythmia as monitored by an electrocardiogram and the pathologic endpoint is a volume of thermally coagulated (dead) myocardium.

Lists of the mechanisms of laser-mediated tissue ablation can be arranged by temperature gradients beginning at high temperature mechanisms requiring huge amounts of heat energy delivered in very short pulses then proceeding to those associated with lower temperatures and longer irradiation times. These are (1) plasma formation, the ionization of molecules at the irradiated surface; (2) tissue vaporization, the phase changes of solid, organic tissue components to liquids and gases; (3) combustion, rapid oxidation of organic tissue components with production of flames and smoke comprised of various gases and very small carbonized particles and inorganic ashes; and (4) explosive tissue fragmentation due to rapid expansion and rupture of water vapor (steam) vacuoles in the heated tissues (“popcorn” effect).

Pathologic evaluation of tissue ablation involves observation and measurement of what is not there, that is, a hole or defect in the tissue at the irradiation site. Also, the defect has to be associated causally with a reasonable thermal history and/or surrounded by other tissue abnormalities that are characteristic of thermal damage and tissue ablation. Otherwise, the naïve (and not-so-naïve) observer could misidentify any old tissue defect as being the “laser photothermal cut” (S. Thomsen and J. Coad, personal experiences, 1979–2008). With proper orientation, measurements of the maximum dimensions the tissue defect and the surrounding zones of thermal tissue damage in the tissue can be done using gross and microscopic observations [19].

2.3.3 Carbon Formation and Carmelization of Tissue

Carbon formation is the heat-mediated reduction of tissue and cellular organic molecules to elemental carbon that forms a very thin black membrane (5–20 microns) covering the walls of the defect or the irradiated surface of the tissue (see Figs. 13.2 and 13.4). The clumped black carbon particles are easily seen and measured in properly stained microscopic tissue sections. Carmelization is the melting, partial reduction and resolidification of tissue sugars. It occurs at slightly lower temperatures than carbonization but, in light microscopic histological sections, is frequently overshadowed by carbonization and staining artifacts induced by tissue desiccation. Carbonization is typically associated with local temperatures in excess of 200°C.

2.3.4 Water Vaporization

Water vaporization, vapor diffusion and tissue desiccation appear as the tissue temperatures approaches 100°C at standard pressure (Fig. 13.4). When generation of water vapor is faster than its rate of diffusion out of the tissue, the vapor can be trapped within the tissue, forming steam vacuoles. If the incident laser irradiation energy continues to be converted to heat energy, pressure rapidly increases, the steam expands within the vacuoles stretching their now desiccated, brittle, thin walls until they rupture either creating bigger vacuoles below the surface or explosions throwing tissue fragments of the vacuole wall from the surface (“popcorn effect”) [20, 24–26].

The temperature history of the “popcorn effect” and explosive vacuole rupture at the tissue surface can be followed using a calibrated thermal camera and a microphone (or ears of the investigators). As irradiation is delivered into the tissues, the surface temperature rises (sometimes up to 130–160°C), the water vapor expands within the vacuoles until they rupture with a loud “pop.” The “pop” marks the rapid escape of the steam into the atmosphere and a marked temperature drop to around 100°C, the temperature of water vapor in the remaining tissues. Depending on the tissue temperatures, tissue water content and heating times, a band-like zone of variably sized, intact and ruptured vacuoles can be seen just below the carbon layer in histologic sections [27]. This zone tends to have distinct and measurable borders with the adjacent thermally coagulated (denatured) tissues when viewed with a light microscope and represents a major zone of water-mediated thermal effects in tissues (Fig. 13.4). Although water vaporization is a complicated thermodynamic process, the pathological effects of tissue vacuole formation, expansion, vacuole size and tissue desiccation can be modeled and incorporated into mathematical models of thermal damage in biological tissues [28].

2.3.5 Thermal Protein Denaturation (Thermal Protein Coagulation)

2.3.5.1 General Comments About Tissue Proteins

Proteins are ubiquitous tissue components that include a myriad of pure proteins, lipoproteins and glycoproteins. These molecules form solids, gels, membranes and dissolved materials within (intracellular) and outside (extracellular) the cells that comprise any tissue [15–18, 29, 30]. Cellular enzymes are proteins that act as catalysts needed for proper, coordinated cell functions required for life. They are critical components of systems responsible for (1) energy production, (2) protein, carbohydrate, lipid and DNA/RNA synthesis and degradation, (3) cell repair, (4) transport of nutrients, water, salts, metabolic wastes and secretory products across cell membranes and (5) signal transduction across membranes inside and outside cells.

Proteins also form the important structural components within the cell which are the intracellular cytoplasmic and nuclear skeletal proteins and the cell motility proteins [17, 31, 32]. The intracellular structural and motility proteins are necessary for (1) organelle and cell support, (2) intracellular and transmembrane material transport, (3) cellular movement and (4) reproduction and regeneration with formation of the mitotic spindle for chromosome sorting and movement and the mechanisms for cell division.

The extracellular structural proteins form intricate scaffolding networks and environments necessary for integration of cells into tissues and for coordinated structural and functional organization of tissues to form organs and the whole organism. The most prevalent extracellular structural proteins are (1) the formed elements, collagens, elastins, fibronectin and neutral, basic and acidic aminoglycans and (2) the various amorphous proteins, aminoglycans and polypetides including mucins, globular carrier proteins, and other cell secretions and products. These elements are mixed with water, salts, sugars and polypeptides to form the semi solid gel in which cells reside, function and travel [15, 16, 18, 33–36].

2.3.6 Thermal Denaturation of Proteins

The direct, primary thermal effect on the intracellular and extracellular proteins, glycoproteins and lipoproteins of cells and tissues is thermal denaturation of these macromolecules [37–40]. Thermal denaturation is the accumulated structural and functional change in heated protein that result from the physical and chemical molecular disequilibrium and phase changes introduced initially by heating and subsequent cooling. Usually, thermally denaturized macromolecules lose their characteristic quaternary, tertiary and secondary molecular configurations due to disruptions of hydrogen and ionic bonds as well as sulfide and other covalent bonds formed among primary amino acids chains, the building blocks of proteins. In the denaturation process, the molecules are reduced from highly organized, characteristic structures with specific functions to amorphous granules of collapsed and ruptured primary chains with no biological function (see Fig. 13.5). Accumulation of these highly scattering granules is responsible for the generation of the usually opaque, firm white or light tan volumes observed in heated target tissues. The transformation of fried egg whites from clear, viscous fluid rich with the protein, albumin, to a white, opaque solid is the classic example of thermal denaturization of protein [40]. The same denaturation processes are responsible for the loss of native birefringence of Type I collagen and the motor proteins, actin and myosin, in the sarcomeres of skeletal and smooth muscle cells, as discussed in Section 13.3 [41–48].

On the other hand, molecular protein denaturation and dehydration can be associated with transparent clearing of highly scattering, opaque white collagen-rich (Type I collagen) tissues such as skin dermis or muscle tendon [49]. Unfortunately, the temperatures/time at temperature thresholds of thermal clearing are much higher than those associated with cell killing; therefore heat cannot be used to “make skin transparent” since second and third degree burns would be produced.

2.3.7 Heating of Cellular Membranes

Cells are membrane-bound bags of proteins, carbohydrates, lipids, water and salts in which are suspended smaller, membrane-bound organelles such as nuclei, endoplasmic reticulum, mitochondria, the Golgi apparatus, secretory vacuoles and phagosomes [15–18, 32, 50]. The organelles compartmentalize, organize and separate different cellular metabolic functions, including energy generation (mitochondria), synthesis of DNA and mRNA (nuclei) and proteins (endoplasmic reticulum), packaging (Golgi apparatus) and storage of secretory products (secretory vacuoles) and degradation products (secondary lysosomes).

Cell membranes are composed of phospholipids oriented into a polar bilayer so that the hypdrophobic portions (fatty acid chains) of the molecule form the inner layers of the membrane and the hydrophilic portions (phosphate or organic esters) form the outer layers to face the watery environments of the extracellular and intracellular spaces. Embedded within and on both surfaces of the cellular membranes are important protein and glycoprotein structures including specific receptor proteins, water pores, ion channels and enzymes. These proteins are important for transport of nutrients, metabolites, salts and water and signal transduction (stimulating and inhibiting) across the membranes.

When heated above physiological temperatures, the hydrogen bonds between the hydrophobic lipids become weaker and the individual phosholipid molecules separate creating defects in the membranes. The water and solutes within and outside the membrane barriers can leak in and out thus disrupting the intracellular and extracellular functional equilibria of cells and the surrounding the environments. If the heating continues and/or the temperatures increase, the defects enlarge, overwhelming the usual repair mechanisms, and the cell dies.

2.4 Mechanisms of Heat-Induced Cell Death

2.4.1 “Heat Sensitive” and “Heat Resistant” Cells and Tissues

Thermal denaturation of intracellular and extracellular proteins and molecular dissociation (melting) of cellular membrane lipids result in changes of molecular structure and loss of cellular function that, if sufficiently severe and/or prolonged, will produce lethal thermal damage and cell death. The tissue temperatures and times at temperature at which these changes occur differ among proteins, lipoproteins, glycoproteins, cell membranes and tissues. Some tissue components are more sensitive to heat elevation than others and denature or dissociate at lower temperatures, thus are considered relatively “heat sensitive.” Those proteins, glycoproteins and lipoproteins that denature at higher temperatures are considered “heat resistant.” A very general order of the range of more heat sensitive to the more heat resistant tissue proteins would be the following: (1) respiratory, energy-generating enzymes of the mitochondria, followed by the (2) intra- and extra-cellular non-structural tissue proteins, glycoproteins and lipoproteins, and lastly, the (3) intra- and extra-cellular structural tissue proteins. Cell membranes dissociate at low temperatures; therefore, the lipoprotein membranes are more heat sensitive than other cellular proteins [50–52].

2.4.2 Minimal Thermal Injury that Leads to Cell Death

To produce the high-energy molecule, adenosine triphosphate (ATP), the respiratory enzymes have to be arranged in a specific order on the inner membrane of the mitochondrion [7, 16, 17]. Therefore, the critical minimal lethal thermal injury to a cell is the rupture and fragmentation of the mitochondrial membrane along with the thermal denaturation of the respiratory enzymes sufficient to produce cell death. However, if the resultant membrane defects are small, they can be repaired and, if few enzymes are denatured, they can be replaced on the membrane by synthesis of new ones. Thus, if sufficiently small, a thermal injury can be considered to be reversible and the cell will survive the injury.

Cell death produced by lethal thermal injury can be due to (1) abrupt physical disruption of cell structure and function and structure by “heat fixation” of proteins in situ, a primary thermal effect, (2) gradual loss of vital cellular functions including disruption and loss of energy production, gradual loss of repair enzyme function and gradual loss of membrane integrity and function, and (3) induction of apoptosis as discussed below. The damage involving gradual loss of vital cellular functions produces severely injured (moribund) cells that can continue to have regions of cytoplasm and nucleoplasm that function for a while, but not at full capacity. These injuries will eventually lead to cellular system failure and death. Therefore, some tests for viability such as cellular dye exclusion and vital stains that rely on the oxidation/reductions functions of certain mitochondrial respiratory enzymes can indicate false “viability” for a time in injured and dying cells. The enzymes may be working for the time being, but the cells and tissues are moribund and eventually will die. Depending on the tissue, organ and animal species, cell and tissue death will occur immediately or within hours, but may take a day or two to become fully manifest by the development of post-mortem necrosis.

The events of isolated cell death by pure apoptotic mechanisms do not lead to post-mortem necrosis [6]. However, in real life and death situations, both post-mortem necrosis and apoptosis can occur in the same tissues and, indeed, may share similar triggering mechanisms [7, 53–56].

The exact temperature/time thresholds for cell death have to be determined for each cell and tissue type. However, some cells are more heat sensitive than others. In general, cancer cells are killed more easily than their normal counterparts. The basic theory of clinical low temperature hyperthermic cancer treatments exploit this temperature/time differential to kill off the cancer cells while sparing the normal ones [57, 58].

2.4.3 Post-Mortem Necrosis

Necrosis, by definition, occurs after cell (and tissue) death and involves the degeneration and disintegration of cells and tissues after death [6]. In laymen’s terms, necrotic tissues and organs are “rotten” tissues and organs. There are generally two histopathologic types of post-mortem necrosis: lytic necrosis and coagulative necrosis; and, frequently, they are intermixed in the same dead tissue. Lytic necrosis, the chemical fragmentation (lysis) of cells, cellular membranes and organelles and extracellular matrix, is due to the extensive and overwhelming effects of proteolytic and lipolytic enzymes [6, 59]. These are enzymes that have either leaked from the ruptured lysosomes of the dead or dying cells themselves or have been secreted by infiltrating inflammatory cells (polymorphonuclear cells). The second type of post-mortem necrosis is coagulative necrosis, the gradual degeneration and collapse of cells and cellular organelles [6]. This necrosis is usually associated with loss of energy production by direct mitochondrial dysfunction or by lack of oxygen delivery to the tissue by blocked blood flow (ischemia). Usually, coagulative necrosis occurs in cells that contain relatively few lysosomes and relatively large numbers of mitochondria such as striated muscle cells of the heart and muscloskeletal system. The two types of post-mortem necrosis occur within thermal lesions with the zone of lytic necrosis being more peripheral to the zone of coagulative post-mortem necrosis since the more peripheral zones receive the infiltrating inflammatory cells with their lytic enzymes first [10, 23] (see Fig. 13.6).

Coagulative post-mortem necrosis is associated with the loss of energy production. The major source of metabolic energy generation occurs in the mitochondria where high energy phosphate bonds of adenosine triphosphate (ATP) are created by chemical reactions catalyzed by the respiratory enzymes. These protein enzymes are found on the inside surface of the folded inner membrane of the mitochondrion arranged in a regular sequential order that allows the efficient functioning of the chain oxidative/reduction reactions that produce ATP [6, 53–56]. Heat-induced mitochondrial membrane rupture will disrupt this highly efficient order and, along with the thermal denaturation of the respiratory enzymes, will lead to insufficient energy production and cell death to be followed by necrosis. However, loss of oxygen due to stoppage of blood flow in the red thermal damage zone can also be an important factor in the generation of coagulative post-mortem necrosis due to ishchemia.

2.4.4 Cell Death Due to Rapid Thermal Coagulation (Heat Fixation)

Rapid elevations of tissue temperature in a short time can produce sudden death by immediate thermal coagulation and precipitation of cellular and extracellular proteins in situ (“heat fixation”). Concomitant extensive membrane rupture leads to immediate membrane depolarization and leakage of water, salts and small molecules out of the cells and their organelles, but their denatured protein contents stay in place. After this catastrophic death, none of the cells are alive, none of their intrinsic enzymes are functional and the heat fixed cells and extracellular matrix will not undergo post-mortem necrosis. Heat fixed tissues are found in the hottest portion of the thermal lesion which usually is the center of the lesion. Because heat fixed tissues resemble “living tissues” in routinely-prepared and stained histopathologic sections viewed with a light microscope, numerous investigators (including some pathologists not aware of this phenomenon) have mistakenly reported the centers of thermal lesions as “alive” (see Figs. 13.6 and 13.7).

Eventually, small volumes of heat fixed dead tissue will be liquefied, organized and removed by inflammatory cells directly invading from viable tissue found at the periphery of the thermal lesion or from blood flowing in open lumens of necrotic blood vessels or in new blood vessels formed in wound granulation tissue. However, if the heat fixed volume is sufficiently large and communication cannot be established with the surrounding viable tissue, the heat fixed tissues will slowly fade away into granules, lose water and frequently will undergo calcification (dystrophic calcification) forming a hard mass surrounded by fibrous scar tissue [10].

2.4.5 Heat Fixation Vs. Pathophysiologic Post-Mortem Coagulation Necrosis: A Diagnostic Conundrum

In photothermal lesions, coagulative post mortem necrosis is to be distinguished from heat fixation which is the immediate (ms to s) result of direct thermal denaturation (thermal coagulation) and fixation of tissue proteins, glycoproteins and lipoproteins in situ. Heat fixed tissues tend to be in the hottest center of the thermal lesion while the post-mortem coagulative necrosis zone is more peripheral [10, 60]. Heat fixed tissues retain patterns of their cellular structures longer than tissues undergoing post mortem necrosis. Heat fixed cells retain distinct differential staining of the cytoplasmic proteins and nuclear DNA for days to weeks while in most cases of post mortem coagulative necrosis, the nuclear DNA disintegrates and disappears within a week. Ultimately, small volumes of heat fixed tissues will undergo lytic necrosis and organization by inflammatory cells invading from the peripheral viable tissues. However, large volumes of heat fixed tissues in some organs will never be penetrated by inflammatory cells or blood vessels and will never undergo lysis or organization. These volumes will be isolated by scar tissue and persist for several months to years [10].

2.4.6 Programmed Cell Death (Apoptosis)

The most critical eukaryotic cell organelles in terms of cell life and death are the mitochondria, intracytoplasmic organelles bound by two membranes that are considered to be the main arbiters of cell life and death [7–9, 53–55, 61]. Mitochondria are not only the sites of the energy production required to sustain life but also they contain the receptors and reactants required for apoptosis. Apoptosis is a physiologic mechanism of cell death in which a series of signal transductions at the mitochondrial or cell membranes initiate an activation cascade of specific mitochondrial and cytoplasmic proteins. These proteins or their precursors have to be in place ready to trigger the energy-requiring chemical reactions that produce characteristic cellular morphologic and functional alterations over time (hours to 2–3 days) that terminate in fragmentation of the cells with the formation of apoptotic bodies, small bits of cell cytoplasm and nucleus. These apoptotic bodies disappear within a few hours after their formation by being ingested (by phagocytosis) into adjacent living cells where the bodies are digested within secondary lysosomes. Theoretically, no cellular contents are released from the cells dying by apoptosis; therefore there are no cytokines released to promote cellular or vascular inflammatory responses by the host. Some cells are genetically programmed to undergo apoptosis more readily than others. These include endothelial cells, lymphocytes, cells of embryologic tissues and some neoplastic cells. To make things even more confusing, the signals that initiate apoptosis can also activate mechanisms of cell death followed by necrosis. Most thermal lesions contain morphological and chemical markers of both death processes.

2.5 Wound Healing

Now that the different mechanisms of cell death and tissue necrosis have been discussed, the next steps the host will initiate will be the resolution of the pathologic effects of the injury, that is, the processes and events of wound healing. These include the activation of the mechanisms of wound healing including (1) organization (2) tissue regeneration, (3) repair and (4) scar tissue formation [6, 62–65].

2.5.1 Organization

Inflammatory cells enter the necrotic volumes upon the resumption of blood flow in damaged yet still open blood vessels or by migration from the surrounding viable tissues beginning about 8–12 h after cell and tissue death. Certain inflammatory cells, the macrophages, function as garbage collectors to clean up (organize) the necrotic debris in preparation for the later stages of wound healing: tissue regeneration, repair and scar formation. Depending on the species, organ and tissue, the organization will begin at about 2–7 days after injury. The debris-collecting macrophages phagocytose (“ingest”) the necrotic debris and carry it around in their cytoplasm in specialized intracellular membrane-bound organelles called secondary lysosomes (phagosomes). The “loaded” macrophages then (1) are incorporated into the wound healing tissues and remain in the local tissues, (2) migrate into the lymphatic vascular channels to land and accumulate in lymph nodes (regional “garbage dumps”), or (3) migrate back into blood vessels to be delivered, metabolized, excreted and/or stored in more distant tissues.

2.5.2 Tissue Regeneration and Tissue Repair

Some parenchymal cells and tissues such as liver cells, the epidermis of the skin and the glandular epithelium of gastrointestinal tract and breast among others are capable of regeneration after necrosis (see Fig. 13.8). As long as the usual tissue connective architecture is intact, the regenerating cells will be guided into the connective tissue scaffolding to form normally functional tissue structures such as liver lobules and digestive mucosal glands. The new functional cells are derived from immature, precursor cells in the surrounding viable tissues. These precursor cells are stimulated to proliferate, migrate and mature in response to numerous growth factors produced by the residual, living connective tissue and inflammatory cells at the periphery of the lesion. However, if the necrotic connective tissue scaffolding collapses, the proliferating parenchymal cells pile up into dysfunctional, disorganized strands or nodules buried in scar tissue. Some adult parenchymal tissues, such as nerve cells of the brain and cardiac muscle cells, do not have readily available, dedicated precursor cells (or very few stem cells that are not easily stimulated to grow and mature) and, in general, are not capable of tissue regeneration. Usually, large volume parenchymal cell regeneration does not occur after large volume necrosis and the tissue repair is mainly by fibrous scar formation to fill in the void.

2.5.3 Granulation Tissue and Scar Formation

Whenever a defect is created by the organization of the necrotic tissues, the void is filled with either regenerating tissues or vascular and fibrous granulation tissue (early wound healing tissues) or a mixture of both (see Fig. 13.9). These granulation repair tissues are the earliest forms of scar tissue and will appear in the wound anywhere from 3–5 to 10 days, again, depending on the species and the tissue type within a species. Over the next days to months the scar tissue will be remodeled (regression of the vascular components and thickening of the collagen fibers) and the resultant maturing fibrous scar will contract and become smaller than the original thermal lesion.

2.6 Role of Heat Shock Proteins (HSP) in Thermal Lesions

HSP include a group of intracellular proteins that, among many other functions, facilitate the assembly of component peptides into the final configuration of functioning proteins [6, 17, 29, 61]. To initiate heat shock protein synthesis, the cell cultures or tissues are exposed to low temperatures (or other low level injury) for relatively short times. This heating can stimulate specific DNA expression and RNA production for heat shock protein synthesis while not killing the cells by direct injury or by triggering apoptosis. Once the DNA expression/RNA molecules are synthesized, they can remain active for some days so that, if the same cultures or tissues are reheated, the heat shock protein synthetic apparatus is in place and ready to produce new facilitators of enzymatic or structural protein repair. Thus, the cultures/tissues “preconditioned” by the initial heating are more “heat resistant” or protected from thermal injury from subsequent heating episodes because they can institute repair processes more readily than “unconditioned” cultures/tissues.

2.7 Mathematical Modeling of Pathophysiologic Thermal Effects

As discussed in great detail in the following Section 13.3, mathematical models estimating pathophysiologic markers (pathologic end points) of thermal damage in cells and tissues that follow first order kinetics can be developed. These can be used to describe the temperature/time thermal history, geometry and extent of clinically significant photothermal effects in laser irradiated cells, tissues and organs. Most of these markers or end points are directly related to primary thermal effects such as protein denaturation and cellular membrane rupture (“melting”). They are created at relatively low temperatures (<100°C) but at variable heating times. And, because some of these pathologic markers form distinct lesions with measurable boundaries, they can be described by rate process models whose kinetic coefficients are experimentally derived. Each end point has a unique set of coefficients; therefore, the end points have to be precisely defined to determine the rate coefficients specific for that particular effect. For example, given equal heating times, the rate coefficients obtained for collagen hyalinization, a protein denaturation process that requires higher temperatures, will be different than the coefficients for cell death due membrane rupture, a lower temperature effect.

3 Thermodynamics and Kinetics of Thermal Damage Processes

The thermal aspects of laser-tissue interaction consist of two governing events: (1) the thermodynamics of temperature elevation, and (2) the kinetics of resulting irreversible thermal alterations in tissues. Kinetic and thermodynamic analyses are very different in style in classical treatments. Kinetic analysis concerns the rate at which processes occur; as, for example, reaction rates in chemical transformation of materials. Thermodynamic analysis is usually concerned only with the thermodynamic state change during a process – that is, the rate at which the change takes place is normally not of interest, only the initial and final values of the thermodynamic state variables. There are six thermodynamic state variables, any two of which fully specify the thermodynamic state – i.e. if any two of them are known, the other four may be found uniquely. Consequently, the several state variables comprise a redundant set. Nevertheless, they each provide a somewhat different view of the thermodynamic state, and are thus useful in turn for studying state changes in different ways. The six thermodynamic variables are: temperature, pressure, density, density’s inverse specific volume, enthalpy and entropy.

3.1 Foundations of Thermal Analysis

The most fundamental of the thermodynamic state variables is temperature; so much so that a whole chapter in this book has been dedicated to its measurement. Two objects are in thermal equilibrium if they are at the same temperature – that is, there will not be a net exchange of heat if the objects are placed in contact with each other. Heat is “energy in transit.” The so-called “Zeroth Law of Thermodynamics” states that: “if object A is in thermal equilibrium with object B, and object C is also in thermal equilibrium with object B, then object A and object C are in thermal equilibrium with each other.”

In the case of laser-tissue interaction, we are interested in the rate of rise of temperature, so the approach in this chapter will be a slightly modified form of thermodynamic analysis: modified to include transient solutions. The thermodynamic processes will be taken to occur under equilibrium or near-equilibrium conditions – that is, state changes occur slowly enough that all thermodynamic variables have the same inter-relationship they would if the rate of change were slowed down even further. Thermal damage kinetics are decidedly non-equilibrium in nature with their rate of change determined by local temperature.

3.1.1 Thermal Energy and Enthalpy

Absorption of laser energy results in an increase in the internal thermal energy, U [J] or u [J · kg⁻¹], of the tissue:

$$U = mcT,u = cT$$

((13.1))

where $m = {\textrm{mass }}[{\textrm{kg}}]$, $c = {\textrm{specific heat }}[{\textrm{J}} \cdot {\textrm{kg}}^{ - 1} \cdot {\textrm{K}}^{ - 1} ]$ and T = temperature [K or °C]. This concept is here formulated in terms of a finite “control volume” of material that consists of enough molecules and/or atoms that a spatial and temporal mean value makes sense. Internal thermal energy in this form describes the mean value of the thermal energies of all of the constituent particles in the control volume at the time it is evaluated. Equation (13.1) is not descriptive in a control volume containing only a very few particles that are widely dispersed with an extremely low probability (i.e. rate) of collision with other particles. A statistical (i.e. quantum mechanical) formulation is required to describe those cases. Our case is for aqueous solutions / suspensions, and tissues where the mean time between collisions is vanishingly small; and in the case under study such mean descriptions are extremely useful.

Internal thermal energy is a form of “potential,” similar to gravitational potential energy or to electrical potential, so it needs a zero reference point. Thermodynamic tables typically assume zero internal thermal energy at a convenient reference temperature. For example, in the classic “Steam Tables” [66] the reference temperature for the internal thermal energy of liquid phase water is 0°C (273.16 K), and the same reference point for water is used for the enthalpy and the entropy, which remain to be discussed. For most refrigerants (freon, etc.) the reference point is often taken to be −40°C (which just happens to be −40°F as well), the spontaneous freezing point for liquid phase water.

If mass crosses the control volume boundary and/or the pressure or density changes inside the control volume, the total stored (i.e. “available”) thermal energy must include “Flow Work” (the pressure-volume product). A more useful description of the available thermal energy is the “enthalpy,” H (J) or “specific enthalpy,” h [J·kg⁻¹]:

$$H = U + PV;\ h = u + Pv$$

((13.2))

where $P {=} {\textrm{pressure }}[{\textrm{Pa}} {=} {\textrm{N}}\cdot{\textrm{m}}^{ - 2} ]$ and ${{V}} {=} {\textrm{volume }}[{\textrm{m}}^3 ]$, $v {=} {\textrm{specific volume}}\break [{\textrm{m}}^3 \cdot {\textrm{kg}}^{-1}] = 1/\rho$, where $\rho = {\textrm{density }}[{\textrm{kg}} \cdot {\textrm{m}}^{ - 3} ]$. In studying the units of the equation here, remember that 1 [J] = 1 [N·m]. Enthalpy may properly be thought of as the total thermal energy stored in the volume that is available to do useful work – i.e. it describes the control volume as a thermal engine.

Example 13.1:

One gram of incompressible tissue absorbs a total of 2 W for 30 s. If the initial temperature is 37 C, calculate the final temperature after 30 s of absorption. Assume that the effective tissue density is 1050 (kg·m⁻³) and specific heat is 4050 (J·kg⁻¹). No mass crosses the tissue boundaries in this example.

Solution:

All of the energy absorbed goes into raising the temperature − Δv, Δm = 0:

$$\begin{aligned}[rl]\Delta U\left( {\textrm{J}} \right)\, &= \,\dot Q_{{\textrm{gen}}} \left( {{\textrm{J}}\,\cdot\,s^{ - 1} } \right)\Delta t\, = \,mc\Delta T \\ \Delta U\, &= \,\left[ 2 \right]\,\left[ {30} \right]\, = \,\left[ {10^{ - 3} } \right]\,\left[ {4050} \right]\Delta T \\ \Delta T\, &= \,\,\frac{{60}}{{4.05}}\,\, = \,\,14.8\,\left( {^ \circ {\textrm{C}}} \right)\end{aligned}$$

When mass crosses the volume boundary, it imports thermal energy according to the net mass inflow rate, and loses thermal energy proportional to the net mass outflow, or efflux:

$$\frac{{\Delta H}}{{\Delta t}} = \frac{{\Delta m_{{\textrm{in}}} }}{{\Delta t}}c_{{\textrm{in}}} T_{{\textrm{in}}} - \frac{{\Delta m_{{\textrm{out}}} }}{{\Delta t}}c_{{\textrm{out}}} T$$

((13.3))

where m _in and m _out refer to the mass inflow and outflow, respectively [kg], c _in and c _out are the respective specific heats $[{\textrm{J}}\cdot{\textrm{kg}}^{-1}\cdot{\textrm{K}}^{ - 1} ]$ and the rate of change in enthalpy is in units of watts $[{\textrm{W}} = {\textrm{J}}\cdot{\textrm{s}}^{-1} ]$. In this formulation we have included only the internal thermal energy contribution, such as would be due to tissue blood perfusion and lymphatic flow. A potential contribution to the total thermal energy, the enthalpy, due to density or pressure change has not been included.

Example 13.2:

For the tissue in Example 13.1, assume that a net blood flow of [0.1g·s⁻¹] crosses the boundaries at an inlet temperature of 37°C. Inlet and outlet blood flow are equal in the tissue. Assume that the blood specific heat is the same as water, 4,186 [J·kg⁻¹]. Calculate the tissue temperature at 30 s, as before.

Solution:

(1)
We will have to make some assumption about the outlet temperature of the blood. The problem has not been specific about how long the blood remains resident in the tissue. However, the reasonable assumption is that the blood leaves the tissue in thermal equilibrium with it (this is equivalent to assuming that the blood circulation is in “thermally small” vessels).
(2)
The temperature rise with no blood flow is substantial, 14.8°C; so, while an accurate solution to this problem requires integrating the transient heat transfer case, we can get a working estimate with a simplified thermodynamic analysis.
$$\begin{aligned}&\Delta U\,mc \left[ {T\, - \,T_0 } \right]\, - \,\dot m_b c_b \left[ {T\, - \,T_0 } \right]\Delta t\, = \,\dot Q_{{\textrm{gen}}} \Delta t \\ &\Delta U\, = \,[1 \times 10^{ - 3} ]\,[4050]\,[T\, - \,37]\,\\ & \qquad\quad + \,[0.1 \times 10^{ - 3} ]\,[4186]\,[T\, - \,37]\,[30]\, = \,[2]\,[30] \\ &\Delta T\left[ {4.05\, + \,3.0 \times 4.186} \right]\, = \,60\left( {\textrm{J}} \right) \\ &\Delta T = \frac{{60}}{{16.6}} = 3.61\;\left( {^\circ {\textrm{C}}} \right)\end{aligned}$$
(3)
Note the strong effect of the addition of blood flow. (See Fig. 13.2. The regional blood flow was stopped by the initial lesion, the thermal coagulation lesion. Therefore, the adjacent tissue was not cooled during the laser irradiation that resulted in more severe damage, the thermal ablation lesion.)

The effect of heat transfer within the volume is to distribute the internal thermal energy uniformly. Of course, in spatially-distributed laser irradiation patterns, adjacent volumes of tissue will have differing temperatures. Consequently, conduction heat transfer can be a dominant thermal phenomenon, especially in long-term low fluence rate laser activation. Conduction heat transfer is a boundary (surface area) phenomenon:

$$\dot {\textit{\textbf q}}_{{\textrm{cond}}} \, = \, - k\frac{{\partial T}}{{\partial n}}{\qquad}\dot {\textit{\textbf q}}_{{\textrm{cond}}} \, = \, - k \nabla T$$

((13.4))

where $\dot q_{{\textrm{cond}}}{\textrm{ = convection heat flux [W}}\cdot{\textrm{m}}^{ - {\textrm{2}}} {\textrm{]}}$ in scalar form, $k = {\textrm{thermal}}\break {\textrm{conductivity }}[{\textrm{W}}\cdot{\textrm{m}}^{ - 1}\cdot{\textrm{K}}^{ - 1} ]$ and n is the normal direction for the surface over which the heat transfer is calculated; $\dot {\textit{\textbf q}}_{{\textrm{cond}}}$ is the vector heat flux, and ∇T is the temperature gradient (the gradient of a scalar field is a vector). The negative sign reflects the direction of conduction heat transfer: from higher temperature to lower temperature, i.e. anti-parallel to the temperature gradient (a vector).

Similarly, if there is heat transfer by convection from an exposed control volume surface, it may be found from:

$$\dot q_{{\textrm{cond}}} \, = \,h\left[ {T\, - \,T_{{\textrm{atm}}} } \right]$$

((13.5))

where $\dot q_{{\textrm{cond}}} {\textrm{ = convection heat flux [W}} \cdot {\textrm{m}}^{ - {\textrm{2}}} {\textrm{]}}$, $h = {\textrm{convection}}\ {\textrm{heat}}\ {\textrm{transfer}}\break {\textrm{coefficient [W}} \cdot {\textrm{m}}^{ - 2} \cdot {\textrm{K}}^{ - 1} {\textrm{]}}$, and T _atm is the atmospheric, or surrounding fluid, temperature [K]. There is also the possibility of thermal radiation from the surface. The direction of convection and radiation heat transfer is normal to the surface (n), so the appropriate surface boundary condition is:

$${\textbf{n}} \cdot \left( {k\nabla T} \right) = h\left[ {T - T_{{\textrm{atm}}} } \right] + \sigma _B \;\varepsilon \left[ {T^4 - T_{{\textrm{atm}}}^4 } \right]$$

((13.6))

where we have now included the contribution of thermal radiation from the surface, as described in Chapter 11 (i.e. $\sigma _B = {\textrm{the Stefan - Boltzmann constant [W}} \cdot {\textrm{m}}^{ - 2} \cdot {\textrm{K}}^{ - 4} {\textrm{]}}$, and ɛ = the surface emissivity, and T is in [K]).

Convection is a special case of conduction heat transfer, where the medium is in motion. Of course, estimation of the convection coefficient becomes quite complex when the fluid near the surface is compressible since the nature of the fluid flow boundary layer (turbulent or laminar) has a very strong influence on the effective convection coefficient. Also, proper formulation of the radiation boundary conditions is even more complex, depending as it does on the shape and coupling factors of surrounding thermal radiation objects, and their temperatures.

Example 13.3:

For the tissue under the same conditions as in Example 13.1, assume that the tissue volume is contained in a 1 cm cube. Assume that the tissue volume is at the surface, and has 5 neighbor tissue volumes below and around it. The thermal conductivity is 0.46 [W m⁻¹ K⁻¹]. Calculate the net conduction heat transfer out of the tissue volume at the end of the activation (30 s) if the final temperature in the 5 adjacent tissue volumes has risen to half of the temperature in this control volume (i.e. $37 + 14.8 = 51.8^ \circ {\textrm{C}}$). Assume that the temperatures represent the values at the centers of the respective control volumes. Neglect inlet and outlet blood flow in the tissue.

Solution:

$$\dot q\left( {{\textrm{W}}\, \cdot \,{\textrm{m}}^{ - 2} } \right)\, = \,k\frac{{\Delta T}}{{\Delta x}}$$

(1)
Each adjacent control volume (CV) has half the temperature rise of this volume, i.e. 7.4°C. The temperature difference between the adjacent CVs is therefore 7.4°C.
(2)
For volumes 1 cm on a side, the distance between adjacent centers is also 1 cm. This means that the temperature gradient is $7.4\,(^ \circ {\textrm{C}} \cdot {\textrm{cm}}^{ - 1} ) = 0.074\,(^ \circ {\textrm{C}} \cdot {\textrm{m}}^{ - 1} )$.
(3)
For the 5 sides the net conduction heat transfer out of the CV is:
$$\dot Q\left( {\textrm{W}} \right)\, = \,[5]\,[10^{ - 4} {\textrm{m}}^2 ]\,[0.46]\,[0.074]\, = \,17\left( {{{\upmu W}}} \right)$$

Example 13.4:

For the same control volume (CV) as in Example 13.3, calculate the heat transfer by convection from the exposed surface. Assume that $h = 25\break({\textrm{W}} \cdot {\textrm{m}}^{ - 2} \cdot {\textrm{K}}^{ - 1} )$ and the room temperature is 23°C.

Solution:

$$\begin{array}{l} \dot q\left( {{\textrm{W}} \cdot {\textrm{m}}^{ - 2} } \right) = h\left[ {T - T_{{\textrm{atm}}} } \right] \\ \\ \dot q(W) = [1 \times 10^{ - 4} {\textrm{m}}^{\textrm{2}} ][25][51/8 - 23^ \circ {\textrm{C}}] = 72({\textrm{mW}}) \\ \end{array}$$

Much more significant than conduction heat transfer.

3.1.2 Entropy: The Second Law of Thermodynamics

The energy balance in the first law of thermodynamics (Chapter 10) represents the “break even” thermal point of view. It assumes that all processes are reversible and that all required energy can be accounted for in terms of the thermal energy. That is, there are no “preferred directions” for energy change or for material transformation. It would be equivalent to considering that all chemical reactions are equally likely to occur in either direction. We need some method for identifying preferred pathways for material transformations, so that we may describe changes that are “more likely,” and the concept of “entropy” fills that need.

3.1.2.1 Classical Thermodynamic Description

As a concept, entropy often gives trouble but it really shouldn’t. In perhaps its simplest form, the specific entropy, s (J·kg⁻¹·K⁻¹), is a measure of randomness or disorder in a system. It is easy to see that a crystalline solid is more ordered than the same substance in liquid or gas phase. As a substance transitions from solid to liquid to gas its entropy increases. It takes energy to create the orderliness of a crystal, so both the enthalpy and the entropy change as a crystal is formed from, say, the liquid phase. Similarly, in chemical reactions the reactants form products because that is more likely than having the products decompose into the reactant atoms or molecules. Hydrogen and oxygen mixtures combine explosively when stimulated to form water; water is an extremely stable molecule that only decomposes to hydrogen and oxygen when considerable energy is added to the system to favor that transition.

Thermodynamic systems (i.e. “heat engines” in the classical sense) are constrained to extract energy from a “thermal energy reservoir” and reject energy to a “thermal energy sink.” That is, there will always be “waste heat” for any thermal engine. This is one of the many statements of the “Second Law of Thermodynamics.” Others include: (1) the entropy of the universe is always increasing, (2) a perpetual motion machine of the second kind is impossible, and (3) any process that would decrease the entropy of a completely isolated system is impossible (a completely isolated system cannot exchange either heat or mass with the outside world). There are others, as well, but this is enough for illustration. All of them boil down to the simple truth that no thermal system can be 100% efficient.

Heat exchange, for example, is always entropy increasing. When heat, dQ [J], is extracted from a thermal energy reservoir at temperature T ₁, the entropy decreases by: $dS_1 = - dQ/T_1 \,[{\textrm{J}} \cdot {\textrm{K}}^{ - 1} ]$. The entropy gained by the extracting object at temperature T ₂ is: $dS_2 = + dQ/T_2$. For conduction heat transfer, it must always be true that $T_1 > T_2$, so the net entropy in the universe, $S_u = dS_1 + dS_2$, has increased because of the heat transfer. This leads to the “Clausius Inequality” [67]:

$$\oint\limits_\Sigma {\frac{{dQ}}{T}} \le 0$$

((13.7))

Here the closed surface integral is used to denote the net entropy change for the system (that is, the system must reject heat to the universe in order to function), and Σ is the closed outer boundary of the thermodynamic system.

A single DNA molecule is very well-ordered, and thus at low entropy compared to the constituent molecules required to form it. This might, at first glance, seem to violate the Second Law, but it doesn’t because the relative “orderliness” of the unused fractions of the reactants is so much less than their original form that the randomness in the universe has increased because of the formation of the DNA molecule. The cell is an open system, and the entropy of the byproducts of metabolism is greater than that of the organized molecules created with them, such as DNA, RNA, ATP and, of course, many others. In addition, every biologic cell generates excess heat of metabolic reactions, the waste heat of the cell as a thermal engine, that must be rejected to the environment. All cells obey the Second Law, just as any heat engine must. If a cell is placed in a completely insulated container, it will die as its temperature increases (essentially without bound) in order to satisfy the Second Law.

Perhaps the most pertinent example for this chapter is the change from liquid phase to vapor phase water (boiling) at constant temperature and pressure – this requires the addition of sufficient thermal energy to increase the entropy of the water to that of the vapor phase. At saturation both the pressure and temperature are constant and both the change in enthalpy, $\Delta h\,[{\textrm{J}} \cdot {\textrm{kg}}^{ - 1} ]$, and the resulting change in entropy, $\Delta s\,[{\textrm{J}} \cdot {\textrm{kg}}^{ - 1} \cdot {\textrm{K}}^{ - 1} ]$, depend on the temperature at which the boiling takes place:

$$\Delta h = T\Delta s$$

((13.8))

In the Steam Tables [66] the somewhat arbitrary reference point for entropy is $s = 0$ at $T = 0^ \circ {\textrm{C }}(273.16\,[{\textrm{K}}])$ – this is the triple point for water at pressure P = 1 atmosphere (i.e. 101.3 [kPa]). At the triple point, the liquid, vapor and solid phase are in thermal equilibrium, so it makes a convenient reference for both the enthalpy and the entropy of water.

Example 13.5:

Calculate the thermal energy $(\Delta h = h_{fg} )$ required to boil 10 g of water at 1 atmosphere pressure and saturation temperature (i.e. 100°C) if you know that the change in entropy is $\Delta s = s_{fg} = 6036\,({\textrm{J/kg/K}})$. Here the “fg” subscript indicates the change in phase from fluid to gas. Compare the answer with the tabulated phase change enthalpy at saturation, $h_{fg} = 2.257 \times 10^6 \,({\textrm{J/kg}})$.

Solution:

$\Delta h = T\Delta s = (100 + 273.16)^*(6036) = 2.252 \times 10^6 \,({\textrm{J/kg}})$ … within round-off error of h _fg
(Note: Don’t forget to use absolute temperature here!)
10 g = 0.01 kg => Energy required: $\Delta h = 2.252 \times 10^4 ({\textrm{J}})$

A similar change in entropy accompanies chemical reactions. The enthalphy change includes Gibb’s Free Energy of Formation, $\Delta g\,[{\textrm{J}} \cdot {\textrm{kg}}^{ - 1} ]$, and the entropy change of the reaction:

$$\Delta h = \Delta g + T\Delta s$$

((13.9))

Remember that some reactions are exothermic and others are endothermic as a result of the relative Gibb’s energy of formation. We will make use of this relation in a later section of this chapter.

3.1.2.2 Statistical Thermodynamic Description

Up to this point the discussion of thermodynamics has been limited to the macroscopic (i.e. classical thermodynamics) point of view: quantities such as enthalpy and internal thermal energy are deterministic macroscopic properties and considered continuous. In classical thermodynamics they may be traded between control volumes in infinitessimally small amounts with no lower limit. The classical thermodynamic relationships are routinely used to describe systems as diverse as steam engines, gas turbines, refrigerators, subsonic and supersonic nozzle expansions and living cells.

But entropy is really a stochastic (random) quantity, as described above, and not deterministic (non-random). We should therefore expect a stochastic (microscopic) description, then, in terms of random variables. This point of view was originally proposed by Boltzmann in the middle of the 19th century, and was quite controversial at the time. Very briefly, the entropy is an “extensive” property; that is, it is the sum of the entropies of all of the particles (molecules or atoms) of a system. Further, the permitted “states” of individual particles of matter are not continuous, but are limited to discretely quantized values. This is a fundamental tenet of quantum mechanics. Systems of many interacting molecules undergo continuous change in their quantum state, exchanging energy in numbers of finite, discrete “quanta,” and in no smaller amount.

If we follow the thermal energy of an individual molecule for a period of time, we would expect to observe a mean value with discrete (i.e. quantum) fluctuations about the mean described by the various statistical moments: mean, standard deviation, variance, and so on. A very highly ordered ensemble of molecules will have a very small standard deviation, and a more disordered ensemble will have a large standard deviation (looking at just the first and second statistical moments for this discussion). The Quantum State Probability, p _i(t) can be used to describe this, where p _i = the fraction of total molecules in state “i” at any time. If the total number of possible quantum states is low, then the p _i values will be high (and the standard deviation low); and the converse. At all times, the total probability = 1, so for N total states:

$$\sum\limits_{i = 1}^N {p_i = 1}$$

((13.10))

The macroscopic property used in the previous discussion is the probability-weighted (expected) value, which is called the mean value – for example, the macroscopic internal thermal energy, u, is the mean value, <u>, of the allowed quantum state molecular thermal energies, u _i. Note that <u> can be continuous even though u _i is discrete since the mean value of a random process need not be an observable value. If there are N total allowed quantum states, then at any instant:

$$u = \sum\limits_{i = 1}^N {p_i } u_i {\textrm{[J}} \cdot {\textrm{kg}}^{ - 1} {\textrm{]}}$$

((13.11))

If a complete thermal system, C, contains constituents A and B, where there are N allowed states for A molecules and M allowed states for B molecules, then the complete system, C, is described by their joint probabilities, $p_{ij} = p_i \cdot p_j$. That is, the joint probability is the product of the individual probabilities when the two random processes are statistically independent. To construct a useful model for the entropy of the thermodynamic system, we use the joint probability in the form of an equivalent function, f (as yet undefined):

$$s = \sum\limits_{i = 1}^N {\sum\limits_{j = 1}^M {p_i \;p_j } } \;f\left( {p_i \,,\;p_j } \right) = \sum\limits_{i = 1}^N {\sum\limits_{j = 1}^M {p_i \;p_j } } \;f\left( {p_i \, \cdot \;p_j } \right) = \sum\limits_{i = 1}^N {p_i f\left( {p_i } \right) + } \sum\limits_{j = 1}^M {p_j f\left( {p_j } \right)}$$

((13.12))

The function f must be such that statistical independence – i.e. Eq. (13.12) – holds irrespective of the values of p _i and p _j. One such function that always satisfies this requirement is the natural logarithm, $f(\arg ) = \ln (\arg )$ – i.e. $\ln \{ AB\} = \ln \{ A\} + \ln \{ B\}$. It can be shown after some development (see, for example, [68], Chapter 6) that the general solution to the governing second order differential equation is $f = C\ln \{ p_i \}$, and since p _i is always ≤1, the constant C must be less than zero so that f is positive-definite. Boltzmann demonstrated that the definition of temperature in the statistical and classical domains would be the same if:

$$s = - k\sum\limits_{i = 1}^N {p_i \;\ln \left\{ {p_i } \right\}} [{\textrm{J}} \cdot {\textrm{kg}}^{ - 1} \cdot {\textrm{K}}^{ - 1} ]$$

((13.13))

where $k = {\textrm{Boltzmann's constant}}(1.380 \times 10^{ - 23} {\textrm{J}} \cdot {\textrm{K}}^{ - 1} )$ and the negative sign is necessary since the quantum state probabilities are constrained to $0 < p_i \le 1$.

If all of the allowed quantum states are equally likely – that is, if p _i is uniformly distributed – and there are W allowed states, then Boltzmann demonstrated that:

$$s = k\;{\textrm{ln}}\left\{ W \right\}$$

((13.14))

A uniformly distributed random variable is one that maximizes the entropy in a system of molecules. Equation (13.14) is the Boltzmann definition of entropy, while Eq. (13.13) is the Gibbs definition of entropy. When first introduced, Eq. (13.14) was an extremely controversial construction; so much so that Boltzmann was vilified by the thermodynamic classicists of the day. His final word on the matter was to use Eq. (13.14) as his epitaph.

In summary, while the detailed thermodynamic state must be described by quantum mechanical statistical thermodynamics, the mean thermodynamic properties are adequate for laser heating calculations unless the distribution of states deviates significantly from the equilibrium distribution. That is to say, classical thermodynamic variables adequately describe the processes unless all of the events occur in extremely short time intervals (i.e. time intervals approaching the mean time between collisions).

The Zeroth Law of Thermodynamics states that “there is a thermal game;” the First Law says that “the best that you can do is to break even,” and the Second Law says that “you can never do that well.” An unstated Third Law is that “you can never get out of the game.”

3.2 Modeling Tissue Thermal Events

3.2.1 Sub-Coagulation Thermal Models

An energy balance for tissue under laser irradiation where metabolic heat and phase change are negligible can be assumed to follow the traditional Bioheat Equation as previously described, and repeated here:

$$\rho c\,\frac{{dT}}{{dt}}\, = \,\dot Q_{{\textrm{gen}}} \, + \,\nabla \, \cdot \,\left( {k\nabla T} \right)\, + \,\dot w\rho c_b \,\left[ {T_b \, - T} \right]\, + \,\dot Q_{{\textrm{met}}} [{\textrm{W}} \cdot {\textrm{m}}^{ - 3} ]$$

((13.15))

where the “b” subscript refers to blood properties, and $\dot w$ is the tissue perfusion $[{\textrm{kg}}_{\textrm{b}} \cdot {\textrm{kg}}_{{\textrm{tissue}}} ^{ - 1} \cdot {\textrm{s}}^{ - 1} ]$. In Chapter 3, the notation for laser heat generation is S; but since S is used for entropy in this chapter we have used $\dot Q$.

3.2.2 Evaporation of Surface Water

Surface water evaporation typically dominates other surface loss terms (convection and radiation) above about 60°C [69]. A useful model for evaporation using Stelling’s Formula has been derived from experimental solar pond data, as described in [70]:

$$W_{{\textrm{evap}}} = \left[ {A_s + B_s U_{{\textrm{inf}}} } \right]\;\left( {P_{{\textrm{sat}}} \left( T \right)\; - \;E_{{\textrm{atm}}} } \right) [{\textrm{m}} \cdot {\textrm{s}}^{ - 1} ]$$

((13.16))

where $W_{{\textrm{evap}}} = {\textrm{surface evaporation rate }}[{\textrm{m}} \cdot {\textrm{s}}^{ - 1} ]$, $A_s = 7.31 \times 10^{ - 11} \,[{\textrm{m}} \cdot {\textrm{Pa}}^{ - 1} \cdot {\textrm{s}}^{ - 1} ]$, $B_s = 1.2 \times 10^{ - 11} \,[{\textrm{Pa}}^{ - 1} ]$, $U_{\inf } {\textrm{ = free stream air}}\ {\textrm{velocity above the surface}}\ {\textrm{[m}} \cdot {\textrm{s}}^{ - {\textrm{1}}} {\textrm{]}}$, $P_{{\textrm{sat}}} (T) = {\textrm{the saturation pressure at tissue temperature}}$, T, [Pa], and $E_{{\textrm{atm}}} = {\textrm{atmospheric water vapor pressure}} = (Hu)P_{{\textrm{sat}}} (T_{{\textrm{inf}}} )\,[{\textrm{Pa}}]$, where Hu = relative humidity (expressed as a fraction). To include this process in model calculations, the evaporation rate, W _evap, is multiplied by the control volume surface area, A [m²], liquid-phase density $\rho \,[{\textrm{kg}} \cdot {\textrm{m}}^{ - 3} ]$, phase change enthalpy $h_{fg} \,[{\textrm{J}} \cdot {\textrm{kg}}^{ - 1} ]$, and divided by the control volume, V [m³], to get the equivalent volumetric power, $Q_{{\textrm{evap}}} \,[{\textrm{W}} \cdot {\textrm{m}}^{ - 3} ]$:

$$\dot Q_{{\textrm{evap}}} \, = \,\frac{{Ah_{fg} \rho W_{{\textrm{evap}}} }}{V} [{\textrm{W}} \cdot {\textrm{m}}^{ - 3} ]$$

((13.17))

Here “f ” indicates the liquid state, and “g” the gas state, as above.

3.2.3 Water Vaporization

When liquid-phase water is vaporized (boiled) we must add the required thermal energy at the boiling point temperature and pressure, T _sat and P _sat, in the form of the vaporization enthalpy, $\Delta h_{fg}$, and as a result the entropy will change, $\Delta s_{fg}$, as well as the density (or specific volume). The density/specific volume change is actually included in the phase change enthalpy:

$$\Delta h_{fg} = T\Delta s_{fg} \; = \Delta u_{fg} + P_{{\textrm{sat}}} \Delta v_{fg}$$

((13.18))

where $\Delta u_{fg}$ is the internal thermal energy of phase ${\textrm{change}} = [c_g - c_f ]T_{{\textrm{sat}}}$, and $\Delta v_{fg}$ is the change in specific volume at the boiling point pressure, P _sat. For example, at 100°C water has (see Example 13.4):

$$\Delta h_{fg} = 2.257 \times 10^6 \,[{\textrm{J}} \cdot {\textrm{kg}}^{ - 1} ],\rho _{{\textrm{sat}}} = 958\,[{\textrm{kg}} \cdot {\textrm{m}}^{ - 3} ],\,\Delta s_{fg} = 6036\,[{\textrm{J}} \cdot {\textrm{kg}}^{ - 1} \cdot {\textrm{K}}^{ - 1} ]$$

Aside from the effect vaporization has on tissue optics, expansion of the tissue water, whether or not phase change takes place, accounts for the internal stresses applied to tissue. In order to include these effects in the energy balance, the left hand side of the energy balance, Eq. (13.15), must be modified. Neglecting perfusion and metabolic heat for the moment:

$$\frac{{\partial \left( {ph} \right)}}{{\partial t}}\, = \,\dot Q_{{\textrm{gen}}} \, + \,\nabla \, \cdot \,\left( {k\nabla T} \right)$$

((13.19))

The enthalpy is the sum of internal thermal energy and the pressure-volume product, as in Eq. (13.2). Note that the thermal model now includes phase change since the pressure-volume product has been included in the enthalpy.

Numerical implementation of this model may follow the same philosophy as that of the low temperature case if two thermodynamic parameters can be evaluated, $\partial P/\partial T$ and $\partial \rho /\partial T$:

$$\left\{ {\rho c\, + \,\frac{{\partial P}}{{\partial T}}\, - \,\frac{P}{\rho }\,\frac{{\partial \rho }}{{\partial T}}} \right\}\,\frac{{\partial T}}{{\partial t}}\, = \,\dot Q_{{\textrm{gen}}} \, + \,\nabla \, \cdot \,\left( {k\nabla T} \right)$$

((13.20))

In free liquid phase water that is not constrained by rigid or visco-elastic tissue structures, $\partial P/\partial T$ is usually negligibly small. However, a fourth order polynomial fit $(r^2 = 1.000)$ derived from “The Thermodynamic Properties of Steam” [66] for the saturation pressure is:

$$\begin{aligned} P_{{\textrm{sat}}} \left( T \right)\, =& \,1.686 \times 10^{ - 9} T^5 \, + \,6.681 \times 10^{ - 8} T^4 \, \\ &+ \,2.411 \times 10^{ - 4} T^3 \, - \,0.03623T^2 \, + \,2.389T\, - \,0.432\end{aligned}$$

((13.21))

where P _sat is in [kPa = kN·m⁻²] and T is in Celsius. In Fig. 13.10 this function deviates from the tabulated data only at temperatures near the freezing point, 0°C. The $\partial P/\partial T$ term may be quickly derived from this polynomial fit.

figure 13_10_100508_2_En — **Fig. 13.10**

The density change term cannot be neglected in free water. This additional term (when phase change is included) accounts for the kinetic energy imparted to expelled water and describes the origin of acoustic waves in tissue, whether or not phase change is significant. For no vaporization, an estimate of $\partial \rho /\partial T$ can be obtained from the saturation density vs. temperature curve under the valid assumption that the liquid phase water density is primarily determined by temperature rather than local pressure. The saturation density curve, a fifth order polynomial fit from 20 to 374 C $(r^2 = 0.998)$ [66] is:

$$\begin{aligned}\rho _{{\textrm{sat}}} \left( T \right)\, = &\,6.918 \times 10^{ - 10} T^5 \, + \,5.594 \times 10^{ - 7} T^4 \, - \,1.650 \\ &\times 10^{ - 4} T^3 \, + \,0.01912T^2 \, - \,1.413T\, + \,1026\end{aligned}$$

((13.22))

where ρ _sat is in [kg·m⁻³] and, again, T is in Celsius. In Fig. 13.11 the polynomial fit only deviates significantly from the tabulated data at temperatures near the critical point, 375°C.

figure 13_11_100508_2_En — **Fig. 13.11**

At temperatures above about 90°C water behavior dominates the observed photo-thermal phenomena. Below the saturation temperature, 100°C at atmospheric pressure, water vaporization is a diffusion-limited surface loss phenomenon (see Eq. (13.16)) – and depends primarily on surface characteristics such as local humidity and temperature dependent mass diffusion coefficients. Above the saturation temperature, the vaporization process becomes a volumetric energy sink in which phase changes generate high local pressures and concomitant tissue stresses. However, the more moderate heating case is of immediate interest in this discussion.

In Fig. 13.10 the saturation pressure at 120°C has increased to 2 atmospheres (200.2 kPa) and climbs precipitously above that. Water vaporization may be included in the modified energy balance of Eq. (13.20):

$$\left\{ {\rho c\, + \,\frac{{\partial P}}{{\partial T}}\, - \,\frac{P}{\rho }\,\frac{{\partial \rho }}{{\partial T}}} \right\}\,\frac{{\partial T}}{{\partial t}}\, = \,\dot Q_{{\textrm{gen}}} \, + \,\nabla \, \cdot \,\left( {k\nabla T} \right)\, - \,\frac{{\partial m}}{{\partial t}}\,\Delta h_{fg} \left( T \right)$$

((13.23))

where ∂m/∂t is the rate of mass vaporization of water per unit volume (kg·s⁻¹·m⁻³) and $\Delta h_{fg} (T)$ is the temperature dependent phase change enthalpy [J·kg^{− 1}] (Fig. 13.12 and Eq. 13.24).

figure 13_12_100508_2_En — **Fig. 13.12**

The temperature dependence of saturation density and saturation pressure may be included by incorporating the appropriate correlation relation from Eqs. (13.21) and (13.22). The temperature dependence of the phase change enthalpy may be included using a similar correlation $(r^2 = 0.995)$ derived from the same tabular data [66]:

$$\Delta h_{fg} \, = \, - 4.143 \times 10^{ - 4} T^4 \, + \,0.247T^3 \, - \,54.3T^2 \, + \,1975T\, + \,2.402 \times 10^6$$

((13.24))

where $\Delta h_{fg}$ is in [J·kg⁻¹] and T is in Celsius. The polynomial fit only deviates significantly from the tabulated data at temperatures near the critical point.

An alternate, and in some ways equivalent, formulation for vaporization was recently presented by [71] in which phase change processes are included in the left hand side of Eq. (13.15) to obtain an effective specific heat, C ^′, where C ^′ combines the first term on the left hand side with the last term on the right hand side of Eq. (13.23):

$$\rho C'\,\frac{{\partial T}}{{\partial t}}\, = \,\rho \left\{ {c\, - \,\Delta h_{fg} \,\frac{{\partial \rho _w }}{{\partial T}}} \right\}\,\frac{{\partial T}}{{\partial t}}\, = \,\dot Q_{{\textrm{gen}}} \, + \,\nabla \cdot \left( {k\nabla T} \right)\, + \,W\rho _b c_b \left[ {T_b \, - \,T} \right]$$

((13.25))

where W is described in their paper as the “density” of the tissue water, and the perfusion heat has been included. This formulation includes the latent heat of vaporization, but not the changes in tissue density or pressure. However, it has to be used with some caution. First, “W” includes much more than the density of water − in fact, W includes the mass fraction of water in the tissue, so W = m in Eq. (13.23). Second, applying the relation directly as given leads to a decidedly aphysical relationship. That is, for equilibrium vaporization at the saturation temperature ${{\partial T} \mathord{\left/ {\vphantom {{\partial T} {\partial t}}} \right. \kern-\nulldelimiterspace} {\partial t}=0}$; consequently, the left hand side of Eq. (13.25) is zero for equilibrium boiling at saturation. The right hand side is certainly not zero – imagine tissue with zero blood flow at uniform temperature (so that both the Fourier conduction term and the blood flow term are zero); $\dot Q_{{\textrm{gen}}}$ is not zero. The energy balancing effect of the ∂m/∂t on the right hand side of Eq. (13.23) has been eliminated, leading to an invalid conclusion from Eq. (13.25).

In a numerical model, one may assume that vaporization occurs at local saturation conditions (equilibrium thermodynamics applies) with rate determined by laser excess energy (that is, energy in excess of that consumed by conduction heat transfer, perfusion heat and other losses) and thermodynamic properties. This is equivalent to assuming that evolved water vapor diffuses readily through the tissue structure, and the pressure does not rise significantly. Under these conditions the left hand side of Eq. (13.23) would be zero – i.e. constant temperature boiling at the saturation temperature.

When laser energy, $\dot Q_{{\textrm{gen}}},$ is added faster than water vapor can diffuse through tissue to maintain liquid saturation at atmospheric pressure, the tissue temperature readily exceeds 100°C when the local tissue pressure rises (and thus lowers the phase change enthalpy). In that case, Eq. (13.23) would have to be used to construct the model. An additional relation would have to be included to complete the model: the rate of pressure rise with water vaporization. The constraints on intra-cellular water are more significant than on extra-cellular water. That is, extra-cellular expanded water and water vapor need only to “percolate” through spaces in the gross structure while intracellular water vapor expands cell volume putting stress on the cell membrane. For either the intra- or extra-cellular tissue water compartment one would need mass diffusion relations to formulate this term.

The applied stresses, both intra-cellular and extra-cellular, create what can be described as a “thin wall pressure vessel problem.” Imagine a spherical pressurized vessel (i.e. cell), such as shown in cross-section in Fig. 13.13, with internal pressure “P” (relative to the external pressure), interior radius “R” and wall thickness, “d.” If we cut the spherical vessel at z = 0, a free-body diagram reveals that the pressure force, $F_p = {\pi }R^2 P$, must be balanced by the total Z-direction force, $F_z [N] = 2\pi Rd\,\sigma$, where $\sigma \,[{\textrm{N}} \cdot {\textrm{m}}^{ - 2} ]$ is the wall stress, and so:

$$\sigma = \frac{{P\;R}}{{2d}}$$

((13.26))

figure 13_13_100508_2_En — **Fig. 13.13**

In many cases the local pressure increases cause rupture in the tissue structure, histologically observable as the “popcorn” effect described in Section 13.2.3.4. Elevated tissue pressures also explain the readily observed tearing phenomena in highly layered tissues, such as arterial vessel wall, when irradiated with highly scattered wavelengths. The effects of high local tissue pressures have been observed in many studies when, for example, highly-scattered argon laser irradiation is used to ablate aorta. The scattering events result in high fluence rates just below the tissue surface and corresponding thermally-induced stresses when the tissue water vaporizes.

Example 13.6:

This example will adapt “thin wall pressure vessel” analysis to calculate the internal pressure of small bubbles generated in liquid phase water. In this case the “surface tension” or surface free energy, $\gamma ({\textrm{N}} \cdot {\textrm{m}}^{ - 1} )$ has the role of the “applied stress,” σ in Eq. (13.26). The surface tension of water is approximately linearly related to the temperature by $\gamma ({\textrm{dynes}} \cdot {\textrm{cm}}^{ - 1} ) = 76.42 - 0.1731\;T(^ \circ {\textrm{C}})$. Calculate the internal pressure of a water vapor bubble 0.1 mm in diameter in water at 90°C.

Solution:

(1)
The pressure forces, F _p, are balanced by the surface tension forces around the circumference of the bubble_γ, similar to the wall stresses in Fig. 13.13. So:
$$\begin{aligned}[rl]F_p \, &= \,P\pi R^2 \, = \,F_\gamma \, = \,\gamma 2\pi R \\[6pt] P\, &= \,\frac{{2\gamma }}{R}\left[ {{\textrm{N}} \cdot {\textrm{m}}^{ - {\textrm{2}}} } \right] \end{aligned}$$
(2)
At $T = 90^ \circ {\textrm{C}}$, $\gamma = 76.4 - 0.1731\,(90) = 60.9\,({\textrm{dynes}} \cdot {\textrm{cm}}^{ - 1} ) = 60.8 \times 10^{ - 7}\break ({\textrm{N}} \cdot {\textrm{m}}^{ - 1} )$
$$P\, = \,\frac{{2[60.8\, \times 10^{ - 7} ]}}{{[0.05\, \times 10^{ - 3} {\textrm{m}}]}}\, = \,0.243\left( {{\textrm{N}} \cdot {\textrm{m}}^{ - {\textrm{2}}} } \right)\, = \,0.243\left( {Pa} \right)$$
(3)
This is a very small internal pressure.

A model based on the energy balance of Eq. (13.23) should use a small time step so that the contributions of local heat transfer may be calculated and summed with the laser power term. Excess energy above that required to maintain constant temperature may be imagined to go into the vaporization process at the local temperature and pressure. The effect of vaporization on the local pressure may be calculated if an acceptable model for tissue mechanical response is available (i.e. the stress-strain relation, and vapor diffusion coefficients).

At extreme heating rates in tissue and/or free water, such as are realized by pulsed excimer laser activations, the water may become metastable. If so, the vaporization is a rate-limited nucleate boiling phenomenon with rate coefficients determined by the supersaturation ratio, P/P _sat, (where P _sat is the saturation pressure, Fig. 13.10) and by the surface free energy of nucleated vapor bubbles in the liquid phase. Nucleate boiling only dominates equilibrium boiling at temperatures approaching the critical temperature. In the excimer laser rapid-ablation case the energy powering the debris plume probably comes from nearly instantaneous water phase changes at highly elevated pressures.

3.2.4 Coagulation Processes

At higher temperatures the desired thermal end point is coagulation, or a significant irreversible alteration in tissue structural proteins – that is, thermal damage. We often use kinetic models in a “volume fraction” or “probability” form (% of tissue damaged) to predict or describe the evolution of thermal damage. Typical damage processes for which the volume fraction kinetic damage model has been used successfully include: (1) tissue necrosis, (2) loss of birefringence in muscle and collagen, and (3) collagen shrinkage. The damage model is based on the assumption that a single damage process is active, which may not hold in the general case where multiple simultaneous processes occur. Still, the model can be used if the damage processes under study have different basic mechanisms and threshold temperatures and can therefore be considered thermodynamically independent. For example, it is hard to imagine how red blood cell coagulation could affect collagen fiber unraveling, so many common damage processes are quite obviously independent in the thermodynamic sense. In practical calculations, we wish to identify particular damage processes that can act as hallmarks of desired therapeutic end points. As we frequently do not have data for the damage process enthalpy, we generally do not include the damage process in the thermodynamic calculations of tissue temperature.

Damage processes in tissue coagulation and necrosis typically follow first order unimolecular reaction kinetics. This is such an important topic that it is reserved for detailed discussion in the following section of the chapter, Section 13.3.3. It is very important to note at this point that tissue damage processes are not phase change processes, despite their rather unfortunate previous description in that fashion [37]. It is easy to see why that is so: all phase changes are reversible, all tissue thermal damage processes are irreversible. Repairing tissue thermal damage requires tremendous amounts of energy during the healing (repair and replacement) process.

3.3 Kinetics of Thermal Damage Processes

Damage processes can often be modeled as first-order rate processes for which two experimentally derived coefficients are sufficient. Thermal damage in this formulation is exponentially dependent on temperature and linearly dependent on time of exposure. The rate process models apply well to the prediction of damage thresholds, and less well as the damage becomes complete since several of the fundamental assumptions are violated. This section reviews the basis for kinetic models of tissue thermal damage.

3.3.1 Theoretical Foundations

The original work on the application of rate process models to thermal damage was reported by Moritz and Henriques in a series of seminal papers entitled “Studies of Thermal Injury” in 1947 [72–75]. In their work, the damage was quantified using a single parameter, Ω, which ranges on the entire positive real axis and is calculated from an Arrhenius integral:

$$\Omega \left( \tau \right) = \int_0^\tau {A\;e^{\left[ {\frac{{ - E_a }}{{R\;T\left( t \right)}}} \right]} \;} dt$$

((13.27))

where A is a frequency factor [s⁻¹], τ the total heating time (s), E _a an activation energy barrier [J · mole⁻¹], R the universal gas constant, $8.3143\,[{\textrm{J}} \cdot {\textrm{mole}}^{ - 1} \cdot {\textrm{K}}^{ - 1} ]$, and T the absolute temperature [K]. One difficulty with this description is that a single damage parameter inherently lumps all damage processes into one global measure, so in multiple damage process cases the lowest temperature process saturates the damage measure early during a laser activation. In this discussion we review the underlying assumptions and origin of the terms in Eq. (13.27) and recast the traditional thermal damage parameter, Ω, into a form suitable for evaluation of multiple-process laser thermal damage effects and for comparison between numerical models and histologic results.

3.3.1.1 Reaction Product Formation Rates

The basis for rate process models of thermal damage may be obtained from chemical reaction kinetics. In a typical reaction process, thermally active reactants jump over an energy barrier to form products, as illustrated in Fig. 13.14. In the figure, E _a is the energy barrier in Eq. (13.27). The collision theory description of ordinary first order bi-molecular reaction kinetics (see, for example, [76]) holds that the reactants are activated by collisions; n* are activated out of n total molecules, and the probability of activation is:

$$\frac{{n^* }}{n} = e^{\left[ {\frac{{ - E_a }}{{RT}}} \right]}$$

((13.28))

figure 13_14_100508_2_En — **Fig. 13.14**

In such a process, activated reactants are considered to form an activated “complex” which may either relax to inactivated single reactants or progress to form product molecules. The complex has some of the properties of an ordinary molecule and is at least temporarily stable. For reactant molecules A and B the activated complex is [AB]*, and the sequence of formation is:

$$\textrm{Reactants} \quad A+B \quad \begin{matrix}{\substack{k_a \\ \longrightarrow}} \\ {\substack{\longleftarrow\\ k_b}}\end{matrix} \quad [AB]^\ast \quad \substack{k \\ \longrightarrow} \quad \textrm{Products}$$

((13.29))

The overall reaction velocity, $k\,[{\textrm{s}}^{ - 1} ]$, determines the rate of formation of product and is related to the equilibrium constant for formation of the activated complex, K*, by:

$$k = \frac{{RT}}{{Nh_P }}\;K^*$$

((13.30))

where

$$K^* = e^{\frac{{ - \Delta G^* }}{{RT}}}$$

and here N is Avogadro’s number $(6.023 \times 10^{23} )$, h _P is Planck’s constant $(6.627 \times 10^{ - 34} \,[{\textrm{J - s}}])$, and ΔG* is the Gibb’s free energy of formation of activated complex [J·mole⁻¹]. As mentioned in the previous section, the free energy of formation is given by:

$$\Delta G^* = \Delta H^* - T\Delta S^*$$

((13.31))

where ΔH* is the enthalpy of activation ${\textrm{[J}} \cdot {\textrm{mole}}^{ - 1} ]$ and ΔS* is the entropy of activation $[{\textrm{J}} \cdot {\textrm{mole}}^{ - 1} \cdot {\textrm{K}}^{ - 1} ]$. The activation entropy is not calculable except for the simplest possible reactions in rarified gases, and is therefore usually determined from experimental measurements of the reaction velocity and activation enthalpy. The activation enthalpy, ΔH*, is determined from the observed activation energy, E _a by:

$$\Delta H^* = E_a - i\;RT$$

((13.32))

where i is 1 for first-order reactions in solution and gases, 2 for second order, and 3 for third order reactions.

3.3.1.2 Unimolecular Process Descriptions

Thermal damage in tissue is a unimolecular process – tissue constituents transition from the native state to the damaged state. Absolute reaction rate theory can also be used to explain the rate of formation for this process if we assume that a time lag exists between molecular activation and denaturation [76]. During this time lag, the molecules may either denature or relax back to the native state, as illustrated in Fig. 13.15. Here, ΔH is the enthalpy (i.e. internal thermal energy) difference between native state and denatured molecules. The relative barriers are such that in the thermal damage of tissue, ΔH*, is almost always smaller than ΔH. So, the activation process may be regarded as reasonably likely, and the probability of denatured tissue relaxing back to native state tissue is near enough to zero that it may be regarded as the impossible event in the absence of an energy-consuming healing process. The rate of damage formation is then proportional to only those molecules that remain activated. For a unimolecular process in the native state C, having an activated state, C*, with velocity constants k _a, k _b, and k ₃:

$$C + C \quad \begin{matrix}{\substack{{k_a} \\ \longrightarrow}}\\ \substack{\longleftarrow \\ {k_b}}\end{matrix} \quad C + C^\ast $$

((13.33a))

$$C^* \quad \substack{{k_3} \\ \longrightarrow} \quad {\textrm{Damaged Molecules}}$$

((13.33b))

figure 13_15_100508_2_En — **Fig. 13.15**

For this process, the rate of disappearance of native state molecules, [C], is given by:

$$- \frac{{d\left[ C \right]}}{{dt}} = k_3 \;\left[ {C^* } \right]$$

((13.34))

where the bracket is used to indicate molar concentration. Generally [C*] is neither known nor calculable; however, at sufficiently low concentrations of C* the steady state principle asserts that for short-lived activated states the rate of formation can be considered equal to the rate of disappearance. So the activated state, [C*], forms at a rate $k_a [C]^2$, relaxes back to inactivated at rate $k_b [C]\,[C^*]$, and denatures at rate $k_3 \,[C^*]$. Consequently:

$$k_a \;\left[ C \right]^2 = k_b \;\left[ C \right]\left[ {C^* } \right] + k_3 \left[ {C^* } \right]$$

((13.35a))

$$\left[ {C^* } \right] = \frac{{k_a \;\left[ C \right]^2 }}{{k_b \;\left[ C \right] + k_3 }}$$

((13.35b))

Of course, we seek an overall reaction velocity, k, which relates [C] to its rate of disappearance:

$$- \frac{{d\left[ C \right]}}{{dt}} = k\;\left[ C \right]$$

((13.36))

There are two limiting cases for Eq. (13.35b). First, the concentration of remaining undamaged material, [C], may be large enough that deactivation at k _b dominates the k ₃ pathway, so $[C^*] \cong [C]\,k_a /k_b$ for which the overall formation rate, $k = k_3 \,(k_a /k_b )$ and a first order process results. Second, if the remaining undamaged material concentration, [C], is small, $k_3 > > k_b \,[C]$ and the process is second order since from Eq. (13.35a) $k \cong k_a /k_3 \,[C]$. In liquid phase systems with appreciable concentrations of native state molecules the first condition should apply, so the first order approximation applies. After a long time of exposure at damaging temperatures such that [C] is very small, $k_3 > > k_b \,[C]$ and a second order process results:

$$- \frac{{d\left[ C \right]}}{{dt}} = k\;\left[ C \right]^2$$

((13.37))

where for simplicity the [C] dependence has been removed from k: $k \cong k_a /k_3$. At such low concentrations the damage process is saturated, so we may ignore that case for the present.

Equation (13.36), then, is a Bernoulli differential equation with the solution:

$$C\left( \tau \right) = C\left( 0 \right)e^{\left\{ { - \int_0^\tau {k\;dt} } \right\}}$$

((13.38))

Equations (13.30) and (13.31) may be used to relate k to $\Delta H^*$ and $\Delta S^*$. It should be noted at this point that the energy barrier, E _a, (Fig. 13.14) is in fact $\Delta H^* + RT$ (Eq. 13.32); however, in practice $\{ \Delta H^* \cong 5 \times 10^5 \} > > \{ RT \cong 3 \times 10^3 \}$, so little error results from assuming that $E_a \cong \Delta H^*$. This approximation may be used to obtain:

$$k = \left( {\frac{{RT}}{{Nh_P }}} \right)\quad e^{\left[ {\frac{{\Delta S^* }}{R} + 1} \right]} \quad e^{\left[ {\frac{{ - E_a }}{{RT}}} \right]} \cong \left( {\frac{{RT}}{{Nh_P }}} \right)\quad e^{\left[ {\frac{{\Delta S^* }}{R}} \right]} \quad e^{\left[ {\frac{{ - \Delta H^* }}{{RT}}} \right]}$$

((13.39a))

$$k \cong A\;e^{\left[ {\frac{{ - \Delta H^* }}{{RT}}} \right]}$$

((13.39b))

The term in parentheses in Eq. (13.39a) suggests that the pre-exponential factor, A, is not constant but is in fact temperature dependent. However, the linear dependence of A on temperature is extremely weak – especially when compared to the exponential dependence in the final term – and its effect is negligible, so that for all practical purposes A may be treated as a constant over the temperature ranges of interest.

3.3.1.3 Arrhenius Formulations in Thermal Damage Studies.

A more useful form of Eq. (13.27) may be obtained by recasting the result into a volume fraction model. In this formulation, as above, C signifies the remaining concentration of native state (undamaged) tissue constituent molecules. Therefore, the physical significance of the traditional damage measure, Ω, is the logarithm of the ratio of the original concentration of native tissue to the remaining native state tissue at time τ:

$$\Omega \left( \tau \right) = \ln \left\{ {\frac{{C\left( 0 \right)}}{{C\left( \tau \right)}}} \right\} = \int_0^\tau {A\;e^{\left[ {\frac{{ - E_a }}{{R\;T\left( t \right)}}} \right]} \;} dt$$

((13.40))

where the frequency factor, A, and energy barrier, E _a, are related to the activation enthalpy and entropy, ΔH* and ΔS*, by Eq. (13.39a).

This form of the damage integral has the advantage that it is more easily compared to quantitative pathologic endpoints such as birefringence loss, collagen damage, or cell survival in culture. Using this description, direct comparisons can be made between computer models of transient thermal fields, T(x,y,z,t), integrated over time and the measured histologic damage. A set of coefficients, A and $E_a \cong \Delta H^*$, is required for each damage process considered in the computer model. For tissue damage processes studied to date, A varies from about 10⁴⁰ to 10¹⁰⁵ [s⁻¹] while E _a ranges from about $1 \times 10^5$ to $9 \times 10^5 \,[{\textrm{J}} \cdot {\textrm{mole}}^{ - 1} ]$. Each damage process is then allowed to progress in parallel, driven by the calculated thermal field. This formulation assumes that the individual processes are thermodynamically independent, a reasonable description for identifiable thermal damage processes. The concentration of each of the damage markers, $C(\tau )_i$, is accumulated; and a distributed field description of the predicted histologic endpoint is generated – for example, the boundary of thermally induced birefringence loss can be predicted. The model predictions are then suitable for comparison to histologic results.

3.3.1.4 Functional Behavior of the Damage Model

The characteristic behavior of the kinetic damage model is that below a threshold temperature the rate of damage accumulation is negligible, and it increases precipitously when this value is exceeded. This behavior is to be expected from the exponential nature of the function. For purposes of discussion, it is useful to define the critical temperature, T _crit, as the temperature at which the damage accumulation rate, dΩ/dt, is 1:

$$\frac{{d\Omega }}{{dt}} = 1 = A\;e^{\left[ {\frac{{E_a }}{{R\;T_{{\textrm{crit}}} }}} \right]}$$

((13.41))

so,

$$T_{{\textrm{crit}}} = \frac{{E_a }}{{R\;\ln \left\{ A \right\}}}$$

For a process with (arbitrary) representative coefficients of $A = 1.0 \times 10^{75}$ and $E_a = 5 \times 10^5$, the critical temperature is 74.8°C. Figure 13.16 illustrates the damage accumulation rate dependence on temperature for the hypothetical example process. Note that the plot is converted from [K] to [°C] for the figure. In Fig. 13.17 the damage accumulation rate is plotted vs. 1/T [K⁻¹] for comparison.

figure 13_16_100508_2_En — **Fig. 13.16**

figure 13_17_100508_2_En — **Fig. 13.17**

Constant temperature exposures of the example process will result in a decrease in concentration of native state material depending on the time of exposure. A constant temperature exposure reduces the integral of Eq. (13.40) to a simple multiplication. Figure 13.18 shows the remaining concentration for this hypothetical example damage process for constant temperature exposures of time τ = 0.1, 1.0, and 10 s. The concentration may be seen to gradually decrease with increasing temperature for fixed exposure times, as expected; the strong exponential nature of the process is evident as well. Applying this model framework relies heavily on identifying independent damage processes that can be quantitatively measured.

figure 13_18_100508_2_En — **Fig. 13.18**

From plots like Fig. 13.18, estimates of A and E _a may be made from the Ω = 1 line (i.e. $C(\tau ) = 36.8\%$). Each curve will give one point on an Arrhenius plot (ln{τ} vs. 1/T) where Ω = 1:

$$\ln \left\{ \tau \right\} = \left( {\frac{{E_a }}{R}} \right)\frac{1}{T} - \ln \left\{ A \right\}$$

((13.42))

Estimates of A and E _a may then be obtained from a least-squares regression fit to the line of Eq. (13.42). Note the necessity of using exposure times that span several orders of magnitude in order to separate the curves sufficiently to give acceptable accuracy in the determination of the kinetic coefficients.

Though the damage parameter, Ω, cannot be measured directly in histologic section, often a clearer picture of the functional behavior can be obtained from it. The exponential dependence of damage on the inverse of absolute temperature means that the temperature required to obtain comparable damage levels is quite sensitive to time of exposure. For example, we may define a threshold temperature, T _TH, as the temperature at which the damage parameter, Ω, is 1 for a given duration, τ. Rearranging Eq. (13.42) gives the threshold temperature:

$$T_{TH} = \frac{{E_a }}{{R\;\left[ {\ln \left\{ \tau \right\} + \ln \left\{ A \right\}} \right]}}$$

((13.43))

Figure 13.19 is a plot of the threshold temperature as a function of duration for the example damage process of Figs. 13.16 and 13.18. Note that threshold temperature is approximately exponentially dependent on duration, as expected, with (in this case) a slope of about −4.65°C/decade. So, while 79.5°C is sufficient to result in Ω = 1 (i.e. 63.2% damage) at 0.1 s, 102.3°C would be required for a 1 μs exposure for this process.

figure 13_19_100508_2_En — **Fig. 13.19**

3.3.2 Experimental Determination of Rate Process Coefficients

Thermal damage kinetic coefficients are usually determined from constant temperature exposures of relatively long duration. Threshold damage results are selected out of a set of damaged tissue samples for analysis from which estimates of A and E _a are obtained.

3.3.2.1 Constant Temperature Exposures

Because of the sensitivity of the damage integral to small changes in temperature, the typical approach in obtaining A and E _a is to expose the tissue to a constant temperature, identify experiments in which the damage, $\Omega = \ln \{ C(0)/C(\tau )\}$ is known and obtain A from the intercept and E _a from the slope of an Arrhenius plot of ln(τ) vs. 1/T for the those experiments. If the temperature is held constant at T (K) Eq. (13.40) becomes:

$$\ln \left\{ \tau \right\} = \left( {\frac{{E_a }}{R}} \right)\frac{1}{T} - \ln \left\{ A \right\} + \ln \left\{ \Omega \right\}$$

((13.44))

For experiments with Ω = 1, ln{Ω} = 0 and a linear regression fit to the data plotted on ln{τ} vs. 1/T axes gives the desired coefficients: intercept $b = - \ln \{ A\}$ and slope $m = (E_a /R)$.

For constant temperature experiments in which Ω is known but not equal to 1, a simple re-arrangement of Eq. (13.44) yields the values required for an Arrhenius plot. In Eq. (13.45) the time of exposure for an equivalent experiment that would have Ω = 1 at the experiment temperature T, i.e. τ ₁, is simply related to the actual experiment conditions − Ω, T and τ – by:

$$\ln \left\{ \tau \right\} - \ln \left\{ \Omega \right\} = \ln \left\{ {\tau _1 } \right\} = \left( {\frac{{E_a }}{R}} \right)\frac{1}{T} - \ln \left\{ A \right\}$$

((13.45))

In practice, one then plots ln{τ ₁} vs. 1/T (K⁻¹) for all useful data points and fits a linear regression line to determine the process coefficients from the slope and intercept.

The usual experimental method is to expose thin slices of tissue to constant temperature in a water bath, by surface application of heated water [72, 77] or on a heated metallic plate for desired time intervals. Table 13.4 lists rate coefficients obtained in various experiments. When using the kinetic models and measured coefficients, it is imperative that an adequate description of the particular end point be given. This is because different end points in the same tissue will have widely varying critical temperatures and damage accumulation rates. For example, in Table 13.4 in addition to the variation in tissues, the thermal damage histologic end points in the various studies differ.

Table 13.4 Experimentally determined Arrhenius rate coefficients

Full size table

3.3.2.1.1 Retina

The measurements of Welch and Polhamus [78] used the diameter of the edge of the visible lesion formed in rhesus monkey retina in vivo under argon laser irradiation as the end point. The temperatures were not measured directly, but were determined in separate experiments on retinas in which a correlation between temperature and radius was established using micro-thermocouples (about 5 μm in diameter) advanced from the posterior surface of the eye to a point just below the retina. The correlates were used to estimate the retinal temperature given laser beam power for durations between 0.1 and 10 s. The critical temperature for these coefficients is 56.0°C.

Takata et al. [79] have used a similar decision criterion for retinal damage for shorter exposure times. They fit the data with three sets of coefficients because a single first order model was not sufficient. It would appear that several parallel damage processes were at work in their study, thus a set of coefficients was required − the usual case is that there will be “breakpoints” in the Arrhenius plots, and the regions require different coefficients to fit the data. The critical temperature for the high temperature set of their coefficients is 59.9°C.

Birngruber et al. [80, 81] estimated a frequency factor and activation energy from consideration of the thermodynamics of protein and enzyme denaturation processes. Their estimates have a critical temperature of 74.5°C.

3.3.2.1.2 Skin

The end point for Ω = 1 in the original studies of Henriques and Moritz corresponded to a continuum of cascaded effects [72, 74]. In their study the skin of pigs was exposed, in vivo, to flowing water at a controlled temperature for exposure times varying over several orders of magnitude. They calibrated their coefficients so that Ω = 0.5 corresponded to the onset of erythema (characterized as “first degree” in their paper). Then Ω = 1 corresponded to a “second degree,” or a partial thickness, burn, and Ω = 10⁴ to a full thickness, or “third degree,” burn. Their published coefficients, $A = 3.1 \times 10^{98}$ and $E_a = 6.28 \times 10^5$, have a critical temperature of 59.7°C. Interestingly, although these coefficients have been used for many years by numerous investigators, they do not fit the original data very well. In a later analysis, Diller and Klutke applied linear regression to their original data for temperatures less than 52°C – for which $A = 1.3 \times 10^{95}$ and $E_a = 6.04 \times 10^5$ – and if the data point at 52°C is included, $A = 0.865 \times 10^{95}$ and $E_a = 6.03 \times 10^5$, virtually the same result [82]. In any event, these new coefficients should be used in future work as they fit the original data much better than those originally published in 1947.

Weaver and Stoll [83] used similar criteria to the original 1947 studies and applied two sets of coefficients, as in Table 13.4 (the upper values are applicable above 50 °C) to match the experimental data. The critical temperature for their highest temperature coefficient set is 59.4°C.

3.3.2.1.3 Egg White and Egg Yolk

Yang et al. [84] exposed egg white and egg yolk to constant temperature (± 0.2°C) in a water bath for varying exposure times. Coagulation was defined as the onset of whiteness (coagulum formation due, apparently, to an increase in scattering in the clear liquid egg white) observed by the naked eye. Water bath temperatures ranged from 60 to 90°C in 4°C increments. At each temperature the time to threshold (onset of observable change) was measured and plotted in accordance with Eq. (13.44) and rate coefficients determined from linear regression, as described. Approximately 3–5 s were required to obtain whitening in the egg white at 70°C; while in the egg yolk 82°C was required at the same exposure time. The Table 13.4 values have a critical temperatures of 76.0°C for egg white and 86.6°C for egg yolk.

3.3.2.1.4 Birefringence Loss in Collagen

Birefringence loss in rat skin collagen was measured at temperatures between 40 and 90°C in 5°C increments for times ranging from 120 to 6000 s [77]. The gross extracted skin was wrapped in aluminum foil and immersed in a water bath. The heated samples were sectioned diagonally to increase their area, mounted on glass slides and stained (H&E) for analysis (see Fig. 13.9). The relative intensity of birefringence was determined using the exposure meter on the microscope camera in the “spot” mode: assuming that reciprocity applies, the intensity is the inverse of the indicated exposure time. This approach was necessary because the birefringence image is very low intensity: normal film exposure times required range up to 20 s in ASA 64 film. Intensity was calculated from data normalized by the background light intensity for each microscope slide, I ₀. Relative birefringence intensity was then calculated by subtracting the fully denatured birefringence intensity, I _d, from the specimen intensity, I _s, and native state (undamaged) intensity, I _n, with all intensities individually normalized by I ₀ before the calculation:

$$B = \frac{{I_s - I_d }}{{I_n - I_d }}$$

((13.45b))

A typical set of histologic sections from these data was presented in Fig. 13.5. The Ω = 1 values were determined from the plot similar to Fig. 13.18 in order to calculate A and E _a. For the estimated Ω = 1 points in this data set, $\ln \{ \tau \} = 36,753\,(1/T) - 104$ with $r^2 = 0.900$, so the calculated values are: $A = 1.606 \times 10^{45} (s^{ - 1} )$ and $E_a = 3.06 \times 10^5 ({\textrm{J}} \cdot {\textrm{mole}}^{ - 1} )$, and $T_{{\textrm{crit}}} = 80^ \circ {\textrm{C}}$.

3.3.2.1.5 Deriving Damage Coefficients from Survival Curves

Another format typically used to present thermal damage data is the cell survival curve, derived from cell culture experiments. The simplest format in which to calculate A and E _a is a plot of surviving fraction vs. time at a fixed temperature. In that case the surviving fraction is simply C(τ), and a straight forward calculation can be done for Ω(τ), from which A and E _a can quickly be extracted by curve fitting.

Alternately, the time required for the population to decrease by a factor of e ⁻¹ may also be used; see Fig. 13.20 [85]. D₀ is the time for the colony formation rate to decrease by e ⁻¹:

$$\frac{S}{{N_0 }} = e^{\frac{{t_0 - t}}{{D_0 }}}$$

((13.46))

where N ₀ = the initial cell count measured at time t ₀, and S = surviving cell count at time t.

figure 13_20_100508_2_En — **Fig. 13.20**

This parameter is plotted for asynchronously dividing Chinese Hamster Ovary (CHO) cells, and for CHO cells in the G1 phase in Fig. 13.20. The G1 phase is cell growth immediately after a new cell is produced. The asynchronous cells have a random distribution of growth phases. CHO cells are common model cell lines used in tumor hyperthermia studies. By Eq. (13.46) the relationship for D₀ reduces to:

$$\ln \left\{ {\frac{{C\left( 0 \right)}}{{C\left( \tau \right)}}} \right\} = 1 = \int_0^{D_0 } {A\;e^{\left[ {\frac{{ - E}}{{RT}}} \right]} } \;dt$$

((13.47))

from which it may quickly be determined that:

$$\ln \left\{ {D_0 } \right\} = \frac{E}{{RT}}\; - \;\ln \left\{ A \right\}$$

((13.48))

and the Arrhenius plot may be constructed as previously described. For the asynchronous CHO cells in this data set at temperatures above the break point (43°C) $A = 2.84 \times 10^{99} ({\textrm{s}}^{ - 1} )$ and $E_a = 6.18 \times 10^5 [{\textrm{J}} \cdot {\textrm{mole}}^{ - 1} ]$, and ${\textrm{T}}_{{\textrm{crit}}} = 51.4^ \circ {\textrm{C}}$.

The usual damage prediction in hyperthermia studies is to express the exposure in terms of “Cumulative Equivalent Minutes” of exposure at the breakpoint temperature, typically very near 43°C in most tissues. The goal in those studies is to ensure cell death in the tumor, so the threshold for “success” is usually taken to be CEM = 30–60 min, depending on the application:

$$CEM\,43 = \sum\limits_{i = 1}^N {R^{\left( {43 - T_i } \right)} \;} t_i$$

((13.49))

where t _i = the time of exposure [minutes] at temperature T _i [°C] and R is derived from a plot like Fig. 13.20. For most tissues above the breakpoint $R \simeq 0.5$, and can be calculated from:

$$\ln \left\{ R \right\} = \frac{{\ln \left\{ {\frac{{D_0 \left( {T_{{\textrm{break}}} } \right)}}{{D_0 \left( T \right)}}} \right\}}}{{\left( {T_{{\textrm{break}}} - T} \right)}}$$

((13.50))

Example 13.7:

This example derives Arrhenius coefficients from the CHO cell survival curve of Fig. 13.20 for temperatures below the break point, 43°C.

Solution:

(1)

Reading lower temperature values for synchronous CHO cells from the plot:

T (°C)	T(K)	D ₀ ⁻¹ (min⁻¹)	D ₀(s)	Ln{D ₀}
43	316.2	0.09	667	6.503
42.5	315.7	2.6 × 10⁻²	2308	7.744
42	315.2	1.6 × 10⁻²	3750	8.230
41.7	314.9	4.8 × 10⁻³	12,500	9.433

(2)
Linear regression fit, ln{D ₀} vs. T ⁻¹
$$\ln \left\{ {D_0 } \right\}\, = \,206,035\left[ {\frac{1}{T}} \right]\, - \,645.1$$
(3)
$\ln \{ A\} = 645.1$ means that $A = 1.46 \times 10^{280}$ (is this a frightening number, or what?)
(4)
$E/R = 206,035$ means that $E = 1.71 \times 10^6 [{\textrm{J}} \cdot {\textrm{mole}}^{ - 1} ]$.

In practice, Eq. (13.49) is the discrete form of a continuous integral for time-varying temperatures. It should be noted that the CEM43 damage calculation yields only an equivalent time of exposure and not a damage probability result. A CEM43 value can be converted into a probability calculation if D ₀(43) is known for the tissue, however, by application of Eq. (13.46).

In summary, Fig. 13.21 compares several of the representative processes selected from Table 13.4 in terms of their relative rates of damage accumulation vs. temperature.

figure 13_21_100508_2_En — **Fig. 13.21**

3.3.2.2 Transient Thermal History Experiments

Problems with the standard constant temperature approach to determining A and E _a are that: (1) with only very few exceptions, the tissue must be excised to perform the experiments, disrupting its normal blood perfusion and activating autolytic (post mortem degeneration and necrosis) processes; (2) no exposure is truly constant temperature, and rise time segments must necessarily be short compared to the total exposure time in order to be negligible; and (3) the methods are essentially impossible to use in laser damage studies since constant temperature exposures are not possible to obtain and the typical small spot sizes mean that substantial thermal gradients confound reliable determination of the temperature history. Working with transient thermal data adds a high level of uncertainty to the damage coefficients since the time of exposure is problematic. Nevertheless, estimates of rate coefficients can be made upon which treatment protocols can be evaluated.

Here we present a new method specifically applicable to transient temperature heating cases [106]. While it does not address the small-spot, steep thermal gradient problem, the new method does solve the transient temperature exposure problem so that shorter exposures may be used in a data set, thus reducing the uncertainty in the estimated damage coefficients. We can rearrange Eq. (13.44) for Ω = 1 experiments to see the basis for the method:

$$\ln \left\{ A \right\} = \left( {\frac{1}{{RT}}} \right)\;E_a - \ln \left\{ \tau \right\}$$

((13.51))

So, for a single constant temperature experiment resulting in Ω = 1, we can plot a straight line in the ln{A} vs. E _a plane, with slope 1/RT and intercept ln{τ}. An ensemble of such lines from several experiments would intersect at the value of ln{A} and E _a that describe the thermal damage process under study. This exercise provides no new information for constant temperature experiments, of course.

However, such a plot does provide a convenient basis for analyzing transient heating experiments. For any experiment in which C(t) or Ω(t) can be directly measured, determining A and E _a is a simple matter of curve fitting. Given an ensemble of experiments for which only the histologic endpoint Ω(τ) and the transient history, T(t), are known, we can assume values for ln{A} and E _a and calculate a relative “Cost” that is minimized when the integral of Eq. (13.40) is evaluated to calculate Ω (A, E _a):

$${\textrm{Cost}} = \ln \left\{ {\frac{{\int_0^\tau {A\;e^{\left[ {\frac{{ - E_a }}{{R\;T\left( t \right)}}} \right]} \;} dt}}{{\Omega \left( \tau \right)}}} \right\}$$

((13.52))

This is a standard optimization technique (Cost has no units). Here the “Cost” has been defined such that its magnitude approaches zero when the assumed values of ln{A} and E _a yield an integral equal to the measured Ω(τ) value.

A two-dimensional gray-scale plot of the magnitude of the Cost on the ln{A} vs. E _a plane using hundreds of assumed values of A and E _a shows that the locus of points for which the |Cost| approaches zero is a straight line (see Fig. 13.22). In Fig. 13.22a, a hypothetical low-energy damage process with $A = 26.5 \times 10^9 [{\textrm{s}}^{ - 1} ]$ and $E_a = 7 \times 10^4 \,[{\textrm{J}} \cdot {\textrm{mole}}^{ - 1} ]$ has been hypothetically heated in transient manner – i.e. with discrete temperatures shown in the plot calculated from a t ^0.5 formula – including 2 s of cooling. The resulting damage value is Ω = 0.9. In Fig. 13.22b the ln{A} vs. E _a plane is scanned for minima in the Cost, as calculated by Eq. (13.52) from the transient curve. In this figure the clarity of the functional behavior is enhanced by plotting the magnitude of (log₁₀{1/Cost}). Other transient heating experiments will yield an ensemble of plots such as Fig. (13.22b), each with differing slope and intercept.

figure 13_22_100508_2_En — **Fig. 13.22**

The slope and intercept of the line of Cost minima in Fig. 13.22b contain extremely useful information. Referring to Eq. (13.51), the slope is $(1/RT_{eq} )$ and the intercept is $\ln \{ \tau _{eq} \}$, where T _eq and τ _eq are, respectively, the temperature and duration of an equivalent constant-temperature experiment for which Ω = 1. It is then a simple matter to plot the ensemble of $(T_{eq} ,\tau _{eq} )$ points on a standard Arrhenius plot, from which A and E _a are obtained in the ordinary fashion, i.e. from the slope and intercept of the linear regression line through them. In practice, the appropriate range to scan in the ln{A} vs. E _a plane can quickly determined by a coarse discrete scan of the values, followed by a fine scan to refine the estimate of T _eq and τ _eq.

This approach has the advantage over other curve fitting approaches that it preserves the underlying physics and reveals processes and relationships. It turns out to be quite critical that accurate values for the temperature be used to calculate the integral of Eq. (13.52). Consequently, an experiment design in which the temperature is uniform over a relatively large area is required so that spatial averaging can be used to reduce uncertainty in both the temperature and the quantitative damage measure. This excludes almost all laser spot approaches to damage process estimation. Note that it is neither necessary nor advantageous to have Ω = 1 to make an Arrhenius plot when using this approach

Example 13.8

This example calculates the equivalent constant temperature experiment for the process described in Fig. 13.22. Here the hypothetical process has been integrated for the transient heating curve that resulted in Ω = 3.78 for the hypothetical thermal damage process. The search plane was estimated using the strategy outlined in the text. The line of convergence in Fig. 13.22b has these points:

Cost	ln{A}	E (kJ/mole)
3.33 × 10⁻³	62.75	181.4
4.05 × 10⁻³	64.25	185.7
1.28 × 10⁻³	65.58	189.6
5.23 × 10⁻³	67.33	194.6
5.04 × 10⁻³	68.75	198.8
3.70 × 10⁻³	70.75	204.5
3.93 × 10⁻³	72.25	208.9
2.73 × 10⁻³	74.33	215.0

The fit for this line is: linear regression

$$\ln \left\{ A \right\}\, = \,0.345E\left( {{\textrm{kJ}} \cdot {\textrm{mole}}^{ - 1} } \right)\, + \,0.201$$

So,

$$\frac{1}{{R\;T_{eq} }} = 0.345{\times}10^{ - 3}$$

$$\ln \left\{ {\tau _{eq} } \right\} = - 0.201$$

Or, $T_{eq} = 75.5^ \circ {\textrm{C}}$ and $\tau _{eq} = 0.818\,{\textrm{s}}$. For the original damage process $(A = 1 \times 10^{75} ,E = 5 \times 10^5 )$ a constant temperature exposure of 0.818 s at 75.5°C results in Ω = 0.677; close to Ω = 1. Note that τ _eq is much shorter than the original experiment time of 5 s because the resulting damage in the simulated experiment gave Ω larger than 1. For experiments that have Ω < 1, τ _eq will be longer than the experiment heating time. The difference between Ω = 0.677 and the expected value of 1 is most likely due to an accumulation of round off errors: note how small 0.201 is compared to ln{A} = 172.7 in Eq. (13.44), for example.

4 Summary

First order kinetic models for tissue damage are useful for predicting trends in laser damage experiments. Unfortunately, there are only a few damage processes for which the frequency factor and energy have been determined in situ – critical temperatures for known processes range from about 50 to 90°C. Nevertheless, these models can be used to make direct comparisons between numerical predictions of laser damage and histologic results, something that cannot, as yet, be achieved any other way. A very careful definition of the particular histologic end point is necessary. Also, to avoid the arbitrariness that characterizes many of the early damage studies, it is essential that a quantitatively measurable damage measure be identified. Birefringence intensity is one such variable; other excellent candidates include, for example, changes in optical properties such as the scattering or absorption coefficients on exposure to elevated temperatures.

In part because of the effects of biologic inhomogeneities and random fluctuations in tissue characteristics, and in part due to the difficulty of resolving small temperature differences, thermal damage data are inherently noisy. However, predicted boundaries in small spot size experiments compare favorably to those observed. This is probably due to the very steep thermal gradients typical of small spot experiments – that is, a rather large error in actual critical temperature may be swamped out by the very steep thermal gradient so that the location of the predicted damage contour may, in fact, agree fairly well with experimental histologic observation. Certainly, even though the results of a particular experiment may eventually prove impossible to duplicate in numerical models, a careful analysis of the trends which one would obtain from, say, changing beam power, spot size and duration can be studied in detail in the numerical model and on a spatial scale similar to that of microscopic observation. Also, the numerical model allows dissection of the transient development of thermal damage – something that cannot be achieved in any other way. So, while there are many uncertainties associated with kinetic models of thermal damage, they can be extremely illuminating and helpful in dosimetry planning.

References

Thomsen S. Pathologic analysis of photothermal and photomechanical effects of laser-tissue interactions. Photochem. Photobiolog., 53(6):825–835 (1991).
Google Scholar
Thomsen S. Identification of lethal thermal injury at the time of photothermal treatment. In: eds. G. Muller and A. Roggan (eds) Intersitial thermotherapy. SPIE, Bellingham, WA, pp. 459–467 (1995).
Google Scholar
Thomsen S. Qualitative and quantitative pathology of clinically relevant thermal lesions. SPIE Crit. Rev., CR75:425–459 (2000).
Google Scholar
Thomsen S. Non-thermal effects in thermal treatment applications of non-ionizing irradiation. Proc. SPIE., 5698:1–14 (2005).
Article ADS Google Scholar
Thomsen S and Coad JE. Developing clinically successful biomedical devices by understanding the pathophysiology of the target tissue: Insights from over 25 years at the microscope. Proc. SPIE., 6440:1–15 (2007).
ADS Google Scholar
Kumar,V, Abbas AK, Fausto N, and Mitchell RN. Robbins basic pathology. Saunders-Elsevier, Philadephia, 8th edition, pp. 1–224 (2007).
Google Scholar
Lee H-C and Wei Y-H. Mitochondrial role in life and death of the cell. J. Biomed. Sci., 7:2–15 (2000).
Article Google Scholar
Letal A and Scorrano L. Laying the foundations of programmed cell death. Cell Death Differ., 13:1245–1247 (2006).
Article Google Scholar
Chan D, Frank S, and Rojo M. Mitochondrial dynamics in cell life and death. Cell Death and Differ., 13:680–684 (2006).
Article Google Scholar
Coad JE. Thermal fixation: A central outcome of hyperthermic therapies. Proc. SPIE, 5698:15–22 (2005).
Article ADS Google Scholar
Metz CN. Fibrocytes: A unique cell population implicated in wound healing. Cell. Mol. Life Sci., 60:1342–1350(2003).
Article Google Scholar
Baum CL and Arpey CJ. Normal cutaneous wound healing: Clinical correlation with cellular and molecular events. Dermatol. Surg., 31:674–686 (2005).
Article Google Scholar
Blakytny R and Jude E. The molecular biology of chronic wounds and delayed healing in diabetes. Diabetic Med., 23:594–608 (2006).
Article Google Scholar
Williams P, Bannister LH, Berry MM et al. (eds). Gray’s anatomy. 38th international edition. Churchill-Livingstone. Edinburg, pp. 1–2092 (1995).
Google Scholar
Fawcett DW. Textbook of histology. Chapman and Hall, New York, 12th edition, pp. 1–964 (1994).
Google Scholar
Ghadiallay FN. Ultrastructural pathology of the cell and matrix, Vols. 1 and 2. Butterworth- Heinemann, Boston, pp. 1–1450 (1997).
Google Scholar
Alberts B, Johnson A, Lewis J et al. Molecular biology of the cell. Garland Science, New York, 4th edition, pp. 1–1463 (2002).
Google Scholar
Junquiera LC and Carneiro J. Basic histology: Text and atlas. McGraw-Hill, Medical Publishing Division, New York, 11th edition, pp. 1–502 (2005).
Google Scholar
Thomsen S. Practical pathology for engineers: How to do the job right the first time. Proc. SPIE, 4754:1–26 (2003).
Article Google Scholar
Welch AJ and van Gemert MJC. (eds). Optical-thermal response of laser-irradiated tissue. Plenum, New York, pp. 1–925 (1995).
Google Scholar
Niemz MH. Laser-tissue interactions: Fundamentals and applications. Springer, Berlin, 3rd edition (2004).
Google Scholar
Thomsen S, Pearce JA, Randeri R, and Chan E. Determination of isotherms of thermal damage. Proc. IEEE/LEOS, 2295–2296 (1995).
Google Scholar
Thomsen S. Mapping thermal injury in biologic tissues using quantitative pathologic techniques. Proc. SPIE., 3595:82–95 (1999).
Article Google Scholar
LeCarpentier GL, Motamedi M, Rastegar S, and Welch AJ. Simultaneous analysis of thermal and mechanical events during cw laser ablation of biological media. Proc. SPIE, 1064:107–113 (1989).
ADS Google Scholar
Verdaasdonk RM, Borst C, and van Gemert MJC. Explosive onset of continuous wave laser tissue ablation. Phys. Med. Biol., 35:1129–1144 (1990).
Article Google Scholar
Gijsbeers GHM, Selten FM, and van Gemert MJC. CW laser ablation velocities as a function of absorption in an experimental one dimentional model. Lasers Surg Med., 11 :287–296 (1991).
Article Google Scholar
Pearce JA and Thomsen S. Rate process analysis of thermal damage. In: AJ Welch and MJD van Gemert (eds) Optical-thermal response of laser irradiated tissue. Plenum, New York, pp. 561–606 (1995).
Google Scholar
Protsenko DE and Pearce JA. Thermo-electrical numerical model of electrosurgical RF cutting. Proc. SPIE, 4247:203–209 (2001).
Article Google Scholar
Nelson DL and M.M. Cox MM. Lehninger principles of biochemistry. W.H. Freeman and Co, New York, 4th edition, pp. 1–1119 (2005).
Google Scholar
Guyton AG and Hall JE. Textbook of medical physiology. Saunders Elsevier, Philadelphia, 11th edition, pp. 1–1116 (2006).
Google Scholar
Bray D. Cell movements: From molecules to motility. Garland, New York, 2nd edition, pp. 1–372 (2001).
Google Scholar
Boal D. Mechanics of the cell. University Press. Cambridge, UK, pp. 1–406 (2002).
Google Scholar
Ramachandran GN and Ramakrishan C. Molecular structure. In: Ramachandran GN and Reddi AH (eds) Biochemistry of Collagen. Plenum, New York. pp. 45–84 (1976).
Google Scholar
Fleischmajer R, Olsen BR, and Kühn K. Biology, chemistry and pathology of collagen. Annals NY Acad. Sci., 460:1–537 (1985) and 580:1–592 (1990).
Article Google Scholar
Uitto J and Perejda AJ. Connective tissue disease: molecular pathology of the extracellular matrix. Marcel Decker, New York, pp. 1–563 (1987).
Google Scholar
Chadwick DJ and Goode JA (eds). The molecular biology and pathology of elastic tissues. Ciba symposium 192. Wiley, New York, pp. 1–351 (1995).
Google Scholar
Flory P and Garrett RR. Phase transitions in collagen and gelatin systems. J. Am. Chem. Soc., 80:4836–4845 (1958).
Article Google Scholar
Lim JJ. Transition temperature and enthalpy change dependence on stabilizing and destabilizing ion in the helix coil transition in native tendon collagen. Biopolymers 15:2371–2381 (1976).
Article Google Scholar
Maitland DJ and Walsh JT Jr. Quantitative measurements of linear birefringence during heating of native collagen. Lasers Surg. Med., 20:310–318 (1997).
Article Google Scholar
Yoshimura H, Viator JA, and Jacques SL. Relationship between damaged fraction and reflected spectra of denaturing tissues. Lasers Surg. Med., 37:308–313 (2005).
Article Google Scholar
Schmidt SJ. Die Doppelbrechung von karyoplasma, zytoplasma und metaplasma. In: Protoplasma-Monographien. Vol. II. Verlag von Gebruder Borntraeger, Berlin, pp. 154–267 (1937).
Google Scholar
Fisher E. The birefringence of striated and smooth muscles. J. Cell. Comp. Physiol., 23:11–130 (1944).
Article Google Scholar
Peckham M and Irvin M. Myosin crossbridge orientation in demembranated muscle fibres studied by birefringence and X-ray diffraction measurements. J. Mol. Biol., 210:113–126 (1989).
Article Google Scholar
Harris P and Heath D. Structure and function of vascular smooth muscle. In: The human pulmonary circulation: Its form and function in health and disease. Churchill Livingstone, New York, pp. 161–182 (1986).
Google Scholar
Hulmes DJS, Miller A, Parry DAD et al. Crystalline regions in collagen fibrils. J. Mol. Biol., 184:473–477 (1985).
Article Google Scholar
Thomsen S, Pearce JA, and Cheong W-F. Changes in birefringence as markers of thermal damage in tissues. IEEE Trans. Biomed. Eng., BME-36:1174–1179 (1989).
Article Google Scholar
Thomsen S. Quantitative morphologic markers of laser thermal injury in cardiovascular muscle. Proc. SPIE, 1878:152–158 (1993).
Article ADS Google Scholar
Thomsen S, Jacques SL, and Flock S: Microscopic correlates of macroscopic optical property changes during thermal coagulation of myocardium. Proc. SPIE, 1202:2–11 (1990).
Article ADS Google Scholar
Thomsen S, Vijverberg H, Huang R, Schwartz J. Changes in optical properties of rat skin during thermal coagulation. Proc. SPIE, 1882:230–236 (1993).
Article ADS Google Scholar
Fielding CJ (ed). Lipid Rafts and Caveolae: From membrane biophysics to cell biology. Wiley-VCH, New York, pp. 1–294 (2006).
Google Scholar
Moussa NA, Tell EN, and Cravalho EG. Time progression of hemolysis of erythrocyte populations exposed to supraphysiological temperatures. Trans. ASME J. Biomech. Eng., 101:176–184 (1979).
Article Google Scholar
Lepock JR, Frey HE, Bayne H, Markus J. Relationship of hyperthermia-induced hemolysis of human erythrocytes to the thermal denaturation of membrane proteins. Biochim. Biophys. Acta, 980:191–201 (1989).
Article Google Scholar
Bernardi, P, Scorrano L, Colonna R, Petronelli V, and Di Lisa F. Mitochondria and cell death: Mechanistic aspects and methodological issues. Eur. J. Biochem., 264:687–701 (1999).
Article Google Scholar
Van Cruchten, S and Van den Brobeck W. Morphological and biochemical aspects of apoptosis, oncosis and necrosis. Anat. Histol. Embryol., 31:214–223 (2002).
Article Google Scholar
Shoshan-Barmatz V, Israelson A, Brdiczda D, and Sheu SS. The voltage-dependent anion channel (VDAC): Function in intracellular signaling, cell life and cell death. Curr. Pharm. Des., 12:2349–2270 (2006).
Article Google Scholar
Kim J-S, He L, Qian T, and Lemasters JJ. Role of the mitochondrial permeability transition in apoptotoc and necrotic death after ischemia/reperfusion injury o hepatocytes. Curr. Mol. Med., 3:527–235 (2003).
Article Google Scholar
Anghieri LJ and Robert J. Hyperthermia in cancer treatment. CRC, Boca Raton (1986).
Google Scholar
van der Zee J. Heating the patient: A promising approach? Ann. Oncol., 13:1173–1184 (2002).
Article Google Scholar
Beynon, R and Bond JS. Proteolytic enzymes. University Press. Oxford, 2nd edition, pp. 1–340 (2001).
Google Scholar
Thomsen S, Schwartz JA, Joseph R, Pearce JA, Rae B, and McMurray TJ. Temperatures associated with thermally induced red blood cell changes in tissues irradiated in vivo. Proc. SPIE, 2130:156–163 (1994).
Article ADS Google Scholar
Milleron RS and Bratton SB. “Heated” debates in apoptosis. Cell. Mol. Life Sci., 64:2329–2333 (2007).
Article Google Scholar
Cohen IK, Diegelmann RF, and Lindblad WJ. Wound healing: Biochemical and clinical aspects. W.B. Saunders, Philadelphia, pp. 1–630 (1992).
Google Scholar
Falanga V. Cutaneous wound healing. Martin Dinitz, London, pp. 1–484 (2001).
Google Scholar
DiPietro LA and Burns AL. Wound healing; Methods and protocols. Humana, Totowa, NJ, pp. 1–453 (2003).
Google Scholar
Grey JE and Harding KG. ABC of wound healing. Blackwell, Malden, MA, pp. 1–47 (2006).
Google Scholar
Keenan JH and Keyes FG. Thermodynamic properties of steam, 1st edition, 19th printing. Wiley, New York (1948).
Google Scholar
Van Wylen GJ. Thermodynamics. Wiley, New York (1964).
Google Scholar
Reynolds WC. Thermodynamics. McGraw-Hill, New York (1970).
Google Scholar
Torres JH, Motamedi M, Pearce JA, and Welch AJ. Experimental evaluation of mathematical models for predicting the thermal response of tissue to laser irradiation. Appl. Opt., 32(4):597–606 (1993).
Article ADS Google Scholar
Brutsaert W. Evaporation into the atmosphere: Theory, history and applications. D. Reidel, Dordrecht, Holland (1982).
Google Scholar
Yang D, Converse MC, Mahvi DM, and Webster JG. Expanding the bioheat equation to include internal water evaporation during heating. IEEE Trans. Biomed. Eng., 54(8):1382–11388 (2007).
Article Google Scholar
Moritz AR and Henriques FC. Studies of thermal injury II. The relative importance of time and surface temperature in the causation of cutaneous burns. Am. J. Pathol., 23:695–720 (1947).
Google Scholar
Moritz AR. Studies of thermal injury III. The pathology and pathogenesis of cutaneous burns: An experimental study. Am. J. Pathol., 23: 915–934 (1947).
Google Scholar
Henriques FC and Moritz AR. Studies of thermal injury in the conduction of heat to and through skin and the temperatures attained therein: A theoretical and experimental investigation. Am. J. Path., 23:531–549 (1947).
Google Scholar
Henriques FC. Studies of thermal injury V. The predictability and significance of thermally induced rate processes leading to irreversible epidermal injury. Arch. Pathol., 43:489–502 (1947).
Google Scholar
Maron SH and Lando JB. Fundamentals of physical chemistry. McMillan, New York (1974).
Google Scholar
Pearce JA, Thomsen S, Vijverberg H, and McMurray T. Kinetics for birefringence changes in thermally coagulated rat skin collagen. Proc. SPIE, 1876:180–186 (1993).
Article ADS Google Scholar
Welch AJ and Polhamus GD. Measurement and prediction of thermal injury in the retina of Rhesus monkey. IEEE Trans. Biomed. Eng., 31:633–644 (1984).
Article Google Scholar
Takata AN et al. Thermal model of laser induced eye damage. Final Report, USAF School of Aerospace Medicine, Brooks AFB TX, Contract F41609-74-C-0005, IIT Research Institute, Chicago, IL (1974).
Google Scholar
Birngruber R. Thermal modeling in biological tissue. In: F Hillenkamp, R Pratesi, and CA Sacchi (eds) Lasers in biology and medicine, Plenum, New York, pp. 77–97 (1980).
Google Scholar
Birngruber R, Hillenkamp F, and Gabel VP. Theoretical investigations of laser thermal retinal injury. Health Phys., 48(6):781–796 (1985).
Article Google Scholar
Diller KR, Pearce JA, and Valvano JW. Bioheat transfer, Chapter 4, Section 4. In: CRC Handb. Therm. Eng., (cat. no. 9581), pp. 4–114, 4–187 (2000).
Weaver JA, and Stoll AM. Mathematical model of skin exposed to thermal radiation. Aerospace Med., 40(1):24 (1969).
Google Scholar
Yang Y, Welch AJ, and Rylander HG. Rate process parameters of albumen. Lasers Surg. Med., 11:188–190 (1991).
Article Google Scholar
Sapareto SA. Physical aspects of hyperthermia. G Nussbaum (ed). American. Institute of Physics, Chapter 1 (1982).
Beckham JT, Mackanos MA, Crooke C, Takahashi T, O’Connell-Rodwell C, Contag CH, and Jansen ED. Assessment of cellular response to thermal laser injury through bioluminescence imaging of heat shock protein 70. Photochem. Photobiol., 79:76–85 (2004).
Article Google Scholar
Diller KR and Klutke GA. Accuracy analysis of the Henriques model for predicting burn injury. Advances in Bioheat and Mass Transfer, AM. Soc. Mech. Eng., New York, 268 (1993).
Google Scholar
Wu YC. A modified criterion for predicting thermal injury. Nat. Bur. Stand. Washington, District of Columbia (1982).
Google Scholar
Fugitt CE. A rate process of thermal injury. Armed Forces Special Weapons Project No. AFSWP-606 (1955).
Google Scholar
Takata AN. Development of criterion for skin burns. Aerospace Med., 45:634–637 (1974).
Google Scholar
VassiliadisA, Christian HC, and Dedrick KG. Ocular laser threshold investigations. Aerospace Med. Rep., F41609-70-0002 (1971).
Google Scholar
Miles CA. Kinetics of collagen denaturation in mammalian lens capsules studied by differential scanning calorimetry. Int. J. Biol. Macromol., 15:265–271 (1993).
Article Google Scholar
Jacques SL and Prahl SA. Modeling optical and thermal distributions in tissue during laser irradiation. Lasers Surg. Med., 6:494–503 (1987).
Article Google Scholar
Gaylor DC. Physical mechanism of cellular injury in electrical trauma. Ph.D. Thesis, Department of Electrical Engineering, Massachusetts Institute of Technology (1989).
Google Scholar
Toner M, Cravalho EG, Gaylor DC, and Lee RC. Cellular mechanisms of thermal injury in electrical trauma. Proc. Ann. Conf. Eng. Med. Biol., 4:1505–1506 (1990).
Google Scholar
Lee RC Gaylor DC, Prakah-Asante K, Bhatt D, and Israel DA. Skeletal muscle cell rupture by pulsed electric fields. IEEE Eng. Med. Biol. Soc. Annu. Conf., 712–714 (1987).
Google Scholar
Jacques SL and Gaeeni MO. Thermally induced changes in optical properties of heart. IEEE Eng. Med. Biol. Mag. 11(Part 4/6):1199–1200 (1989).
Google Scholar
Agah R, Gandjbakhche AH, Motamedi M, Nossal R, and Bonner RF. Dynamics of temperature dependent optical properties of tissue: dependence on thermally induced alteration. IEEE Trans. Biomed. Eng., 43(8):839–846 (1996).
Article Google Scholar
Flock ST, Smith L, and Waner MD. Quantifying the effects on blood of irradiation with four different vascular-lesion lasers. Proc. SPIE, 1882: 237–242 (1993).
Article ADS Google Scholar
Jacques SL, Newman C, and He XY. Thermal coagulation of tissues: Liver studies indicate a distribution of rate parameters not a single rate parameter describes the coagulation process. Proc. Winter Annu. Meeting Am. Soc. Mech. Eng. (1991).
Google Scholar
Matthewson K, Coleridge-Smith P, O’Sullivan JP, Northfield TC, and Bown SG. Biological effects of intrahepatic neodymium:yttrium-aluminum-garnet laser photocoagulation in rats. Gastroenterology, 93:550–557 (1987).
Google Scholar
Jacques SL, Motamedi M, and Rastegar S. Computer simulation of laser coagulation of prostate: A guide to dosimetry. Am Soc. Lasers Med. Surg. Lasers Surg. Med., Suppl 5, abstract 311 (1993).
Google Scholar
Skinner MG, Everts S, Reid AD, Vitkin IA, Lilge L, and Sherar MD. Changes in optical properties of ex vivo rat prostate due to heating. Phys. Med. Biol., 45:1375–1386 (2000).
Article Google Scholar
He X, McGee S, Coad JE, Schmidlin F, Iaizzo PA, Swanlund DJ, Kluge S, Rudie E, and Bischof JC. Investigation of the thermal and tissue injury behaviour in microwave thermal therapy using a porcine kidney model. Int. J. Hyperthermia, 20:567–593 (2004).
Article Google Scholar
Pop M, Molckovsky A, Chin L, Kolios MC, Jewett MA, and Sherar MD. Changes in dielectric properties at 460 kHz of kidney and fat during heating: Importance for radio-frequency thermal therapy. Phys. Med. Biol., 48:2509–2525 (2003).
Article Google Scholar
Pearce JA and Thomsen S. Arrhenius model thermal damage coefficients for birefringence loss in rabbit myocardium. Proc. ASME Int’l. Mech. Eng. Conf. Expos., IMECE2003-41986 (2003).
Google Scholar

Download references

Author information

Authors and Affiliations

500 Discovery View Drive, Sequim, WA, USA
Sharon Thomsen
Department of Electrical and Computer Engineering, The University of Texas, Austin, TX, USA
John A. Pearce

Authors

Sharon Thomsen
View author publications
You can also search for this author in PubMed Google Scholar
John A. Pearce
View author publications
You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sharon Thomsen .

Editor information

Editors and Affiliations

Dept. Biomedical Engineering, University of Texas, Austin, University Station 1, Austin, 78712, Texas, USA
Ashley J. Welch
Academic Medical Centre, Laser Centre, University of Amsterdam, Meibergdreef 9, Amsterdam, 1105 AZ, Netherlands
Martin J.C. van Gemert

Rights and permissions

Reprints and permissions

Copyright information

About this chapter

Cite this chapter

Thomsen, S., Pearce, J.A. (2010). Thermal Damage and Rate Processes in Biologic Tissues. In: Welch, A., van Gemert, M. (eds) Optical-Thermal Response of Laser-Irradiated Tissue. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-8831-4_13

Download citation

DOI: https://doi.org/10.1007/978-90-481-8831-4_13
Published: 10 November 2010
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-8830-7
Online ISBN: 978-90-481-8831-4
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)

Publish with us

Policies and ethics

Thermal Damage and Rate Processes in Biologic Tissues

Abstract

Similar content being viewed by others

Variable heat shock response model for medical laser procedures

An analytical solution of the bioheat model in a spherical tissue due to laser irradiation

Energy-Based Devices: Radiofrequency and High-Intensity Focused Ultrasound

Keywords

1 Introduction

2 Pathophysiology and Pathogenesis of Photothermal Lesions

2.1 General Principles

2.1.1 Primary Thermal Injury

2.1.2 Secondary (Delayed) Thermal Injury

2.1.3 Identification of Thermal Lesions

2.1.4 Factors Associated with Thermal Lesion Size, Development and Progression

2.1.5 Cells: Basic Units of Life and Death

2.2 Zones of Thermal Damage: Gross and Histologic Pathologic Features

2.2.1 Heat Source Volumes and Thermal Gradients

2.2.2 Ex Vivo and In Vivo Heating: No Survival

2.2.3 In Vivo Heating with Short Term Survival

2.2.4 In Vivo Heating with Prolonged Survival

2.3 Mechanisms of Thermal Injury

2.3.1 Primary (Direct) Thermal Injury

2.3.2 Tissue Mass Ablation

2.3.3 Carbon Formation and Carmelization of Tissue

2.3.4 Water Vaporization

2.3.5 Thermal Protein Denaturation (Thermal Protein Coagulation)

2.3.5.1 General Comments About Tissue Proteins

2.3.6 Thermal Denaturation of Proteins

2.3.7 Heating of Cellular Membranes

2.4 Mechanisms of Heat-Induced Cell Death

2.4.1 “Heat Sensitive” and “Heat Resistant” Cells and Tissues

2.4.2 Minimal Thermal Injury that Leads to Cell Death

2.4.3 Post-Mortem Necrosis

2.4.4 Cell Death Due to Rapid Thermal Coagulation (Heat Fixation)

2.4.5 Heat Fixation Vs. Pathophysiologic Post-Mortem Coagulation Necrosis: A Diagnostic Conundrum

2.4.6 Programmed Cell Death (Apoptosis)

2.5 Wound Healing

2.5.1 Organization

2.5.2 Tissue Regeneration and Tissue Repair

2.5.3 Granulation Tissue and Scar Formation

2.6 Role of Heat Shock Proteins (HSP) in Thermal Lesions

2.7 Mathematical Modeling of Pathophysiologic Thermal Effects

3 Thermodynamics and Kinetics of Thermal Damage Processes

3.1 Foundations of Thermal Analysis

3.1.1 Thermal Energy and Enthalpy

Example 13.1:

Solution:

Example 13.2:

Solution:

Example 13.3:

Solution:

Example 13.4:

Solution:

3.1.2 Entropy: The Second Law of Thermodynamics

3.1.2.1 Classical Thermodynamic Description

Example 13.5:

Solution:

3.1.2.2 Statistical Thermodynamic Description

3.2 Modeling Tissue Thermal Events

3.2.1 Sub-Coagulation Thermal Models

3.2.2 Evaporation of Surface Water

3.2.3 Water Vaporization

Example 13.6:

Solution:

3.2.4 Coagulation Processes

3.3 Kinetics of Thermal Damage Processes

3.3.1 Theoretical Foundations

3.3.1.1 Reaction Product Formation Rates

3.3.1.2 Unimolecular Process Descriptions

3.3.1.3 Arrhenius Formulations in Thermal Damage Studies.

3.3.1.4 Functional Behavior of the Damage Model

3.3.2 Experimental Determination of Rate Process Coefficients

3.3.2.1 Constant Temperature Exposures

3.3.2.1.1 Retina

3.3.2.1.2 Skin

3.3.2.1.3 Egg White and Egg Yolk

3.3.2.1.4 Birefringence Loss in Collagen

3.3.2.1.5 Deriving Damage Coefficients from Survival Curves

Example 13.7:

Solution: