Vascular Adaptation and Mechanical Homeostasis at Tissue, Cellular, and Sub-cellular Levels

Abstract

Blood vessels exhibit a remarkable ability to adapt throughout life that depends upon genetic programming and well-orchestrated biochemical processes. Findings over the past four decades demonstrate, however, that the mechanical environment experienced by these vessels similarly plays a critical role in governing their adaptive responses. This article briefly reviews, as illustrative examples, six cases of tissue level growth and remodeling, and then reviews general observations at cell-matrix, cellular, and sub-cellular levels, which collectively point to the existence of a “mechanical homeostasis” across multiple length and time scales that is mediated primarily by endothelial cells, vascular smooth muscle cells, and fibroblasts. In particular, responses to altered blood flow, blood pressure, and axial extension, disease processes such as cerebral aneurysms and vasospasm, and diverse experimental manipulations and clinical treatments suggest that arteries seek to maintain constant a preferred (homeostatic) mechanical state. Experiments on isolated microvessels, cell-seeded collagen gels, and adherent cells isolated in culture suggest that vascular cells and sub-cellular structures such as stress fibers and focal adhesions likewise seek to maintain constant a preferred mechanical state. Although much is known about mechanical homeostasis in the vasculature, there remains a pressing need for more quantitative data that will enable the formulation of an integrative mathematical theory that describes and eventually predicts vascular adaptations in response to diverse stimuli. Such a theory promises to deepen our understanding of vascular biology as well as to enable the design of improved clinical interventions and implantable medical devices.

Introduction

It has been believed, at least since the time of Galileo Galilei (1564–1642), that mechanical factors strongly influence biological form and function. Indeed, the general concept that mechanics influences biology was articulated clearly over a century ago by J. Wolff (1835–1902), who suggested that the trabecular architecture of bone is dictated primarily by directions of maximum tension that arise in response to external loading. Nevertheless, it has been much more recent that we have learned that mechanics influences biology down to the level of altered gene expression, which in many cases appears to contribute to a “mechanical homeostasis” at multiple length and time scales. Within vascular mechanobiology, for example, two of the first studies to provide direct correlations between altered mechanical stimuli and altered gene expression were reported in the mid-1970s.

Rosen et al. [1] hypothesized that “chronically elevated shearing stresses imposed on the endothelium by blood flow can induce subsequent permeability alternations through a mechanism similar to that of the delayed-prolonged inflammatory response.” To test this hypothesis, they cultured bovine endothelial cells on cover slips and subjected the cells to different levels of flow-induced shear stresses in a parallel plate experimental system. They found 2.3–3.7-fold increases (relative to no-flow controls) in the production of histidine decarboxylase in response to a 1.5 h exposure to mean shear stresses ranging from ∼0.3 to 0.6 Pa (Note: decarboxylation of the amino acid histidine yields histamine, which can affect endothelial permeability and smooth muscle contractility). Soon thereafter, Leung et al. [2] were motivated thus: “Recent investigations have revealed consistent qualitative and quantitative relationships between the composition of arterial walls and estimates of medial stress. These findings suggest that physical forces related to pressure and flow direct medial cell biosynthesis, thereby modulating structural adaptations to hemodynamic changes.” To test this hypothesis, they cultured vascular smooth muscle cells on sheets of elastin and subjected the cells to 10% cyclic uniaxial stretching for 2 days. They found that cyclic strain increased the production of collagens I and III, as well as hyaluronate and chondroitin sulfate, each by 3–5-fold relative to unstretched controls. Hence, both an altered mechanical stress (flow-induced shear) and an altered mechanical strain (substrate-induced stretching) correlate well with changes in the cellular production of important soluble and insoluble constituents (Fig. 1).

Fig. 1

Schema of two of the early observations that revealed a correlation between altered mechanical stimuli (either wall shear stress τ _w or cyclic in-plane stretch λ(t)) and altered gene expression in vascular endothelial cells (EC) and vascular smooth muscle cells (SMC). From Humphrey [40], with permission from Springer

Subsequent to these two seminal studies, countless others confirmed and extended the basic observation that altered mechanical stimuli can mediate altered gene expression by vascular cells, which in turn appears to contribute to a mechanical homeostasis. Indeed, a far reaching, ubiquitous nature of mechano-stimulated growth and remodeling within the vasculature was realized by the 1990s [3–10] and continues to expand to this day (cf. [11]).

Whereas mechanisms by which mechanics influences vascular biology exist at molecular and cellular levels, clinical presentations of vascular adaptation, disease, and responses to treatment often manifest at tissue and organ levels. Hence, there is a need to understand the role of mechanics in physiology and pathophysiology, not just molecular and cell biology. This article reviews diverse examples of vascular pathophysiology and mechanobiology, and then highlights common homeostatic tendencies across length and time scales relevant to tissue, cellular, and sub-cellular levels. By homeostasis, of course, we mean processes that attempt to maintain preferred form and function. In general, we imagine such biochemomechanical processes to result via a change in mass (i.e., growth, with negative growth called atrophy), a change in structure (i.e., remodeling), or both.

Clinical Manifestations

The blood vessel is no longer considered to be simply a non-thrombogenic passive conduit for blood flow. Rather, it is increasingly viewed as a continually adapting, physically and chemically interdependent network of elements with the common goal of maintaining optimal function in response to continually changing hemodynamic and metabolic conditions. Malek and Izumo [4]

Abdominal aortic aneurysms, arterial adaptations to microgravity, arteriogenesis, arteriovenous malformations, compensatory enlargement in atherosclerosis, cerebral vasospasm, dissecting ascending aortas, effects of exercise, senescence or smoking on arteries, hypertension, intracranial saccular aneurysms, Marfan syndrome, microvascular regression, pulmonary hypertension, remodeling of vein grafts, restenosis following intravascular stenting—the list of examples of vascular growth and remodeling in normal processes, disease progression, adaptations for altered environments, and responses to injury or treatment goes on and on (Fig. 2). Whereas the interested reader is referred to other review articles for a discussion of some of the many mechano-stimulated vascular adaptations [9, 12–22], here let us consider but two illustrative examples.

Fig. 2

Schema of diverse normal processes, progressive diseases, vascular adaptations, and responses to injury that each appear to reflect a ubiquitous growth and remodeling process, most often consistent with an underlying mechanical homeostasis. From Humphrey [18], with permission from Springer

Cerebral Aneurysms

Intracranial saccular aneurysms are focal dilatations of the arterial wall that occur at or near bifurcations in the circle of Willis, the primary network of arteries that supplies blood to the brain. The natural history of these lesions is believed to consist of three critical phases: pathogenesis, enlargement, and rupture. The genesis is the subject of considerable debate, but it is generally agreed that unique mechanical features of cerebral arteries play fundamental roles. Cerebral arteries do not have an external elastic lamina; they have sparse medial elastin, lack supporting perivascular tissue, and have structural irregularities at the apex of their bifurcations [23–25]. Such factors appear to render these arteries susceptible to a local weakening under the persistent action of hemodynamic loads, particularly in hypertension. Rupture occurs when local wall stress exceeds wall strength; fortunately, the rate of rupture tends to be <0.1% per year, which suggests that these lesions tend to be structurally robust. Although precise failure mechanisms remain unclear, rupture usually occurs at the fundus (pole) despite the neck often being thinner [23]. This finding was long perplexing, but mechanical calculations now suggest that maximum intramural tensile stresses typically occur near the fundus if the lesion does not remodel optimally [26, 27].

Just as mechanical factors play important roles in the genesis and rupture of aneurysms, they also play important roles in the enlargement. Saccular aneurysms enlarge from an initially small out-pouching of the arterial wall (which likely arises due to the loss of elastin, then smooth muscle) to lesions often having diameters of 5–10 mm and consisting primarily of fibrillar collagen. Diverse molecular, cellular, and biomechanical studies now suggest that this enlargement results in large part from mechano-stimulated growth and remodeling (Fig. 3). Indeed, Peters et al. [28] said it well: “aneurysmal dilation results in a highly dynamic cellular environment in which extensive wound healing and tissue/extracellular matrix remodeling are taking place.” As in other arterial adaptations (e.g., responses to sustained alterations in blood flow, blood pressure, or axial extension), there is a significant increase in matrix metalloproteinase (MMP) activity, the synthesis of extracellular matrix proteins, particularly collagens I and III by resident fibroblasts, and apoptosis of smooth muscle cells in saccular aneurysms [25, 29–32], all hallmarks of significant growth and remodeling. In particular, it appears that new collagen fibers are synthesized and organized under stress in alternating layers (as in a multi-ply engineering composite material), and along great circle trajectories [33]. Biomechanical calculations suggest further that this evolving microstructure tends to return intramural stresses back toward baseline values following the increase in stress that occurs due to the early loss of elastin and smooth muscle [34]. In other words, although long believed to be a “disease process,” it appears that the enlargement of a cerebral aneurysm may be a protective, adaptive response to an initial insult that is driven in large part by mechanical stimuli and tends to yield a surprisingly robust structure having intramural stresses near normal values. Yet, many lesions rupture (∼32,000 per year in the United States alone), which may result from local imbalances in the removal (via MMPs) and production (via synthesis by fibroblasts) of collagen, which in turn is likely influenced by the changing hemodynamic loads within the lesion as it enlarges. There is, therefore, a pressing need to understand better the inter-relationships among the hemodynamics, wall mechanics, matrix biology, and cell biology.

Fig. 3

Schema of an intracranial saccular aneurysm emphasizing that tissue-level mechanical loads (pressure, P, and flow-induced wall shear stress τ _w) influence cellular-level tensions (T ₁ and T ₂) and thus structural adaptations via an altered regulation of collagen deposition (influenced by the tensions and growth factors, GF) and degradation via matrix metalloproteinases (MMPs). From Humphrey [18], with permission from Springer

Cerebral Vasospasm

Rupture of intracranial aneurysms is the leading cause of non-traumatic subarachnoid hemorrhage. For those patients who survive to hospitalization, the period 3–7 days following rupture is critical because during this time, there is often a marked short-term reduction in the lumen of, and diminished blood flow within, major cerebral arteries in the vicinity of the hemorrhage. This so-called cerebral vasospasm is believed to contribute to a delayed distal cerebral ischemia or infarction, thus rendering it the leading cause of morbidity and mortality in these patients. Although vasospasm remains enigmatic, much has been learned [see 25, 35–39], and it now appears that it is yet another manifestation of arterial growth and remodeling in altered vasoactive states [40]. In particular, the organizing extravascular clot increases the concentration of vasoconstrictors (e.g., serotonin, thromboxane, and thrombin) presented to the arterial wall while increasing the concentration of nitric oxide scavengers (e.g., reactive oxygen species, oxyhemoglobin). This initial shift toward constrictors causes an acute, likely maximal, net vasoconstriction (∼30% reduction in diameter in most cerebral arteries). The presence of multiple mitogens [e.g., platelet-derived growth factor, transforming growth factor-beta (TGF-β), and vascular endothelial growth factor] also appears to promote the proliferation of medial smooth muscle cells and adventitial fibroblasts in the evolving vasoconstricted states, as well as possibly to cause changes in the smooth muscle that favors a leftward shift in the active force–length curve (i.e., a decrease in the diameter at which maximum contraction occurs). Whereas cell proliferation and the synthesis of fibrillar collagens may increase the thickness of the wall, and consequently increase the structural stiffness as revealed by altered pressure–diameter curves, the leftward shift in the active force–length curve may exacerbate the initial vasoconstriction (gradually to greater than ∼50% reduction in diameter). An extreme constriction, in turn, may disrupt endothelial cells due to corrugation of the intima, which can cause endothelial dysfunction (thus the transient loss of a wall shear stress induced production of nitric oxide (NO) and prostacyclin, two vasodilators) and possibly increase endothelial permeability, while up-regulating multiple adhesion and chemotactic molecules (e.g., vascular cell adhesion molecule-1 and monocyte chemoattractant protein-1) that promote both inflammatory (involving cytokines such as the interleukins and tissue necrosis factor-alpha) and degradatory (via macrophages and their production of MMPs) responses by sequestering and activating lymphocytes and mononuclear phagocytes. Together, therefore, the extravascular blood clot and disruption of the endothelium (Fig. 4) appear to set into motion a heightened growth and remodeling of the arterial wall in evolving vasoconstricted states. This leads to a significantly narrowed lumen and thickened wall (i.e., a stiffer structure), as well as a marked increase in collagen mass fraction (i.e., a stiffer material). Such a net change in geometry, active and passive properties, and perivascular constraints (possibly including an increased intracranial pressure and a mass effect due to the organizing clot) appears to be dominated by the chemobiology early on. As the clot dissolves and the endothelium recovers, however, it appears that the altered biomechanics, particularly a decreased intramural stress (which may promote atrophy of the wall; [41]) and an increased wall shear stress (which may stimulate the production of the vasodilator NO; [6]), both relative to normal/homeostatic values, may allow the mechanobiology to dominate again, such that a flow-induced growth and remodeling in vasodilated states can restore the lumen, geometry, and properties toward normal [42]. Clinically, such recovery may take 4 weeks or more, if the patient survives the severe effects that may initiate between 3 and 7 days and peak between 5 and 12 days following rupture. In summary, as noted by Mayberg et al. [35], “cerebral arteriopathic changes after subarachnoid hemorrhage may represent one component of a common vascular response to injury, which may be mediated by similar processes” and “structural changes may act in concert with contractile mechanisms to alter normal physiologic responses and maintain a narrowed lumen.”

Fig. 4

Schema of a constricted and remodeled arterial wall in cerebral vasospasm due in large part to the many molecules that are presented to the wall by the extravascular clot, but also to associated alterations in the mechanical loading. The nomenclature is: endothelial cell (EC), smooth muscle cell (SMC), fibroblast (FB), myofibroblasts (myo-FB), white blood cell (WBC), monocyte (M), platelet (PL), macrophage (MΦ), endothelin-1 (ET-1), thromboxane (TXA₂), platelet-derived growth factor (PDGF), vascular endothelial growth factor (VEGF), tissue necrosis factor-alpha (TNF-α), interleukin-1,6 (IL-1,-6), hemoglobin (Hb), reactive oxygen species (ROS), matrix metalloproteinase (MMP), and von Willebrand factor (vWF). From Humphrey [40], with permission from Springer

We considered here but two examples of arterial growth and remodeling in disease, yet perusal of the literature reveals common findings across the many different cases of vascular development, adaptation, disease progression, and responses to injury or clinical treatment [18]. We conclude, therefore, that mechanical stimuli have a profound effect on vascular biology, physiology, and pathophysiology, and there is a pressing need to understand better the associated mechanobiology and biomechanics.

Basic Concepts of Continuum Biomechanics

The mechanical motions, which take place in an animal body, are regulated by the same general laws as the motions of inanimate bodies. T. Young (1773–1829)

Simply put, mechanics is the study of responses of materials and structures to applied loads; thus, biomechanics can be defined as the development, extension, and application of mechanics for purposes of understanding better the influence of applied loads on the structure, properties, and function of living things. Mechanics is a remarkably diverse and storied field within physics, encompassing the discrete mechanics of I. Newton (1642–1727) and J. Lagrange (1736–1813), the continuum mechanics of L. Euler (1707–1783) and A. Cauchy (1789–1857), the statistical mechanics of J. Maxwell (1831–1879) and L. Boltzmann (1844–1906), the quantum mechanics of M. Planck (1858–1947) and W. Heisenberg (1901–1976), the relativistic mechanics of A. Einstein (1879–1955), and even modern thermodynamics. Herein, however, we focus on the classical theory of isothermal continuum mechanics, in which it is assumed that quantities of interest can be averaged locally, over appropriately chosen neighborhoods or representative volume elements, in order to provide meaningful point-wise descriptions of material behavior independent of any direct knowledge of molecular level interactions. A general rule of thumb is that continuum mechanics can be useful if the characteristic length scale of the microstructure is much less than the characteristic length scale of the physical problem. For example, continuum mechanics can be used equally well to study stresses in an artery, wherein diameters of load bearing proteins such as elastin and collagen fibers (μm) are much less than overall wall thickness (mm), and stresses in an isolated cell, wherein diameters of load bearing cytoskeletal proteins such as actin and microtubules (nm) are much less than cell thickness (μm). As noted by Truesdell and Noll [43], however, “Whether the continuum approach is justified, in any particular case, is a matter, not for the philosophy or methodology of science, but for the experimental test...” The last four decades reveal that continuum biomechanics has been very successful in describing the mechanical behavior of tissues and cells subjected to diverse conditions of interest as well as in estimating the stresses and strains that they experience (see e.g. [18, 44–47]).

Five general postulates govern the mechanics of continua: Balance of Mass, Balance of Linear Momentum, Balance of Energy, Balance of Angular Momentum, and the Entropy Inequality. That is, each of these five postulates must be satisfied (explicitly or implicitly) in every problem in continuum mechanics, which as suggested centuries ago by T. Young, and many others both before and since, includes responses by living and nonliving things alike. Whereas the first three of these postulates provide the so-called equations of motion (including equilibrium equations as special cases of linear momentum balance), which allow us to compute spatial and temporal changes in quantities of interest, the last two of these postulates provide theoretical restrictions on constitutive relations, that is, they ensure that descriptors of material behavior are physically meaningful (e.g., angular momentum balance requires that relations for the Cauchy stress be symmetric). Note, therefore, that a constitutive relation describes the response of a material to applied loads, which depends of course upon the internal make-up or constitution of the material (e.g., it distinguishes between the behavior of collagen and elastin or between actin and microtubules). It cannot be overemphasized that constitutive relations describe behaviors over conditions of interest; they do not describe the material itself. As a case in point, we have different constitutive relations for water depending on whether it is in its solid (ice), liquid (water), or gaseous (steam) form, that is, depending on the temperature and pressure of interest. Similarly, multiple constitutive relations will be useful for describing the biochemomechanical behavior of the same tissue, cell, or sub-cellular structure depending on conditions of interest—we should thus be careful not to argue over which relation is necessarily best (provided each is physically reasonable and does not violate fundamental requirements of mechanics), knowing that different relations can be equally useful under different conditions.

Among the earliest descriptions of material behavior in mechanics was that by R. Hooke (1635–1703), who also gave us the word “cell” in biology. Hooke described the response of simple linear springs as follows: the force varies as the extension, which in mechanics is written as f = kδ, where f is the applied force, k is the stiffness of the spring, and δ ≡ x − x _o is the extension of the spring (i.e., difference between the current x and original x _o length). It was realized later, however, that “force” and “extension” are not useful concepts for quantifying mechanical behaviors of “bulk materials,” thus they gave way to new concepts of “stress” and “strain” (formalized during the period 1757–1852; see [43]). Stress is often believed to be a measure of a force acting over an oriented area and strain is often believed to be a measure of changes in lengths or angles. Actually, however, stress and strain are much more general concepts; they are locally averaged mathematical (rigorously defined in terms of tensor operations), not physical, quantities that are defined point-wise in continua according to different definitions [18]. These concepts will be addressed in more detail below, but here it is important to note two related terms. “Residual stress” denotes stresses that exist in a body in the absence of external loads; in the classical context, this requires separate regions of self-equilibrating tensile and compressive stresses in the body.¹ One of the most important recent discoveries in arterial mechanics is that arteries are residually stressed (reported independently by Y.C. Fung and R.N. Vaishnav at conferences in 1983), which is revealed easily by excising a thin arterial ring and introducing a radial cut—the artery will spring open as it relieves the residual stresses. These stresses tend to homogenize the distribution of blood pressure induced stresses across the arterial wall [48], and thereby support the likelihood that there exists a preferred (homeostatic) mechanical state. “Prestress” often denotes stresses in a body that exist in a normal state in the absence of what might be considered as additional external loads; prestresses need not self-equilibrate, they are often dictated by normal displacement and traction boundary condition.² Arteries are axially prestressed, which probably results during development and depends largely on intramural elastin [49, 50]; this similarly appears to contribute to preferred mechanical states that may differ from vessel to vessel and in disease conditions [51, 52]. We shall see below that prestress is also recognized as an important aspect of biomechanics at cellular and sub-cellular levels.

We often think in biomechanics of three primary types of constitutive relations between measures of applied forces and resulting motions: stress depending directly on deformation, stress depending on a rate of deformation, and stress depending on a history of deformation. Proper relations respect the aforementioned basic postulates as well as other guiding principles [43]. Of course, converse relations can be written whereby deformations depend on stress, stress rates, or stress histories. Regardless, the first of these classes of relations is typically used to describe a solid-like “elastic” behavior, the second to describe a fluid-like “viscous” behavior, and the third to describe a combined solid-fluid-like “viscoelastic” behavior (which can be modeled in some cases by combining solid-like and fluid-like descriptors). Because soft tissues consist of insoluble extracellular matrix proteins in a viscous interstitial fluid and cells consist of insoluble cytoskeletal proteins in a viscous cytosol, viscoelastic behaviors are to be expected, even though elastic descriptors have often proven useful under many conditions. Again, the goal is to identify the simplest and yet physically realistic constitutive relation for the problem at hand.

Analogous to Hooke’s law for linear springs (f = kδ), probably the most widely known stress–strain relation for solid-like behaviors is the 1D linearly elastic form σ = Eε (also known as Hooke’s law, even though Hooke had no concept of stress or strain; see Appendix for more discussion), where σ is the true or Cauchy stress (obtained from the actual force acting over the actual oriented area), E is the Young’s modulus (a measure of material stiffness), and ε is the linearized or infinitesimal extensional strain (useful if both the strains and the rigid body rotations are small). Stiffness is an important constitutive property of a material; it is defined as a change in stress with respect to a change in strain (e.g., a local slope from stress–strain data or dσ/dε if the functional relationship is known). Although often not obvious at first sight, an analogous 1D relation can be used to describe small strain, linearly viscoelastic behaviors in response to sinusoidal loading at frequency ω, as, for example, σ(t) = E ^*(ω)ε(t), where E ^*(ω) = E _S(ω) + iE _L(ω) is the so-called complex modulus, with E _S the (elastic) storage modulus, E _L the (dissipative) loss modulus, and $i = {\sqrt { - 1} }$ (Note: E _L is zero if stress and strain are in phase, that is, if there is no viscous dissipation in the system, in which case this relation recovers the simple Hooke’s law. Moreover, an analogous relation can be written in shear as τ(t) = G ^*(ω)γ(t), where τ is the shear stress, G ^* is the complex modulus, and γ is the shear strain, with the complex shear modulus given in terms of storage and loss terms). Alternatively, constitutive relations for linearly viscoelastic behaviors are often postulated based on results for “mechanical analog models” consisting of a finite number of linear (elastic) springs and linear (viscous) dashpots arranged in series and/or in parallel (e.g., Maxwell and Kelvin–Voigt models). Linearized constitutive relations have proven very useful in the study of engineering materials, including many glassy polymers and metals undergoing small strains (often ε < 0.002), and elegant mathematical theories are available for solving associated initial-boundary value problems (see e.g. [53–55]). Nevertheless, it was recognized long ago that, other than bone and teeth, few biological materials exhibit a linear behavior under small strain [56, 57], and it is for this reason that soft tissue and cell biomechanics did not truly come into their own until the mid-1960s (see discussion in [44]), that is, until after development of a rational formulation of nonlinear field theories (cf. [43]) and methods for solving the associated initial-boundary value problems (cf. [58–60]). We shall return to such analyses below. For example, what may have appeared to be an esoteric mathematical formulation in Green and Adkins [58] enabled Chuong and Fung [48] to realize the importance of residual stresses in arteries as discussed below, and knowledge of the general theory of mixtures in Truesdell and Noll [43] allowed Mow et al. [61] to show its great utility in tissue mechanics.

Mechanical Homeostasis

Nature seems to operate always according to an original plan, from which she departs with regret and whose traces we come across everywhere. D’Azyr (c. 1784)

Tissue Level

It appears that Thoma [62] was the first to report that a sustained change in mechanical stimulus, an altered volumetric blood flow, within a segment of the vasculature gives rise to predictable adaptive responses. Specifically, he observed that the caliber of an artery tends to increase in regions of increased blood flow and decrease in regions of decreased blood flow. Murray [63] examined this observation mathematically within the context of an optimization problem. He reasoned that a blood vessel should have the largest caliber possible in order to reduce the resistance to flow and thus the workload on the heart; on the other hand, he reasoned that a blood vessel should have the smallest caliber possible, for there is a considerable metabolic burden to maintain large amounts of blood. Defining a cost function that accounted for these opposing needs, Murray showed that an optimal mathematical solution required that the cube of the luminal radius must be proportional to volumetric flowrate. Among others, Zamir [64] recognized that this “Murray’s law” could be interpreted as a “constancy of wall shear stress” if the mean wall shear stress is given by the solution of a steady, fully developed, laminar, one-dimensional, incompressible flow of a Newtonian fluid within a rigid circular tube (a rough approximation for an artery): τ _w  = 4μQ/πa ³, where μ is the viscosity of blood at high shear rates (∼3.5 cP), Q is the volumetric flowrate (e.g., in ml/min), and a is the deformed luminal radius. That is, if a ³ remains proportional to Q for a fixed viscosity, as suggested by Murray’s simple analysis, then τ _w must remain constant. This is a good example of how observations (Thoma) lead to theory (Murray), which in turn leads to experimentally testable hypotheses. Theoretical frameworks can thus play the same role in biology as in chemistry and physics.

Although the target value of wall shear stress varies from vessel to vessel and from animal to animal (often ∼1.5 Pa in large arteries; see [65, 66]), it is now generally accepted that most arteries regulate their lumen to maintain constant the wall shear stress at a preferred (homeostatic) value. See Pries et al. [7] for a similar discussion of the pressure–shear hypothesis for the microvasculature. Langille et al. [67] provided important insight into how flow-induced adaptations can be accomplished in arteries. Briefly, they found that in response to a decrease in wall shear stress, arteries decrease their caliber acutely via a vasoactive response (first 3 days); if the altered flow is sustained, the change in caliber is maintained by remodeling the extracellular matrix and smooth muscle of the wall at the new diameter (after ∼ 14 days), which shifts both the passive pressure–diameter response and the active length–tension response; periods between these two adaptations are characterized by partial vasoactive and partial remodeling responses. In all cases, it appears that the feedback control is driven by the attempt to maintain constant the wall shear stress, which as noted by Rosen et al. [1] and many others since (see e.g. [4, 6, 68]) governs endothelial production of a host of vasoactive and mitogenic factors that play direct roles in the adaptation (e.g. [69–71]).

The mean circumferential stress in a cylindrical blood vessel is given by Laplace’s relation, σ _θ  = Pa/h, where P is the transmural pressure, a is again the deformed luminal radius, and h is the deformed wall thickness (Note: although often derived for thin-walled tubes, this mean value of stress holds for thick-walled tubes exhibiting linear or nonlinear material behavior; [72]). Values of a/h tend to be 7–10 in large normal arteries, thus yielding values of stress of the order of 100–150 kPa (which is ∼5 orders of magnitude greater than mean wall shear stress) at mean arterial pressure. Recalling that the tendency toward a constant wall shear stress governs the value of the radius, a tendency toward a constant mean circumferential stress in response to changes in transmural pressure would thus require appropriate changes in wall thickness. Although wall thickness changes slightly during vasoactive responses (due to an isochoric constraint; [44]), marked changes in thickness require changes in smooth muscle (hyperplasia or hypertrophy) or extracellular matrix (deposition or degradation) mass. Wolinsky and Glagov [73] used Laplace’s relation to interpret observations that pointed strongly to a mechano-control of aortic structure in both development and disease. They found that, in mammals ranging from mice to rabbits to dogs to pigs to humans, the number of elastic lamellae in the aorta increases in proportion with diameter, which in turn increases in proportion to wall thickness. They concluded that “the average tension per lamellar unit of aortic media was remarkably independent of species and very nearly constant” (∼2 N/m), thus suggesting a constancy of circumferential stress during development. Wolinsky [74] later suggested that in hypertension there is a “possibility that tension, either directly or indirectly, provides the stimulus for elaboration of these [elastin and collagen] fibrous proteins” that contribute to thickening of the wall (i.e., by thickening existing lamellae, not increasing in their number). Indeed, it now appears that such thickening works over a period of a few weeks to maintain the circumferential stress near its preferred value [75, 76].

Conversely, decreased wall thickness occurs over periods of 21 days in response to decreased transmural pressures, as, for example, when the aorta is cuffed [41]. Indeed, such stress-mediated changes in wall thickness appear to explain the “compensatory enlargement” described by Glagov et al. [77], whereby the media thins due to a stress shielding by stiff atherosclerotic plaques. In other words, an initially increased circumferential stress appears to promote a thickening that restores circumferential stress toward normal and an initially decreased stress appears to promote an atrophy that also returns this stress toward normal. A similar atrophy due to decreased stress is well known in bone; compressive stresses also play a key role in bone and cartilage, but its role in vascular mechanics is largely unknown.

The mean axial stress in a cylindrical blood vessel is given by σ _z = f/πh(2a + h), where f is the total applied axial force (result again valid for thin or thick wall vessels). If constancy of wall shear stress primarily governs the luminal radius a, and constancy of circumferential stress primarily governs the wall thickness h, then a constancy of axial stress would govern the internal axial force f, which is not directly measurable in vivo. Induced alterations in axial stress should correlate with increases or decreases in axial stretch, the normal value of which (prestretch) appears to be set during development. Values of axial prestretch are observed easily when a vessel is excised. It has been shown, for example, that the prestretch in carotid arteries in normal canines, rabbits, and rats tends to be 1.56, 1.62, and 1.69, respectively (see discussion in [18]); prestretch in the aorta differs from about 1.2 near the arch to 1.5 in the abdominal aorta [78]. Hence values of these finite prestretches vary from species to species and with location along the vascular tree, but they (or the associated stresses) appear to be preferred nonetheless. Note, too, that hypertensive arteries tend to retract less than matched normotensive arteries [51], thus revealing complex couplings between circumferential and axial effects.

Jackson et al. [79] reported very important findings with regard to growth and remodeling in response to changes in axial extension (or stress). Briefly, they showed in rabbit carotid arteries that a surgically imposed 22% increase in axial extension (above the normal value of 1.62) accelerated proliferation rates for endothelial cells and smooth muscle cells, induced “unprecedented” accumulations of extracellular matrix, and heightened MMP activity. Together, these changes reduced the in vivo stretch from 97% at surgery to 72% at 3 days and to 67% at 7 days (not significantly different from the normal value near 62%); that is, “the stretched artery grew into its new length” independent of changes in circumferential stress or wall shear stress. Indeed, Gleason et al. [80] showed in organ culture studies that overextended mouse carotid arteries began to grow longer and thereby restore their axial prestretch (stress) toward normal in only 2 days. Gleason and Humphrey [81] showed that such changes in response to altered axial stretch can be accounted for qualitatively by a mathematical model of arterial growth and remodeling that similarly accounts for changes in response to altered flow and pressure [82, 83], thus supporting the hypothesis that structural changes are manifestations of common underlying mechano-controlled processes. Complicating this issue, however, is the observation by Jackson et al. [84] that significant reductions in axial stretch tend to cause an artery to become tortuous, which is not ameliorated by growth and remodeling processes over a 5-week study period, but is prevented entirely by the MMP-inhibitor doxycycline. There remains, therefore, more to learn about mechanically induced growth and remodeling in the vasculature in response to altered axial loads as well as altered blood flow and pressure.

In summary, the arterial wall appears to change its geometry and microstructure (and hence material properties) in response to sustained alterations in mechanical loading (Fig. 5) so as to maintain constant diverse mechanical parameters [16, 18, 44], with the stress defined by Cauchy appearing to be a useful correlate in many cases. In other words, there appears to be a tissue-level mechanical homeostasis that governs cases ranging from normal vascular development, adaptations to sustained changes in flow, pressure, or axial extension, disease processes, and responses to injury or clinical treatment. Because vascular cells (primarily endothelial, smooth muscle, and fibroblast) are responsible for such processes, one can easily imagine a similar mechanical homeostasis at the cellular level. Let us now consider observations from the literature that support such a hypothesis.

Fig. 5

Schema of three of the primary loads that normally act on an artery: flow-induced wall shear stress τ _w, pressure-induced circumferential (or, hoop) wall stress σ _θ , and axial load-induced axial wall stress σ _z. This reminds us that the arterial wall, and the cells within, experiences complex, multiaxial mechanical stimuli, and that function is likely optimized simultaneously relative to all aspects of this loading. For this reason, basic morphological measurements should always include changes in radius, thickness, and length

Cell-Matrix Level

Elsdale and Bard [85] showed the utility of using cells seeded in collagen gels as model systems to study cell–matrix interactions. Bell et al. [86] showed that fibroblasts remodel hydrated collagen gels by compaction and exclusion of water, and suggested that they accomplish this via a “tension exerted by cell processes on the lattice.” Harris et al. [87] subsequently observed that different types of cells exert different degrees of traction on their substrate that are well in excess of the tractions needed for migration; indeed, less motile cells (e.g., fibroblasts) exerted greater tractions on the matrix than did more motile cells (e.g., macrophages). Consistent with observations by Bell and colleagues, they suggested that these tractions (distinct from contractions generated by muscle cells) enable cells to “rearrange and re-pack collagen” and thereby to organize the extracellular matrix as needed in morphogenesis, wound healing, and adaptation. Tomasek and Hay [88] later emphasized the importance of the actin cytoskeleton in such processes. Excellent reviews on this important topic can be found in Tomasek et al. [89] and Grinnell [90], but let us focus on two observations that are particularly relevant to our discussion here.

Delvoye et al. [91] quantified force generation by fibroblasts seeded within 3D collagen matrices. Briefly, they allowed the cells to compact mechanically constrained matrices over a 6–12 h culture period, and thereby generate an “endogenous tensile force,” which they reported to be ∼5 nN per cell. When they subsequently changed the overall level of force by externally loading or unloading the matrix, they found that the fibroblasts relaxed or contracted to restore the overall force nearly to the original endogenous value (within 1 h). They suggested that “an equilibrium existed in the mechanical relationship of the cells with their support.” Brown et al. [92] confirmed these results, although they reported the endogenous force to be 0.5 nN per cell and that different periods were needed for the cells to restore the preferred mechanical state when perturbed from it. Subtle differences between studies are expected, of course, depending on initial collagen and cell density, type of cell, composition of the culture media, percent change in mechanical loading, and so forth. Nevertheless, as suggested by Brown and colleagues, these general observations suggest the existence of a “tensional homeostasis,” which they defined as “the control mechanism by which fibroblasts establish a tension within their extracellular matrix and maintain its level against opposing influences of external loading.”

Notwithstanding the utility of in vitro model systems consisting of cells embedded in prepared matrices, a particularly nice finding in intact, living arterioles was reported by Martinez-Lemus et al. [93]. Briefly, they isolated first-order arterioles and induced a sustained contraction using norepinephrine. They found that the overall diameter “adapted” to the new, contracted state after just 4 h. That is, whereas the initial contraction shortened the circumferentially oriented smooth muscle cells below their initial lengths, the cells readjusted their lengths (or, stress/strain) back toward baseline values over the 4 h period. This time-dependent relengthening manifested as an increased overlap between the cells. The authors concluded that smooth muscle cells “undergo a mechanoadaptation process involving ‘length autoregulation’ that would be energetically favorable for maintenance of a reduced diameter.” Not surprisingly, rapid changes in actin filament length have been proposed as a means by which smooth muscle cells can adapt quickly to changes in length [94], but obviously cell–cell, cell–matrix, and matrix–matrix changes can also play important roles [95].

Whereas many cell–gel culture studies from the early 1970s to the mid-1990s established that diverse cell types control the force (or more precisely, the stress) acting on the extracellular matrix in which they reside, the outstanding question was, why? Consider, therefore, the provocative observation of Grinnell [96]: “As long as the tissue is under mechanical stress, cell proliferation and biosynthetic activity will persist. Once mechanical stress is relieved ... cells will switch to a non-proliferative phenotype and begin to regress even in the continued presence of growth factors.” That is, it seems that maintenance of a preferred extracellular mechanical environment by cells is self serving—particular levels of stress promote health in a broad class of cell types (which can be classified as mechanocytes). Of course, if cells seek to maintain constant a particular mechanical state of the extracellular matrix in which they reside, a natural corollary is that they should likewise seek to maintain constant a particular internal mechanical state. This is, in effect, an extension of Newton’s third law: for every action, there is an equal and opposite reaction. Indeed, Brown et al. [92] suggested that cells exert tension on the extracellular matrix through their cytoskeleton, in which there may exist a “residual internal tension” due to “an internal balance between contractile forces of actin-myosin motor elements in microfilaments and compressive loading onto the microtubular elements which maintain cell shape... The resultant forces of these balanced cytoskeletal functions would be applied to the matrix through integrin attachment plaques to the ECM.” Let us thus consider related cellular and sub-cellular level observations.

Cell Level

Among others, Ingber [97] helped to renew interest in the importance of the mechanical properties of the cytoskeleton in many cellular processes, including cell spreading, migration, division, and even apoptosis. A particularly important finding in this regard was reported by Pourati et al. [98]. Whether by disrupting integrin attachments to the fibronectin substrate or by mechanically severing cytoskeletal filaments in isolated adherent endothelial cells, they showed qualitatively that a preexisting tension (or prestress) exists in cells that depends on an intact actin network but is independent of ATP and Ca²⁺ dependent processes. The same laboratory subsequently showed, using a laser nanoscissor to sever individual stress fibers in adherent endothelial cells, that actomyosin motor activity can also be important in controlling cytoskeletal prestress [99]. The latter article did not cite the former article, and thus did not discuss potential contributions by passive versus active prestresses, but it appears that the culture media differed in the two studies (e.g., serum-free in the study showing the importance of passive prestress and 10% fetal calf serum in the study showing the importance of active prestress). Because cells are highly responsive to both their chemical and mechanical environment, both factors must be considered carefully when interpreting such observations (cf. [91]). We can probably conclude, however, that cells possess multiple means to achieve the same end, which endows them with remarkable ability to respond to diverse perturbations from preferred chemical and mechanical environments.

Wang et al. [100] showed further that the prestress in cells relates directly to overall cell stiffness, even if “changes in cell stiffness in these cells were accounted for by induced changes in the prestress in the actin lattice that are secondary to changes in actomyosin force generation.” They reported prestresses in human airway smooth muscle cells from 0.5 to 2 kPa for values of cell stiffness from 0.1 to 0.5 kPa. Using an atomic force microscope to estimate stiffness in adherent fibroblasts, Mizutani et al. [101] confirmed that overall cell stiffness initially increases with step increases in stretch (cf. [98]). In addition, they showed that this stiffness tended to return back toward baseline values over a period of hours and that the converse occurred when cells were subjected to a step decrease in stretch. The authors concluded that “a fibroblast maintains an optimal level of intracellular stiffness,” thus supporting the aforementioned concept of a cell–matrix tensional homeostasis even at the cellular level. The restored stiffness (over 2 h, in 10% calf serum) in the Mizutani et al. study was due to altered actomyosin interactions, not “reconstruction of the actin network.” Again, therefore, we can infer that cells likely use different mechanisms in different situations in an attempt to maintain constant their prestress, or overall stiffness, as well as their extracellular mechanical environment.

Sub-Cellular Level

Realizing the difficulty of measuring cytoskeletal prestress directly, Costa et al. [102] devised a clever experiment to quantify the associated prestretch of actin stress fibers in adherent endothelial cells. Briefly, they cultured cells on prestretched membranes for up to 28 h (in 2% serum), after which they released the stretch in the membrane and measured the value of stretch at which the stress fibers began to buckle. Reasoning that the thin stress fibers would buckle when unloaded, they estimated fiber prestretch to be between 15 and 26%. A very important related observation showed that such studies must be performed quickly, for buckled stress fibers could disassemble and begin to reassemble in an unbuckled configuration within 1 min. We return to this important observation below. Here, simply note that using a direct method of measurement, Deguchi et al. [103] reported that the prestretch of stress fibers in adherent vascular smooth muscle cells is similarly about 21%, with a preexisting tensile force of about 10 nN (cf. a breaking force of 377 nN) in an individual actin bundle [104].

At this juncture, consider a popular conceptual and mechanical analog model of the cytoskeleton called tensegrity (Note: Wang et al. [100] give the following definition, “Structures in which the prestress is balanced predominantly by internal struts are called tensegrity structures”). Ingber [105] and colleagues suggest that the cytoskeleton should be treated as a self-equilibrating collection of prestressed elements, some of which are in tension (e.g., actin) and some of which are in compression (e.g., microtubules). They suggest further that a distinguishing characteristic of this class of structures is that “resistance to shape distortion, as expressed by the system’s stiffness, must increase in nearly direct proportion to the magnitude of the prestress” [100]. Although such a relationship can be consistent with calculations based on a tensegrity structure, it is likewise consistent with basic continuum concepts of tissue elasticity put forth much earlier by Fung [106]. For example, a 1D exponential stress–stretch relation inferred by Fung directly from experiments on a collagenous membrane yields precisely a linear relationship between stiffness (dσ/dλ) and stress (σ), namely

$$ \sigma = \frac{c} {b}{\left( {{\text{e}}^{{b(\lambda - 1)}} - 1} \right)}\,\,\, \to \,\,\,\frac{{{\text{d}}\sigma }} {{{\text{d}}\lambda }} = c + b\sigma $$

(1)

where σ is stress (or prestress), λ is stretch (or prestretch), and c, b are material parameters (recalling the linearized strain ε that appears in Hooke’s law, note that ε = λ − 1 in 1D, but one cannot relate directly the linearized strain tensor and the general stretch tensor). Studies on isolated actin filaments and actin bundles reveal a general exponential-like stress–stretch relation [103, 107] similar to that of most soft tissues [44], thus suggesting that this simple exponential stress–stretch relation could hold for actin filaments as well, and thereby could help explain the observation that cell stiffness correlates with cell prestress independent of the need to invoke a particular mechanical analog model of the cytoskeleton.

Notwithstanding the potential importance of compression elements in the mechanics of the cytoskeleton, Wang et al. [100] estimated that “compression of MTs [microtubules] balances 14% of the prestress, with the rest being balanced by stresses in the cell substrate to which the cell is adherent.” This very important observation reminds us that adherent cells are indeed prestressed, not residually stressed, and traction or displacement boundary conditions must be accounted for carefully at the basal surface. A similar situation should be expected for cells embedded in a 3D matrix. Hence, consider the mechanical role played by integrins or, in particular, focal adhesions (cf. Fig. 6). Briefly, focal adhesions consist of a cluster of integrins (i.e., heterodimeric transmembrane proteins) and multiple linker proteins (e.g., vinculin, paxillin, α-actinin, talin, and focal adhesion kinase) that physically connect the extracellular matrix to the cytoskeleton, particularly the actin network (see [108] for a detailed review of focal adhesions). A nice demonstration of the essential role played by focal adhesions in force generation/transmission from cell to matrix was given by Saez et al. [109]. They found that agonist-induced increases in contractile forces in smooth muscle cells correlated with the rapid recruitment of focal adhesion linker proteins to the plasma membrane (a 3–4-fold increase after 5 min of stimulation with 10⁻⁴ M acetylcholine), particularly near the periphery of the cell. They suggested that this facilitated the “transfer of mechanical stresses from the cytoskeleton across the cell surface.” More generally, however, they suggested further that “reorganization of the connections between the actin cytoskeleton and the extracellular matrix may enable smooth muscle cells to adapt their structure to the mechanical conditions present in their external environment at the time of activation.”

Fig. 6

Schema of possible ways that cells sense and respond to mechanical loads, including through cell–cell interactions (cadherins), growth factor receptors (with growth factor availability influenced by mechanical loads), G-proteins, and ion channels. Of particular interest here, however, is mechanotransduction via the extracellular matrix (ECM)—integrin (αβ)—cytoskeleton (CSK) axis, including sites of focal adhesion complexes (FAC) consisting of clusters of integrins. As can be seen, altered mechanical stimuli can result in altered cell function and properties

Whereas the study of Saez and colleagues revealed the importance of focal adhesions in transferring mechanical stresses from within the cell to the matrix (inside-out), Cunningham et al. [110] showed that focal adhesions similarly play an important role in the transfer of mechanical stresses from the matrix to the cell interior (outside-in). They found that an increased cyclic stretch (7% equibiaxial at 0.25 Hz) imposed on adherent smooth muscle cells via an underlying deformable substrate resulted in a rapid increase in focal adhesion associated paxillin and vinculin. Interestingly, this increase in linker proteins was transient, peaking after 1–5 min of stretching but returning toward baseline levels after 15–60 min of stretching. The rapidity of the increase in focal adhesion area was due to the recruitment of soluble proteins from the cytoplasmic store, thus avoiding the time delay required for frank protein synthesis. Cunningham and colleagues speculated that “new binding sites for vinculin are exposed by mechanical strain,” thus facilitating the transient increase soon after the initiation of cyclic strain. They did not address the subsequent return toward baseline values after longer durations of stretch, however, which obviously hints to the possibility of a rapid mechanical adaptation consistent with remarks by Saez et al. [109]. Again, we will return to this issue below.

Balaban et al. [111] showed further that the cross-sectional area occupied by focal adhesions increases directly with the force applied locally by an adherent cell on the substrate. That is, the cell appears to maintain constant the stress acting at a focal adhesion, with the preferred value on the order of 5 nN/μm² (i.e., 5 kPa). Given measured focal adhesion areas of the order of 1–10 μm², this means that forces exerted at single focal adhesions can be of the order of 10–30 nN, with forces per cytoskeletal filament estimated to be on the order of 1 pN (recall, too, the aforementioned estimate by Deguchi et al. [103] of 10 nN per stress fiber). They concluded that “the force applied by the cell on its substrate is closely linked to the assembly of the adhesion sites.” Tan et al. [112] confirmed this general finding using a different experimental system, but noted that the constant stress of about 4 kPa held for focal adhesions >1 μm² in area; smaller clusters were capable of generating larger forces. A logical extension of these two studies was reported by Goffin et al. [113]. Briefly, they defined supermature focal adhesions (suFA) having a linear dimension >7.5 μm, and suggested that suFA have different molecular compositions than regular focal adhesions, and that they support an approximately 3-fold larger stress (∼12 kPa). In fact, they suggested that suFA were needed to support the higher forces generated in actomyosin contractions involving α-smooth muscle actin (α-SMA) in myofibroblasts. Note, therefore, that α-SMA was found to associate with pre-existing actin stress fibers only in the presence of suFA and sufficiently stiff substrates (>16 kPa), perhaps suggesting a sequential response by cells when generating increasingly larger forces. Goffin and colleagues concluded that “FA size limits the maximum tension developed in stress fibers, as well as α-SMA recruitment” and speculated that “it is conceivable that stress fiber contractile function is rapidly adapted to a new mechanical challenge by modulating its molecular composition.”

Given the reported values of 4–5 kPa at normal focal adhesions, it is interesting to note that Kolodney and Wysolmerski [114] reported that steady state (over 7 days) isometric stress generation by non-muscle cells in collagen gels, with 10% fetal calf serum, was a comparable 4.5 kPa for chick embryo fibroblasts and 6.1 kPa for human umbilical vein endothelial cells. This endogenous stress generation decreased significantly when actin filaments were disrupted (cytochalasin D), but transiently increased 2-fold when microtubules were disrupted (nocodazole) with a return toward baseline after a few hours. Finally, they noted that these endogenous values could be increased to ∼10 kPa when the cells were stimulated with thrombin (0.2–1 U/ml), again comparable to the 12 kPa at suFAs. For purposes of comparison, Kolodney and Wysolmerski noted that reported stress generation in skeletal and smooth muscle cells can be as high as 200–700 kPa (recall, too, that homeostatic values of intramural stress in arteries appear to be 100–150 kPa; [18, 44]).

The reader interested in the role of focal adhesions in cell mechanobiology should consult the review by Discher et al. [115]; they emphasize the importance of the stiffness of the substrate/matrix in controlling many cell functions, including migration speed and strength of adhesion, which in turn have important implications for tissue level development, disease, and regeneration. For example, they report that cells cultured on soft gels (1 kPa stiffness) tend to have weak focal adhesions, whereas cells on stiff gels (30–100 kPa stiffness) tend to have strong focal adhesions. Likewise, they note that weaker traction forces are exerted on soft gels and stronger traction forces are exerted on stiff gels, with substrate strain tending to be nearly constant (∼3–4%) independent of substrate stiffness, hence suggesting that cells may seek to maintain constant the substrate/matrix strain. Although it is clear that cells sense, and likely seek to control, both their local external and internal mechanical environment, whether stress, strain, or some other continuum metric is the best correlate in any particular case remains unclear. Recalling, however, that constitutive relations describe behaviors of a material under particular conditions, not the material itself, stress and strain can be equally useful correlates for different conditions of interest or for the interpretation of different classes of experiments [116]. Indeed, a discussion of underlying mechanobiological mechanisms will ultimately need to be independent of mathematically defined continuum quantities such as stress and strain, and will need to focus on molecular level metrics such as conformational changes or inter-atomic forces (or energies). Nevertheless, continuum metrics are, and will continue to be, very useful correlates, particularly as we begin to address much needed multi-scale models to relate pathophysiology to the underlying mechanobiology.

Finally, let us return to the issue of rapid cytoskeletal remodeling in cellular adaptations to altered mechanical conditions. Na et al. [117] subjected vascular muscle cells to cyclic 10% equibiaxial stretch and, similar to Cunningham et al. [110], found a transient increase, relative to the unstretched controls, in focal adhesion associated paxillin and vinculin after 1–2 min of stretching but a return toward normal after 30 min of cyclic stretching. In addition, they used atomic force microscopy to show that this transient increase and then decrease in focal adhesion area correlated with a transient increase and then decrease in cell stiffness (which can correlate directly with cytoskeletal prestress [100], which in turn relates to cytoskeletal prestretches). Na and colleagues suggested that this finding implied a rapid adaptation to the new mechanical environment that was achieved by an assembly (polymerization) of new stress fibers at the new state and a disassembly (depolymerization) of the original stress fibers that were stretched beyond their preferred value (recall that Costa et al. [102] similarly reported a rapid disassembly and reassembly of stress fibers in response to an altered substrate strain). In other words, a graded transient increase in stiffness could be explained by an increased stretching of original filaments (cf. Eq. 1) plus the assembly of new, less stretched, filaments that are associated with new focal adhesions. Because the new filaments could be assembled at the original value of prestress (cf. [102, 103]), disassembly of the original filaments and associated loss of focal adhesion area could restore all stresses back toward original preferred values at the new mechanical state. In other words, a rapid assembly of filaments and focal adhesions having original values and disassembly of filaments and focal adhesions that were perturbed from their original mechanical state could return the cell to its original mechanical state, a process that would be consistent with a mechanical homeostasis or adaptation. It is interesting that a similar process was hypothesized at the tissue level [118] based on evidence collagen deposition and degradation.

Matrix Turnover

Given that many cellular and sub-cellular level studies suggest a central role for rapid growth (i.e., change in mass) and/or remodeling (i.e., change in structure) of functional cytoskeleton and integrins in mechanical homeostasis, recall that mechano-stimulated tissue level changes in geometry and properties likewise involve significant growth and remodeling of the extracellular matrix (e.g. [73]). As an illustrative example, consider briefly the production, organization, and degradation of fibrillar collagen in arteries in response to changes in mechanical loading; similar findings have been reported for elastin and the many proteoglycans found within the matrix (cf. [119]). Recall from Fig. 1 that Leung et al. [2] showed many years ago that the rate of production of fibrillar collagen by smooth muscle cells increases with increased mechanical stretching, a finding that has been confirmed many times (e.g. [120, 121]). Also Brown et al. [92] suggested that “mechanical forces would be important in regulating the amount and the orientation of newly deposited collagen ... increases in perceived strain would stimulate collagen production rates in fibroblasts, eventually falling again as the collagen matrix shielded the resident cells from the strain of applied loads.”

Production of functional fibrillar collagen is complex, involving multiple extracellular modifications in addition to usual transcriptional, translational, and post-translational processes [122–124]. Nevertheless, simple multi-step models based on first-order kinetics (e.g. [125, 126]) appear to capture salient features. Whereas rates of production of vascular collagen decrease dramatically from high levels in development to low basal levels in maturity (Note: the half-life of vascular collagen in maturity appears to be ∼70 days; [9]), slightly increased rates (up to 4.5-fold above basal values) can be recovered in maturity in cases of disease, injury, or altered mechanical loading (e.g. [127–129]). In addition to the amount and orientation of newly deposited collagen, one of the most important and yet overlooked issues is the value of stress/strain at which collagen is deposited (which in analogy to the cytoskeletal literature could be called a prestress or prestrain). Humphrey and Rajagopal [118] conjectured that “new constituents are likely deposited under stress, but this stress need not equal the stress in the neighboring, pre-existing constituents.” Tomasek et al. [89] put it this way in the context of wound healing: “Matrix shortening and increased stiffness indicates that the resident cells have locked a tension into the collagen structure.” That is, if indeed cells seek to maintain constant a preferred mechanical state in extracellular matrix, it should be expected that they would not just remodel extant matrix in an attempt to maintain or restore normalcy, but also deposit new matrix at the preferred value. Nevertheless, values of this deposition stress/stretch have yet to be measured; they have only been estimated based on mathematical modeling (e.g. [34, 82]), which reminds us of the pressing need for revealing experiments.

Meshel et al. [130] did not measure the deposition strain (or stress) of collagen, but they showed how fibroblasts can actively organize individual collagen fibers and probably establish an endogenous tension in a fiber. They found that cells extend lamellipodia, bind to a collagen fiber via integrins (e.g., α ₂ β ₁), retract the fiber via the action of non-muscle myosin heavy chain II-B, and then release the fiber such that it remains in the new configuration. Repetitive “hand-over-hand” cycles can move collagen fibers up to 2.5 μm/min and thereby introduce significant deformations and thus tensions. They conclude that “cells typically must bend collagen fibers to move them, which requires significant force, and fibres under tension are moved in the same way.” Related studies on elastin by Kozel et al. [131] and Czirok et al. [132] show that newly synthesized elastic fibers appear to be oriented and stretched (up to 50%) via cell-mediated deposition processes, again suggesting that cells can put an endogenous tension into newly deposited fibers (for more on vascular elastin, see [133–136]). Finally, Canty et al. [137] and Canty and Kadler [138] discovered a plasma membrane structure in chick embryonic fibroblasts that they call a “fibropositor.” They suggest that “collagen fibrils are continuous from the ECM through the lumen of the fibropositor. The sidewalls of the lumen appear to grip the collagen fibrils...and deposit the fibrils into hexagonally packed bundles.” It is easy to imagine, therefore, that cells can deposit fibrils under tension, literally by “work[ing] on the collagen they have secreted, crawling over it and tugging on it – helping to compact it into sheets and draw it out into cables” (Alberts et al. [139, p. 1101]).

Maintaining and adapting the extracellular matrix requires delicate balances between deposition and degradation. Although cells use multiple means to degrade matrix, the primary class of secreted molecules are the MMPs. For details on MMPs, and their primary inhibitors (tissue inhibitors of MMPs, or TIMPS), see the excellent reviews by Nagase et al. [140] and Newby [141]. Of particular relevance herein, MMP-1 is one of three “collagenases” that act primarily on fibrillar collagen, whereas MMP-2 and MMP-9 are “gelatinases” that act primarily on elastin, fibronectin, and denatured fibrillar collagen. Given that collagen and elastin are the primary structural proteins in the extracellular matrix of blood vessels, these three MMPs play particularly important roles in vascular biology and pathophysiology, although many other members of the family of MMPs are important. MMPs are secreted in an inactive, latent form (proMMPs) that must subsequently be activated. Cell-mediated production/inhibition and activation/inactivation of MMPs are achieved via both chemical and mechanical means [142], the latter both directly and indirectly. For example, Prajapati et al. [143] suggested that both the “duration and orientation of mechanical loads determine fibroblast cyto-mechanical activation,” including the production of MMPs. Interestingly, they also suggested that “cells were equally stimulated [by] an increase and decrease in matrix tension.” Alternatively, NO inhibits the induction of MMP-9 by vascular smooth muscle cells, and TGF-β inhibits the induction of MMP-1 [142]. Of course, the production of NO by endothelial cells is controlled in part by wall shear stress [144] and the production of TGF-β by smooth muscle cells is controlled in part by wall stretch [121], hence establishing an indirect link between altered mechanical stimuli and MMPs.

In summary, evidence suggesting the importance of mechanical homeostasis across tissue, cellular, and sub-cellular levels is diverse and irrefutable. Nevertheless, there remains a pressing need for much better quantification so that functional relationships can be established, which in turn will allow the development of a theory of vascular growth and remodeling and the associated mathematical modeling so that we can achieve predictive capability.

Need for Mechanical Dose Response Curves

Many of the attempts to apply mathematics to physiological problems were faulty because the empirical data were insufficient. C. Bernard (1865)

Recall again the seminal studies by Rosen et al. [1] and Leung et al. [2], depicted in Fig. 1, that demonstrated a strong correlation between mechanical stimuli and altered gene expression and thereby motivated many related studies. Although such studies reveal the widespread importance of vascular mechanotransduction, many do not admit detailed quantifications per se for they focus on but a few conditions—for example, no flow versus flow, or no stretch versus a single level of cyclic stretch. It is suggested here, therefore, that there is a pressing need for “mechanical dose response curves” that represent altered cellular activity as a quantifiable function of mechanical stimuli. Let us consider four excellent examples here.

Frangos et al. [145] studied the production of prostacyclin (PGI₂) by cultured human endothelial cells in response to different durations of steady and pulsatile wall shear stress imposed using a parallel plate flow chamber. PGI₂ is a potent vasodilator and the most potent endogenous inhibitor of platelet aggregation; its half-life is only ∼3 min. They found qualitatively similar results for steady and pulsatile flows, but that production was higher in response to pulsatile flow. They quantified the rate of production of PGI₂ in response to cyclic wall shear stress τ _w (from 0.8 to 1.2 Pa at 1 Hz) via the equation rate = D + BCe ^−Ct where D = 0.161 ng/min 10⁶cells, B = 1.86 ng/10⁶cells, and C = 0.135 min⁻¹. Plotting this relation reveals that following a sudden initiation of shear stress (at t = 0, which could represent the case of removing a vascular clamp or implanting a vein graft), the rate of production of PGI₂ decreases dramatically but soon (∼20–30 min) reaches a steady state value. Integrating this rate yields a cumulative production of A + Dt + B(1 − e ^−Ct). These investigators did not study the effects of different levels of shear stress, however; thus the main finding is the rapid achievement of a steady state rate.

Malek and Izumo [4] studied the production of endothelin-1 (ET-1) by cultured bovine aortic endothelial cells in response to steady and pulsatile shear stresses imposed using a cone and plate device. ET-1 is one of the most potent vasoconstrictors and is mitogenic for vascular smooth muscle. In response to a steady 1.5 Pa shear stress, cells began to align along the direction of flow after 6 h and were fully aligned by 18–24 h. This overall response was slow compared to mRNA expression for ET-1, which began to decrease within 1 h of initiating the 1.5 Pa shear stress and reached steady state (∼80% lower than baseline) within 2–4 h. Note that studies at different flows and with different viscosity media confirmed that shear stress, not flow, correlated best with ET-1 production, and complementary tests showed that neither turbulence nor 20% cyclic wall stretch were factors. Finally, they showed that mRNA expression of ET-1 decreased ∼70% in response to changes in shear from 0 to 2.0 Pa. Although they did not parameterize the associated changes, data at 0, 0.3, 0.8, 1.5, and 2.0 Pa are consistent with a decreasing sigmoidal relation between mRNA and wall shear stress applied for 4 h (Fig. 7). This is a very important result.

Fig. 7

Measured endothelial-derived endothelin-1 (solid circles) as a function of the level of flow-induced wall shear stress experienced by the cultured cells. A possible functional relationship is given by the equation $A(1 - {\text{e}}^{{ - \beta \tau ^{2}_{{\text{w}}} }} ) + D,$ though it is clear that more data are needed to ensure a robust fit. Data analysis by A. Valentin

Uematsu et al. [144] similarly studied the expression of mRNA for endothelial cell derived nitric oxide synthase (eNOS) by cultured bovine aortic endothelial cells in response to steady wall shear stresses imposed using a parallel plate flow chamber. eNOS is essential for converting L-arginine to L-citrulline plus nitric oxide, a potent vasodilator and inhibitor of smooth muscle proliferation. eNOS mRNA was increased slightly by 3 h after initiating a steady wall shear stress of 1.5 Pa, but increased thereafter with maximum values achieved by 18 h. They showed further that eNOS mRNA increased approximately 3-fold for mean values of wall shear increasing from 0 to 0.12, 0.3, and 1.5 Pa. This relation can similarly be described by a sigmoidal equation (Fig. 8), thus revealing a qualitative similarity with the ET-1 results.

Fig. 8

Measured endothelial-derived nitric oxide synthase expression (solid circles) as a function of flow-induced wall shear stress experienced by cultured cells. The solid line is given by the equation $A(1 - {\text{e}}^{{ - \beta \tau ^{2}_{{\text{w}}} }} ) + D.$ Data analysis by A. Valentin

Li et al. [121] showed that vascular smooth muscle cells increase their production of angiotensin-II (ANG-II) and TGF-β in response to increased mechanical stretch, where ANG-II and TGF-β are promoters of collagen synthesis [124]. For example, basal values of ANG-II and TGF-β1 production were 13 pg/ml and 0.05 ng/ml before stretch, but 55 pg/ml (4.2-fold increase) and 0.65 ng/ml (13-fold increase) when stretched cyclically at 10% for 24 h at 0.5 Hz. Figure 9 shows the overall relation between collagen production and 0, 5, 10, and 20% stretch for 24 h (cf. Fig. 1). Again we see a sigmoidal mechanical dose response curve (cf. Figs. 6 and 7), consistent with the plateau in collagen synthesis at 10% stretch reported by Li et al.

Fig. 9

Normalized collagen synthesis as a function of in plane stretch (solid circles) experienced by cultured smooth muscle cells. The solid line is given by the equation $A(1 - {\text{e}}^{{ - a\tau ^{2}_{{\text{w}}} }} ) + D.$Data analysis by A. Valentin. Compared to Fig. 1, we see that results at two mechanical states can reveal the existence of a mechano-control, but results at multiple states are needed to establish the functional relationship

These four examples are not the only reports wherein altered cellular responses were measured as a function of multiple values of the mechanical stimulus, but there are surprisingly few studies in the literature that allow one to begin to identify “mechanical dose response curves.” Moreover, even in cases such as we considered here, more data are needed to ensure robust parameter estimation. Knowing the importance of chemical dose response curves in vascular pharmacology, it is suggested that mechanical dose response curves are equally important and we should begin to focus on functional, not just statistical, analyses of data. Of course, determination of mechanical dose response curves having physiologic relevance will be complicated by the tensorial nature of stress and strain (i.e., multiple components at each point relative to each coordinate system of interest) and the multiaxial loading that exists in vivo. Although chemical dose response curves (typically based on scalar concentrations) are clearly less complex in principle, possible synergistic or antagonistic actions of diverse molecules must be quantified as well for blood vessels and vascular cells are typically exposed simultaneously to multiple molecules or classes of molecules (cf. Fig. 4).

Need for Mathematical Modeling

The success of reductionist and molecular approaches in modern medical science has led to an explosion of information, but progress in integrating information has lagged ... Mathematical models provide a rational approach for integrating this ocean of data, as well as providing deep insight into biological processes. 1998 NIH Bioengineering Consortium Report

Tissue Level

Theories of tissue level growth and remodeling must be consistent with the basic postulates of mechanics as well as accepted biomechanical analyses, including constitutive formulations and stress analyses. As noted above, linear theories of elasticity [53, 55] and viscoelasticity [54, 146] are elegant and they often yield mathematically tractable initial-boundary value problems, yet experience has demonstrated that nonlinear field theories [43, 147] are generally essential in soft tissue mechanics [44, 60]. Indeed, Fung [106] put it this way: “The main difficulty lies in the customary use of [the] infinitesimal theory of elasticity to media which normally exhibit finite deformations.” A simple example is the aforementioned case of residual stress in arteries. Although discovered independently by R.N. Vaishnav and Y.C. Fung in 1983 (see discussion in [18]), Vaishnav invoked the principle of superposition of solutions that holds in linear elasticity, whereas Fung appropriately used results for sequential motions from nonlinear elasticity (cf. [58]). Vaishnav’s calculations suggested that the calculated small residual stresses (∼3 kPa compression in the inner wall and 3 kPa tension in the outer wall) would not significantly modify the in vivo stresses that arise due to blood pressure (i.e., the early prediction of monotonically decreasing transmural values of stress that start at ∼600 kPa at the inner wall). In contrast, Fung’s calculations showed that because of the strong nonlinearities, this same residual stress actually tended to homogenize the transmural stresses, decreasing their originally computed values of ∼600 kPa to ∼100 kPa, the value that is generally accepted today. In other words, strong nonlinearities due to large deformations and complex material behavior can yield important, counter-intuitive findings. Although certain linearizations can be useful, as, for example, in stability analyses or computations of fluid–solid interactions [148, 149], one should always consider the exact (nonlinear) relations when possible.

General approaches to modeling organ or tissue level growth and remodeling can be classified in four categories: the aforementioned optimization approach introduced by Murray [63], the use of reaction–diffusion equations introduced by Turing [150] and formalized by Murray [151], the concept of kinematic growth introduced by Skalak [152] and formalized by Rodriguez et al. [153], and the concept of mass–stress relations introduced by Fung [44] and formalized by Humphrey and Rajagopal [18]. For a brief review of most of these ideas, see Taber [8]. One of the key contributions of the work of Skalak is that it appropriately brought biological growth within the purview of nonlinear continuum biomechanics. Nevertheless, it is suggested here that despite kinematic growth models providing reasonable results for diverse vascular adaptations (see [154–157]), they describe consequences of growth, not the processes by which growth and remodeling occur. That is, in Skalak’s approach, one typically prescribes evolution equations for changes in geometry or strain as a function of the difference in stress from a homeostatic value; such changes continue until stress is restored to normal and “optimal” growth kinematics are achieved. As suggested by Humphrey and Rajagopal [118], however, growth and remodeling in maturity need not be optimal nor return stresses to homeostatic values. Rather, differences in tissue-level stress (or any other convenient metric) from homeostatic appear to control rates of cell proliferation, apoptosis, and the synthesis and degradation of extracellular matrix; evolution relations should thus focus on mass production, removal, and organization, which often yield an effective adaptation but may yield an ineffective maladaptation.

Briefly, Humphrey and Rajagopal [118] suggested that because soft tissues and cells consist of diverse constituents having different mass fractions, orientations, material properties, natural (i.e., stress-free) configurations, and rates of turnover, an associated theory of growth and remodeling should “track local balances or imbalances in the continual production and removal of individual constituents, the thermomechanical state in which the constituents are formed, and how these constituents are organized.” This can be accomplished by building on the continuum theory of mixtures (cf. [43]), which requires that each constituent obey fundamental postulates of mechanics such as balance of mass and momentum. Mathematical details and illustrative results of this approach can be found in Baek et al. [34] or Humphrey [40]. For a brief discussion of some of the mathematical aspects of this approach, please see Appendix.

We conclude this section by noting that the continuum theory of mixtures has also proven very useful in other areas of soft tissue mechanics, particularly in describing effects of solid–fluid interactions within tissues that give rise to viscoelastic phenomena [61]. Because mixture theory allows one to consider many aspects of tissue mechanics that cannot be addressed by classical (materially uniform) theories, it may well be that some past criticisms of continuum biomechanics in the literature would not have been voiced had the investigators been familiar with (cf. [43, 158]) this potentially powerful approach.

Cellular Level

When we speak of “cell growth,” we can consider either a change in cell number (i.e., population) or a change in cell mass that may be associated with changes in intracellular composition and structure (i.e., remodeling). The former is very important (cf. [159]), particularly in designing tissue and cell cultures and studying cancer, yet our focus is the latter. Mathematical models of cell growth and remodeling should build on, but not be limited by, prior advances in cell mechanics, for which there is a growing literature as revealed by the following reviews: Zhu et al. [160], Stamenovic and Ingber [161], Bao and Suresh [162], Huang et al. [163], Heidemann and Wirtz [164], and Lim et al. [165]. Indeed, despite the incredible complexity of cell constitution and associated mechanical behaviors, one finds that many simple mechanical analog models, network models, and (materially uniform) continuum models continue to find wide usage. For example, most studies using the atomic force microscope (AFM) to estimate cell stiffness (see e.g. [166–168]) are interpreted via the classical indentation solution by H. Hertz (ca, 1882), which is based on assumptions of linearly elastic, homogeneous, isotropic, small strain material behavior. Mijailovich et al. [169] similarly employed a linearly elastic, homogeneous, isotropic constitutive relation (with Young’s modulus E = 3 kPa and Poisson’s ratio υ = 0.49) to interpret magnetic bead experiments on isolated cells using a finite element model. Mack et al. [170] studied the effects of discrete tractions applied at the basal surface of a cell (i.e., at focal adhesions) using a finite element model based on a linearly viscoelastic, homogeneous, isotropic Maxwell model (i.e., a linear spring and dashpot in series, which at long times models a fluid). Conversely, Kumar et al. [99] suggested that the observed time-dependent retraction of severed stress fibers in isolated cells can be interpreted using a Kelvin–Voigt model (parallel linear spring and dashpot). Finally, the aforementioned 1D linear stress–strain relation in terms of a storage and a loss modulus has been used widely to interpret diverse experiments on cells (e.g. [171–173]), and has given rise to a provocative interpretation of cell mechanics in terms of the concept of soft glassy rheology [174].

Linear, isotropic, and 1D constitutive relations for materially uniform bodies simplify analysis for they yield simpler initial-boundary value problems in terms of a smaller number of material parameters, yet limitations must be kept in mind for vascular applications. For example, most experiments reveal a nonlinear relationship between stress and strain for vascular cells or their cytoskeletal filaments [103, 107] when tested over physiologic ranges relative to an appropriate natural configuration. Linearized strains are sensitive to rigid body rotations, which are associated with any shearing motion, and they simply add in sequential motions; in contrast, exact measures of strain are insensitive to rigid body motions and they arise from multiplicative operations in sequential motions (which we found to be critical in tissue-level residual stress analyses and which we expect to be important in cell mechanics given that cytoskeletal prestretch can reach 20% in the basal state prior to imposing further loads). With regard to interpretation of AFM experiments, Costa and Yin [175] show explicitly that the Hertz (linearized) formulation can yield substantial errors in data analysis partly because of nonlinear effects near the tip of the indenter. Similarly, with regard to viscoelasticity, Maxwell models cannot capture measured creep responses by cells, and Kelvin–Voigt models cannot capture usual stress relaxation behavior. Moreover, a related issue noted by Desprat et al. [173] is that the viscoelastic response of cells tends to be characterized by a continuous relaxation spectrum, not a discrete spectrum such as those associated with simple mechanical analog models consisting of a finite number of springs and dashpots. In this regard, it is interesting to note that just as tissue and cell level stiffness–stress relations tend to be very similar (Eq. 1), a continuous relaxation spectrum is also characteristic of soft tissues and has been modeled as such for years [44].

Inasmuch as actin filaments, microtubules, and intermediate filaments have different nonlinear material properties [176] and turnover rates [139], that they may have different prestresses [105], and that they may have different orientations [139] suggests that a materially nonuniform (mixture) nonlinear theory may hold promise in cell mechanics just as it has in tissue mechanics. Indeed, as in tissue level responses (cf. [61]), viscoelastic characteristics exhibited by cells can arise, in part, from interactions between solid constituents (e.g., stress fibers) and a viscous fluid (cytosol), hence reminding us that continuum mixture theories offer some advantages over classical viscoelastic models by addressing mechanisms of dissipation. See, for example, the very nice mixture model of cell mechanics by Herant et al. [177] for cell phagocytosis (remembering that different approaches or particular constitutive assumptions may be equally good for different problems, as, for example, modeling cell spreading, proliferation, migration, phagocytosis, and synthetic activity in response to altered loading). Albeit a first generation model, Na et al. [117] showed further that constrained mixture models (multiple solid constituents are constrained to move together, but can have different properties) can incorporate time-dependent changes in mass fractions of the cytoskeletal filaments (cf. [178, 179]) as well as different orientations and “deposition stresses” for individual filaments. Yet, such efforts to model cellular growth and remodeling can be complicated by many factors that have not been addressed herein, as, for example, cell heterogeneities (e.g. [180–182]) and complex cell-to-cell interactions, including direct mechanical coupling (e.g. [183]) as well as autocrine (e.g. [184]) and paracrine (e.g. [185]) interactions. Again, therefore, there is a pressing need for additional data. For example, changes in wall shear stress cause endothelial cells to change their production of many different classes of molecules that diffuse within the wall and affect smooth muscle and fibroblast activity. Clever co-culture systems will be needed to quantify such interactions, and associated models of co-existing structural and non-structural constituents will need to be developed. Nevertheless, despite the predominant use of mechanical analog models and classical continuum models based on linear, homogeneous, isotropic, materially uniform elastic or viscoelastic behaviors, “the general framework of continuum mechanics is sufficiently broad” [177] to incorporate many of the complexities that arise in cell biology, remembering of course that the constitutive relations should be formulated with the conditions of interest in mind and that a single model should not be expected to hold for all cases.

Sub-Cellular Level

Much of the attention directed toward sub-cellular growth and remodeling has appropriately focused on models of polymerization/depolymerization kinetics for the primary cytoskeletal constituents. For example, Flyvbjerg et al. [186] and Edelstein-Keshet [187] consider, respectively, the kinetics of microtubule self-assembly and the kinetics of actin polymerization including bundling via actin cross-linking proteins. Such studies show, among other things, that it is both natural and important to combine kinetic modeling with continuum thermomechanical modeling; again, the continuum theory of mixtures offers promise in this regard. In particular, although such studies are motivated by single molecule kinetics and mechanics, the “large population of interacting components which can diffuse, bind, or unbind” reveals the utility of modeling “mean fields” [187]. Toward this end, note that one of the key assumptions in a continuum mixture theory is that of affine deformations (i.e., deformations at large scales reflect mean deformations at small scales). As noted by Storm et al. [188], however, “Although there is no a priori reason to believe the affine approximation is valid, recent theoretical and experimental studies suggest that it is a good approximation for densely cross-linked filaments of high molecular weight.” There is, therefore, a pressing need to include in our analyses the fundamental roles of cytoskeletal cross-linkers [189, 190], not just to base models on rheological experiments on dilute solutions of cytoskeletal filaments.

The utility of combining kinetics, mechanics, and thermodynamics is illustrated further by studies such as Sept and McCammon [191], on actin filament nucleation, and Van Buren et al. [192], on microtubule assembly. Although the latter uses simple mechanical ideas (e.g., modeling microtubule filament bending based on ideas from introductory strength of materials, that is, in terms of a “flexural rigidity” EI, where E is Young’s modulus and I is the second moment of area), extension to nonlinear models may be straightforward. There is, nonetheless, a key challenge to include simultaneously the contributions by, and possible interactions between, the actin filaments, microtubules, and intermediate filaments [193]. Indeed, such is the case for cell–matrix and tissue level modeling as well—we must describe better the contributions by and interactions between the many different constituents that define each length and time scale of relevance to vascular growth and remodeling. Of course, multiscale modeling will benefit from continued advances in molecular dynamics modeling and systems biology (e.g., network models of signaling pathways; [194]), agent based modeling of multi-cellular networks [195], and continuum models at cell, tissue, and organ levels [196], which for the vasculature must include coupled hemodynamics and wall mechanics.

Closure

To-day, thanks to the great development and powerful support of the physico-chemical sciences, study of the phenomena of life, both normal and pathological, has made progress which continues with surprising rapidity. C. Bernard (1865)

Although written nearly a century and a half ago, this observation by the physiologist Bernard continues to ring true. Nevertheless, it is suggested here that we must increase our efforts to collect the data that are necessary for developing the next generation of constitutive relations and to advance our mathematical models of the diverse vascular adaptations that are important clinically. For example, with regard to the topic at hand, the biologist A.K. Harris put it this way, “without the aid of mechanicians, and others skilled in simulation and modeling, developmental biology will remain a prisoner of our inadequate and conflicting physical intuitions and metaphors” [197]. Developmental biology, of course, likely holds many of the clues needed to understand better the adaptations and maladaptations that occur in maturity, hence increased attention must be given to developmental biomechanics. Similarly, Tomasek et al. [89] noted that part of the reason that cell mediated remodeling of the extracellular matrix is not understood better in wound healing and pathological contracture is because of “the limited account taken of the inescapable mechanical laws.” Understanding the biochemomechanics of wound healing would similarly help immensely in understanding normal growth and remodeling. Of course, we must not forget that the goal of such understanding is improving health care delivery. As noted by R. Altman (Director of Symbios—A National Center for Biomedical Computing at Stanford University), mathematical simulations (i.e., modeling) aid greatly in the engineering of “novel drugs, drug delivery systems, synthetic tissues, medical devices, and surgical interventions,” each of which promises to continue to play a vital role in health care delivery.

In summary, it was not the goal of this article to provide an exhaustive review of all articles that address mechanobiology and biomechanics related to vascular adaptation; separate reviews could be written for each of the levels of organization considered herein. Rather, the primary goal of this article was to highlight some of the key articles that reveal a remarkable consistency in mechanical homeostasis across many length and time scales that contributes to vascular adaptation, to point the reader to appropriate review articles in each of these areas, to draw attention to the need for better functional quantification of “mechanical dose response curves,” and to highlight the need for theories of growth and remodeling across multiple length scales that will permit meaningful mathematical modeling, with nonlinear mixture theories deserving increased attention.

Although it is clear that much is known, many issues remain unresolved. With respect to multiscale modeling, a simple example is that homeostatic stresses within the arterial wall are of the order of 100 kPa, yet adherent cell and cell-gel studies suggest that homeostatic stresses at focal adhesions and in the cytoskeleton may be on the order of 5–10 kPa. One possible explanation for this order of magnitude difference is that “cells are stress-shielded by the matrix that they deposit and remodel” [89]. If this is the case, we will need to determine how to model cell-mediated responses based on large scale computations of hemodynamics and wall mechanics at the tissue or organ level—that is, we will need to know how the cell interrogates the mechanical state of the wall in which it resides. Many other central questions can be enumerated similarly.

In closing, I wish to quote from the foreword by Y.C. Fung [198] in the inaugural issue of the journal Biomechanics and Modeling in Mechanobiology. To put this into historical context, however, first note that J. Flamsteed (1646–1719), the first British Astronomer Royal, is known for his precise astronomical observations, yet we tend to think first of I. Newton (1642–1727) who explained these data within a consistent mathematical theory (Newton’s laws); M. Faraday (1791–1867), one of the most prolific experimentalists in all of physics, is known for his seminal work in electromagnetism, yet we tend to think first of J.C. Maxwell (1831–1879) who gave us a unifying mathematical theory (Maxwell’s equation). These and similar examples reveal the central role of observation and experiment, but also the power of a unifying mathematical theory. As suggested in this review, much has been learned about vascular mechanobiology and pathophysiology via clever observations and experiments, yet much of our understanding remains as pieces of a large puzzle that has yet to be assembled. There is a pressing need for a consistent mathematical theory of vascular adaptation across multiple length and time scales that can help us to assemble this puzzle and to achieve better predictive capability. Indeed, as Fung wrote in his foreword: “An exasperated biomechanicist may demand: ‘Show me a compact law that summarizes the mechanical consequences of genetic activity.’ Yes, that is our wish. Someday, somebody may give us something as beautiful as Newton’s laws, Maxwell’s equation, quantum theory, or relativity for gene functions. Until then, let us enjoy the work.”

Appendix

For details on mathematical modeling relevant to continuum biomechanics, the interested reader is referred to Holzapfel [147], Humphrey [18], and Cowin and Doty [47] for both an introduction and an extended list of references. Here, however, let us note a few basics. Whereas the 1D Hooke’s law for linearly elastic behavior can be written as σ = Eε, a generalized 3D Hooke’s law can be written as σ _ij  = C _ijkl ε _kl (using the Einstein summation convention, whereby repeated indices are summed 1 to 3 unless otherwise indicated and free indices can take values from 1 to 3, thereby yielding the different components of stress, stiffness, and strain). Of course, σ _ij ≡σ _ji (i≠j) by angular momentum balance (one of the two basic postulates that restricts constitutive relations). Recalling that material stiffness is obtained via ∂σ _ij /∂ε _kl , C _ijkl represents 81 components of the “stiffness matrix,” in which 21 components are non-zero and independent for general anisotropy, with 9, 5, and 2 components representing orthotropy (three preferred directions), transverse isotropy (a single preferred direction), and isotropy (i.e., the same behavior in all directions, which can be accounted for via a single Young’s modulus and Poisson’s ratio). Note, too, that the entropy inequality (the other basic postulate that restricts constitutive relations) implies the existence of a “stored energy function” for elastic processes; it can be shown for Hooke’s law that this energy is given by W = 0.5C _ijkl ε _ij ε _kl , whereby σ _ij  = ∂W/∂ε _ij (with C _ijkl  = C _klij by general symmetry arguments) in this hyperelastic approach. As noted above, although the linearized theory of elasticity (both a linearized stress–stress relation and a linearized strain measure) is mathematically elegant, it has long been believed [57] and continually confirmed [44] that it is not appropriate for most problems of vascular mechanics.

The reason for noting these few results from linear elasticity is to emphasize the need to generalize stress–strain relations to 3D (cf. Fig. 5), to account for anisotropy in behavior (e.g., behaviors differ in radial, circumferential, and axial directions in arteries; [18, 44]), and to emphasize that the entropy inequality restricts the form of the constitutive relation in terms of a stored energy function W. As it turns out, a stored energy function can play a basic role in extending elastic descriptors to viscoelastic (cf. [199]) and growth and remodeling [34] models. Whereas the linearized strain is determined from the deformation gradient tensor F (the fundamental measure of motion, including deformation and rigid body rotations) via ε = 0.5(F + F ^T−2I), where I is the identity tensor, an exact nonlinear measure of strain is the Green strain E = 0.5(F ^T · F−I). It is easy to show, therefore, that the linearized measure is sensitive to rigid body motions, whereas the nonlinear measure is properly insensitive to rigid body motions (because F = R · U, where R is rigid body motion and U describes deformation). It is also easy to show in a simple 1D case that the deformation gradient tensor reduces to a single component of stretch, namely F → λ, that the 1D ε = λ − 1 and E = 0.5(λ ² − 1). Of course, this E can also be written as E = 0.5(λ + 1)(λ − 1), whereby for small strains, λ∼1 and thus E = 0.5(∼2)(λ − 1) ∼ λ − 1, or ε∼E, hence justifying the use of the linearized measure in cases of small deformations (F ∼ I). Recall, however, that the axial prestretch alone often ranges from 1.2 (aortic arch) to 1.7 (rat carotid), and the prestretch of cytoskeletal proteins appears to be ∼1.2 even in the basal state. The linearized strain introduces a 9% error at a stretch of 1.2 and larger errors at higher stretches.

Finally, analogous to the result from the entropy inequality for Hooke’s law, a general stress–strain relation for nonlinear material behavior and finite strains is

$$ \varvec{\upsigma} = \frac{1} {{\det {\mathbf{F}}}}{\mathbf{F}} \cdot \frac{{\partial W({\mathbf{E}})}} {{\partial {\mathbf{E}}}} \cdot {\mathbf{F}}^{{\text{T}}} \,\,\, \leftrightarrow \,\,\,\sigma _{{ij}} = \frac{1} {{\det {\mathbf{F}}}}F_{{iA}} F_{{jB}} \frac{{\partial W}} {{\partial E_{{AB}} }} $$

where we note that the Cauchy stress tensor σ is defined independent of the specific material behavior or the range of strains and is thus common to both theories. Of course, in the case of small strains, F ∼ I (e.g., F _iA ∼δ _iA , the Kronecker delta) and ε ∼ E, whereby the linear elastic result can be recovered as a special case. Again, however, the assumption of small strains is not appropriate for blood vessels and the material behavior is nonlinear (cf. Eq. 1), hence necessitating the use of the nonlinear theory. Although fewer exact solutions are available in the nonlinear theory (cf. [200]) than in the linear theory (cf. [53]), once one is familiar with the nonlinear theory it is conceptually as simple as the linear theory and yet not subject to the same limitations.

Finally, as noted above, given an appropriate nonlinearly elastic framework, extension to nonlinearly viscoelastic and general growth and remodeling mechanics can be straightforward. For example, for nonlinear viscoelasticity we can write ([199])

$$ \varvec{\upsigma} = \frac{1} {{\det {\mathbf{F}}}}{\mathbf{F}} \cdot {\left( {{\int_{ - \infty }^t {G(t - \tau )\frac{\partial } {{\partial \tau }}{\left( {\frac{{\partial W}} {{\partial {\mathbf{E}}}}} \right)}d\tau } }} \right)} \cdot {\mathbf{F}}^{{\text{T}}}, $$

where G(t − τ) is a stress relaxation (viscoelastic) function. Finally, for a growth and remodeling theory, wherein individual constituents can be produced and removed at different rates as well as have different stiffnesses, orientations, and stress-free states, we can write [34]

$$ \varvec{\upsigma} = \frac{1} {{\det {\mathbf{F}}}}{\mathbf{F}} \cdot \frac{\partial } {{\partial {\mathbf{E}}}}{\left( {{\sum\limits_{k = 1}^n {W^{k} } }} \right)} \cdot {\mathbf{F}}^{{\text{T}}} , $$

where the superscript k (= 1,2,...,n) denotes the kth structurally significant constituent in the tissue or cell. Employing a rule of mixtures, we have suggested that

$$ W^{k} (t) = \frac{{\rho ^{k} (0)Q^{k} (t)}} {\rho }\hat{W}^{k} ({\mathbf{E}}^{k}_{{n(0)}} (t)) + {\int_0^t {\frac{{m^{k} (\tau )}} {\rho }} }q^{k} (t - \tau )\hat{W}^{k} ({\mathbf{E}}^{k}_{{n(\tau )}} (t)){\text{d}}\tau . $$

Here, ρ ^k(t) is the mass density of constituent k, ρ = ∑ρ ^k(t) is the overall mass density of the tissue or cell (often constant), Q ^k(t) ∈ [0,1] is a non-dimensional function that reveals what fraction of the material produced at and before time t = 0 survives to time t, with Q ^k(0)≡1, and $\hat{W}^{k}$ is the energy stored elastically in constituent k, which depends on ${\mathbf{E}}^{k}_{{n(\tau )}}$, the Green strain tensor associated with each constituent relative to its individual natural configuration n(τ). To account for the production and loss of individual constituents at different times, m ^k(τ) is a mass density production and q ^k(t − τ) is an associated survival function that ensures that a constituent contributes to load bearing only until it is degraded, which because of the convolution integral is reminiscent of viscoelasticity, wherein a constituent contributes more the more recently it was produced. Although this theory is a straightforward extension of classical ideas, the primary challenge is to determine appropriate functional forms for the mass density production, survival function, and stored energy for each constituent, with production and removal depending on mechanotransduction mechanisms. Again, therefore, there is a clear need for more data, particularly on the time course of cell and matrix turnover.

Given the generality of the constrained mixture approach, it is useful to consider a few specific cases to build intuition. If the altered loading after time t = 0 is constant, caused only by a step change in the current configuration, then using exponential decays for the survival functions recovers simple cases as considered in Humphrey [201] and Gleason et al. [82]. Finally, note that at t = 0, prior to a perturbation in loading that elicits a growth and remodeling response, one recovers a classical rule of mixtures relation for the stored energy $W = {\sum {\phi ^{k} \hat{W}^{k} } }$ (with $\phi ^{k} = \rho ^{k} /\rho$ the usual mass fractions), which yields the classical relation for stress, which in turn yields the classical single constituent relation when k = 1 (even Hooke’s law if so desired). In other words, this framework recovers multiple special cases as it should. See Humphrey [40] for more on a theory of growth and remodeling for tissues and cells, including a discussion of parallel reaction–diffusion equations for non-structurally significant constituents in the mixture, knowing of course that such modeling is in its infancy and will require continuing advances as more data become available. The main point here is that “the general framework of continuum mechanics is sufficiently broad” [177] to incorporate many of the complexities that arise in cell biology, thus we should not restrict ourselves to often more familiar linearized theories for materially uniform bodies or simple mechanical analog models unless more sophisticated approaches suggest that such simplifications capture the salient features.