Principles and Instrumentation

Abstract

Four major experimental techniques exist in the field of near infrared spectroscopy (NIRS). The simplest one is continuous-wave spectroscopy (CWS) in which a light of constant intensity is injected into tissue, and then the attenuated transmitted/reflected light signal is measured at a distance from the light source. The CWS technique has the limitation of obtaining only changes in optical density. More elaborate approaches such as spatially resolved spectroscopy (SRS), time-resolved spectroscopy (TRS), and phase-modulated spectroscopy (PMS) have certain advantages. The principles of these techniques are described in the chapter.

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1.1 Light Absorption and Light Scattering

Quantification of chromophore concentration is based on the Beer-Lambert law [1–3]. In Fig. 1.1 the absorption coefficient μ _a is defined as

$$ dI=-{\mu_{\mathrm{ a}}}\ I\ dl, $$

(1.1)

where dI is the change in intensity I of light moving along an infinitesimal path dl in a homogeneous medium. Integration over a thickness l (mm) yields

$$ {I_1} = {I_0}\exp (-{\mu_{\mathrm{ a}}}\;l), $$

(1.2)

where I ₀ is the incident light intensity. This equation is also expressed as base 10 logarithms as

$$ {I_l}={I_0}\ {10^{{-c\varepsilon l}}}. $$

(1.3)

Where c is the concentration of the compound, and ε is the molar absorption coefficient.

Fig. 1.1

Light transmission through a nonscattering medium

Transmission T is defined as the ratio of the transmitted light intensity to the incident light intensity, I ₁/I ₀. The optical density (OD) is given by

$$ \mathrm{ OD} = \log 10(1/T) = \log 10({I_0}/{I_1}). $$

(1.4)

When OD and ε are expressed as base 10 logarithms, the following expression is obtained from Eqs. 1.2, 1.3, and 1.4:

$$ \mathrm{ OD} = c\varepsilon l = {\log_{10 }}(\mathrm{ e})\ {\mu_{\mathrm{ a}}}l, $$

(1.5)

where log₁₀(e) is 0.43429.

Scattering of light in biological tissue is caused by refractive index mismatches at boundaries, such as cell membranes and organelles. The area that contributes to scattering is called the effective cross-section, as depicted in Fig. 1.2. The scattering coefficient μ _s (mm⁻¹) is expressed as the cross-sectional area (mm²) per unit volume of the medium (mm³). When the scattered photon does not return to the incident axis, μ _s can be defined as

Fig. 1.2

Effective cross-section related to the scattering coefficient

$$ I = {I_0}\exp (-{\mu_{\mathrm{ s}}}l). $$

(1.6)

The scattering path length, defined as 1/μ _s, is the expected value of distance that a photon travels between scattering events and defines the distance that reduces incident light I ₀ to I ₀/e due to scattering. The total attenuation coefficient, μ _t is defined as the sum of the absorption and scattering coefficients, and 1/μ _t is called the mean free path. When a photon is incident along the direction i, the angular probability of the photon being scattered in the direction s is given by the phase function f(i,s). The phase function is conventionally expressed as a function of the cosine of the scattering angle. The anisotropy can be represented as the mean cosine of the scattering angle, and the anisotropy factor g is defined as

$$ g=\int_{-1}^1 {\cos \theta\ f(\cos \theta )d\cos \theta }. $$

(1.7)

When g = 0 scattering is isotropic. Moreover, when g = 1 the incident light travels in a straight line. In contrast, when g = −1 complete backward scattering is observed. Biological tissues arestrongly forward-scattering media (0.69 < g < 0.99). The reduced scattering coefficient (${{\mu^{\prime}}_s}$) is defined using the anisotropy factor as follows:

$$ {{\mu^{\prime}}_s} = {\mu_s}(1-g). $$

(1.8)

The reduced scattering coefficient can be interpreted as representing the equivalent isotropic scattering coefficient and is used for the diffusion theory or Monte Carlo simulation when assuming isotropic scattering.

1.2 Optical Properties of Tissue

Hemoglobin is an iron-containing protein in red blood cells. One mole of deoxygenated hemoglobin (Hb) binds with four moles of oxygen to become oxygenated hemoglobin (HbO₂). The absorption spectra of oxygenated hemoglobin and deoxygenated hemoglobin [4] are shown in Fig. 1.3. The curves of the two hemoglobins intersect at about 800 nm, and the crossing point is called the isosbestic point. Since myoglobin and hemoglobin have similar absorption spectra, it is not easy to distinguish concentrations with spectroscopy. The separation of absorbers is also described in §1.4.

Fig. 1.3

Absorption spectra of oxyhemoglobin and deoxyhemoglobin

The absorption coefficient for water [5] is shown in Fig. 1.4. The absorption of water is small at wavelengths between about 200 and about 900 nm. Considering all components related to absorption in biological tissues, measurements at wavelengths between 680 and 950 nm are particularly suitable for spectroscopy.

Fig. 1.4

Absorption spectrum of water

Although many values for the optical properties of muscle and the overlying tissues (fat and skin) have been reported, there are significant differences in the results depending on the method of tissue preparation (fresh, saline-immersed, frozen, or thawed) and the theoretical analysis (diffusion theory, adding–doubling, Monte Carlo lookup tables). In Table 1.1 the optical properties of muscle, fat, dermis, and epidermis at wavelengths between 630 and 850 nm are given.

Table 1.1 Optical properties of muscle, fat, bone, dermis, and epidermis

1.3 Near-Infrared Spectroscopy

Spectroscopic measurement of in vivo tissue was first studied by Nicolai et al. [15] in 1932. They examined the optical characteristics of hemoglobin. The first practical ear oximeter for aviation use was developed by Millikan [16] ten years later. In 1949 Wood and Geraci [17] modified this instrument to obtain absolute values of oxygen saturation of arterial blood. The basic idea of this instrument was used to manufacture ear oximeters that were used in clinical settings until the 1970s. However, they did not have sufficient measurement stability for continuous monitoring of oxygen saturation because the calibration procedures were based on various extraneous assumptions. In 1974 Aoyagi et al. [18] presented a new idea called pulse oximetry, which utilizes the pulsation of arteries. This allowed for accurate measurement of oxygen saturation of arterial blood without the influence of factors other than arterial blood. Pulse oximetry is now used in clinical medicine throughout the world. As mentioned above, oxygen saturation of arterial blood mainly reflects gas exchange occurring in the lungs and is an important factor for respiratory care. However, from the viewpoint of metabolism in tissues, measurement of blood oxygenation within the capillaries of each tissue is desirable.

In 1977, utilizing the relatively high penetration of near-infrared light, Jöbsis demonstrated that it was possible to measure attenuation spectra across the head of a cat, thereby obtaining information about tissue oxygenation [19]. In near-infrared spectroscopy (NIRS), tissue oxygenation is determined by analyzing the reflected or transmitted light intensity. However, NIRS using transmitted light is not suitable for clinical measurement, because it is difficult to detect transmitted light in adult human tissues; thus, reflection techniques are most commonly used nowadays. Several clinical studies on NIRS were conducted during the 1980s, including those by Ferrari et al. [20], Brazy et al. [21], Chance et al. [22], and Tamura et al. [23]. They detected changes in concentrations of oxy- and deoxyhemoglobin and in the cytochrome c oxidase redox state. They also demonstrated that NIRS is a useful, noninvasive technique for rapid detection of changes in tissue oxygenation.

Four major experimental techniques exist in the field of NIR spectroscopy, as shown in Fig. 1.5. The simplest one is continuous-wave spectroscopy (CWS) in which light of constant intensity is injected into tissue, and then the attenuated light signal is measured at a distance from the light source. The CWS technique has the limitation of obtaining only changes in optical density. More elaborate approaches are spatially resolved spectroscopy (SRS), time-resolved spectroscopy (TRS), and phase-modulated spectroscopy (PMS). Table 1.2 shows the advantages and disadvantages of the four measurement methods. The principles of these techniques are described in the following section.

Fig. 1.5

Various techniques used in tissue oximetry employing NIRS

Table 1.2 Advantages and disadvantages of CWS, SRS, TRS, and PMS

1.4 Continuous-Wave NIRS

In NIRS, scattered light is detected at a distance from the light source, and tissue oxygenation is determined from the change in absorption coefficients of a tissue using the basic equations of conventional oximetry. Oximetry is the colorimetric measurement of the degree of oxygen saturation. Assuming that changes in light absorption are mainly due to changes in blood oxygenation or volume, [HbO₂] and [Hb] can be determined as follows. Change in the absorption coefficient of a tissue Δμ _a is expressed as

$$ \Delta {\mu_{\mathrm{ a}}} = \varepsilon_{\lambda}^{{\mathrm{ Hb}{{\mathrm{ O}}_2}}}\Delta [\mathrm{ Hb}{{\mathrm{ O}}_2}]+\varepsilon_{\lambda}^{\mathrm{ Hb}}\Delta [\mathrm{ Hb}] $$

(1.9)

where $\varepsilon_{\lambda}^{{\mathrm{ Hb}{{\mathrm{ O}}_2}}}$ and $\varepsilon_{\lambda}^{\mathrm{ Hb}}$ are molar absorption coefficients of HbO₂ and Hb at wavelength λ, respectively. For example, $\varepsilon_{\lambda}^{\mathrm{ Hb}}$ at a wavelength of 760 nm is 0.1674 OD mM⁻¹ mm⁻¹, as reported by Matcher et al. [4]. In the above equations, $\varepsilon_{\lambda}^{\mathrm{ Hb}}$ of 0.385 mM⁻¹ mm⁻¹ (= 0.1674 × ln10) is used because Matcher et al. defined OD as the logarithm to base 10 and μ _a is defined in base e. The two unknowns, Δ[HbO₂] and Δ[Hb], are obtained from measurements at two wavelengths. The NIRS instruments usually use a combination of wavelengths between 680 and 950 nm. These wavelengths are usually chosen to be around an isosbestic point (805 nm) of HbO₂ and Hb – for example, 770/830, 760/840, and 690/900 nm. When the difference of two wavelengths is large, changes in intensity due to wavelength are easily obtained, but the change in optical path length would not be ignored. The following equations are solved on assumption that the path lengths of each wavelength are same:

$$ \Delta [\mathrm{ Hb}{{\mathrm{ O}}_2}] = \frac{1}{k}\left( {\varepsilon_2^{\mathrm{ Hb}}\Delta {\mu_{\mathrm{ a}1}}-\varepsilon_1^{\mathrm{ Hb}}\Delta {\mu_{\mathrm{ a}2}}} \right), $$

(1.10)

$$ \Delta [\mathrm{ Hb}] = \frac{-1 }{k}\left( {\varepsilon_2^{{\mathrm{ Hb}{{\mathrm{ O}}_2}}}\Delta {\mu_{\mathrm{ a}1}}-\varepsilon_1^{{\mathrm{ Hb}{{\mathrm{ O}}_2}}}\Delta {\mu_{\mathrm{ a}2}}} \right), $$

(1.11)

$$ k = \varepsilon_1^{{\mathrm{ Hb}{{\mathrm{ O}}_2}}}\varepsilon_2^{\mathrm{ Hb}}-\varepsilon_1^{\mathrm{ Hb}}\varepsilon_2^{{\mathrm{ Hb}{{\mathrm{ O}}_2}}}. $$

(1.12)

The change in absorption, Δμ _a, can be determined by various NIRS techniques, such as CWS, SRS, PMS, and TRS. The CWS method, which is the simplest one, only enables determination of change in absorption. In contrast, the optical properties in absolute values can be obtained by using SRS, PMS, or TRS. However, with any technique it is difficult to quantify the concentration other than that of hemoglobin, such as myoglobinin muscle tissue, cytochrome oxidase, carboxy-hemoglobin, and methemoglobin. Schenkman et al. [24, 25] reported a method for quantification of myoglobin and hemoglobin using the difference in peak position around 760 nm at near-infrared wavelengths. A composite myoglobin–hemoglobin peak would be slightly shifted by the absorption of hemoglobin derivatives and path length for each wavelength. Alternatively, some researchers have used NMR to observe the distinct Mb and Hb signals and then apply the results to determine Mb and Hb contributions in the NIRS spectra [26, 27].

Further studies with accurate quantification of small-quantity absorbers, mentioned above, scattering coefficient (or path length) for each wavelength, and combination of spectroscopic techniques could potentially form a basis for developing key technologies to measure the myoglobin and hemoglobin contribution to the NIRS signal.

In CW-NIRS, change in the optical density – defined by ΔOD = ln(R ₀/R), where R ₀ and R are intensities of backscattered light at a reference state (usually taken at the start of measurement) and during measurement, respectively – is measured. Assuming that the scattering coefficient does not change during measurement, we can determine Δμ _a using the modified Beer–Lambert law: ΔOD = Δμ _a d, where d = ∂OD/∂μ _a, which is defined as the differential path length and is equal to the mean optical path length.

CW-NIRS systems measure only ΔOD, and at least two different wavelengths are usually employed to obtain spectral information. Relative changes of HbO₂ and Hb are continuously monitored utilizing Eqs. 1.10 and 1.11. The CW method is advantageous because it is highly sensitive, enabling a data sampling rate of less than a second, economical, and can be miniaturized to the extent of a multipoint monitor even for imaging. The assembly of a multiwavelength, multisource, multidetector imager for brain function and the circuit diagram are depicted in Fig. 1.6 [28].

Fig. 1.6

The 16-channel “CW imager” is illustrated in (a) giving circuit constants and component values for one channel in (b)

Silicon photodiodes and multiwavelength light-emitting diodes (LEDs) are used as detectors and the near-infrared light source, respectively. They are held by elastic bands at a source–detector distance of 3 cm. The penetration depth of light and the spatial resolution are about 1.5 and 2 cm, respectively. Because digital gain control is used, the system can readily be controlled over a 20-dB range to equalize 16-channel signals. The output is then connected to a multiplexer (MUX) switch, which is synchronized with the flashing of the LED, so that one wavelength is sampled by separate integrating capacitors, which gives an RC charging time constant. The stored signal in this capacitor can be updated step by step by using the MUX switch. The storage capacitor is then oversampled by an analog-to-digital converter (ADC) at 250 samples/s to avoid aliasing. The temporal resolution of oxygenation measurement is ≥0.3 s.

Brain activation was monitored during the anagram test, which required the subject to identify a five-letter word from some items. The imager pad is centered on the nose bridge and symmetrically attached to the forehead, eyebrow to hairline, and temple to temple. The covering area is corresponding to Brodmann’s areas 9 and 10, which are the part of the frontal cortex in the human brain. The subject’s signals are illustrated in Fig. 1.7. The data show postsolution hyperemia and brief deoxygenation prior to problem solving and prolonged hyperoxygenation thereafter.

Fig. 1.7

Raw waveform data obtained during an anagram test. Dashed line: blood volume; solid line: oxygenation. The decision that the anagram has been solved is indicated by the dashed vertical line labeled “Key press”

Although the CW method gives only relative values, it is sufficient for many cases, such as studies of the functional activity of the brain [29–32] or interventional studies for testing reactions on drugs or changes in treatment.

1.5 Spatially Resolved NIRS

Patterson et al. [33] proposed that the effective attenuation coefficient μ _eff of tissues can be obtained by measuring the spatial profile of the intensity of backscattered light as a function of the distance from the light source using a large source–detector separation. They showed that the intensity of reflected light R can be expressed as follows:

$$ R\propto \frac{{\exp (-{\mu_{\mathrm{ eff}}})}}{{{\rho^2}}}, $$

(1.13)

where ${\mu_{\mathrm{ eff}}} = \sqrt{{3{\mu_{\mathrm{ a}}}\left( {{\mu_{\mathrm{ a}}}+{{{\mu^{\prime}}}_{\mathrm{ s}}}} \right)}}$.

In the case of CW spectroscopy, the local reflectance I(ρ) at position ρ is expressed as an integral of R(ρ, t) over time, and OD is defined as the negative of the logarithm of I(ρ).

Thus, OD is expressed by

$$ \begin{array}{lll} OD=-\ln I(\rho ) \hfill \\= -\ln ({z_0}/2\boldsymbol{\pi} )+\frac{3}{2}\ln \left( {{\rho^2}+{z_0}^2}\right)-\ln \left( {1+{\mu_{\mathrm{ eff}}}{{{\left({{\rho^2}+{z_0}^2} \right)}}^{{{1 \left/ {2} \right.}}}}}\right)+{\mu_{\mathrm{ eff}}}{{\left( {{\rho^2}+{z_0}^2}\right)}^{{{1 \left/ {2} \right.}}}}. \hfill \\ \end{array} $$

(1.14)

In the measurement of human tissues, if ρ ² ≫ z ₀ ² is assumed, μ _eff ρ ≫1, the following equation is derived.

$$ \mathrm{ OD} = -\ln \left( {{{{{z_0}}} \left/ {{2\pi}} \right.}} \right)+3\ln (\rho )-\ln ({\mu_{\mathrm{ eff}}}\rho )+{\mu_{\mathrm{ eff}}}\rho. $$

(1.15)

Differentiating OD with respect to ρ and assuming that μ _a ≪ μ _s′ yield the following relation [34, 35]:

$$ \frac{{\partial \mathrm{ OD}}}{{\partial \rho }} = -\frac{\partial }{{\partial \rho }}\ln I(\rho ) = \frac{2}{\rho }+\sqrt{{3{\mu_a}{\mu_s}^{\prime}}}, $$

$$ \therefore {\mu_a}{\mu_s}^{\prime} = \frac{1}{3}{{\left( {\frac{{\partial \mathrm{ OD}}}{{\partial \rho }}-\frac{2}{\rho }} \right)}^2}. $$

(1.16)

where ∂OD/∂ρ is the local gradient of attenuation with respect to the source–detector separation.

In the first-order approximation, ${{\mu^{\prime}}_{\mathrm{ s}}}$ can be assumed to be constant within a narrow wavelength region of NIR light. Then the relative concentration changes of HbO₂ and Hb are derived from the following equation:

$$ \left[ {\begin{array}{lll} {{{{\left( {\frac{{\partial \mathrm{OD}({\lambda_1})}}{{\partial \rho }}} \right)}}^2}-\frac{2}{\rho }} \\ {{{{\left( {\frac{{\partial \mathrm{OD}({\lambda_2})}}{{\partial \rho }}} \right)}}^2}-\frac{2}{\rho }} \\ {{{{\left( {\frac{{\partial \mathrm{OD}({\lambda_3})}}{{\partial \rho }}} \right)}}^2}-\frac{2}{\rho }} \\ \end{array}} \right]=3{{\mu^{\prime}}_{\mathrm{s}}}\;\left[ {\begin{array}{*{20}{c}} {{\varepsilon_{{\mathrm{Hb}{{\mathrm{O}}_2}}}}({\lambda_1})} \\ {{\varepsilon_{{\mathrm{Hb}{{\mathrm{O}}_2}}}}({\lambda_2})} \\ {{\varepsilon_{{\mathrm{Hb}{{\mathrm{O}}_2}}}}({\lambda_3})} \\ \end{array}\;\;\;\left. {\begin{array}{*{20}{c}} {{\varepsilon_{\mathrm{Hb}}}({\lambda_1})} \\ {{\varepsilon_{\mathrm{Hb}}}({\lambda_2})} \\ {{\varepsilon_{\mathrm{Hb}}}({\lambda_3})} \\ \end{array}} \right]} \right.\left[ {\begin{array}{*{20}{c}} {\;\;{C_{{\mathrm{Hb}{{\mathrm{O}}_2}}}}} \\ {{C_{\mathrm{Hb}}}} \\ \end{array}} \right]. $$

(1.17)

Moreover, the tissue oxygenation index (TOI) is calculated using the following equation:

$$ \mathrm{ TOI}=\frac{{{C_{{\mathrm{ Hb}{{\mathrm{ O}}_2}}}}}}{{\;{C_{{\mathrm{ Hb}{{\mathrm{ O}}_2}}}}+{C_{\mathrm{ Hb}}}}}\times 100\ (\%). $$

(1.18)

A commercially available instrument using SRS for the measurement of hemoglobin saturation has been developed by Hamamatu Photonics, as shown in Fig. 1.8 [35].

Fig. 1.8

Spatially resolved spectroscopy system (NIRO-200NX) with an optical probe consisting of photodiodes and a light source

A NIRO 300 was incorporated into an established multimodal monitoring system, enabling recording of cerebral hemodynamic changes during carotid endarterectomy (CEA) [36]. Brief periods of cerebral ischemia often occur during cross-clamping of the internal carotid artery (ICA) during surgery. Multimodal monitoring consists of frontal cutaneous laser-Doppler flowmetry (LDF) and transcranial Doppler mean flow velocity (FV) measurements of the ipsilateral middle cerebral artery. Typical data obtained during CEA are shown in Fig. 1.9. Sequential clamping was performed on the external carotid artery (ECA) before ICA clamping. The measurements obtained by LDF can be seen to fall only when the ECA clamp is applied. In this case the drop in FV is seen to be specific to ICA clamping, similar to the drop in the TOI. On insertion of an ICA vascular shunt, FV, and TOI were restored to values approaching baseline levels.

Fig. 1.9

Data obtained from a patient during elective carotid endarterectomy (CEA). Vertical lines demonstrate time of application of vascular clamps

1.6 Time-Resolved NIRS

In TRS temporal changes in the reflected light intensity are measured after irradiation of a picosecond pulse, thereby giving a distribution of the total path length of a photon traveling in the scattering medium [6, 37–39]. This technique can be used to determine the absorption coefficient and the reduced scattering coefficient of tissues. A method for determining the absorption and scattering coefficients is based on a curve fitting between measured data and a theoretical curve obtained by diffusion theory. When a semiinfinite medium is assumed and the zero-boundary condition is applied, the reflectance R at a source–detector separation ρ and time t is given [40] by

$$ R(\rho, t) = {{(4\pi cD)}^{-3/2 }}{{\left( {{\mu_a}+{\mu_s}^{\prime}} \right)}^{-1 }}{t^{-5/2 }}\exp \left( {-\frac{{{\rho^2}+{{{\left( {{\mu_a}+{\mu_s}^{\prime}} \right)}}^{-2 }}}}{4cD}\frac{1}{t}} \right)\exp (-{\mu_a}ct). $$

(1.19)

Taking the natural logarithm on both sides of Eq. 1.19 and assuming that ρ ≫ ${{\left( {{\mu_{\mathrm{ a}}}+{{{\mu^{\prime}}}_{\mathrm{ s}}}} \right)}^{-1 }}$,we obtain the following equation:

$$ \ln \left[ {R(\rho, t)} \right]=\;\kappa -\frac{5}{2}\ln (t)-\left( {ct+\frac{{3{\rho^2}}}{4ct }} \right){\mu_{\mathrm{ a}}}-\frac{{3{\rho^2}}}{4ct }{\mu_{\mathrm{ s}}}^{\prime}, $$

(1.20)

where κ = −ln(4πcD)3/2 − ln(μ _a + μ _s′). Simple mean least-squares-fitting algorithms can be used to determine μ _a and μ _s′ from experimental data. A typical waveform of time-resolved measurement is shown in Fig. 1.10.

Fig. 1.10

Time-resolved waveform of the incident short pulse and reflectance at a 40-mm separation

A TRS system (TRS-20) uses the time-correlated single-photon counting (TCPC) method to measure the temporal profile of the detected photons (see Fig. 1.11). The system [41] consists of a three-wavelength (759, 797, and 833 nm) light pulse source (PLP: Picosecond Light Pulser, Hamamatsu Photonics KK, Hamamatsu, Japan), which generates light pulses with a peak power of about 60 mW, pulse width of 100 ps, pulse rate of 5 MHz, and an average power of 30 μW for one wavelength. For the detection, a photomultiplier tube (PMT, H7422-50MOD, Hamamatsu Photonics KK) was used in photon-counting mode. The timing signals were received and accumulated by a TRS circuit that consists of a constant-fraction discriminator, a time-to-amplitude converter, an ADC, and histogram memory. The PLP emits three-wavelength light pulses in turn, and the light pulses are guided into one illuminating optical fiber by a fiber coupler (CH20G-D3-CF, Mitsubishi Gas Chemical Company Inc., Japan). A single optical fiber (GC200/250L, FUJIKURA Ltd., Japan) with a numerical aperture (NA) of 0.21 and a core diameter of 200 μm was used for illumination. An optical bundle fiber (LB21E, Moritex Corp., Japan) with an NA of 0.21 and a bundle diameter of 3 mm was used to collect diffuse light from the tissues. TRS-20 has two sets of PLP and TCPC detectors, enabling the independent measurement of two portions.

Fig. 1.11

Schematic diagram of a time-resolved spectroscopy system

TRS allows for determination of relative light intensity, mean optical path length, transport scattering coefficient (${{\mu^{\prime}}_{\mathrm{ s}}}$), and μ_a. The intensity can be obtained by integrating the temporal profiles, and the modified Beer–Lambert law uses this information to calculate absorbance changes. The mean optical path lengths were calculated from the center of gravity of the temporal profile [42]. The calculations of the intensity, absorbance change, and mean path length are model independent. Applying the diffusion equation for semiinfinite homogeneous media with zero-boundary conditions in reflectance mode into all observed temporal profiles, we obtained the values of ${{\mu^{\prime}}_{\mathrm{ s}}}$ and μ _a using the nonlinear least-squares method [43].

If it is assumed that absorption in the 700–900 nm range arises from absorption of HbO₂, Hb, and water, ${\mu_{{\mathrm{ a}\lambda }}}$ of the measured wavelengths: λ (759, 797, and 833 nm) is expressed as shown in simultaneous Eq. 1.21 [44, 45]:

$$ \begin{array}{lll}{\mu_{\mathrm{a}759\mathrm{nm}}}={\varepsilon_{{\mathrm{H}\mathrm{b}{{\mathrm{O}}_2}759\mathrm{nm}}}}{C_{{\mathrm{H}\mathrm{b}{{\mathrm{O}}_2}}}}+{\varepsilon_{\mathrm{H}\mathrm{b}759\mathrm{nm}}}{C_{\mathrm{H}\mathrm{b}}}+{\mu_{{\mathrm{a}{{\mathrm{H}}_2}\mathrm{O}759\mathrm{nm}}}}\hfill \\{\mu_{\mathrm{a}797\mathrm{nm}}}={\varepsilon_{{\mathrm{H}\mathrm{b}{{\mathrm{O}}_2}797\mathrm{nm}}}}{C_{{\mathrm{H}\mathrm{b}{{\mathrm{O}}_2}}}}+{\varepsilon_{\mathrm{H}\mathrm{b}797\mathrm{nm}}}{C_{\mathrm{H}\mathrm{b}}}+{\mu_{{\mathrm{a}{{\mathrm{H}}_2}\mathrm{O}797\mathrm{nm}}}}\hfill \\{\mu_{\mathrm{a}833\mathrm{nm}}}={\varepsilon_{{\mathrm{H}\mathrm{b}{{\mathrm{O}}_2}833\mathrm{nm}}}}{C_{{\mathrm{H}\mathrm{b}{{\mathrm{O}}_2}}}}+{\varepsilon_{\mathrm{H}\mathrm{b}833\mathrm{nm}}}{C_{\mathrm{H}\mathrm{b}}}+{\mu_{{\mathrm{a}{{\mathrm{H}}_2}\mathrm{O}833\mathrm{nm}}}},\hfill \\ \end{array} $$

(1.21)

where ${\varepsilon_{{\mathrm{ m}\lambda }}}$ is the molar absorption coefficient of substance m at wavelength λ, and C _m is the concentration of substance m. After subtracting water absorption from μ_a at each wavelength, assuming that the volume fraction of the water content was constant, [HbO₂] and [Hb] were determined using the least-squares-fitting method.

The total concentrations of Hb (tHb) and tissue oxygen saturation (SO₂) were calculated as follows:

$$ \mathrm{ tHb} = \mathrm{ H}{{\mathrm{ b}}_{{{{\mathrm{ O}}_2}}}}+\mathrm{ Hb}, $$

(1.22)

$$ \mathrm{ S}{{\mathrm{ O}}_2} = \frac{{[\mathrm{ Hb}{{\mathrm{ O}}_2}]}}{{[\mathrm{ tHb}]}}\times 100. $$

(1.23)

The hemodynamics of the brain were monitored by TRS during a coronary artery bypass grafting surgery using an artificial heart-and-lung machine [45]. Figure 1.12 shows the time course of changes in [HbO₂], [Hb], and [tHB] in (A) and SO₂ and the internal jugular vein oxygen saturation (SjvO₂) in (B). When extracorporeal circulation was started by the pump, HbO₂ and tHb decreased rapidly. At the end of extracorporeal circulation, those values returned to the initial levels. SO₂ estimated by TRS was found to be nearly the same as SjvO₂ before and after extracorporeal circulation, but during circulation they behaved differently in this case.

Fig. 1.12

Fluctuations of SO₂ and SjvO₂ were separated during extracorporeal circulation in one patient

Figure 1.13 shows the correlation between the hematocrit (Hct) values of arterial blood and tHb in 9 patients. The correlation among patients was high (r ² = 0.63), showing that tHb measured by TRS has good linearity with Hct.

Fig. 1.13

Relationship between tHb by TRS-10 and hematocrit (Hct) in nine patients

1.7 Phase-Modulated NIRS

Phase-modulated (frequency domain) measurements were first reported by Chance in 1949 [46]. Pulse code or phase modulation gives mean time delay between source and detector. The time delay is related to light scattering and absorption, including biological signals. Figure 1.14 shows an example of the phase shift obtained by a computer simulation. Different types of equipment were developed and classified by the multiplicity of wavelengths and the type of phase detection. Homodyne systems [47], heterodyne systems [48, 49], and network analyzers [50] were used for measuring the phase (see Fig. 1.15). The performance of both homodyne and heterodyne detection systems was examined, and it was suggested that a homodyne system has advantages in terms of simplicity of construction and execution, while a heterodyne system has high precision and the possibility of low-frequency phase detection [51].

Fig. 1.14

Phase shift between the incident light (dashed line) and scattered light through tissues (solid lines) at 70 MHz of modulation frequency

Fig. 1.15

Schematic diagram of phase-modulated homodyne (a) and heterodyne (b) NIRS systems

The supporting theoretical background is the diffusion equation. An analytical solution was obtained on the basis of an assumption that the modulation frequency (ω/2π) is much smaller than the typical frequency of scattering processes (i.e., $\mathrm{ c}{{\mu^{\prime}}_s}$, where c is the speed of light in a medium). The condition for this assumption is satisfied by most biological tissues in an NIR spectral region for modulation frequencies up to 1 GHz. Fishkin and Gratton obtained the following expressions [52]:

$$ \ln (\rho {U_{\mathrm{ dc}}}) = -\rho \sqrt{{\frac{{{\mu_{\mathrm{ a}}}}}{D}}}+\ln \left( {\frac{S}{{4\boldsymbol{\pi} D}}} \right), $$

(1.24)

$$ \ln (\rho {U_{\mathrm{ ac}}}) = -\rho A\cos \left( {\frac{\theta }{2}} \right)+\ln \left( {\frac{kS }{{4\pi D}}} \right), $$

(1.25)

$$ \phi = \rho A\sin \frac{\theta }{2}, $$

(1.26)

$$ \theta ={\tan^{-1 }}\left( {\frac{\omega }{{{\mu_{\mathrm{ a}}}c}}} \right),\kern0.5em A={{\left( {\frac{{\mu_{\mathrm{ a}}^2{c^2}+{\omega^2}}}{{{c^2}{D^2}}}} \right)}^{1/4 }}, $$

(1.27)

where φ is the phase shift of a detected signal relative to an excited signal, U _dc is the direct current (dc) component of the photon density, U _ac is the amplitude of the alternating current (ac) component of the photon density, S is the source strength (in photons per second), and k is defined as the ratio of the ac to the dc components of the intensity. Fantini et al. [53] developed practical instrumentation based on diffusion theory with multidistance detection. As mentioned above, various phase-modulated NIRS systems have been developed. A typical commercial phase-modulated NIRS system is depicted in Fig. 1.16. OxiplexTS (ISS Inc., Champaign, IL) is a tissue oximeter that includes light sources (690 and 830 nm), with a multidistance emitter array, and one detection channel using PMT. The detection type is heterodyne with an offset frequency of several kHz at a 110-MHz modulation frequency. Various optical probes can be coupled to the system for specific medical research applications.

Fig. 1.16

Phase-modulated NIRS system, ISS Oxiplex (ISS Inc.)

Appendices

Problem

• 1.1. How can the weak photocurrent of an Si photodiode be converted to a voltage signal on continuous-wave NIRS or spatially resolved NIRS?

Further Reading

Demrow B (1971) Op amps as electrometers or — the world of fA. Anal Dial 5(2): 48–49

Frenzel LE (2007) Accurately measure nanoampere and picoampere currents. Electron Design Strat News, Feb 15

Hutchings MJ, Blake-Coleman BC (1994) A transimpedance converter for low-frequency, high-impedance measurements. Meas Sci Technol 5(3):310–313

Rako P (2007) Measuring nanoamperes. Electron Design Strat News, Apr 26