_{2}are calculated to be close for an ischemia type protocol where both Hb and HbO

_{2}are assumed to have equal magnitude but opposite concentration changes. The minimum estimation errors are found for the 700/825- and 725/825-nm pairs for this protocol.

## 1.

## Introduction

Near infrared spectroscopy (NIRS) is increasingly used as an optical noninvasive method to monitor the changes in tissue oxygenation in brain,^{1, 2, 3} breast,^{4} and particularly in muscle tissues.^{5, 6, 7} Continuous wave near-infrared spectroscopy (cw-NIRS) is based on a steady state technique where the changes in the detected light intensities at multiple wavelengths are converted to concentration changes of oxygenation sensitive chromophores. Typically, cw-NIRS is used in muscle physiology studies to calculate oxygen consumption and blood flow values. Spatially resolved spectroscopy^{8} along with frequency and time domain techniques are other NIRS methods^{9, 10} that have the capability of quantifying absolute concentrations.

NIRS techniques suffer inaccuracies for the heterogonous tissue structures when the homogeneous medium assumption is made for the sake of simplicity.^{11, 12} In fact, there are solutions based on complex layered models
^{11, 12, 13, 14, 15, 16, 17, 18, 19, 20} for NIRS. The degree of inaccuracy because of the homogeneous medium assumption depends on the region of interest, geometry, optical coefficients of the structures in the tissue, source-detector distance, and the choice of NIRS technique.
^{11, 12, 14, 21, 22, 23, 24, 25, 26, 27} Hence, the estimated parameter (i.e., absorption coefficient change) could be related to a layer’s (or to combination of layers) property, or it may not be related to any property of any one of those layers at all.^{12, 14, 28}

Muscle tissue has superficial skin and fat layers. A Fat layer has varying thicknesses between subjects and has a lower absorption coefficient than the underlying muscle layer, masking the muscle’s optical parameters, hence making it difficult to determine the optical coefficients and quantify concentration changes in the lower muscle layer. It has been shown experimentally that adipose tissue causes sensitivity and linearity problems,
^{25, 29, 30, 31, 32, 33, 34, 35} underestimation of oxygen consumption^{36} in muscle cw-NIRS measurements in which modified Beer-Lambert law (MBLL) with a homogeneous medium assumption is used. These problems are mainly related to the so-called partial volume effect, which refers to the fact that hemodynamic changes occur in a volume smaller than that assumed by homogeneous medium assumption.^{23, 24, 37}

Crosstalk in NIRS measurements refers to the measurement of chromophore concentration change although no real change happens for that chromophore but for other chromophores’ concentrations.^{23, 24, 38, 39, 40, 41} This is caused again by the homogeneous medium assumption with the use of mean optical path length instead of wavelength-dependent partial optical path length in the tissue layer of interest where the concentration changes occur (i.e., muscle or gray matter in the brain). There are detailed studies on the analysis of the crosstalk effect for brain measurements, while as we know, there is only one study of Iwasaka and Okada^{42} on the crosstalk effect for muscle measurements, where the analysis was done for a fixed fat thickness of
$4\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$
.

The effect of adipose tissue layer on cw-NIRS measurements with the homogeneous medium assumption using MBLL is investigated in our study by Monte Carlo simulations for a two-wavelength system. Simulations were performed for a homogeneous layered skin-fat-muscle heterogeneous tissue model with varying fat thickness up to $15\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ . The wavelengths are in $675\text{-}\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}775\text{-}\mathrm{nm}$ range for the first wavelength and in $825\text{-}\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}900\text{-}\mathrm{nm}$ range for the second wavelength, and in total 24 wavelength pairs were used. For the considered wavelengths and fat thicknesses, mean partial path lengths in the three layers and detected light intensities were found. An error analysis for estimated concentration changes was analyzed by partitioning the error into an underestimation term for a real change in muscle layer and a crosstalk term, where the aims are the investigation of the fat layer thickness effect and a search for wavelength pairs that result in low errors. An error analysis for a particular measurement protocol of vascular occlusion is also discussed.

## 2.

## Theory

## 2.1.

### Homogeneous Medium Assumption

The cw-NIRS technique relies on the MBLL to convert detected light intensity changes into concentration changes of chromophores. For a single light absorber in a homogeneous medium, light attenuation is given by^{1}

## Eq. 1

$${\mathrm{OD}}^{\lambda}=\mathrm{ln}({I}_{o}\u2215I)={\u03f5}^{\lambda}c{\mathrm{DPF}}^{\lambda}r+{G}^{\lambda},$$^{43}from 0 to ${\mu}_{a}^{\lambda}$ . Nevertheless, MBLL formulation can still be used to determine concentration changes for small absorption changes for which $\u27e8{L}^{\lambda}\u27e9$ remains nearly constant.

^{22, 43, 44}Light scattering change is another issue.

^{44}

For tissues where the main light absorbers are Hb and $\mathrm{Hb}{\mathrm{O}}_{2}$ ,

## Eq. 2

$$\Delta {\mathrm{OD}}^{\lambda}=({\u03f5}_{\mathrm{Hb}}^{\lambda}\Delta \left[\mathrm{Hb}\right]+{\u03f5}_{\mathrm{Hb}{\mathrm{O}}_{2}}^{\lambda}\Delta \left[\mathrm{Hb}{\mathrm{O}}_{2}\right]){\mathrm{DPF}}^{\lambda}r,$$## Eq. 3

$$\Delta {\left[\mathrm{Hb}\right]}_{\mathrm{MBLL}}=\frac{({\u03f5}_{\mathrm{Hb}{\mathrm{O}}_{2}}^{{\lambda}_{2}}\Delta {\mathrm{OD}}^{{\lambda}_{1}}\u2215{\left(\mathrm{DPF}\right)}^{{\lambda}_{1}})-({\u03f5}_{\mathrm{Hb}{\mathrm{O}}_{2}}^{{\lambda}_{1}}\Delta {\mathrm{OD}}^{{\lambda}_{2}}\u2215{\left(\mathrm{DPF}\right)}^{{\lambda}_{2}})}{r({\u03f5}_{\mathrm{Hb}{\mathrm{O}}_{2}}^{{\lambda}_{2}}{\u03f5}_{\mathrm{Hb}}^{{\lambda}_{1}}-{\u03f5}_{\mathrm{Hb}{\mathrm{O}}_{2}}^{{\lambda}_{1}}{\u03f5}_{\mathrm{Hb}}^{{\lambda}_{2}})},$$## Eq. 4

$$\Delta {\left[\mathrm{Hb}{\mathrm{O}}_{2}\right]}_{\mathrm{MBLL}}=\frac{({\u03f5}_{\mathrm{Hb}}^{{\lambda}_{1}}\Delta {\mathrm{OD}}^{{\lambda}_{2}}\u2215{\left(\mathrm{DPF}\right)}^{{\lambda}_{2}})-({\u03f5}_{\mathrm{Hb}}^{{\lambda}_{2}}\Delta {\mathrm{OD}}^{{\lambda}_{1}}\u2215{\left(\mathrm{DPF}\right)}^{{\lambda}_{1}})}{r({\u03f5}_{\mathrm{Hb}{\mathrm{O}}_{2}}^{{\lambda}_{2}}{\u03f5}_{\mathrm{Hb}}^{{\lambda}_{1}}-{\u03f5}_{\mathrm{Hb}{\mathrm{O}}_{2}}^{{\lambda}_{1}}{\u03f5}_{\mathrm{Hb}}^{{\lambda}_{2}})}.$$Note that for the considered muscle measurements, [Hb]
$\left(\left[\mathrm{Hb}{\mathrm{O}}_{2}\right]\right)$
refers to combined concentrations of deoxyhemoglobin and deoxymyoglobin (oxyhemoglobin and oxymyoglobin) since hemoglobin and myoglobin have very similar absorption spectra.^{45}

## 2.2.

### Underestimation Error and Crosstalk

For muscle cw-NIRS measurements, a more realistic tissue model should contain skin, fat, and muscle tissue layers. Measured optical density change can be written as^{22}

## Eq. 5

$$\Delta {\mathrm{OD}}^{\lambda}=\Delta {\mu}_{a,s}\u27e8{L}_{s}^{\lambda}\u27e9+\Delta {\mu}_{a,f}\u27e8{L}_{f}^{\lambda}\u27e9+\Delta {\mu}_{a,m}\u27e8{L}_{m}^{\lambda}\u27e9,$$## Eq. 6

$$\Delta {\mathrm{OD}}^{\lambda}\cong ({\u03f5}_{\mathrm{Hb}}^{\lambda}\Delta {\left[\mathrm{Hb}\right]}_{m}+{\u03f5}_{\mathrm{Hb}{\mathrm{O}}_{2}}^{\lambda}\Delta {\left[\mathrm{Hb}{\mathrm{O}}_{2}\right]}_{m})\u27e8{L}_{m}^{\lambda}\u27e9,$$^{24}

## Eq. 7

$$\Delta {\left[\mathrm{X}\right]}_{\mathrm{MBLL}}={U}_{\mathrm{X}}\Delta {\left[\mathrm{X}\right]}_{m}+{C}_{\mathrm{X}}\Delta {\left[\mathrm{O}\right]}_{m}$$## Eq. 8

$${U}_{\mathrm{Hb}}=\frac{{\u03f5}_{\mathrm{Hb}}^{{\lambda}_{1}}{\u03f5}_{\mathrm{Hb}{\mathrm{O}}_{2}}^{{\lambda}_{2}}{l}^{{\lambda}_{1}}-{\u03f5}_{\mathrm{Hb}}^{{\lambda}_{2}}{\u03f5}_{\mathrm{Hb}{\mathrm{O}}_{2}}^{{\lambda}_{1}}{l}^{{\lambda}_{2}}}{{\u03f5}_{\mathrm{Hb}}^{{\lambda}_{1}}{\u03f5}_{\mathrm{Hb}{\mathrm{O}}_{2}}^{{\lambda}_{2}}-{\u03f5}_{\mathrm{Hb}}^{{\lambda}_{2}}{\u03f5}_{\mathrm{Hb}{\mathrm{O}}_{2}}^{{\lambda}_{1}}},$$## Eq. 9

$${U}_{\mathrm{Hb}{\mathrm{O}}_{2}}=\frac{{\u03f5}_{\mathrm{Hb}}^{{\lambda}_{1}}{\u03f5}_{\mathrm{Hb}{\mathrm{O}}_{2}}^{{\lambda}_{2}}{l}^{{\lambda}_{2}}-{\u03f5}_{\mathrm{Hb}}^{{\lambda}_{2}}{\u03f5}_{\mathrm{Hb}{\mathrm{O}}_{2}}^{{\lambda}_{1}}{l}^{{\lambda}_{1}}}{{\u03f5}_{\mathrm{Hb}}^{{\lambda}_{1}}{\u03f5}_{\mathrm{Hb}{\mathrm{O}}_{2}}^{{\lambda}_{2}}-{\u03f5}_{\mathrm{Hb}}^{{\lambda}_{2}}{\u03f5}_{\mathrm{Hb}{\mathrm{O}}_{2}}^{{\lambda}_{1}}},$$## Eq. 10

$${C}_{\mathrm{Hb}}=\frac{{\u03f5}_{\mathrm{Hb}{\mathrm{O}}_{2}}^{{\lambda}_{1}}{\u03f5}_{\mathrm{Hb}{\mathrm{O}}_{2}}^{{\lambda}_{2}}}{{\u03f5}_{\mathrm{Hb}}^{{\lambda}_{1}}{\u03f5}_{\mathrm{Hb}{\mathrm{O}}_{2}}^{{\lambda}_{2}}-{\u03f5}_{\mathrm{Hb}}^{{\lambda}_{2}}{\u03f5}_{\mathrm{Hb}{\mathrm{O}}_{2}}^{{\lambda}_{1}}}({l}^{{\lambda}_{1}}-{l}^{{\lambda}_{2}}),$$## Eq. 11

$${C}_{\mathrm{Hb}{\mathrm{O}}_{2}}=\frac{{\u03f5}_{\mathrm{Hb}}^{{\lambda}_{1}}{\u03f5}_{\mathrm{Hb}}^{{\lambda}_{2}}}{{\u03f5}_{\mathrm{Hb}}^{{\lambda}_{1}}{\u03f5}_{\mathrm{Hb}{\mathrm{O}}_{2}}^{{\lambda}_{2}}-{\u03f5}_{\mathrm{Hb}}^{{\lambda}_{2}}{\u03f5}_{\mathrm{Hb}{\mathrm{O}}_{2}}^{{\lambda}_{1}}}({l}^{{\lambda}_{2}}-{l}^{{\lambda}_{1}}),$$^{46}In summary, the magnitude of underestimation and crosstalk terms depend on the wavelength dependence of specific absorption coefficients, choice of ${\mathrm{DPF}}^{\lambda}$ factors, which are used instead of unavailable $\u27e8{L}_{m}^{\lambda}\u27e9\u2215r$ .

A common definition for crosstalk is the ratio of the estimated concentration change of the chromophore X for which no change happens to the estimated concentration change of the chromophore O for which real change is induced,^{40, 46} denoted as
${C}_{\mathrm{O}\to \mathrm{X}}$
. According to this definition and previous formulation,
${C}_{\mathrm{Hb}{\mathrm{O}}_{2}\to \mathrm{Hb}}$
and
${C}_{\mathrm{Hb}\to \mathrm{Hb}{\mathrm{O}}_{2}}$
are

## 12.

## Eq. 13

$${E}_{\mathrm{MBLL}}=100\times (\Delta {\left[\mathrm{X}\right]}_{\mathrm{MBLL}}-\Delta {\left[\mathrm{X}\right]}_{m})\u2215\Delta {\left[\mathrm{X}\right]}_{m}\%.$$## 3.

## Methods

## 3.1.

### Tissue Model

For the simulations, three homogeneously layered skin-fat-muscle heterogeneous model is used. Skin thickness is taken to be
$1.4\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$
and muscle thickness is infinite. Reduced scattering coefficients of the three tissues and absorption coefficients of skin and adipose tissues are taken from Simpson
^{47} For the muscle tissue, the absorption coefficient is calculated with the equation

## Eq. 14

$${\mu}_{a,m}^{\lambda}={\mu}_{a,w}^{\lambda}{V}_{w}+\left[t\mathrm{Hb}\right][{\u03f5}_{\mathrm{Hb}{\mathrm{O}}_{2}}^{\lambda}\mathrm{St}{\mathrm{O}}_{2}+{\u03f5}_{\mathrm{Hb}}^{\lambda}(1-\mathrm{St}{\mathrm{O}}_{2})]+{\mu}_{a,b},$$^{48, 49}The ${\mu}_{a,w}^{\lambda}$ values are taken from the study of Hollis.

^{50}The background absorption coefficient of muscle tissue ${\mu}_{a,b}$ is taken as $0.072\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{-1}$ so that the calculated ${\mu}_{a,m}^{798\phantom{\rule{0.3em}{0ex}}\mathrm{nm}}$ equals the experimentally found

*in vitro*value of Simpson

^{47}since absorption at this isobestic point is unaffected by the oxygen saturation of the hemoglobin. Table 1 lists the absorption and reduced scattering coefficients of the three layers used in the simulations.

## Table 1

Optical properties of the skin, fat and muscle tissue layers used in the simulations (for log base e ).

λ (nm) | μa (cm−1) | μs′ (cm−1) | ||||
---|---|---|---|---|---|---|

Skin | Fat | Muscle | Skin | Fat | Muscle | |

675 | 0.232 | 0.097 | 0.321 | 24.81 | 12.24 | 8.53 |

700 | 0.191 | 0.089 | 0.254 | 23.17 | 12.03 | 8.08 |

725 | 0.172 | 0.089 | 0.243 | 21.99 | 11.87 | 7.89 |

750 | 0.165 | 0.092 | 0.288 | 20.97 | 11.67 | 7.69 |

760 | 0.159 | 0.093 | 0.306 | 20.53 | 11.61 | 7.50 |

775 | 0.146 | 0.087 | 0.291 | 19.91 | 11.50 | 7.21 |

800 | 0.127 | 0.083 | 0.284 | 19.07 | 11.36 | 6.99 |

825 | 0.121 | 0.085 | 0.309 | 18.24 | 11.12 | 6.78 |

850 | 0.122 | 0.086 | 0.343 | 17.57 | 11.09 | 6.60 |

875 | 0.122 | 0.091 | 0.368 | 16.98 | 10.97 | 6.43 |

900 | 0.134 | 0.125 | 0.393 | 16.30 | 10.88 | 6.32 |

## 3.2.

### Monte Carlo Simulations

In a Monte Carlo simulation of photon propagation in biological tissues, a stochastic model was constructed in which rules of photon propagation were modeled in the form of probability distributions.^{51} In the simulation, photons were launched with initial direction along
$z$
axis (the axis perpendicular to tissue layers) from a point source. For a photon traveling in layer
$i$
, which has absorption coefficient
${\mu}_{a,i}$
, scattering coefficient
${\mu}_{s,i}$
, and reduced scattering coefficient
${\mu}_{s,i}^{\prime}$
[which is equal to
$(1-g){\mu}_{s,i}$
, where
$g$
is the mean cosine of the single scattering phase function and is called anisotropy factor], successive scattering distances are selected using a random variable
$l=-\mathrm{ln}\left(R\right)\u2215{\mu}_{s,i}^{\prime}$
, with
$R$
having a uniform distribution over (0,1]. The remaining scattering length
$\Delta {l}_{i}$
for photons crossing tissue boundary from medium
$i$
to medium
$j$
is recalculated by
$\Delta {l}_{j}=\Delta {l}_{i}{\mu}_{s,i}^{\prime}\u2215{\mu}_{s,j}^{\prime}$
. Isotropic scattering is utilized using principle of similarity.^{52} Scatter azimuthal angle was uniformly distributed over the interval
$[0,2\pi )$
. Fresnel formulas are used for reflection or transmission at the boundaries.^{51}

Total distance traveled in layer
$i$
by a photon
$\left({L}_{i}\right)$
was found by summing scattering lengths taken in this layer. Photon propagation was continued until it escapes the medium or travels
$220\phantom{\rule{0.3em}{0ex}}\mathrm{cm}$
in length
$\left(10\phantom{\rule{0.3em}{0ex}}\mathrm{ns}\right)$
. For those reaching the surface, exit (survival) weight
$\left(w\right)$
is calculated using Lambert-Beer law as
$w={w}_{0}\phantom{\rule{0.3em}{0ex}}\mathrm{exp}[-{\sum}_{i}\left({L}_{i}{\mu}_{a,i}\right)]$
, with
${w}_{0}$
accounting for reflections and refractions at the boundaries encountered by the particular photon when there are refractive index mismatches.^{22} Because of the symmetry of the medium considered, photons reaching a ring (thickness is
$\mathrm{d}r$
, distance from center of ring to the light source is
$r$
) were taken as the photons reaching the detector. The mean partial path length in medium
$i$
$(\u27e8{L}_{i}\u27e9)$
for the detected photons was found using the formula
$\u27e8{L}_{i}\u27e9={\sum}_{j=1}^{N}{L}_{i,j}{w}_{j}\u2215\left({\sum}_{j=1}^{N}{w}_{j}\right)$
, where
${L}_{i,j}$
is the total path length taken in medium
$i$
by detected photon
$j$
with weight
${w}_{j}$
, and
$N$
is total number of detected photons. Refractive indices of air and tissue layers were taken to be 1 and 1.4, respectively.^{53} Each simulation was performed using
$5\times {10}^{7}$
photons and the
$\mathrm{d}r$
thickness is taken to be
$0.5\phantom{\rule{0.3em}{0ex}}\mathrm{cm}$
.

## 4.

## Results

## 4.1.

### Path Lengths and Detected Light Intensity

We performed Monte Carlo simulations to calculate the mean partial path lengths for the 11 distinct wavelengths given in Table 1. Note that $\u27e8{L}_{i,r,{h}_{f}}^{\lambda}\u27e9$ represents the mean partial path length in layer $i$ ( $s$ , $f$ , or $m$ for skin, fat, and muscle, respectively, as used in Sec. 2.2), for a source-detector distance $r$ (in centimeters), at fat thickness ${h}_{f}$ (in millimeters) and wavelength $\lambda $ . Also $\u27e8{L}_{i,r,{h}_{f}}\u27e9$ denotes the $\text{mean}\pm \text{standard}$ deviation of the mean partial path length in layer $i$ computed over all wavelengths.

The term $\u27e8{L}_{m}^{\lambda}\u27e9$ is the most important variable affecting the underestimation error and crosstalk, as shown in Fig. 1 . The value of $\u27e8{L}_{m,3.0,{h}_{f}}^{\lambda}\u27e9$ decreased linearly with a higher slope for $0\u2a7d{h}_{f}\u2a7d7\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ , while the slope decreased for ${h}_{f}>7\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ . The value of $\u27e8{L}_{m,3.0,0}\u27e9\phantom{\rule{0.3em}{0ex}}$ is $11.5\pm 1.20\phantom{\rule{0.3em}{0ex}}\mathrm{cm}$ and that of $\u27e8{L}_{m,3.0,7}\u27e9$ is $2.35\pm 0.43\phantom{\rule{0.3em}{0ex}}\mathrm{cm}$ . Above $10\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ of fat thickness, $\u27e8{L}_{m,3.0,{h}_{f}}^{\lambda}\u27e9$ decreased much more slowly but eventually approached null, where $\u27e8{L}_{m,3.0,15}\u27e9=0.20\pm 0.04\phantom{\rule{0.3em}{0ex}}\mathrm{cm}$ . It was possible to infer a considerable wavelength-dependent variability in $\u27e8{L}_{m,3.0,{h}_{f}}^{\lambda}\u27e9$ . The value of $\u27e8{L}_{m,3.0,{h}_{f}}^{\lambda}\u27e9$ was found to increase from $675\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}725\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ , while it had a decreasing trend from the $725\text{-}\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}900\text{-}\mathrm{nm}$ range. This finding can be explained by the wavelength dependence of the optical properties of muscle and fat tissues given in Table 1. The coefficient of variation ( $\mathrm{CV}=\text{standard}$ deviation/mean) of $\u27e8{L}_{m,3.0,{h}_{f}}^{\lambda}\u27e9$ values over 11 wavelengths increased from 11% at ${h}_{f}=0\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ to 23% at ${\mathrm{h}}_{f}=15\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ .

The value of $\u27e8{L}_{\mathrm{s},3.0,{h}_{\mathrm{f}}}^{\lambda}\u27e9$ was found to be the least varying mean partial path length with respect to ${h}_{f}$ variation with values ranging from $1.78\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}2.39\phantom{\rule{0.3em}{0ex}}\mathrm{cm}$ having a maximum at around ${h}_{f}=6\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}7\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ for all considered wavelengths. In contrast to $\u27e8L_{m}^{\lambda}{}_{,r,{h}_{f}}\u27e9$ , $\u27e8{L}_{f,r,{h}_{f}}^{\lambda}\u27e9$ and mean path length increased with increasing ${h}_{f}$ as expected. The value of $\u27e8{L}_{\mathrm{f},3.0,{h}_{f}}\u27e9$ ranged from $1.84\pm 0.13\phantom{\rule{0.3em}{0ex}}\mathrm{cm}$ at a $1\text{-}\mathrm{mm}$ fat thickness to $21.77\pm 1.24\phantom{\rule{0.3em}{0ex}}\mathrm{cm}$ at ${h}_{f}=15\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ , while the mean path length ranged from $13.03\pm 1.26\phantom{\rule{0.3em}{0ex}}\mathrm{cm}$ at ${h}_{f}=0\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ to $24.17\pm 1.30\phantom{\rule{0.3em}{0ex}}\mathrm{cm}$ at ${h}_{f}=15\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ . The mean path length had a decreasing trend with local peaks at either 700 or $725\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ and either 775 or $800\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ .

An increase in the fat layer thickness caused an increase in the detected light intensity. These increases in the detected light intensities for the 11 wavelengths expressed as $\text{mean}\pm \text{standard}$ deviation were $74\pm 28$ , $272\pm 97$ , and $537\pm 184\%$ at ${h}_{f}=4$ , 8, and $15\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ , respectively, with respect to detected intensities at ${h}_{f}=0\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ $(r=3.0\phantom{\rule{0.3em}{0ex}}\mathrm{cm})$ .

With increase in source-detector distance, $\u27e8{L}_{m}^{\lambda}\u27e9$ and mean path length increased, while detected light intensity decreased. In particular, $\u27e8{L}_{m,4.0,0}\u27e9=15.31\pm 1.65\phantom{\rule{0.3em}{0ex}}\mathrm{cm}$ , and $\u27e8{L}_{m,4.0,7}\u27e9=4.31\pm 0.75\phantom{\rule{0.3em}{0ex}}\mathrm{cm}$ .

## 4.2.

### Underestimation Error

Underestimation errors were calculated for a two-wavelength cw-NIRS system under varying fat thicknesses. The two wavelengths were chosen to fall before and after the isobestic point at around $800\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ . Hence, there were 24 wavelength pairs ${\lambda}_{1}\u2215{\lambda}_{2}$ , where ${\lambda}_{1}$ is between 675 and $775\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ and ${\lambda}_{2}$ is between 825 and $900\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ . DPF was taken to be wavelength independent with a value of 4.37 found for ${h}_{f}=0$ and $\lambda =800\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ . Underestimation error for the pair ${\lambda}_{1}\u2215{\lambda}_{2}$ is denoted by ${E}_{\mathrm{X},r,{h}_{f}}^{{\lambda}_{1},{\lambda}_{2}}$ , where the first subscript refers to the chromophore, the second and (if present) third subscripts refer to source-detector distance (in centimeters), and the ${h}_{f}$ value (in millimeters), respectively. For the all considered ${\lambda}_{1}\u2215{\lambda}_{2}$ pairs, ${E}_{\mathrm{X},r,{h}_{f}}$ showed $\text{mean}\pm \text{standard}$ deviation of the absolute values of the underestimation errors ${E}_{\mathrm{X},r,{h}_{f}}^{{\lambda}_{1},{\lambda}_{2}}$ .

Figures 2a and 2b show ${E}_{\mathrm{Hb},3.0,{h}_{f}}$ and ${E}_{\mathrm{Hb}{\mathrm{O}}_{2},3.0,{h}_{f}}$ along with minimum errors for ${E}_{\mathrm{Hb},3.0,{h}_{f}}^{{\lambda}_{1},{\lambda}_{2}}$ and ${E}_{\mathrm{Hb}{\mathrm{O}}_{2},r,{h}_{f}}^{{\lambda}_{1},{\lambda}_{2}}$ . The $725\u2215900\text{-}\mathrm{nm}$ pair gives the minimum values for ${E}_{\mathrm{Hb},3.0,{h}_{f}}^{{\lambda}_{1},{\lambda}_{2}}$ except at ${h}_{f}=0\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ , for which the $700\u2215825\text{-}\mathrm{nm}$ pair gives the minimum error. The $675\u2215825\text{-}\mathrm{nm}\phantom{\rule{0.3em}{0ex}}$ pair gives the minimum error for ${\mathrm{E}}_{\mathrm{Hb}{\mathrm{O}}_{2},r,{h}_{f}}^{{\lambda}_{1},{\lambda}_{2}}$ from ${h}_{f}=0\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ up to and including $10\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ , and at higher ${h}_{f}$ values, the $760\u2215825\text{-}\mathrm{nm}$ pair is the minimum error producing pair. Both the errors ${E}_{\mathrm{Hb},3.0,{h}_{f}}$ and ${E}_{\mathrm{Hb}{\mathrm{O}}_{2},3.0,{h}_{f}}$ exhibited a steep increase in the fat thickness range $<5\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ and a decreasing slope beyond this value. Interestingly, ${E}_{\mathrm{Hb},3.0,{h}_{f}}$ began at a lower value compared to ${E}_{\mathrm{Hb}{\mathrm{O}}_{2},3.0,{h}_{f}}$ but had a larger slope in this range. As expected, ${E}_{\mathrm{Hb},3.0,{h}_{f}}$ and ${E}_{\mathrm{Hb}{\mathrm{O}}_{2},3.0,{h}_{f}}$ approached a complete underestimation error (100%) at ${h}_{f}=15\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ . For the no-fat-thickness case, ${E}_{\mathrm{Hb},3.0,0}$ was $6.1\pm 3.5\%$ and ${E}_{\mathrm{Hb}{\mathrm{O}}_{2},3.0,0}$ was $28.9\pm 5.8\%$ . The slopes of the least-squares fits to the absolute values of underestimation errors in ${h}_{f}=0\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}5\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ range were $11.5\%\u2215\mathrm{mm}$ $({R}^{2}=0.94)$ for $\mid {E}_{\mathrm{Hb},3.0}^{{\lambda}_{1},{\lambda}_{2}}\mid $ and $9.1\%\u2215\mathrm{mm}$ $({R}^{2}=0.91)$ for $\mid {E}_{\mathrm{Hb}{\mathrm{O}}_{2,3.0}}^{{\lambda}_{1},{\lambda}_{2}}\mid $ .

There is wavelength pair dependency in the underestimation errors. The value of ${E}_{\mathrm{Hb},3.0}^{{\lambda}_{1},{\lambda}_{2}}$ decreased in magnitude for an increase in ${\lambda}_{2}$ , while that of ${E}_{\mathrm{Hb}{\mathrm{O}}_{2},3.0}^{{\lambda}_{1},{\lambda}_{2}}$ increased, for fixed ${\lambda}_{1}$ at a given ${h}_{f}$ . This change of variation over ${\lambda}_{2}$ was higher for ${E}_{\mathrm{Hb}{\mathrm{O}}_{2},3.0}^{{\lambda}_{1},{\lambda}_{2}}$ . The variation of ${\lambda}_{1}$ —for fixed ${\lambda}_{2}$ at a given ${h}_{f}$ —led to a high range of change for ${E}_{\mathrm{Hb},3.0}^{{\lambda}_{1},{\lambda}_{2}}$ , where 700 and $725\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ lead to lower errors. Underestimation errors for ${h}_{f}=2\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ are given in Table 2 to show wavelength pair effect. The wavelength pair dependency of underestimation errors decreasd with ${h}_{f}$ increase. CV values of absolute underestimation errors were 56.5% (20.0%) at ${h}_{f}=0\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ and 0.3% (0.4%) at ${h}_{f}=15\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ for $\mid {E}_{\mathrm{Hb},3.0}^{{\lambda}_{1},{\lambda}_{2}}\mid $ $(\mid {E}_{\mathrm{Hb}{\mathrm{O}}_{2},3.0}^{{\lambda}_{1},{\lambda}_{2}}\mid )$ over the considered ${\lambda}_{1}\u2215{\lambda}_{2}$ pairs.

## Table 2

Underestimation errors EHb,3.0,2λ1,λ2 (in percentages) and EHbO2,3.0,2λ1,λ2 (in percentages) for the considered λ1∕λ2 pairs.

λ1 (nm) | 675 | 700 | 725 | 750 | 760 | 775 | |
---|---|---|---|---|---|---|---|

${\lambda}_{2}$ (nm) | |||||||

825 | 32.4 | 21.6 | 18.0 | 28.7 | 32.4 | 29.2 | |

${E}_{\mathrm{Hb},3.0,2}^{{\lambda}_{1},{\lambda}_{2}}$ (%) | 850 | 32.0 | 21.2 | 17.4 | 27.6 | 31.1 | 26.9 |

875 | 31.8 | 20.9 | 16.8 | 26.8 | 30.2 | 25.4 | |

900 | 31.6 | 20.5 | 16.1 | 26.0 | 29.4 | 23.9 | |

825 | 36.5 | 38.1 | 40.4 | 38.2 | 37.2 | 39.2 | |

${E}_{\mathrm{Hb}{\mathrm{O}}_{2},3.0,2}^{{\lambda}_{1},{\lambda}_{2}}$ (%) | 850 | 41.4 | 43.0 | 45.6 | 43.9 | 43.1 | 46.0 |

875 | 44.7 | 46.4 | 49.2 | 47.7 | 47.0 | 50.5 | |

900 | 47.1 | 49.0 | 52.1 | 50.8 | 50.1 | 54.3 |

For longer source-detector distance of $4.0\phantom{\rule{0.3em}{0ex}}\mathrm{cm}$ , errors are lower. Here, ${E}_{\mathrm{Hb},4.0,0}$ and ${E}_{\mathrm{Hb}{\mathrm{O}}_{2},4.0,0}$ were $5.7\pm 2.3$ and $26.7\pm 5.7\%$ , respectively. The slopes of the least-squares fits in the $0\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}5\text{-}\mathrm{mm}$ fat thickness range are $9.9\%\u2215\mathrm{mm}$ $({R}^{2}=0.89)$ for $\mid {E}_{\mathrm{Hb},4.0}^{{\lambda}_{1},{\lambda}_{2}}\mid $ and $8.0\%\u2215\mathrm{mm}$ $({R}^{2}=0.88)$ for $\mid {E}_{\mathrm{Hb}{\mathrm{O}}_{2},4.0}^{{\lambda}_{1},{\lambda}_{2}}\mid $ . Again above ${h}_{f}=10\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ , ${E}_{\mathrm{Hb},4.0}$ $\left({E}_{\mathrm{Hb}{\mathrm{O}}_{2},4.0}\right)$ became very high, with values above $87.3\pm 2.3\%$ $(92.1\pm 1.5\%)$ .

## 4.3.

### Crosstalk Analysis

Crosstalk was calculated using Eqs. 12a, 12b for the two-wavelength system represented by ${C}_{\mathrm{O}\to \mathrm{X},r,\left({h}_{f}\right)}^{{\lambda}_{1},{\lambda}_{2}}$ , where the superscripts refer to particular wavelength pair and first, second, and third (if present) subscripts represent crosstalk type, source-detector distance (in centimeters), and ${h}_{f}$ value (millimeters), respectively. Crosstalk was computed for the same ${\lambda}_{1}\u2215{\lambda}_{2}$ pairs in underestimation error computations. DPF was assumed to be taken as wavelength independent, for which case crosstalk defined by Eq. 12 resulted in DPF independence. Not that ${C}_{\mathrm{O}\to \mathrm{X},r,\left({h}_{f}\right)}$ represents $\text{mean}\pm \text{standard}$ deviation of absolute values of crosstalk $\mid {C}_{\mathrm{O}\to \mathrm{X},r,\left({h}_{f}\right)}^{{\lambda}_{1},{\lambda}_{2}}\mid $ for the all ${\lambda}_{1}\u2215{\lambda}_{2}$ pairs.

In general, ${C}_{\mathrm{Hb}{\mathrm{O}}_{2}\to \mathrm{Hb},3.0}^{{\lambda}_{1},{\lambda}_{2}}$ had positive values, while ${C}_{\mathrm{Hb}\to \mathrm{Hb}{\mathrm{O}}_{2},3.0}^{{\lambda}_{1},{\lambda}_{2}}$ had negative values. The minimum-error-producing pairs for ${C}_{\mathrm{Hb}{\mathrm{O}}_{2}\to {\mathrm{Hb}}_{3.0}}^{{\lambda}_{1},{\lambda}_{2}}$ were the $675\u2215825\text{-}\mathrm{nm}$ pair at ${h}_{f}=0\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ up to including $5\phantom{\rule{0.3em}{0ex}}\mathrm{mm},$ the $760\u2215825\text{-}\mathrm{nm}$ pair at ${h}_{f}=6$ ,7, and $9\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ ; and the $675\u2215850\text{-}\mathrm{nm}$ pair at other ${h}_{f}$ values. Also ${C}_{\mathrm{Hb}\to \mathrm{Hb}{\mathrm{O}}_{2,3.0}}^{{\lambda}_{1},{\lambda}_{2}}$ had the minimum errors for the $760\u2215825\text{-}\mathrm{nm}$ pair at ${h}_{f}=0$ ,1,2,4,5,6,7,8,9, and $10\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ , for the $675\u2215825\text{-}\mathrm{nm}$ pair at ${h}_{f}=3\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ ; for the $675\u2215850\text{-}\mathrm{nm}$ pair at ${h}_{f}=11$ ,12,13, and $14\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ ; and for the $750\u2215825\text{-}\mathrm{nm}$ pair at ${h}_{f}=15\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ . The values ${C}_{\mathrm{Hb}{\mathrm{O}}_{2}\to \mathrm{Hb},3.0}$ (about 9.5%) and ${C}_{\mathrm{Hb}\to \mathrm{Hb}{\mathrm{O}}_{2},3.0}$ (about 14.2%) were nearly constant in the ${h}_{f}=0\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}3\text{-}\mathrm{mm}$ range, as shown in Fig. 3 While in the ${h}_{f}=3\text{-}\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}14\text{-}\mathrm{mm}$ ${C}_{\mathrm{Hb}{\mathrm{O}}_{2}\to \mathrm{Hb},3.0}$ increased up to $25.0\pm 34.9\%$ , ${C}_{\mathrm{Hb}\to \mathrm{Hb}{\mathrm{O}}_{2},3.0}$ showed an increasing trend in the ${h}_{f}=3\u2013\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}10\text{-}\mathrm{mm}$ range, with ${C}_{\mathrm{Hb}\to \mathrm{Hb}{\mathrm{O}}_{2},3.0,10}=20.3\pm 10.2\%$ . The slopes of the least-squares fits in these respective ${h}_{f}$ ranges to the absolute crosstalk values were $1.4\%\u2215\mathrm{mm}$ $({R}^{2}=0.1)$ for $\mid {C}_{\mathrm{Hb}{\mathrm{O}}_{2}\to \mathrm{Hb},3.0}^{{\lambda}_{1},{\lambda}_{2}}\mid $ and 0.9%/mm $({R}^{2}=0.1)$ for $\mid {C}_{\mathrm{Hb}\to \mathrm{Hb}{\mathrm{O}}_{2},3.0}^{{\lambda}_{1},{\lambda}_{2}}\mid $ .

In Table 3 , crosstalk values are given for ${h}_{f}=\text{-}0$ -, 5-, 10-, and $15\text{-}\mathrm{mm}$ values for all wavelength pairs. Similar to the increase seen in the mean values the standard deviations of absolute crosstalk over considered wavelength pairs showed dramatic increases as the fat thickened. The ${C}_{\mathrm{Hb}{\mathrm{O}}_{2}\to \mathrm{Hb},3.0}^{{\lambda}_{1},{\lambda}_{2}}$ had CV values of 64.9, 82.3, and 159.2% at ${h}_{f}$ values of 0, 5, and $15\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ , respectively. The ${C}_{\mathrm{Hb}\to \mathrm{Hb}{\mathrm{O}}_{2},3.0}^{{\lambda}_{1},{\lambda}_{2}}$ had lower CV values of 31.0, 47.0, and 57.6% at ${h}_{f}=0$ , 5, and $15\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ . However, ${C}_{\mathrm{Hb}\to \mathrm{Hb}{\mathrm{O}}_{2},3.0}^{{\lambda}_{1},{\lambda}_{2}}$ had higher magnitudes in general. Examining the results from Table 3, we can observe that both absolute values of crosstalk are less than 11% for pairs $675\u2215825$ , $675\u2215850$ , $675\u2215875$ , $750\u2215825$ , $760\u2215825$ , $760\u2215850$ , and $775\u2215825\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ for ${h}_{f}<10\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ . In addition to these pairs, ${C}_{\mathrm{Hb}{\mathrm{O}}_{2}\to \mathrm{Hb},3.0,{h}_{f}}^{{\lambda}_{1},{\lambda}_{2}}$ had low crosstalk values also for pairs $675\u2215900$ , and $700\u2215825\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ . Higher crosstalk magnitudes where computed for the choice of a higher ${\lambda}_{2}$ for a fixed ${\lambda}_{1}$ at a given ${h}_{f}$ .

## Table 3

Crosstalk values CHb→HbO2,3.0,hfλ1,λ2 (in percentages) and CHbO2→Hb,3.0,hfλ1,λ2 (in percentages) for different λ1∕λ2 pairs and hf=0 , 5, 10, and 15mm .

Fat thickness | λ1 (nm) | CHb→HbO2,3.0,hfλ1,λ2 (%) | CHbO2→Hb,3.0,hfλ1,λ2 (%) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

675 | 700 | 725 | 750 | 760 | 775 | 675 | 700 | 725 | 750 | 760 | 775 | ||

${\lambda}_{2}$ (nm) | |||||||||||||

825 | $-7.8$ | $-14.3$ | $-16.2$ | $-9.1$ | $-5.1$ | $-7.9$ | 1.3 | 4.2 | 8.8 | 5.0 | 2.6 | 6.8 | |

$0\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ | 850 | $-10.8$ | $-16.2$ | $-17.9$ | $-12.3$ | $-9.2$ | $-12.1$ | 2.2 | 5.7 | 11.9 | 8.4 | 6.0 | 13.3 |

875 | $-13.1$ | $-18.0$ | $-19.7$ | $-14.8$ | $-12.1$ | $-15.0$ | 2.9 | 7.0 | 14.5 | 11.2 | 8.7 | 18.9 | |

900 | $-15.7$ | $-20.6$ | $-22.4$ | $-17.8$ | $-15.1$ | $-18.5$ | 3.6 | 8.3 | 17.4 | 14.2 | 11.6 | 25.3 | |

825 | 4.1 | $-16.3$ | $-20.9$ | $-7.6$ | $-2.4$ | $-9.1$ | $-0.6$ | 4.9 | 12.7 | 4.1 | 1.2 | 8.1 | |

$5\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ | 850 | $-4.0$ | $-20.0$ | $-23.8$ | $-14.1$ | $-10.3$ | $-16.3$ | 0.7 | 7.7 | 18.8 | 9.9 | 6.8 | 20.3 |

875 | $-8.7$ | $-22.9$ | $-26.4$ | $-18.1$ | $-14.9$ | $-20.5$ | 1.8 | 10.1 | 24.1 | 14.9 | 11.5 | 31.1 | |

900 | $-12.0$ | $-25.7$ | $-29.3$ | $-21.4$ | $-18.4$ | $-24.3$ | 2.6 | 11.9 | 28.8 | 19.1 | 15.4 | 41.3 | |

825 | 10.3 | $-18.2$ | $-24.5$ | $-5.9$ | 2.4 | $-9.6$ | $-1.3$ | 5.7 | 16.2 | 3.0 | $-1.1$ | 8.7 | |

$10\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ | 850 | $-3.0$ | $-24.1$ | $-29.0$ | $-16.3$ | $-10.9$ | $-20.4$ | 0.5 | 10.4 | 27.1 | 12.2 | 7.3 | 29.0 |

875 | $-9.6$ | $-27.8$ | $-32.1$ | $-21.6$ | $-17.2$ | $-25.6$ | 2.0 | 14.1 | 36.6 | 19.9 | 14.2 | 47.8 | |

900 | $-18.3$ | $-34.0$ | $-38.0$ | $-29.4$ | $-25.9$ | $-33.6$ | 4.5 | 20.9 | 56.3 | 34.8 | 27.4 | 93.6 | |

825 | 17.7 | $-15.1$ | $-21.8$ | $-2.0$ | 8.6 | $-6.5$ | $-2.1$ | 4.5 | 13.5 | 1.0 | $-3.6$ | 5.5 | |

$15\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ | 850 | 2.8 | $-21.6$ | $-26.8$ | $-13.4$ | $-6.4$ | $-18.1$ | $-0.5$ | 8.6 | 23.2 | 9.3 | 3.9 | 23.9 |

875 | $-6.9$ | $-27.0$ | $-31.6$ | $-21.1$ | $-15.7$ | $-25.7$ | 1.4 | 13.4 | 35.4 | 19.0 | 12.5 | 48.2 | |

900 | $-24.8$ | $-39.2$ | $-42.8$ | $-36.1$ | $-32.7$ | $-40.4$ | 7.1 | 30.2 | 88.3 | 58.8 | 46.1 | 209.4 |

Crosstalk values for a source-detector distance of $r=4.0\phantom{\rule{0.3em}{0ex}}\mathrm{cm}$ results in slightly smaller values. At ${h}_{f}=0$ , 5, 10, and $15\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ , ${C}_{\mathrm{Hb}{\mathrm{O}}_{2}\to \mathrm{Hb},4.0,{h}_{f}}$ was $9.0\pm 5.6$ , $11.5\pm 9.2$ , $19.3\pm 17.6$ , and $22.6\pm 27.4\%$ , and ${C}_{\mathrm{Hb}\to \mathrm{Hb}{\mathrm{O}}_{2},4.0,{h}_{f}}$ was $14.1\pm 4.4$ , $15.3\pm 7.5$ , $20.1\pm 10.0$ , and $20.6\pm 11.0\%$ , respectively.

## 5.

## Discussion

We showed that the presence of a fat tissue layer causes underestimation error and crosstalk problems in cw-NIRS muscle measurements and that these problems are fat-thickness dependent. The main cause of these problems is the homogeneous medium assumption in the MBLL calculations with the use of a constant path length instead of fat thickness and wavelength-dependent mean partial path length in the muscle layer. The fat layer has a lower absorption coefficient than the underlying muscle layer and it has been shown^{30, 32, 33, 54} that as the fat layer thickens, probed volume by NIRS system also increases (the “banana” gets fatter). However, as the banana gets fatter, probed muscle volume decreases (
$\u27e8{L}_{m}^{\lambda}\u27e9$
decreases). Thicker fat layer leads to an increase in
$\u27e8{L}^{\lambda}\u27e9$
and
$\u27e8{L}_{f}^{\lambda}\u27e9$
and detected light intensity for the considered wavelengths in the
$675\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}900\text{-}\mathrm{nm}$
range, as shown in Sec.
4.1. Similar findings were reported in the literature such as the inverse relation between
$\u27e8{L}_{m}\u27e9$
and
${h}_{f}$
found by simulation studies
^{25, 30, 31, 34, 35, 54, 55, 56} and by theoretical investigations.^{55} Higher detected light intensities have been also reported for thicker fat layer.^{32, 33, 54, 57}

There is also a strong wavelength dependency of
$\u27e8{L}_{m}^{\lambda}\u27e9$
. The concentration of
$\mathrm{Hb}{\mathrm{O}}_{2}$
(taken as 70%) is higher than [Hb], and for longer wavelengths,
${\u03f5}_{\mathrm{Hb}{\mathrm{O}}_{2}}^{\lambda}$
is higher, which result in
${\mu}_{a,m}^{\lambda}$
increasing, leading to a decrease in
$\u27e8{L}_{m}^{\lambda}\u27e9$
and
$\u27e8{L}^{\lambda}\u27e9$
for longer wavelengths. In experimental studies, wavelength dependency has been reported^{58, 59, 60} only for the DPF factor, since it is impossible to measure and isolate
$\u27e8{L}_{m}^{\lambda}\u27e9$
from a layered structure. Duncan
^{61} reports DPF values of
$4.43\pm 0.86$
$(5.78\pm 1.05)$
at
$690\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
, and
$3.94\pm 0.78$
$(5.33\pm 0.95)$
at
$832\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
in the forearm (calf) for
$r=4.5\phantom{\rule{0.3em}{0ex}}\mathrm{cm}$
. In the same study, a significant female/male difference in the DPF values was shown, with values of
$4.34\pm 0.78$
for females and
$3.53\pm 0.55$
for males in the forearm at
$832\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
. For
$r>2.5\phantom{\rule{0.3em}{0ex}}\mathrm{cm}$
, DPF has been shown to be almost constant by van der Zee, ^{60} where it was also stated that a female/male difference was present with mean DPF values of
$5.14\pm 0.43$
for females versus
$3.98\pm 0.46$
for males at
$761\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
in the adult calf, but no difference was observed in the adult forearm (both DPF are
$3.59\pm 0.32$
). A general trend of DPF decrease in
$740\text{-}\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}840\text{-}\mathrm{nm}$
range was also found by Essenpreis, ^{58} although no significant female/male difference was observed. In these studies, a female/male difference was attributed to fat/muscle ratio differences, although statistics concerning fat thicknesses were not present about the subjects in the studies.

In this study, we investigated the error in the estimation of the concentration changes using MBLL with homogeneous medium assumption under two headings: an underestimation error and crosstalk. We showed that fat thickness has a strong effect on both. The means of both absolute underestimation errors and absolute crosstalk over the considered wavelength pairs were calculated to be high for thick fat layer, as stated in Sec. 4.2, 4.3. As stated, a decrease of $\u27e8{L}_{m}^{\lambda}\u27e9$ with increased ${h}_{f}$ and the use of a fixed DPF value in MBLL calculations because of the homogeneous medium assumption leads to rise in underestimation error. Crosstalk depends on $\u27e8{L}_{m}^{\lambda}\u27e9$ but not the used DPF value when a wavelength-independent DPF is used. The wavelength dependency of ${\u03f5}_{\mathrm{Hb}{\mathrm{O}}_{2}}^{\lambda}$ and ${\u03f5}_{\mathrm{Hb}}^{\lambda}$ as well as the difference between them also affect crosstalk.

The choice of wavelength pair had a significant impact on the errors. The variability in the absolute underestimation errors for different wavelength pairs is higher for low fat thickness values while the variability in the absolute crosstalk for different wavelength pairs increases with increasing fat thickness. The means of absolute underestimation errors and absolute crosstalk were found to be higher for ${E}_{\mathrm{Hb}{\mathrm{O}}_{2},3.0,{h}_{f}}$ and ${C}_{\mathrm{Hb}\to \mathrm{Hb}{\mathrm{O}}_{2},3.0,{h}_{f}}$ . These findings are related to wavelength dependency of $\u27e8{L}_{m}^{\lambda}\u27e9$ and specific absorption coefficients. Note $\u27e8{L}_{m}^{\lambda}\u27e9$ has a decreasing trend at longer wavelengths and ${\u03f5}_{\mathrm{Hb}}^{\lambda}$ $\left({\u03f5}_{\mathrm{Hb}{\mathrm{O}}_{2}}^{\lambda}\right)$ is higher (lower) for wavelengths less than $798\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ , the isobestic point. In more detail, the reason for a higher underestimation error of ${E}_{\mathrm{Hb}{\mathrm{O}}_{2},3.0,{h}_{f}}$ with respect to ${E}_{\mathrm{Hb},3.0,{h}_{f}}$ can be explained by $\Delta {\mathrm{OD}}^{{\lambda}_{2}}$ $(\propto \u27e8{L}_{m}^{{\lambda}_{2}}\u27e9)$ being more heavily weighted by the real concentration change of $\Delta {\left[\mathrm{Hb}{\mathrm{O}}_{2}\right]}_{m}$ in the muscle layer than $\Delta {\left[\mathrm{Hb}\right]}_{m}$ . In the MBLL equations, measured $\Delta {\mathrm{OD}}^{\lambda}$ ’s are assumed to be proportional to DPF $\times r$ instead of unavailable $\u27e8{L}_{m}^{\lambda}\u27e9$ . Wrongly used DPF $\times r$ overestimates the $\u27e8{L}_{m}^{\lambda}\u27e9$ (leading to underestimation error for concentration change), however, the degree of path length overestimation is higher for longer wavelength since $\u27e8{L}_{m}^{\lambda}\u27e9$ decreases with wavelength. Hence, the path length overestimation because of homogeneous medium assumption is higher for measured optical density change $\Delta {\mathrm{OD}}^{{\lambda}_{2}}$ leading to more underestimation error for $\Delta {\left[\mathrm{Hb}{\mathrm{O}}_{2}\right]}_{\mathrm{MBLL}}$ .

There is one previous study on crosstalk for muscle cw-NIRS measurements by Iwasaki and Okada.^{42} This analysis was done for a fixed fat thickness of
$4\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$
, a two-wavelength system was assumed,
${\lambda}_{2}$
was fixed at
$830\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
, and
$r$
was taken as 2.0 or
$4.0\phantom{\rule{0.3em}{0ex}}\mathrm{cm}$
. The
$720\u2215830\text{-}\mathrm{nm}$
and
$780\u2215830\text{-}\mathrm{nm}$
pairs were found to be the favorable pair selections resulting in minimal crosstalk. In our study, the
$775\u2215825\text{-}\mathrm{nm}$
pair also gave low crosstalk values along with the
$750\u2215825$
- and
$760\u2215825\text{-}\mathrm{nm}$
pairs, for both
${C}_{\mathrm{Hb}{\mathrm{O}}_{2}\to \mathrm{Hb}}^{{\lambda}_{1},{\lambda}_{2}}$
and
${C}_{\mathrm{Hb}\to \mathrm{Hb}{\mathrm{O}}_{2}}^{{\lambda}_{1},{\lambda}_{2}}$
. Iwasaki and Okada^{42} found negative
${C}_{\mathrm{Hb}{\mathrm{O}}_{2}\to \mathrm{Hb}}^{{\lambda}_{1},{\lambda}_{2}}$
values and positive
${C}_{\mathrm{Hb}\to \mathrm{Hb}{\mathrm{O}}_{2}}^{{\lambda}_{1},{\lambda}_{2}}$
values; however, we calculated not only opposite signs but also different magnitudes. These could be due to choice of muscle absorption coefficients, the values in this study range between 2.1 to 3.7 times higher than the values used in our study. We also looked at the effect of fat thickness variation on crosstalk and found a rise in the mean of absolute crosstalk values over the considered wavelength pairs for an increase in fat thickness. Moreover, other
${\lambda}_{2}$
values were studied, up to
$900\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
. There was an increase in crosstalk amplitudes for an increase in
${\lambda}_{2}$
for values higher than
$825\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
for a fixed
${\lambda}_{1}$
at a given
${h}_{f}$
value. The absolute values of
${C}_{\mathrm{Hb}{\mathrm{O}}_{2}\to \mathrm{Hb},3.0}^{{\lambda}_{1},{\lambda}_{2}}$
and
${C}_{\mathrm{Hb}\to \mathrm{Hb}{\mathrm{O}}_{2},3.0}^{{\lambda}_{1},{\lambda}_{2}}$
were calculated to be less than 11% for the
$675\u2215825$
,
$675\u2215850$
,
$675\u2215875$
-,
$750\u2215825$
-,
$760\u2215825$
-,
$760\u2215850$
-, and
$775\u2215825\text{-}\mathrm{nm}$
pairs for
${h}_{f}<10\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$
.

Arterial occlusion is employed in cw-NIRS measurements to estimate muscle oxygen consumption. In this case, ideally blood volume remains constant, while $\Delta {\left[\mathrm{Hb}{\mathrm{O}}_{2}\right]}_{m}$ decreases and $\Delta {\left[\mathrm{Hb}\right]}_{m}$ increases in equal magnitudes in the probed volume. Using Eq. 13, the estimation errors were found to be $10.6\pm 5.2$ , $30.7\pm 4.6$ , and $54.6\pm 4.1$ % for $\Delta {\left[\mathrm{Hb}\right]}_{\mathrm{MBLL}}$ and $15.1\pm 4.3$ , $34.3\pm 3.7$ , and $57.1\pm 3.2\%$ for $\Delta {\left[\mathrm{Hb}{\mathrm{O}}_{2}\right]}_{\mathrm{MBLL}}$ at ${h}_{f}=0$ , 2, $4\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ respectively, computed over 24 wavelength pairs ( $\mathrm{r}=3.0\phantom{\rule{0.3em}{0ex}}\mathrm{cm}$ , $\mathrm{DPF}=4.37$ ) These estimation errors for the two chromophores are closer compared to the differences between underestimation errors (Sec. 4.2) due to the crosstalk. The estimation error for $\Delta {\left[\mathrm{Hb}\right]}_{\mathrm{MBLL}}$ is higher than the underestimation error ${E}_{\mathrm{Hb},3.0,{h}_{f}}$ , while estimation error of $\Delta {\left[\mathrm{Hb}{\mathrm{O}}_{2}\right]}_{\mathrm{MBLL}}$ is lower than the underestimation error ${E}_{\mathrm{Hb}{\mathrm{O}}_{2},3.0,{h}_{f}}$ . For this protocol, the minimum estimation errors were found for the $700\u2215825$ - and $725\u2215825\text{-}\mathrm{nm}$ pairs. For a fixed ${\lambda}_{2}$ , the estimation errors for the occlusion protocol were found to be low for choice of 700 or $725\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ as ${\lambda}_{1}$ , while for fixed ${\lambda}_{1}$ , errors rise for an increase in ${\lambda}_{2}$ , for both $\Delta {\left[\mathrm{Hb}{\mathrm{O}}_{2}\right]}_{\mathrm{MBLL}}$ and $\Delta {\left[\mathrm{Hb}\right]}_{\mathrm{MBLL}}$ .

The error analysis in this study showed the clear failure of the homogenous medium assumption and the requirement to correct cw-NIRS measurements even for low fat thickness values, although it was stated that correction may not be required for less than
$5\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$
fat thickness by Yang
^{57} There are already several proposed approaches for cw-NIRS measurement corrections, in particular for
$m\stackrel{\u0307}{V}{\mathrm{O}}_{2}$
. Several investigators^{25, 32, 55} have proposed correction algorithms using theoretically determined
$\u27e8{L}_{m}\u27e9$
. Niwayama
^{56, 62, 63} combined the results of simulations and experiments (for
$\u27e8{L}_{m}\u27e9$
, detected light intensities, and experimental sensitivities) to obtain correction curves for
$m\stackrel{\u0307}{V}{\mathrm{O}}_{2}$
. Utilizing these corrections, the variance of the experimental
$m\stackrel{\u0307}{V}{\mathrm{O}}_{2}$
results were reduced,^{56, 63} moreover, a higher correlation was found between
$m\stackrel{\u0307}{V}{\mathrm{O}}_{2}$
values measured by
${}^{31}\mathrm{P}$
-NMR and corrected
$m\stackrel{\u0307}{V}{\mathrm{O}}_{2}$
values measured^{62} by cw- NIRS. Yet another correction algorithm was proposed by the same group in which a relationship between detected light intensity and measurement sensitivity was utilized as an empirical technique to reduce the variance in
$m\stackrel{\u0307}{V}{\mathrm{O}}_{2}$
findings due to fat thickness.^{32, 33, 64} Yang
^{57} proposed a correction for intensity of cw-NIRS measurements using a polynomial fit to detected intensity change with fat thickness. Lin
^{65} used a neural-network-based algorithm for spatially resolved reflectance, first to find the optical coefficients of the top layer and then that of the layer below, assuming the top layer thickness is known. There are also broadband cw-NIRS techniques. One method orthogonalizes the spectra collected at a long source-detector distance
$\left(r\right)$
to the spectra collected at a short
$r$
and maps to the long
$r$
space.^{66, 67} Another one uses multiple detectors and the derivative of attenuation with respect to distance, utilizing a particular wavelength sensitive to fat thickness.^{68, 69}

Figure 4
shows four cw-NIRS measurement sensitivity curves. The first curve from our study is the calculated
$\Delta {\left[\mathrm{Hb}{\mathrm{O}}_{2}\right]}_{\mathrm{MBLL}}$
computed for the ischemia protocol (for unit magnitude and opposite
$\Delta {\left[\mathrm{Hb}\right]}_{\mathrm{m}}$
and
$\Delta {\left[\mathrm{Hb}{\mathrm{O}}_{2}\right]}_{m}$
) using
$\mathrm{DPF}=4.37$
, at a
$750\u2215850\text{-}\mathrm{nm}$
pair
$(r=3.0\phantom{\rule{0.3em}{0ex}}\mathrm{cm})$
. The computed
$\Delta {\left[\mathrm{Hb}\right]}_{\mathrm{MBLL}}$
for the same conditions (not shown) has a slightly higher sensitivity. The sensitivity curve of Niwayama
^{63} is proposed for muscle measurement correction by dividing the calculated concentration changes by itself—given by
$\mathrm{exp}\phantom{\rule{0.2em}{0ex}}{[-\left(\right({h}_{f}+{h}_{s})\u22158.0)}^{2}]$
, using the
$760\u2215840\text{-}\mathrm{nm}$
pair for *r*=3.0 cm, we take
${h}_{s}$
to be
$1.4\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$
. The curve of Niwayama
^{63} indicates higher sensitivity than the one our curve predicts. For the computed
$\Delta {\left[\mathrm{Hb}{\mathrm{O}}_{2}\right]}_{\mathrm{MBLL}}$
, taking a lower DPF value of 4.0 (the value used in the Niwayama
^{63}) leads to a higher sensitivity. Yet another curve is derived from the experimental resting state oxygen consumption curve of van Beekvelt
^{36} [
$m\stackrel{\u0307}{V}{\mathrm{O}}_{2}=0.18\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}-0.14\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{10}(hf+hs)$
ml of
${\mathrm{O}}_{2}\phantom{\rule{0.3em}{0ex}}{\mathrm{min}}^{-1}\phantom{\rule{0.2em}{0ex}}100\phantom{\rule{0.2em}{0ex}}{\mathrm{g}}^{-1}$
, used
$\mathrm{DPF}=4.0\phantom{\rule{0.3em}{0ex}}\mathrm{cm}$
,
$r=3.5\phantom{\rule{0.3em}{0ex}}\mathrm{cm}$
, three wavelengths a
$770\u2215850\u2215905\text{-}\mathrm{nm}$
system, we take
${h}_{s}$
as
$1.4\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$
) by normalizing it to its value at a
$0\text{-}\mathrm{mm}$
fat thickness. The study had 78 volunteers with highest fat (plus skin) thickness of
$8.9\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$
(approximating a
$7.5\text{-}\mathrm{mm}$
fat thickness), hence shown up to
${h}_{f}=8\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$
. It is closer to our curve for low-fat-thickness values
$(<4\phantom{\rule{0.3em}{0ex}}\mathrm{mm})$
but presents higher sensitivity for higher fat thickness values and becomes closer to the curve of Niwayama
^{63} van Beekvelt
^{36} reports a 50% decrease in experimentally found oxygen consumption
$\left(m\stackrel{\u0307}{V}{\mathrm{O}}_{2}\right)$
for fat thickness (including skin) in a range from
$5\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}10\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$
. Niwayama
^{56, 63} reports of a roughly 50% decrease in cw-NIRS measurement sensitivity for a twofold increase in fat (including skin) thickness, but the range for fat thickness is not given. In our study, we calculated a nearly 55% decrease in the
$\Delta {\left[\mathrm{Hb}{\mathrm{O}}_{2}\right]}_{\mathrm{MBLL}}$
and
$\Delta {\left[\mathrm{Hb}\right]}_{\mathrm{MBLL}}$
for the ischemia protocol at
$750\u2215850$
and
$775\u2215850\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
(the closest pairs to the wavelengths used in the mentioned studies) for
${h}_{f}$
increase from
$3\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}6\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$
, while the decrease becomes nearly 34% for
${h}_{f}=2\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}4\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$
, and 70% for
${h}_{f}=4\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}8\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$
.

MBLL calculations are based on a linear approximation for the relationship of optical density change to absorption coefficient change, which leads to deviations for large concentration changes, as shown by Shao
^{70} The presence of the fat layer deteriorates the linearity of measurement characteristics investigated by Lin
^{25} In our study, we assumed small concentration changes. In quantitative studies aimed at oxygen consumption calculations, concentration change rates within small time scales during ischemia are typically used. In the experimental study of Ferrari, ^{71} a difference of
$\Delta \left[\mathrm{Hb}{\mathrm{O}}_{2}\right]-\Delta \left[\mathrm{Hb}\right]$
was computed for ischemia alone and for ischemia with maximal voluntary contraction. For these measurements, desaturation rates were computed with constant DPF and with changing DPF values found using time-resolved spectroscopy with the same experiment protocols. Similar rate values were calculated within short time scales.

The effect of fat layer thickness on cw-NIRS measurements is very explicit and dominant; note, however, that partial path length values, detected intensities, underestimation errors, and crosstalk are all subject to both intrasubject and intersubject variability because of optical coefficients’ variability of tissue layers, variability in physiological status, muscle anatomy differences, and anisotropy in the skin^{72} and in the muscle.^{73}

An increase in the source-detector distance leads to lower errors because of increased
$\u27e8{L}_{m}\u27e9$
, however, signal-to-noise ratio (SNR) also decreases since detected intensity decreases leading to a trade-off. It may be possible to discover an optimal source-detector distance based on optimization of SNR maximization and error minimization,^{35, 54} by also taking into account the fat thickness of the subject.

## 6.

## Conclusion

The fat layer influence on muscle cw-NIRS measurements based on MBLL calculations with homogeneous medium assumption was investigated for both underestimation error and crosstalk using Monte Carlo simulations for a two-wavelength system. Although the computed values of underestimation errors and crosstalk are dependent on the “true” optical coefficients of the tissue layers, and hence could change for each subject, an explicit finding is that the mean values of the absolute underestimation errors and absolute crosstalk computed over the considered wavelength pairs increase for the thicker of the fat layer. The means of absolute underestimation errors
${E}_{\mathrm{Hb}{\mathrm{O}}_{2},3.0,{h}_{f}}$
and absolute crosstalk
${C}_{\mathrm{Hb}\to \mathrm{Hb}{\mathrm{O}}_{2},3.0,{h}_{f}}$
over the considered wavelength pairs were found to be higher, while due to the crosstalk, the estimation errors for the concentration changes of the two chromophores were calculated to be closer for the ischemia protocol. These errors also depended on the wavelength pair selection for the two-wavelength system with greater impact on the crosstalk. This dependency of wavelength leads to the fact that correction algorithms should be dependent on the choice of wavelengths, although different wavelength combinations can have very similar sensitivities. The measurement of the fat thickness values and providing information about it should become a standard routine, as suggested by van Beekvelt
^{74} for the cw-NIRS measurements.

## Acknowledgments

This study was supported by the Boğaziçi University Research Fund through projects 04X102D and 04S101 and by Turkish State Planning Organization through projects 03K120250 and 03K120240. The doctoral fellowship for Ömer Şayli by TÜBİTAK (The Turkish Scientific & Technological Research Council) is gratefully acknowledged.

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