2.1.

Modeling

To study and describe the reflectance of the tissue, the reflectance was modeled for a two-layered structure, where the upper layer mimics the adipose tissue (thickness $\mathrm{ATT}⩽16\mathrm{mm}$ ) and the lower layer represents muscle (infinite thickness). The main difficulty for any model is the selection of the appropriate absorption and scattering properties, expressed by ${\mu }_a$ and ${\mu }_s^{\prime }$ .

In-vitro measurements by Simpson ⁴⁷ showed values of ${\mu }_s^{\prime }$ in the range of $2{\mathrm{mm}}^{-1}$ for skin, 1.1 $(\pm 0.3){\mathrm{mm}}^{-1}$ for adipose subdermis tissue, and $0.6{\mathrm{mm}}^{-1}$ for abdominal muscle. Similarly, Bashkatov ⁴⁸ report on ${\mu }_s^{\prime }$ of adipose tissue and found in-vitro values in the range of 1.20 $(\pm 0.35){\mathrm{mm}}^{-1}$ for the spectral range considered here. Van Veen ⁴¹ measured the scattering coefficient of lipid prepared from pig lard with time-domain methods and reported values decreasing from $0.7\text{to}0.45{\mathrm{mm}}^{-1}$ for the wavelength range from $600\text{to}1100\mathrm{nm}$ . There are many data published of bulk optical properties in vivo, which differ widely, depending on the tissue interrogated, the experimental technique employed, and the analysis method used. Torricelli ³⁴ measured transport scattering coefficients in the range of about $0.8\text{to}0.9{\mathrm{mm}}^{-1}$ for the arm and slightly higher values for the abdomen. Casavola ⁴⁹ report values of ${\mu }_s^{\prime }$ in the range of $0.5{\mathrm{mm}}^{-1}$ for arm measurements. Matcher, Cope, and Delpy⁴⁵ obtained scattering coefficients of about 0.68 $(\pm 0.08){\mathrm{mm}}^{-1}$ for the forearm and 0.95 $(\pm 0.07){\mathrm{mm}}^{-1}$ for calf at $800\mathrm{nm}$ . In all reports, ${\mu }_s^{\prime }$ is slightly decreasing with wavelength.^{34, 45, 47} Similarly, published values of the absorption coefficients of tissue vary widely. Van Veen ⁴¹ provided, the absorption spectra of purified lipid [see Fig. 1c], which coincides well with in-vitro data.⁴⁷ Matcher, Cope, and Delpy⁴⁵ obtained absorption coefficients of about 0.023 $(\pm 0.004){\mathrm{mm}}^{-1}$ for the bulk tissue of the forearm and 0.017 $(\pm 0.005){\mathrm{mm}}^{-1}$ for calf at $800\mathrm{nm}$ . This is in line with other work.^{34, 50}

Based on these literature values, the following parameters were chosen for the simulations, with ${\mu }_a$ and ${\mu }_s^{\prime }$ set independently for muscle (index M) and lipid (index L) layers: ${\mu }_{a,M}=0.005$ , 0.01, and $0.02{\mathrm{mm}}^{-1}$ , and ${\mu }_{s,M}^{\prime }=0.7$ , 1.0, and $1.3{\mathrm{mm}}^{-1}$ . The same scattering properties were used for the lipid layer $\left({\mu }_{s,L}^{\prime }\right)$ . The wavelength-dependent absorption coefficient of the lipid layer ${\mu }_{a,L}$ was taken from the spectra of pure lipid ${\mu }_{a,\text{lipid}}$ [Fig. 1c], assuming that the adipose tissue layer is pure lipid.⁴¹ To account for additional chromophores, an additional offset ${\mu }_{a,\text{Loffset}}$ between 0.0 and $0.006{\mathrm{mm}}^{-1}$ in steps of $0.002{\mathrm{mm}}^{-1}$ was included: ${\mu }_{a,L}={\mu }_{a,\text{lipid}}+{\mu }_{a,\text{Loffset}}$ . Therefore, all parameters besides ${\mu }_{a,L}$ were assumed to be wavelength independent. The refractive index was set to $n=1.4$ for both layers. Reflectance was calculated for source–detector distances $\rho$ up to $45\mathrm{mm}$ .

To compare the experimental OLS data with the simulations, reflectance spectra were calculated for the range $850\text{to}970\mathrm{nm}$ by inclusion of the spectra ${\mu }_{a,\text{lipid}}\left(\lambda \right)$ . These spectra were converted into attenuation spectra and analyzed as described for the experimental attenuation data, i.e., the second derivative with respect to wavelength was computed with the OLS obtained in analogy to Eq. 2. To mimic the finite spectral resolution of the experimental setup (about $10\mathrm{nm}$ ), the simulated attenuation spectra were smoothed with a gliding average filter. Initially, a Monte Carlo model was employed for this calculation, which is rather computationally time consuming when reflectance spectra were required.⁵¹ Finally, all calculations were done with an analytical solution for the two-layer model based on diffusion approximation.^{38, 39} For this purpose, an executable program was kindly provided by Kienle.⁵² For a limited set of tissue parameters, the agreement of Monte Carlo and the analytical method was confirmed.

3.1.

Experimental Lipid Measurements

In Figs. 2, 3, 4 , the correlation plots of the optical lipid signal OLS with ultrasound ${\mathrm{ATT}}_{\mathrm{US}}$ , calliper skin thickness (CST), and MRI-based ${\mathrm{ATT}}_{\mathrm{MR}}$ are shown with ATT in the range from $<1\text{to}16\mathrm{mm}$ . In Fig. 2a, each point represents data from a single subject with different symbols for the five muscle groups. There is a clear nonlinear relationship between OLS and ${\mathrm{ATT}}_{\mathrm{US}}$ . For the estimation of prediction accuracy in the optical estimation of ATT, the data of Fig. 2a were fitted by the function

Fig. 2

Correlation of optical lipid signal (OLS) and ultrasound ${\mathrm{ATT}}_{\mathrm{US}}$ for five different muscle groups on 20 subjects. The solid line represents an exponential fit (see text for details). (a) Data of all 20 subjects (insert: histogram of residuum with a mean of absolute residuum of $0.84\mathrm{mm}$ and mean relative error of 28%). The magnitudes of the experimental errors (see text for details) are plotted at the lower and upper end of the fitted curve. (b) Mean $(\pm \mathrm{SD})$ values for the different muscle groups, with the same exponential fit as in (a).

Fig. 3

Correlation of optical lipid signal (OLS) and calliper skin thickness (CST) for five different muscle groups of 20 subjects. The solid line represents a fit (see text). Insert: histogram of residuum with absolute and relative errors of $1.2\mathrm{mm}$ and 93%, respectively.

Fig. 4

Correlation of optical lipid signal (OLS) and MRI-based ${\mathrm{ATT}}_{\mathrm{MR}}$ . The solid line represents an exponential fit (see text) of the experimental data. Right insert: histogram of residuum. Left insert: example of an MRI of the thigh with the bright outer lipid regions (thickness: $4\text{to}14\mathrm{mm}$ ).

To allow a comparison with published work, the correlation coefficient $R$ was evaluated: when ${\mathrm{ATT}}_{\mathrm{US}}$ was directly correlated with OLS, $R=0.92$ was obtained. Taking the nonlinearity into account by scaling ${\mathrm{ATT}}_{\mathrm{US}}$ by Eq. 4 including $p_1$ and $p_2$ , gave $R=0.95$ .

When averaging the experimental data for each of the five muscle groups, the mean values $(\pm \mathrm{SD})$ shown in Fig. 2b were obtained. There is a clear difference in mean ATT value for the different muscles with large intersubject variations. As can be expected, for the thigh (vastus lateralis and vastus medialis), the ATT is larger than for the calf (gastrocnemius) and the arm. The fitted line from the data of Fig. 2a correlates well with these mean values. The explanation for the nonlinear relationship between ${\mathrm{ATT}}_{\mathrm{US}}$ and OLS can be found in the optical path length in the lipid layer, which depends both on its thickness and the underlying muscular tissue. A detailed analysis is presented next within the theoretical model. The error bars given at the lower and upper end of the ${\mathrm{ATT}}_{\mathrm{US}}$ range of Fig. 2a are based on experimental uncertainties in determining the ultrasound and OLS (see next for details).

The same OLS data were correlated with calliper skin thickness (CST), which is shown in Fig. 3. CST is limited to the range $⩽7\mathrm{mm}$ , and as said in Sec. 2, no reading could be obtained for 24.2% of the samples. We have no explanation for this failure rate, but suspect that it is due to limitations in the calliper instrumentation and the underlying approach.

For a limited number of measurements, CST is small $(<1\mathrm{mm})$ with corresponding high OLS values. Again the data were fitted according to Eq. 4 (with $p_1=5.429\cdot {10}^{-3}\mathrm{OD}\cdot {\mathrm{nm}}^{-2}$ and $p_2=0.1876{\mathrm{mm}}^{-1}$ ) giving the histogram of the residuum shown in the insert. The prediction errors are ${\epsilon }_a=1.2\mathrm{mm}$ and ${\epsilon }_r=93\%$ .

When taking a calliper reading it must be assumed that the sum of both skin and adipose tissue is measured. Therefore it seems obvious that an offset has to be included that takes into account the skin thickness. Equation 4 was therefore modified to include a third fitting parameter as offset, $\mathrm{OLS}=p_1\cdot [1-{\exp }^{(-p_2\cdot \mathrm{REF}-p_3)}]$ . When fitting this function to the experimental data, no marked difference was obtained in terms of residuum and prediction errors ${\epsilon }_a$ and ${\epsilon }_r$ . Calliper CST and ${\mathrm{ATT}}_{\mathrm{US}}$ were correlated (data not shown), and assuming that both measurements depend linearly on the lipid layer, a linear regression was fitted, giving prediction errors ${\epsilon }_a=1.2\mathrm{mm}$ and ${\epsilon }_r=43\%$ (again, for the 75.8% of the samples where a reading could be obtained). To permit a better comparison of the methods, the optical lipid signal and ultrasound readings were evaluated (OLS versus ${\mathrm{ATT}}_{\mathrm{US}}$ ) for the same limited dataset, giving prediction errors of ${\epsilon }_a=0.7\mathrm{mm}$ and ${\epsilon }_r=31\%$ .

For the separate experimental dataset based on MRI, the correlation with OLS is shown in Fig. 4. An example MRI is shown in the insert where the nine vitamin E markers can be seen, and the approximate measurement positions for the optical spectra are numbered (O1 to O9). The regions rich in lipid are revealed by high signal intensity. The smallest ${\mathrm{ATT}}_{\mathrm{MR}}$ values $\left(2\mathrm{mm}\right)$ are larger than for the ${\mathrm{ATT}}_{\mathrm{US}}$ data, as all measurements were taken from the leg [compare with Fig. 2b]. Fitting the data according to Eq. 4 $(\mathrm{REF}={\mathrm{ATT}}_{\mathrm{MR}})$ gives parameters $p_1=7.443\cdot {10}^{-3}\mathrm{OD}\cdot {\mathrm{nm}}^{-2}$ and $p_2=0.0979{\mathrm{mm}}^{-1}$ . The prediction errors are ${\epsilon }_a=0.9\mathrm{mm}$ and ${\epsilon }_r=16\%$ . The results of the prediction errors are summarized in Table 1 .

Table 1

Absolute and relative prediction errors ( εa and εr ) for the data shown in Figs. 2, 3, 4. In column 1 (OLS versus ATTUS ), the whole dataset of 240 readings was included. Columns 2, 3, and 4 are based on the data where calliper readings could be obtained (see text for details). For column 5, an independent dataset was analyzed.

3.2.

Estimation of Errors

The prediction error ${\epsilon }_a$ of ATT based on the OLS is on the order of $0.7\text{to}1.2\mathrm{mm}$ , which brings up the question of experimental limitations for each measurement modality. The ${\mathrm{ATT}}_{\mathrm{US}}$ values were generated by an analysis of three readings taken from three different images, with the mean of these three readings taken as ${\mathrm{ATT}}_{\mathrm{US}}$ for each sample. The standard deviation of the three readings was also calculated, which had a mean of $0.26\mathrm{mm}$ when averaging over all samples. Similarly, for the optical attenuation spectra of the three detectors, the OLS was calculated independently with the mean plotted in Figs. 2, 3, 4. The standard deviation of OLS for the three detectors was evaluated for each measurement site that had a mean value of $4.14\cdot {10}^{-4}\mathrm{OD}\cdot {\mathrm{nm}}^{-2}$ . These two mean values of the standard deviations represent a lower limit for the attainable residuum. In Fig. 2a, these inherent experimental errors are plotted for two arbitrary points at the lower and upper ${\mathrm{ATT}}_{\mathrm{US}}$ range of the fitted curve. It was tested whether this is the dominant error with the hypothesis that a larger standard deviation of the three independent measurements is correlated with a larger residuum. To this end, the standard deviation in the three OLS values was correlated with the residuum of Fig. 2a for all 240 samples. Similarly, the standard deviation of the three ultrasound readings was correlated with the residuum. These plots (data not shown) showed no clear correlation and therefore gave no indication that the residuum for the data of Fig. 2a is predominantly due to these experimental limitations.

The main limitation of the calliper readings is likely to be due to differences in tissue structure and composition that determine the mechanical properties, which is beyond the validation or assessment outlined here. The limited CST range $(<7\mathrm{mm})$ makes it difficult to compare the findings of CST versus ${\mathrm{ATT}}_{\mathrm{US}}$ and OLS versus ${\mathrm{ATT}}_{\mathrm{US}}$ . When taking ATT readings from the MRIs, similar errors for the ultrasound images resulted. Additional errors are likely to have occurred as the subjects had to be repositioned after taking MRIs in the MR coil to readings of optical spectra.

3.3.

Crosstalk of Optical Lipid Signal with Hemoglobin

A further concern is that the hemoglobin concentration and oxygenation saturation of both the adipose and the muscle tissue influence the OLS due to variations in the tissue attenuation spectra. To clarify this point experimentally, data acquired during an exercise protocol were reanalyzed. In Fig. 5, data from a four-hour period is shown monitoring oxygenation during incremental cycle exercise periods. The optical lipid signal has a mean value $\mathrm{OLS}=0.0027\mathrm{OD}\cdot {\mathrm{nm}}^{-2}$ with variations of about $\pm 5\%$ . The incremental exercise resulted in strong alterations of blood flow and oxygenation (for details see Geraskin ⁴⁰) and subsequently in large attenuation changes. The time course of attenuation at $760\mathrm{nm}$ (see Fig. 5) varies by about 0.35 OD (corresponding to more than a factor of 2 in intensity) and 0.25 OD for $\lambda =930\mathrm{nm}$ . For three points in times $t_1$ , $t_2$ , and $t_3$ at various phases (power loads) of the first exercise period, the attenuation spectra are shown in the insert. There is no apparent correlation of OLS and attenuation. Similar variations in OLS were observed in 18 other datasets acquired during the same exercise paradigm [mean of ${\mathrm{ATT}}_{\mathrm{US}}$ 5.5 $(\pm 3.5)\mathrm{mm}$ ]. The mean of the coefficient $R$ for the correlation between OLS and $\mathrm{A}\left(760\mathrm{nm}\right)$ was 0.37 $(\pm 0.20)$ . The coefficient of variation CV in OLS averaged over these 18 long datasets was 3.8 $(\pm 2.9)\%$ . Based on the fitted line shown in Fig. 2, this variation of 3.8% in OLS corresponds to an uncertainty in ATT of about $0.17\mathrm{mm}$ (at $\mathrm{ATT}=4\mathrm{mm}$ ). This number can be taken as an upper limit for the crosstalk of hemoglobin and OLS.

Fig. 5

Long-time monitoring of the OLS on the thigh (vastus lateralis) during three phases of an incremental exercise protocol separated by rest periods. The insert shows the attenuation spectra at three times ( $t_1$ , $t_2$ , and $t_3$ ) during various stages of the exercise. At the bottom, the attenuation A for $\lambda =760\mathrm{nm}$ is shown (right-hand scale, with arbitrary offset). The exercise periods are marked by E1, E2, and E3 (corresponding to 21, 15.4, and 11.9% ${\mathrm{O}}_2$ in inhaled air; see text for details).

With a similar intent to check for a crosstalk of different blood concentrations, the measurement on the forearm were done twice with the arm in a lower position (about the height of the heart) and by holding the arm upward. When comparing the OLS readings during upper and lower positions, no significant difference in the OLS was observed.

3.4.

Influence of Adipose Tissue Thickness on Hemoglobin and $\mathrm{S}{\mathrm{O}}_2$

For the 240 measurements, the hemoglobin parameters oxyHb, deoxyHb, tHb, and $\mathrm{S}{\mathrm{O}}_2$ were plotted as a function of ${\mathrm{ATT}}_{\mathrm{US}}$ (Fig. 6 ), separately for each muscle group. There is a clear decrease in the concentrations and an increase in $\mathrm{S}{\mathrm{O}}_2$ with increasing ${\mathrm{ATT}}_{\mathrm{US}}$ for all sites. This trend is different in quality, as expressed by the correlation coefficient and in the slope for the different muscle groups. The correlation of tHb and ${\mathrm{ATT}}_{\mathrm{US}}$ is strongest for values derived from vastus lateralis and vastus medialis ( $R=0.93$ and 0.79, respectively), while weak for the forearm. Similarly, the correlation of $\mathrm{S}{\mathrm{O}}_2$ and ${\mathrm{ATT}}_{\mathrm{US}}$ is best for the large muscle groups of the thigh and lowest for the forearm.

Fig. 6

Concentration of total hemoglobin tHb (left), deoxyHb (middle: solid symbols) and oxyHb (middle: open symbols), and oxygen saturation $\mathrm{S}{\mathrm{O}}_2$ (right) as a function of the adipose tissue thickness ${\mathrm{ATT}}_{\mathrm{US}}$ ( $n=40$ for each site). Values are separately plotted for the different muscle groups, with the correlation coefficient $R$ and the slope $m$ stated.

3.5.

Comparison with Tissue Model

The tissue model was employed to further shed light on the experimental findings with the objectives to explain the magnitude of the OLS and its nonlinear relationship with ATT, its dependence on source-detector distance $\rho$ , and to estimate the influence of the optical properties of the tissue $({\mu }_s^{\prime },{\mu }_a)$ . Furthermore, the experimental condition of a finite bandwidth of the spectrometer has to be evaluated. Here the two-layer model described in Sec. 2 was employed with the upper and lower layer being lipid and muscle. In Fig. 7a the calculated optical lipid signal is plotted as a function of ATT with fixed, wavelength independent values ${\mu }_{a,M}=0.01{\mathrm{nm}}^{-1}$ , ${\mu }_{s,L}^{\prime }=1{\mathrm{mm}}^{-1}$ , and ${\mu }_{s,M}^{\prime }=1{\mathrm{mm}}^{-1}$ . The absorption of the lipid layer ${\mu }_{a,L}$ is assumed to be that of pure lipid. It is apparent that the source-detector distance $\rho$ has a strong influence on OLS. When ATT is small compared to $\rho$ , OLS increases approximately linearly with ATT, while for ATT $>\rho ∕3$ it levels off. The form of this function reflects the dependence of OLS on the photon penetration into the tissue (depth sensitivity profile), which increases with $\rho$ . As a consequence, when a lipid thickness of above $10\mathrm{mm}$ needs to be determined, a source-detector distance of $⩾25\mathrm{mm}$ is advisable. With a similar argument, optical measurements of thicker adipose layers are likely to have larger absolute errors when $\rho$ is not appropriate.

Fig. 7

OLS calculated from a two-layer tissue model (upper lipid layer: index L, lower muscle layer: index M) as a function of ATT. (a) Absorption coefficient of muscle was fixed at ${\mu }_{a,M}=0.01{\mathrm{mm}}^{-1}$ , and the scattering coefficients of both layers were set to ${\mu }_s^{\prime }=1{\mathrm{mm}}^{-1}$ . The source-detector distance $\rho$ was varied from $6\text{to}42\mathrm{mm}$ . (b) OLS as a function of ATT for different scattering coefficients of both muscle and lipid layer by averaging over source-detector distances $\rho$ from $29\text{to}34\mathrm{mm}$ .

To better compare the modeled functions with the experimental conditions, all further analysis was done by averaging the OLS signals for a detector geometry from $\rho =29\text{to}34\mathrm{mm}$ . This is shown in Fig. 7b, where the influence of the transport scattering coefficient is scrutinized. Scattering of lipid was varied between ${\mu }_{s,L}^{\prime }=0.7{\mathrm{mm}}^{-1}$ and $1.3{\mathrm{mm}}^{-1}$ , with the higher scattering giving a larger OLS. Again, this is explicable by the photon path length in lipid that scales with scattering. This is different from the muscle layer, where the effect of variations from ${\mu }_{s,M}^{\prime }=0.7\text{to}1.3{\mathrm{mm}}^{-1}$ is small, with the higher muscle scattering giving a slightly higher OLS. Unsurprisingly, the effect of muscle scattering is smallest for large ATT, as the relative photon path length in the muscle layer gets smaller with larger ATT.

The influence of the absorption coefficient ${\mu }_a$ on OLS is depicted in Fig. 8, where it is assumed that the absorption of the lipid layer is that of pure lipid added by a wavelength independent offset. The largest offset chosen here $\left(0.006{\mathrm{mm}}^{-1}\right)$ is in the range of the likely muscle absorption coefficient and is much higher than what can be expected for adipose tissue. For this offset, the change in OLS is about 13% at the highest ATT.

Fig. 8

(a) OLS for $\rho =29\text{to}34\mathrm{mm}$ as function of ATT calculated for different levels of the absorption offset ${\mu }_{a,L,\text{offset}}$ in the lipid layer. In the model, the other parameters were fixed: ${\mu }_{a,M}=0.01{\mathrm{mm}}^{-1}$ , ${\mu }_{s,L}^{\prime }=1{\mathrm{mm}}^{-1}$ , ${\mu }_{s,M}^{\prime }=1{\mathrm{mm}}^{-1}$ , and $\rho =29\text{to}34\mathrm{mm}$ . (b) Influence of muscle absorption coefficient ${\mu }_{a,M}$ on OLS when ${\mu }_{s,L}^{\prime }=0.7{\mathrm{mm}}^{-1}$ , ${\mu }_{s,M}^{\prime }=0.7{\mathrm{mm}}^{-1}$ , and ${\mu }_{a,L,\text{offset}}=0.002{\mathrm{mm}}^{-1}$ . A spectrometer resolution of 1, 10, 15, and $20\mathrm{nm}$ was mimicked by spectral smoothing. The experimental values are based on the data shown in Fig. 2.

For a comparison of the experimental findings shown in Figs. 1, 2, 3, 4, 5, the model was modified to take into account the finite resolution of the optical spectrometer employed for the measurement of the attenuation spectra. To simulate this resolution, the modeled tissue reflectance spectra were smoothed by a gliding average filter with bandwidths of 1, 10, 15, and $20\mathrm{nm}$ [Fig. 8b]. To assess the agreement between experimental data and model, the most likely scattering and absorption properties have to be included. Here the following values were assumed: ${\mu }_{s,M}^{\prime }=0.7{\mathrm{mm}}^{-1}$ , ${\mu }_{s,L}^{\prime }=0.7{\mathrm{mm}}^{-1}$ , and ${\mu }_{a,L\text{offset}}=0.002{\mathrm{mm}}^{-1}$ . Though these values are somewhat arbitrary, a vindication is given in Sec. 2. Furthermore, the absorption coefficient of the muscle layer was varied by a factor of 4 $({\mu }_a=0.005–0.02{\mathrm{mm}}^{-1})$ . From the traces of Fig. 8b, it is apparent that the effect of ${\mu }_{a,M}$ on the OLS versus ATT relationship is small, which again is explicable by the small photon path length in muscle. The larger bandwidth considerably reduces the OLS signal as the lipid absorption feature at $930\mathrm{nm}$ [compare with Fig. 1c] is smoothed. To allow a comparison with the experimental data, the 240 OLS values shown in Fig. 2a were grouped according to their corresponding ${\mathrm{ATT}}_{\mathrm{US}}$ values, with equally spaced intervals of $1\mathrm{mm}$ . For each interval, the mean $(\pm \mathrm{SD})$ of the experimental OLS values are plotted in Fig. 8b. There is an overall agreement between the experimental data (spectrometer bandwidth $\approx 10\mathrm{nm}$ ) and the model (for the modeled $\text{bandwidth}=10\mathrm{nm}$ ). However, it must be added that the uncertainties in the true scattering parameters of the lipid tissue have a large effect on the model predictions.