1.

Introduction

Imaging with near-infrared diffuse optical imaging (DOI) in the neurosciences has seen increased interest over the past $15\text{years}$ .^{1, 2, 3, 4} In the commonly used continuous-wave (CW) version,^{5, 6} DOI can only measure relative changes in oxy- $\left(\mathrm{Hb}{\mathrm{O}}_2\right)$ and deoxyhemoglobin (HbR) concentrations. On the contrary, frequency domain (FD)^{7, 8, 9, 10} and time domain (TD) ^{11, 12, 13, 14, 15, 16, 17, 18, 19} technologies enable absolute measurements of the medium’s optical properties.^{20, 21, 22} This is particularly useful to calibrate brain activation and quantify the underlying hemodynamic processes within the brain. For example, multimodal studies (e.g., optical-MRI fusion²³) need quantitative information to estimate the cerebral metabolic rate of oxygen $\left(\mathrm{CMR}{\mathrm{O}}_2\right)$ . In addition, the blood-oxygen level dependant (BOLD) signal depends both on cerebral blood flow (CBF) and $\mathrm{CMR}{\mathrm{O}}_2$ . However, this relation is not straightforward, and a calibration constant must be estimated. The change in CBF can be measured separately by arterial spin labeling fMRI so the only two unknowns are the calibration constant and $\mathrm{CMR}{\mathrm{O}}_2$ . To estimate $\mathrm{CMR}{\mathrm{O}}_2$ , one must measure the BOLD signal at two different CBF values but without altering the $\mathrm{CMR}{\mathrm{O}}_2$ . This must be done during hypercapnic periods using two different levels of $\mathrm{C}{\mathrm{O}}_2$ pressure.²⁴ However, TD measurements could provide an alternative to this procedure.

Brain optical properties have been measured previously in vitro,²⁵ but physiological factors make those measurements distinct from the in vivo situation. Among those distinctions is the swelling of the mitochondria, which results in structural changes, and the small fluctuations of the brain temperature, which change the hemoglobin solubility in the blood. Moreover, previous studies performing in vivo measurements reported noticable intersubject variability.^{8, 26} These results preclude the generalization of single-subject measured values to other subjects and confirm the importance of obtaining these values individually for quantitative imaging.

In previous work, baseline optical measurements with a TD system using multidistances and a homogeneous model has been used to fit the data and estimate the optical parameters.^{26, 27} A major drawback is that this homogeneous model does not distinguish between hemoglobin concentrations in the scalp and those of the cerebral tissues, because it does not account for the layered structure of the head. The superficial layer, the skin and skull, limit the accuracy of DOI, since light is also absorbed and scattered in these regions, which are not part of the cerebral cortex. Moreover, the skin layer is subject to a physiology that may cause interference in the process of recovering cerebral activity. Several methods based on multi-distance measurements have been developed to overcome this problem.^{28, 29, 30, 31} In a multimodal study combining position emission tomography and TD optical imaging, it was shown that the contributions from these superficial layers are reduced significantly, even using a homogeneous model, when source-detector distances are increased beyond $4\mathrm{cm}$ .³² Such measurements require a large signal-to-noise ratio, which was not available with the system used in our experiment.

Separately, analytical models have been developed by solving the diffusion equation and its boundary conditions for a two-layered medium.^{22, 33, 34, 35} These models, validated with Monte Carlo simulations, showed adequate efficiencies at recovering the parameters if the thickness of the first layer of the model was known a priori. In vivo measurements with a two-layered model also have been reported using a FD system,⁸ and as expected, clear distinctions between scalp and cerebral tissues properties were made. The goal here is to provide further confirmation of the above FD results (Choi) with an independent TD technique.

In this work, we report intra- and extracerebral hemoglobin concentrations recovered on individual subjects with a time-resolved system using a two-layered analytical model for the first time. The measurements were taken with four wavelengths (690, 750, 800, and $850\mathrm{nm}$ ) at four distances: 10, 15, 25, and $30\mathrm{mm}$ . All wavelengths were fit simultaneously with a two-layered analytical model for the absorption and reduced scattering coefficient of both layers. Concentrations were then computed with the recovered absorption coefficients. We observed a large variability between subjects. Results were compared to the literature, and differences between time and frequency measurements for the oxygen saturation in the skin and skull layer were observed.

2.

Theory

Light propagation in a turbid medium is described by the radiative transport equation. This equation can be further approximated by the diffusion equation when the medium is highly scattering. Analytical solutions have been developed in the literature for simple geometries and homogeneous media.³⁶ Analytical ^{35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46} and numerical^{33, 47} solutions also have been developed for multilayered media. Only the solutions used in our fitting procedure are described in this section.

3.

Methodology

A system consisting of four pulsed lasers (PicoQuant, Berlin, Germany) operating at 690, 750, 800, and $850\mathrm{nm}$ , and four single-photon counting photomultiplier tubes (PMTs), was used to collect the light. A 4:1 combiner temporally interlaced the four wavelengths in a single fiber. Four detection fibers were placed at 10, 15, 25, and $30\mathrm{mm}$ , respectively, from the source fiber on a rubber patch. The latter was then placed on the forehead of the subject. Measurements were taken for both the left and right hemisphere. Photons were collected for a period of $300\text{seconds}$ to generate sufficient statistics, i.e., a total count of more than ${10}^4$ photons.

Five healthy adult subjects were recruited for this experiment. Every one previously passed a magnetic resonance imaging (MRI) exam (Siemens MAGNETOM Trio 3T, Malvern, Pennsylvania). We used the anatomical T1 image to measure the thickness of the scalp, skull, and cerebrospinal fluid (CSF) layers with the software SPM5 (UCL, London, UK). For the optical measures, the subjects were seated on a comfortable chair and were asked to stay calm during the acquisition. The experiment took place in a dark room to reduce the noise on the PMTs.

The fitting procedure used a nonlinear optimization routine (MATLAB function lsqcurvefit) with parameters ${\mu }_a$ , ${\mu }_s$ , and the amplitude factor $A$ to get the best fit of the theoretical model to the experimental temporal point spread functions (TPSFs). The 16 TPSFs (four distances and four wavelengths) were first fit individually with a homogeneous model, Eqs. 1, 2, with three parameters: ${\mu }_a$ , ${\mu }_s$ , and the amplitude factor $A$ . The TPSFs were then fit simultaneously four by four (four distances for each wavelength) using the two-layered model, Eqs. 6, 7, 8, using eight parameters ( ${\mu }_a$ and ${\mu }_s$ for each layer and an amplitude factor for each TPSF). For each subject, the initial values for these two-layered fits were taken to be the values obtained by the previous homogeneous fits at $10\mathrm{mm}$ and the linear fit of the tail of the reflectance at $30\mathrm{mm}$ for the first and second layer, respectively. The temporal range for the fit was from 20% of the maximum prior to the maximum peak, to 0.5% after the peak. Similar limits are found in the literature.⁵⁵

The TPSFs measured with the apparatus were the convolutions of the TPSF due to the optical properties of the scattering medium with the instrument response function. Since each PMT had a characteristic response function, we measured each separately with a simple setup: the source and detector fibers faced each other and were separated by a thin piece of scattering paper to prevent PMT saturation and fill the whole numerical aperture of the detection fiber.⁵⁶ In the fitting procedure, each theoretical TPSF function of the medium was convolved with the experimental instrument response function (IRF) before comparison with the experimental TPSF. When IRF measurements are done well, this procedure avoids including a jitter delay time in the fitting parameters.

Finally, hemoglobin concentrations were recovered by inverting Eq. 9 for each layer of the model.

4.

Results

4.1.

Simulation of the Recovery Procedure

The fitting procedure was first tested on four simulated data sets with different values for the first and the second layer. These sets were generated using a $100\times 100\times 100\mathrm{mm}$ volume with $0.5\times 0.5\times 0.5\mathrm{mm}$ voxels. The thickness of the first layer was $10\mathrm{mm}$ , and the distance between the source and the detector was $30\mathrm{mm}$ . The simulations were made using a three-dimensional Monte Carlo code.⁵⁷ We simulated a peak amplitude of ${10}^4$ photons, giving a photon noise of 1% at the peak, which is comparable to the experimental peak noise $(<1\%)$ . Results in Table 1 show less than an 11.6% error on the recovery of the absorption coefficient of the first layer, and less than a 3.1% error for the second layer, by using the multi-distance fitting procedure. In these recoveries, we assumed that the thickness of the first layer was known. Table 1 shows that the linear fit of the tail of the TPSF logarithm gave a better estimate of the absorption coefficient than using a homogeneous model.

Table 1

Absorption coefficients recovered with the two-layered model from four Monte Carlo data sets for both layers of the model. The thickness of the first layer was 10mm . Absorption coefficients were recovered from a single TPSF with source-detector distances of 10 and 30mm , and also from a multi-distance fit (10, 15, 25, and 30mm ). Results obtained using a homogeneous model are also shown for comparison.

Data set	μa1 (mm−1)		Error (%)	μa2 (mm−1)		Error (%)	μa (mm−1)
	Real	Recovered		Real	Recovered		Homogenous	Slope
$10\mathrm{mm}$
1	0.00809	0.0081	0.0894	0.0171	0.0154	9.53	0.0112	0.0204
2	0.00887	0.00807	9.01	0.0167	0.015	10.5	0.0112	0.0203
3	0.00849	0.00853	0.432	0.0172	0.0154	10.1	0.0117	0.0208
4	0.0101	0.0103	1.62	0.0204	0.0167	18.2	0.0137	0.0231
$30\mathrm{mm}$
1	0.00809	0.00881	8.88	0.0171	0.0176	3.06	0.0132	0.0158
2	0.00887	0.00884	0.284	0.0167	0.0173	3.07	0.0131	0.0156
3	0.00849	0.00924	8.87	0.0172	0.0176	2.61	0.0135	0.016
4	0.0101	0.0108	6.45	0.0204	0.0214	5.11	0.0156	0.0181
4 TPSFs
1	0.00809	0.00903	11.6	0.0171	0.0165	3.09
2	0.00887	0.00889	0.228	0.0167	0.0164	1.93
3	0.00849	0.0094	10.8	0.0172	0.0167	2.6
4	0.0101	0.0108	6.34	0.0204	0.0201	1.44

Separately, we simulated a complete Monte Carlo data set of 16 TPSFs, four source-detector distances (10, 15, 25, and $30\mathrm{mm}$ ) for four different wavelengths (690, 750, 800, and $850\mathrm{nm}$ ) with absorption coefficients computed from given hemoglobin concentrations in the superficial layer and in the brain. The concentrations chosen were 33 and $10\mu M$ in the superficial layer, and 65 and $22\mu M$ in the brain for $\mathrm{Hb}{\mathrm{O}}_2$ and HbR, respectively. The thickness of the first layer was also $10\mathrm{mm}$ in these simulations. Recovered concentrations are shown in Fig. 1 . Using the two-layered model, the errors in the recovered concentrations were less than 10% and 2% in the superficial layer and in the brain, respectively, when the thickness of the first layer was known. However, as shown in Fig. 1, the homogeneous model underestimated the hemoglobin concentrations in the brain by a factor of 30%.

Fig. 1

Hemoglobin concentrations and $\mathrm{S}{\mathrm{O}}_2$ recovered in the superficial layer and in the brain on simulated data.

4.2.

Thickness Sensitivity

To test the sensitivity of the procedure to the thickness of the first layer, hemoglobin concentrations were recovered from the same simulated data set used in the previous section but assuming different thicknesses, between 8 and $12\mathrm{mm}$ , for the first layer. Errors in the $\mathrm{Hb}{\mathrm{O}}_2$ , HbR, and total hemoglobin (HbT) concentrations, and the oxygen saturation $\left(\mathrm{S}{\mathrm{O}}_2\right)$ recovered with this procedure, are shown in Fig. 2 . These results show that the accuracy of the recovery depends on the assumed thickness. For an error of $\pm 1\mathrm{mm}$ , the errors in the recovered hemoglobin concentrations in the brain were between 10% and 15%. On the other hand, the $\mathrm{S}{\mathrm{O}}_2$ error stayed under 2%.

Fig. 2

Errors on the recovered concentrations related to the assumed thickness of the first layer. $\mathrm{Hb}{\mathrm{O}}_2$ , HbR, HbT, and $\mathrm{S}{\mathrm{O}}_2$ errors are shown for data simulated with a first layer thickness of $10\mathrm{mm}$ .

We also observed that the error in the hemoglobin concentrations in the brain due to an error in the estimation of the thickness of the first layer were always higher than the errors in the superficial layer. This was due to the brain’s low contribution to the TPSFs for source-detector separations between 1 and $3\mathrm{cm}$ , so a small variation in the thickness resulted in a large relative variation in the brain’s contribution to the TPSFs. This also explains the fact that underestimating the thickness of the superficial layer resulted in smaller errors for the hemoglobin concentration in the brain, as shown in Fig. 2, compared to an overestimation. However, when the thickness of the first layer was well estimated, the errors were similar in the superficial layers and in the brain.

Figure 2 also shows the sign of the error in the recovered concentrations. For an overestimation of the first-layer thickness, the contribution of the brain to the TPSF decreased, which resulted in an overestimation of the brain’s hemoglobin to fit the theoretical model. As such, this overestimation produced an underestimation of hemoglobin in the superficial layer. On the other hand, if one underestimates the skin, skull, and CSF thicknesses, the contribution of the brain to the TPSF increases. This results in an underestimation of brain hemoglobin concentrations while superficial layers ones are less sensitive.

In practice, the thickness of the first layer enclosing skin, skull, and CSF can be determined very precisely using a high-resolution MRI scan. The error on the measurement is of the order of the voxel size. Here, the voxels were $1{\mathrm{mm}}^3$ with a $3\mathrm{T}$ magnet, meaning that the maximum errors on the hemoglobin concentrations were expected to be no more than 15%.

4.3.

In Vivo Measurements of Baseline Values

A typical example of the recovery procedure for in vivo data is presented in Fig. 3 . The first step consisted of recovering the thickness of the model’s first layer from an anatomical MRI image that comprised skin, skull, and CSF. After the fits were done with Eq. 1, the final step consisted of inverting Eq. 9 to recover hemoglobin concentrations. Detailed results for all subjects are presented in Table 2 for left-and right-side measurements on the forehead. For comparison, concentrations computed with absorption coefficients recovered with a homogeneous model are shown in Table 3 . A group average was also performed with concentrations recovered using the two-layered model, and results are presented in Table 4 .

Fig. 3

Typical procedure applied to recover the hemoglobin concentrations. (a) Recovery of the thickness of the first layer enclosing skin, scalp, and CSF on the MRI image. (b) Recovery of the reduced scattering and absorption coefficients for both layers using the fitting procedure. (c) Recovery of the chromophore concentrations with the inversion of Eq. 9.

Table 2

Hemoglobin concentrations measured on the left and right hemisphere for each subject computed with absorption coefficients recovered using a two-layered model.

Subject	HbO2 (μM)		HbR (μM)		HbT (μM)		SatO2 (%)
	Superficial	Brain	Superficial	Brain	Superficial	Brain	Superficial	Brain
1 left (2 layers)	4.2	19.1	1.33	15.9	5.53	35	75.9	54.5
1 right (2 layers)	2.03	24.5	3.25	16.4	5.28	40.9	38.4	59.8
2 left (2 layers)	5.87	33.7	3.4	23	9.26	56.7	63.3	59.5
2 right (2 layers)	4.35	32.7	4.26	20.5	8.6	53.2	50.5	61.5
3 left (2 layers)	15.9	39.6	11.1	21.8	27	61.4	58.9	63.7
3 right (2 layers)	17.7	43.4	11.8	20	29.6	63.4	60	68.4
4 left (2 layers)	27.1	35.6	13.9	26.4	41	62	66.1	57.4
4 right (2 layers)	25.8	36.7	13.9	22.7	39.7	59.4	65.1	61.8
5 left (2 layers)	10.4	21.2	5.48	17	15.9	38.2	65.4	55.4
5 right (2 layers)	12.7	23.7	4.49	15.3	17.2	39	73.9	60.8

Table 3

Hemoglobin concentrations measured on the left and right hemisphere for each subject computed with absorption coefficients recovered using a homogeneous model for the superficial layer and the linear fit of the TPSF slope for the brain. Superficial layer concentrations correspond to measurements at 10mm , while brain concentrations correspond to measurements at 30mm .

Subject	HbO2 (μM)		HbR (μM)		HbT (μM)		SO2 (%)
	Superficial	Brain	Superficial	Brain	Superficial	Brain	Superficial	Brain
1 left (1 layer)	8.08	22.5	3.14	17.6	11.2	40.1	72	56.1
1 right (1 layer)	15.3	24.4	5.68	17.2	21	41.7	72.9	58.7
2 left (1 layer)	22.9	31.5	5.99	18.7	28.9	50.1	79.3	62.8
2 right (1 layer)	19.9	31.6	7.7	17	27.6	48.6	72.1	64.9
3 left (1 layer)	19.2	16.9	8.28	25.9	27.5	42.8	69.9	39.6
3 right (1 layer)	28	31.6	13.4	22.4	41.4	54	67.7	58.5
4 left (1 layer)	19.2	33.9	11.9	17.7	31.2	51.6	61.7	65.7
4 right (1 layer)	23.8	39.7	21.7	18.3	45.4	58	52.3	68.5
5 left (1 layer)	12	24.8	10.1	18.8	22.2	43.6	54.2	56.8
5 right (1 layer)	12.2	25.9	7.87	17.2	20.1	43.1	60.8	60

Table 4

Average hemoglobin concentrations for all the subjects. Concentrations were computed using the two-layered model.

Chromophore	Left		Right		Average
	Superficial	Brain	Superficial	Brain	Superficial	Brain
$\mathrm{Hb}{\mathrm{O}}_2$	12.7	29.8	12.5	32.2	12.6	31
$\left(\mu \mathrm{M}\right)$	$\pm 9.25$	$\pm 9.12$	$\pm 9.77$	$\pm 8.31$	$\pm 8.97$	$\pm 8.32$
HbR	7.05	20.8	7.54	19	7.29	19.9
$\left(\mu \mathrm{M}\right)$	$\pm 5.3$	$\pm 4.34$	$\pm 4.92$	$\pm 3.05$	$\pm 4.83$	$\pm 3.67$
HbT	19.7	50.7	20.1	51.2	19.9	50.9
$\left(\mu \mathrm{M}\right)$	$\pm 14.4$	$\pm 13$	$\pm 14.4$	$\pm 10.9$	$\pm 13.6$	$\pm 11.3$
$\mathrm{S}{\mathrm{O}}_2$	65.9	58.1	57.6	62.5	61.7	60.3
(%)	$\pm 6.24$	$\pm 3.66$	$\pm 13.6$	$\pm 3.43$	$\pm 10.9$	$\pm 4.06$

5.

Discussion

Tables 2, 4 show a clear distinction between the superficial layer and brain for hemoglobin baseline concentrations (as expected). This distinction is in agreement with results recovered using FD system (Choi ⁸). The $\mathrm{Hb}{\mathrm{O}}_2$ and HbR concentrations were lower in the superficial layer than in the brain. The hemoglobin in the first layer was probably concentrated in the skin, which is highly irrigated with blood vessels; but this layer also included CSF, which contains low hemoglobin concentrations. However, our model did not validate this assumption. A three-layered model has been developed,⁴² but to our knowledge, no results were reported. Our hemoglobin concentrations, computed with the homogeneous model, were overestimated in the superficial layer—as expected from the Monte Carlo simulation. However, the difference was lower for brain hemoglobin estimations, because the linear fit of the TPSF slopes gave a better estimate of the absorption coeffcient than the fit with the homogeneous analytical expression, as shown in Table 1. Our results suggest that the layered structure of the head should be taken into account to extract hemoglobin concentrations for the brain and to confirm the need for using a two-layered model.

We also observed a large intersubject variability for hemoglobin concentrations, shown in Table 2. This fact was also reported both in Choi ⁸ and Comelli, ²⁶ thus reinforcing the importance of baseline physiology measurements before interpreting activation maps of individual subjects. Simulation results from Table 2 and Fig. 2 show that the uncertainty in the estimated hemoglobin concentrations were less than 15% with our recovery procedure, which is due primarily to the estimation of the thickness of the superficial layer. This uncertainty alone cannot explain our observed variability between subjects. Differences in anatomical structure may explain this variability, but the subject’s baseline metabolism at the moment of the experiment may also play a role. Physical activity, for example, just before the measurement raises the blood circulation in the body, which may alter the hemoglobin concentration of the brain tissue. Moreover, the two-layered model does not separate the CSF layer from the skin and skull, which can cause changes in light distribution. Also, the model assumes that all the boundaries are parallel and flat, which is obviously not the case in reality. Finally, other background absorbers can contribute to absorption, but their contributions are considered very low according to the literature.^{32, 50, 51, 52}

By looking at the $\mathrm{S}{\mathrm{O}}_2$ variations in the brain, we see in Table 2 that these variations are only 4% between subjects. This result was also reported in Choi Figure 2 shows that the errors in the oxygen saturation are low $(<2\%)$ even if the assumption for the superficial layer thickness is bad. This makes sense, because the partial volume effect⁵ reduces the magnitude of the estimated concentration. However, with a good choice of wavelengths, we can reduce the crosstalk and preserve the relative magnitudes for $\mathrm{Hb}{\mathrm{O}}_2$ and HbR,⁵⁸ which gives good $\mathrm{S}{\mathrm{O}}_2$ values.

6.

Conclusion

We reported baseline hemoglobin concentrations measured in human intra- and extracerebral tissue using a TD system with a two-layered analytical model for the first time. We showed that hemoglobin concentrations recovered with TD systems differ when a two-layered model is used instead of a homogeneous one. Clear distinctions were obtained between the superficial layer $(\text{skin}+\text{skull}+\mathrm{CSF})$ and the brain. A large intersubject variability was observed, as previously reported in the literature. The uncertainty for hemoglobin concentrations was estimated to be less than 15% for simulations dominated by uncertainty in the layer thickness, suggesting that the observed difference between subjects was real.