_{p}, for the photons in order to get good estimates of the absorption coefficient, µ

_{a}. The optical path length was found to be well fitted to the equation l

_{p}=a+b ln(I

_{s})+c ln(µ

_{a})+d ln(I

_{s})ln(µa), where I

_{s}is the reflected light intensity for scattering alone (i.e., zero absorption). Although coefficients a-d calculated in this study are specific to the probe used here, the general form of the equation should be applicable to similar probes.

## 1.

## Introduction

The brain is a most fascinating and complex organ, containing numerous chromophores such as blood, cytochromes, neuromelanin, and lipofuscin. The spectral profile of blood is highly dependent on the oxygenation level and the amount of cytochrome c oxidase reflects the activity of the tissue,^{1} whereas melanin is characteristic to certain structures and the amount of lipofuscin is an indicator of tissue damage.^{2} Stereotactic neurosurgery, such as implantation of deep brain stimulation (DBS) electrodes^{3} or radio-frequency lesioning (ablation),^{4} provides an opportunity to study these chromophores very locally, *in vivo*, and without adding any discomfort or risk to the patient, by the use of diffuse reflectance spectroscopy.^{5, 6}

At the Department of Biomedical Engineering, Linköping University, probes have been developed for optical measurements in the brain during stereotactic neurosurgery. These have been used for optical studies with diffuse reflectance spectroscopy and laser Doppler flowmetry during creation of paths for DBS electrodes.^{5, 7, 8} These measurements, and those by others,^{6, 9, 10} have shown that the reflected light intensity at
$780\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
can differentiate between gray matter, light gray matter (e.g., the thalamus), and white matter. The chromophore content has not yet been studied in detail, however. A difficulty with reflectance spectroscopy, compared to standard transmittance spectroscopy through thin samples, is that the apparent optical path length,
${l}_{\mathrm{p}}$
(in millimeters), is unknown and varies with wavelength and tissue composition. An estimation of
${l}_{\mathrm{p}}$
is thus required in order to reliably analyze diffuse reflectance spectroscopy data.

Monte Carlo (MC) simulation^{11} is a valuable tool for simulation of light transport in biological tissue. The technique utilizes random walk for a large number of photons and is considered as the gold standard for simulations in the field. MC simulations have for example been used to estimate
${l}_{\mathrm{p}}$
for transcranial near-infrared spectroscopy^{12} and to calculate expected reflected intensity and look-ahead distance in brain tissue.^{13, 14} Another method for solving the unknown path length is to use optical phantoms with known optical properties for inverse determination of the absorption and scattering coefficients in tissue.^{15} Such phantoms, spanning appropriate ranges of values for optical absorption and scattering, have been used for inverse determination of chromophore content in, e.g., the heart.^{16}

The aim of this study is to develop an appropriate mathematical model for extraction of the absorption coefficients from intracellebral diffuse reflectance spectra, measured during DBS surgery. This will allow for determination of the chromophore content in the brain.

## 2.

## Material and Method

## 2.1.

### Probe and Spectroscopy System

The probe studied consist of a $190\text{-}\mathrm{mm}$ -long metal tube with a diameter of $2.2\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ that decreases to $1.6\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ at the tip. It is used to create trajectories for DBS electrodes prior to their insertion. For spectroscopy, the probe contains two parallel, adjacent optical fibers with a core diameter of $200\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ , a cladding diameter of $230\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ , and a numerical aperture of 0.22 (Fig. 1 ). The emitting fiber is connected to a white halogen lamp (AvaLight-Hal-S, Avantes BV, The Netherlands) and the receiving fiber to a spectrometer (AvaSpec-2048-2, Avantes BV, The Netherlands). The lamp contains a shutter that allows the light to be shuttered without turning it off. A white polytetrafluoreten reference tile (WS-2, Avantes BV, The Netherlands) is used to normalize the light intensity so that it only shows the characteristics of the illuminated tissue, not the sensitivity of the spectrometer or the intensity $\left({I}_{0}\right)$ of the lamp, which both are highly nonlinearly dependent on the wavelength.

## 2.2.

### Monte Carlo Simulations

The optical fibers of the probe were modeled as illuminating and viewing a homogeneous domain of brain tissue. Models were made for white and gray matter at wavelengths between 480 and
$900\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
. The scattering coefficients,
${\mu}_{\mathrm{s}}$
$\left({\mathrm{mm}}^{-1}\right)$
, and anisotropy factors,
$g$
; were taken from human *ex vivo* data^{17} (see Table 1
). The Henyey–Greenstein phase function^{18} was used to randomize scattering angles based on
$g$
. From the simulations, a pure scattering intensity,
${I}_{\mathrm{s}}$
; was calculated as

^{19}

## Table 1

Used optical properties for MC simulations and corresponding μs′ .

Tissue | Wavelength(nm) | μs (mm−1) | g | n | μs′ (mm−1) |
---|---|---|---|---|---|

Gray matter | 480 | 11 | 0.89 | 1.36 | 1.21 |

Gray matter | 580 | 9.7 | 0.89 | 1.36 | 1.07 |

Gray matter | 640 | 9.0 | 0.89 | 1.36 | 0.99 |

Gray matter | 700 | 8.2 | 0.9 | 1.36 | 0.82 |

Gray matter | 780 | 7.8 | 0.90 | 1.36 | 0.78 |

Gray matter | 900 | 6.6 | 0.9 | 1.36 | 0.66 |

White matter | 480 | 43 | 0.80 | 1.38 | 8.60 |

White matter | 580 | 41 | 0.83 | 1.38 | 6.97 |

White matter | 640 | 41 | 0.84 | 1.38 | 6.56 |

White matter | 700 | 40 | 0.86 | 1.38 | 5.60 |

White matter | 780 | 38 | 0.87 | 1.38 | 4.94 |

White matter | 900 | 33 | 0.88 | 1.38 | 3.96 |

UHT milk | 7801 | 6.6 | 0.756 | 1.36 | 1.61 |

## 2.3.

### Fitting of Scattering Intensity and Apparent Optical Path Length

${I}_{\mathrm{s}}$
and
${\mu}_{\mathrm{s}}^{\prime}$
for gray and white matter were fitted, using the Levenberg–Marquardt algorithm, to the expression
$\alpha \cdot {\lambda}^{-\beta}$
, where
$\alpha $
and
$\beta $
are fitted coefficients. This relation is generally used for the reduced scattering coefficient of biological scatterers in the visible and near-infrared range.^{20} Matlab 7.9 (Mathworks Inc., Natick, Massachusetts, USA) was used for the fitting. Weighted errors,
$E$
, for the fits were calculated according to

With ${I}_{\mathrm{S}}$ and $I$ calculated from Eqs. 1, 2, an apparent optical path length, ${l}_{\mathrm{p}}$ , was calculated as

## Eq. 4

$${l}_{\mathrm{p}}=\frac{\mathrm{ln}\left({I}_{\mathrm{s}}\right)-\mathrm{ln}\left(I\right)}{{\mu}_{\mathrm{a}}}.$$^{15, 21}but ${I}_{\mathrm{s}}$ and ${l}_{\mathrm{p}}$ was not assumed to be polynomials of the reduced scattering coefficient, ${\mu}_{\mathrm{s}}^{\prime}={\mu}_{\mathrm{s}}\cdot (1-g)$ , as ${l}_{\mathrm{p}}$ also is affected by ${\mu}_{\mathrm{a}}$ .

^{22}Instead, the apparent optical path length from the simulations were plotted against ${I}_{\mathrm{s}}$ and ${\mu}_{\mathrm{a}}$ in order to determine a more appropriate fit, ${l}_{\mathrm{p}}({I}_{\mathrm{s}},{\mu}_{\mathrm{a}})$ .

## 2.4.

### Phantom Measurements

Twenty-five phantoms consisting of ultrahigh temperature-treated (UHT) milk (1.5% fat, Arla Foods, Sweden), water-soluble blue ink (Coloris, Germany), and water were created. Milk concentrations were varied in steps of 20% between 20 and 100%, and the ink was added in doubling concentrations between
$1\u22154000$
and
$1\u2215250$
, corresponding to peak absorption values from
$0.23\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}3.67\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1}$
.^{23} The milk was assumed to be a purely scattering medium and the ink a purely absorbing medium. One row of phantoms without ink was also made in order to provide measurements of the pure scattering spectra,
${I}_{\mathrm{s}}$
. Diffuse reflectance spectroscopy measurements were performed with the optical brain probe in random order, at room temperature, on the same day as the phantoms were created. Normalized spectra were calculated by

A reference simulation of the reflected light intensity at $780\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ for UHT milk was made in order to provide a reference level for the intensity. The ink was diluted to 10% in order to ease the mixing process. However, this causes minor fluctuations in milk concentration between the phantoms. In order to compensate for this, ${I}_{\mathrm{s}}$ from the inkless phantom with the same milk concentration was scaled for each ink-containing phantom so that its intensity between 815 and $900\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ was equal to the intensity, $I$ , from the phantom. The scaling factor was taken as the mean $I$ between 815 and $900\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ divided by the mean ${I}_{\mathrm{s}}$ between 815 and $900\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ .

As for the results from the simulations, a regression model for ${l}_{\mathrm{p}}$ was made. Only the intensity at the peak absorption wavelength $\left(633\phantom{\rule{0.3em}{0ex}}\mathrm{nm}\right)$ of the ink was used for the regression here though.

## 2.5.

### Application on Clinical Data

Finally, the method obtained from the simulations and phantom measurements was tested on two spectra from cortical gray matter and subcortical white matter, obtained during DBS surgery in an earlier study.^{5} As for the phantom measurements,
$I$
was calculated with Eq. 5 and the mean intensity at
$780\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
from subcortical white matter was used to define
$I=1$
. After thoroughly cleaning the optical fibers, white and dark spectra had been obtained right after the surgery without turning off the lamp or disconnecting the probe in between.

Using linear regression, the calculated absorption spectra were fitted to chromophores that can be expected to be found in the brain: Deoxyhaemoglobin,^{24} oxyhaemoglobin,^{24} lipofuscin,^{25} eumelanin,^{26} and reduced and oxidized forms of cytochromes c and aa3.^{27, 28, 29} If any chromophore yielded a negative regression coefficient, then the chromophore with the largest negative coefficient was removed and the regression was redone.

## 3.

## Results

## 3.1.

### MC Simulations

The fits of the reduced scattering coefficient used for the simulations were proportional to ${\lambda}^{-0.96}$ for gray matter (mean ${E}^{2}=0.0027$ ) and to ${\lambda}^{-1.16}$ for white matter (mean ${E}^{2}=0.0046$ ). The corresponding fits to ${I}_{\mathrm{s}}$ were proportional to ${\lambda}^{-1.03}$ for gray matter (mean ${E}^{2}=0.0019$ ) and ${\lambda}^{-0.80}$ for white matter (mean ${E}^{2}=0.0035$ ). There was an obvious curvature of the residuals versus wavelength for ${I}_{\mathrm{s}}$ from white matter though, and it was found to fit better to a linear decrease with wavelength (mean ${E}^{2}=0.0002$ ). A linear regression in the range $780\u2013900\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ , where ${\mu}_{\mathrm{a}}$ can be assumed to be low or at least approximately constant, was thus assumed for ${I}_{\mathrm{s}}$ in white matter [Fig. 2 ] ${I}_{\mathrm{s}}$ increased linearly with ${\mu}_{\mathrm{s}}^{\prime}$ for gray matter but nonlinearly with the higher ${\mu}_{\mathrm{s}}^{\prime}$ for white matter [Fig. 2]. Simulated ${I}_{\mathrm{s}}$ $({\mu}_{\mathrm{a}}=0\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1})$ was $\sim 4\mathrm{\%}$ higher than simulated $I$ from gray matter at ${\mu}_{\mathrm{a}}=0.02\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1}$ and $\sim 10\mathrm{\%}$ higher than simulated $I$ from white matter at ${\mu}_{\mathrm{a}}=0.08\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1}$ . The simulations thus indicate that ${I}_{\mathrm{s}}$ should be $\sim 7\mathrm{\%}$ higher than measured $I$ in average for brain tissue.

By plotting ${l}_{\mathrm{p}}$ versus the variables ${I}_{\mathrm{s}}$ and ${\mu}_{\mathrm{a}}$ , it was found that a logarithmic transformation of the variables gave a fairly linear relation (Fig. 3 ). The following regression model was thus fitted:

## Eq. 6

$${l}_{\mathrm{p}}=a+b\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\left({I}_{\mathrm{s}}\right)+c\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\left({\mu}_{\mathrm{a}}\right)+d\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\left({I}_{\mathrm{s}}\right)\mathrm{ln}\left({\mu}_{\mathrm{a}}\right),$$## Eq. 7

$${\mu}_{\mathrm{a}}[a+b\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\left({I}_{\mathrm{s}}\right)]+{\mu}_{\mathrm{a}}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\left({\mu}_{\mathrm{a}}\right)[c+d\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\left({I}_{\mathrm{s}}\right)]=\mathrm{ln}\left({I}_{\mathrm{s}}\right)-\mathrm{ln}\left(I\right),$$## Table 2

Coefficients for Eqs. 4, 5 and corresponding goodness of fit, R2 .

Coefficient | From MC simulations | From phantommeasurements |
---|---|---|

$a$ | 0.813 | 0.838 |

$b$ | $-0.198$ | $-0.298$ |

$c$ | $-0.145$ | $-0.289$ |

$d$ | 0.0522 | 0.0377 |

${R}^{2}$ (%) | 99.1 | 96.2 |

## 3.2.

### Phantoms

From the MC simulation of the milk, the intensity level was calculated to 0.38 of white matter intensity at $780\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ for pure milk. Examples of reflectance spectra from the phantoms are presented in Fig. 5 .

When applying Eq. 7 to the reflectance spectra from the phantoms, a decent fit of ${\mu}_{\mathrm{a}}$ was found for the highest ink concentrations but lower concentrations yielded overestimations, which were higher for the lower the ink concentration [Figs. 5 and 4 ]. Fitting the results at $633\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ (peak absorption) from the phantoms to Eq. 6 yielded coefficients $b$ and $c$ with higher magnitude (Table 2). Fitting errors when using the coefficients from the phantoms on the phantom results are presented in Fig. 4 (mean ${E}^{2}=0.0040$ ).

## 3.3.

### Clinical Data

On the basis of the simulations and phantom measurements, the following method was used to calculate the absorption coefficient in brain tissue based on a measurement with the optical probe:

1. Calculate a linear regression against ${\lambda}^{-1}$ for gray matter or $\lambda $ for white matter from the brain spectrum in the interval $780\u2013900\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ and extrapolate it downward in wavelength to $480\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ in order to obtain ${I}_{\mathrm{s}}$ .

2. Scale ${I}_{\mathrm{s}}$ by a factor 1.04 for gray matter or 1.1 for white matter in order to compensate for the nonzero absorption in the interval $780\u2013900\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ . Solve Eq. 7 numerically for each wavelength in the interval $480\u2013900\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ using the coefficients $a\u2013d$ obtained from the simulations or phantom measurements in order to obtain the absorption coefficient spectrum.

3. Fit the spectra of known chromophores, as described earlier, to the calculated absorption spectrum by the use of linear regression with the constraint that no chromophore concentration can be negative.

## 4.

## Discussion

In this study, a method for estimation of the absorption coefficient in brain tissue based on diffuse reflectance spectra has been developed. The coefficients $a\u2013d$ in Eqs. 6, 7 are depending on the fiber configuration and are thus not directly applicable to other optical probes. The equations should however be valid, in general, and could be used for similar probes if the corresponding coefficients are estimated for them.

A reliable estimate of
${\mu}_{\mathrm{a}}$
is important if reliable quantification of the chromophores in the tissue is to be achieved. Chromophores that could be of interest to quantify are, e.g., oxy- and deoxyhaemoglobin, lipofuscin, neuromelanin, and cytochromes. Haemoglobin content can give information about blood volume in the tissue. Caution should be taken regarding this when using an invasive probe, such as the one investigated in this study, because it is likely to affect the blood distribution in the tissue when inserted. The pressure from the insertion could press out blood from the vessels in the tissue, and bleeding could cause the presence of blood between the probe and the tissue. The estimated blood concentrations of 0.54% in the cortex and 0.36% in the white matter in this study are a lot lower than values from positron emission tomography (PET) studies of
$5.5\pm 0.6\phantom{\rule{0.3em}{0ex}}\mathrm{ml}\u2215100\phantom{\rule{0.3em}{0ex}}\mathrm{g}$
and
$2.1\pm 0.56\phantom{\rule{0.3em}{0ex}}\mathrm{ml}\u2215100\phantom{\rule{0.3em}{0ex}}\mathrm{g}$
, respectively, which should equal the percentage under the approximation that
$1\text{\hspace{0.17em}}\mathrm{ml}$
tissue has a mass of
$1\phantom{\rule{0.3em}{0ex}}\mathrm{g}$
. Some of the difference could also be explained by the short fiber distance, which usually only will view the microvascular blood fraction, though. Haemoglobin content can also give the tissue blood oxygen saturation by calculating the ratio of oxygenated to total blood volume fraction. The example in this paper of an *in vivo* spectrum from the brain cortex shows a surprising lack of oxyhaemoglobin. Why this is and whether this is common in the brain would be of great interest to investigate further. According to PET studies, the oxygen extraction fraction in brain tissue should be
$\sim 40\mathrm{\%}$
;^{30} thus, an oxygen saturation
$\sim 60\mathrm{\%}$
would be expected in this case, assuming the probe primarily views venous microvascular blood. One possible contribution to this discrepancy could be arteriovenous shunting, which has been indicated in dog brain, especially during hypercapnia.^{31} If some arterial blood is shunted directly to the veins without delivering oxygen to the tissue, then it could cause the venous blood to have higher oxygen content than the microvascular blood.

Lipofuscin is a common chromophore that is usually overlooked in the field of biomedical optics. It consists of biological garbage that the lysosomes of the cells have not been able to digest and it accumulates with age in long-lived cells that undergo little or no cell division (e.g., neurons and cardiac and skeletal myocytes). A high amount of lipofuscin indicates high oxidative damage, is also likely to further hamper digestion of damaged structures, and adds more oxidative damage.^{2} Neuromelanin exists in certain structures of the brain (e.g., the substantia nigra). The substantia nigra lies just beneath the subthalamic nucleus (STN), which is a common target for DBS,^{32} and the presence of melanin in a measured spectrum could thus indicate a too deep position for a DBS electrode that is supposed to be in the STN. Cytochromes c and aa3 (a.k.a. cytochrome c oxidase) are important components for the generation of adenosine triphosphate in the mitochondria, and cytochrome aa3 concentration is indicative of how active the neurons are.^{1} In this study’s cortex spectrum, the concentrations for cytochromes c and aa3 were estimated to 2.2 an
$4.0\phantom{\rule{0.3em}{0ex}}\mu \mathrm{M}$
, respectively. Navarro
^{33} have made *in vitro* measurements of cytochrome c and aa3 concentrations in human frontal cortex. In Parkinsonian brains, they found a mitochondrial concentration of
$11.8\pm 0.4\phantom{\rule{0.3em}{0ex}}\mathrm{mg}\phantom{\rule{0.3em}{0ex}}\text{protein}\u2215\mathrm{g}$
brain cortex and cytochrome concentrations of
$0.23\pm 0.02\phantom{\rule{0.3em}{0ex}}\mathrm{nmol}\u2215\mathrm{mg}$
mitochondrial protein and
$0.17\pm 0.01\mathrm{nmol}\u2215\mathrm{mg}$
mitochondrial protein for cytochromes c and aa3, respectively. Assuming that
$1\text{\hspace{0.17em}}\mathrm{g}$
brain cortex equals
$1\phantom{\rule{0.3em}{0ex}}\mathrm{ml}$
, this should correspond to volume concentrations of
$\sim 2.6\phantom{\rule{0.3em}{0ex}}\mu \mathrm{M}$
for cytochrome c and
$2.0\phantom{\rule{0.3em}{0ex}}\mu \mathrm{M}$
for cytochrome aa3.

The simulated ${I}_{\mathrm{s}}$ for gray brain matter was found to be proportional to ${\lambda}^{-1}$ . The reduced scattering coefficients used for the brain matter simulations (Table 1) were also approximately proportional to ${\lambda}^{-1}$ . ${I}_{\mathrm{s}}$ for white brain matter was better described by a linear relation to wavelength, however. The reason for the difference can be seen in Fig. 2: ${I}_{\mathrm{s}}$ increases linearly with ${\mu}_{\mathrm{s}}^{\prime}$ for the lower values in gray matter, but the increase is slowed down for the higher values in white matter. The behavior of myelin-rich light gray matter, such as the globus pallidus or the lateral part of the thalamus, can unfortunately not be determined by this, but it seems reasonable to expect that it will behave as a mixture of gray and white matter.

The modeling is based on a single fiber distance because that is the kind of probe that we currently use in the brain. It would be beneficial to include at least one larger fiber separation in order to get more reliable estimates of
${I}_{\mathrm{s}}$
. Such a setup would allow for more objective separation of the effect of scattering and absorption on the reflected intensity because the relative impact of scattering and absorption is different for different source-detector separations.^{34} It will be particularly difficult to distinguish between melanin content and potential errors in the estimation of
${I}_{\mathrm{s}}$
when using just one fiber separation, because melanin has a smoothly decreasing absorption curve that neither has any characteristic peaks nor can be assumed to be zero anywhere in the spectral range
$480\u2013900\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
. This limitation is probably less severe for tissue that does not contain melanin. Melanin in the cortex spectra could be expected if the melanin-containing leptomeninges^{35} (arachnoidea and pia mater) are pushed ahead of the probe a bit. The fitting of melanin to white matter in Fig. 7 is dubious though, and the fit also includes a baseline term that is difficult to explain. It is always possible that the absorption coefficients used in the simulations are too high and that
${I}_{\mathrm{S}}$
should be closer to
$I$
in the infrared range. This could be suspected based on the very flat residual spectra in Figs. 6 and 7. It is also possible that there is some unknown chromophore absorbing infrared light. Absorption from lipid^{36} and water is often included when using near-infrared spectroscopy with large fiber distances, but because their
${\mu}_{\mathrm{a}}$
is
$<0.01\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1}$
in the range
$600\u2013900\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
, they should be neglible here. What could be of interest to investigate, however, is whether myelin, whose abundance causes the high
${\mu}_{\mathrm{s}}^{\prime}$
in white matter, has any noticeable absorption in the visible and near-infrared range.

Comparing to *in vitro* data (Table 3
), the
${\mu}_{\mathrm{a}}$
spectrum for white matter calculated by Yaroslavsky
^{17} from measurements on human *in vitro* samples has a low-varying level of
$\sim 0.08\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1}$
above
$500\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
and does not show any obvious melanin content. Their samples were rinsed from blood prior to measurements, and consequently, their spectra show little influence from blood absorption. Their gray matter spectrum show a more evident decrease in absorption with wavelength in the visible range. A similar tendency can be seen in spectra by Gebhart, ^{37} with the biggest difference that blood still is present in their spectra.

## Table 3

Comparison of absorption coefficient, μa , to in vitro values from literature.

Tissue | Wavelength(nm) | This study | Yaroslavsky 17 (rinsed from blood) | Gebhart 37 |
---|---|---|---|---|

Gray matter | 480 | 0.37 | 0.06 | 0.23 |

Gray matter | 560 | 0.26 | 0.04 | 0.26 |

Gray matter | 580 | 0.19 | 0.03 | 0.21 |

Gray matter | 640 | 0.05 | 0.01 | 0.09 |

Gray matter | 780 | 0.02 | 0.02 | 0.05 |

Gray matter | 900 | 0.02 | 0.03 | 0.05 |

White matter | 480 | 0.35 | 0.11 | 0.19 |

White matter | 560 | 0.21 | 0.09 | 0.24 |

White matter | 580 | 0.19 | 0.09 | 0.20 |

White matter | 640 | 0.09 | 0.08 | 0.08 |

White matter | 780 | 0.06 | 0.08 | 0.06 |

White matter | 900 | 0.06 | 0.10 | 0.07 |

It is important to note that
${\mu}_{\mathrm{a}}$
has an effect on
${l}_{\mathrm{p}}$
that in no way is neglible compared to the effect of scattering. The effect of
${\mu}_{\mathrm{a}}$
on
${l}_{\mathrm{p}}$
has nevertheless often been omitted in studies when estimating
${\mu}_{\mathrm{a}}$
by a modified Beer–Lambert law.^{21, 38, 39} Using a light transport model such as Eq. 6 should greatly improve the estimates of
${l}_{\mathrm{p}}$
and thus also the estimates of
${\mu}_{\mathrm{a}}$
. Matcher
^{40} have, e.g., pointed out that the impact of
${\mu}_{\mathrm{a}}$
on
${l}_{\mathrm{p}}$
dampens higher absorption coefficients compared to lower ones, which is in agreement with the negative value of coefficient
$c$
obtained in this study. The tendency for Eq. 7 to overestimate low
${\mu}_{\mathrm{a}}$
indicates that the impact of
${\mu}_{\mathrm{a}}$
on
${l}_{\mathrm{p}}$
actually is greater than implied by the simulations, as can be seen by the greater magnitude of coefficient
$c$
obtained from the phantom measurements. The coefficients from the phantom measurements are probably more reliable than those from the simulations as the latter may be affected by errors in the MC model assumptions regarding, e.g., geometry, phase function, and fiber characteristics. There are possible errors with the phantoms, too, though. Although assumed to be purely absorbing, the ink used seems to have a small scattering component, which probably is causing some overestimation of its
${\mu}_{\mathrm{a}}$
. Similarly, there is probably some absorption in the milk itself. Also,
${I}_{\mathrm{s}}$
from pure UHT milk is lower than from white brain matter and the phantoms thus do not cover the entire range of interest. A more scattering liquid, such as Intralipid,^{41} might provide a better base for the phantoms.

It can be seen in Figs. 3 and 4 that the apparent optical path length, ${l}_{\mathrm{p}}$ , is not completely linearly dependent on the logarithms of the absorption and pure scattering intensity. Higher order terms could thus be added in order to get more accurate modeling. This may be worthwhile if more fibers at further distances are added. The greater slope on the lower ${I}_{\mathrm{s}}$ values also indicates that the magnitude of coefficient $c$ from the phantoms may be too high for white matter because the higher white matter intensities are not covered by the phantom intensities. In particular, one should be very cautious if ${I}_{\mathrm{s}}$ and ${\mu}_{\mathrm{a}}$ are very high because Eq. 6 then can give negative, and thus unphysical, estimates of the optical path length. That could happen if a blood vessel is ruptured for example.

In conclusion, the proposed method will give an approximate quantitative measurement of the absorption coefficient and should thus allow for quantification of chromophores in the brain. This study also elucidates the importance of taking the impact of the absorption coefficient into account when estimating the apparent photon path lengths.

## Acknowledgments

The author thanks his colleague Tobias Lindbergh, for valuable tips and discussion. The study was supported by NovaMedTech (funded by the European Union and Tillväxtverket, the Swedish Agency for Economic Growth), the Swedish Governmental Agency for Innovation Systems (Vinnova), the Swedish Foundation for Strategic Research (SSF), and the Swedish Research Council (VR, Dnr. 311-2006-7661).

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