## Abstract

Measurement of tissue optical absorption and (transport) reduced scattering coefficients (μ* _{a}* and μ

*', respectively) is fundamental to many applications of light in medicine and biology. We report a handheld fiberoptic probe to determine these coefficients by measuring the diffuse reflectance at multiple source-collector distances, which allows for a larger dynamic range than a single source-collector separation. Diffusion theory and*

_{s}*a priori*knowledge of the spectral shape of μ

*and μ*

_{a}*' are used in a forward model of the diffuse reflectance. The dynamic range and accuracy of this method were evaluated using Monte Carlo simulations, phantom experiments and tissues*

_{s}*in vivo*.

© 2010 OSA

## 1. Introduction

Accurate measurement of tissue optical properties, specifically the absorption (μ* _{a}*) and transport (reduced) elastic scattering (μ

*') coefficient and their spectral dependence, is central to many diagnostic and therapeutic optical techniques. For example, the outcome of treatments such as photodynamic therapy [1] and laser interstitial thermal therapy [2] depends on the light dose distribution in the target tissue (such as tumor), which is governed by these optical properties at the treatment wavelength. Similarly, diagnostic procedures can exploit the changes in the optical properties due to disease, since these properties are sensitive to changes in tissue microstructure (collagen fibers, nuclei and other particulates) and chromophores (hemoglobin, water and others) [3,4].*

_{s}There are a variety of methods to determine these optical properties. One broad class involves the measurement of the diffuse reflectance and transmittance of an excised tissue sample of known thickness in an integrating sphere setup [5]. This requires *ex vivo* tissue samples, adding uncertainty due to distorting factors such as deoxygenation, loss of blood and the effects of tissue handling (*e.g.* cryofreezing) [6]. Hence, there has been considerable effort to develop minimally-invasive techniques that can be used *in vivo*, especially for clinical applications. These are generally based on either fiberoptics in contact with, or a non-contact detector in close proximity to, the tissue. Frequency-domain [3,7] and time-resolved diffuse reflectance [8] are among these *in vivo* methods, where the dynamic migration of photons through tissue is measured. Steady-state fluence rate [1] or radiance measurements [2,9] have also been advanced for interstitial measurements. Spatially-resolved, steady-state diffuse reflectance measurements represent another technique, where in general a source fiber launching light into the tissue is set at varying distances from several collector fibers, such that the optical properties may be derived from the spatially-resolved reflectance measurements [10,11]. A spatial-frequency domain method has been explored as a corollary to this technique, where the tissue is imaged with spatially-modulated illumination [12].

The aforementioned techniques typically rely upon large sampling volumes and long acquisition times. One typical time-domain diffuse reflectance system has source-collector fiberoptic distances of 15-20 mm, with acquisition times from 20 to 30 s [8], while another frequency-domain system for non-invasive measurements of breast tumors has a fiberoptic separation of 25 mm [3]. The spatially-modulated reflectance technique of Cuccia *et al*. takes images with sinusoidal frequencies of up to 0.63 mm^{−1} projected on the tissue, thus integrating over tissue surface areas ~1 cm^{2} [12]. The analogous spatially-resolved diffuse reflectance technique requires source-collector distances that vary over several mm [11,13]. While large volume and/or wide field measurement of the optical properties is desired in many applications (such as bulk measurements in breast) and where measurement time is not a major constraint, there are many clinical situations where rapid, localized measurements are desired, such as during interstitial procedures, operating in a surgical field, during endoscopy and performing measurements on small anatomical structures.

To address this need, spectrally-constrained diffuse reflectance methods have been developed that allow the use of a single fiberoptic source-collector pair (Fig. 1
). The source fiber delivers broadband white light and the diffuse reflectance spectrum is detected by the collector fiber located at a distance, *r*. Since there is only one reflectance measurement per wavelength, *λ*, solving for μ* _{a}* and μ

*' relies upon spectral constraints,*

_{s}*i.e.*applying

*a priori*knowledge of the shapes of μ

*(*

_{a}*λ*) and μ

*'(λ) in a forward model, which can then be used to solve for the absolute coefficient values. Recent efforts using this approach have explored the use of very small source-collector distances, <1 mm [14]. This results in tissue-sampling volumes ~1 mm*

_{s}^{3}and fast acquisition times (<1 sec to ~seconds).

In addition, these small reflectance probes can be implemented in interstitial format. A major advantage over the fluence rate and radiance interstitial techniques is that the source-collector distance is fixed and known (within the housing of the reflectance probe); by contrast, fluence rate and radiance techniques generally have the source fiber and the collector fiber independently inserted into the tissue, which is a major source of uncertainty in determining the tissue optical properties and requires an independent imaging technique to determine the fiber positions.

In the present work we address one of the outstanding problems with the spectrally-constrained diffuse reflectance method, namely the limited dynamic range of μ* _{a}* and μ

*' that can be measured with a single source-collector distance. Here, three source-collector distances (260, 520 and 780 μm) were used. Since each distance spans a unique range over which μ*

_{s}*and μ*

_{a}*' can be measured, overlap of the reflectance measurements at the three distances extends the dynamic range beyond that of each distance separately. This approach is distinct from the aforementioned spatially-resolved diffuse reflectance techniques in that the multiple source-collector distances are used to expand the dynamic range of optical properties measurement, not to constrain the solution. We believe that this approach has distinct advantages over other reported methods, in particular the extended range of optical properties over which it is valid and the ability to make rapid, highly localized measurements of these properties which is advantageous in many applications.*

_{s}## 2. Theory

#### 2.1 Using diffusion theory and spectral constraints to extract optical properties

The absorption spectrum can be modeled as a linear combination of the separate chromophore contributions. In the visible spectrum, it is commonly expressed using hemoglobin concentration and an oxygen saturation term [10,15]:

*(*

_{a}^{oxyHb}*λ*) and µ

*(*

_{a}^{deoxyHb}*λ*) are the wavelength-dependent absorption coefficients of oxygenated hemoglobin and deoxygenated hemoglobin, respectively, in units of [cm

^{−1}·L/g].

*f*is the total hemoglobin concentration [g/L] and

_{Hb}*StO*

_{2}is the oxygenation fraction. Only the significant chromophores should be included in order to have an accurate absorption model. In this work, optical properties in the visible light range are measured, with hemoglobin being dominant. Other absorbers, such as beta carotene in breast tissue [16], should be included if their concentration is significant. Water absorption is neglected here, since the diffuse reflectance is measured in the range 450-850 nm where water is optically clear relative to hemoglobin.

The reduced scattering coefficient spectrum of bulk tissue has been shown to fit well to a power law dependence on wavelength [10,14,15] given by

*A*and

*k*are constants.

*A priori*knowledge of the shapes of these spectra can be combined with a forward model of the diffuse reflectance. Non-linear fitting can then be applied to extract the free parameters,

*f*,

_{Hb}*StO*

_{2},

*A*and

*k*, from which μ

*(*

_{a}*λ*) and μ

*'(*

_{s}*λ*) can be computed using Eqs. (1) and (2). Our approach was to use the well-known diffusion theory equation for steady-state diffuse reflectance,

*R*, as the forward model, with the assumption of homogeneous optical properties in the volume of light interrogation [11]. In this embodiment,

_{DT}*r*is then fixed and μ

*(*

_{a}*λ*) and μ

*'(*

_{s}*λ*) are wavelength-dependent:

*z*

_{0}= 1/µ

*'*

_{s}_{,}and

*r*

_{1}

^{2}=

*z*

_{0}

^{2}+

*r*

^{2}and

*r*

_{2}

^{2}= (

*z*

_{0}+ 2

*z*)

_{b}^{2}+

*r*

^{2}. The parameters

*z*

_{0},

*r*

_{1},

*r*

_{2},

*z*and µ

_{b}*are then all wavelength-dependent.*

_{eff}*z*depends on µ

_{b}*, µ*

_{a}*' and the internal reflection parameter к = (1 +*

_{s}*r*)/(1-

_{id}*r*) due to index mismatch between tissue and the external medium. The extrapolated boundary distance is given by

_{id}*z*= 2κ

_{b}*D*, where

*D*is the diffusion constant given by

*D*= (3µ

*')*

_{s}^{−1}. To quantify the index mismatch at the boundary,

*r*has been empirically determined as

_{id}*r*= −1.44

_{id}*n*

_{rel}^{−2}+ 0.71

*n*

_{rel}^{−1}+ 0.67 + 0.0636

*n*[17]. Refractive index matching indices at the tissue surface yields κ = 1, and we have used this condition since the external medium is that of ink-blackened epoxy surrounding the probe fibers, which has approximately the same refractive index as tissue. The effective attenuation coefficient is expressed as µ

_{rel}*= (3µ*

_{eff}*µ*

_{a}*')*

_{s}^{1/2}, and μ

*(*

_{a}*λ*) and μ

*'(*

_{s}*λ*) are given by Eqs. (1) and (2).

The diffusion constant, *D,* has been variously cited as (3μ* _{s}*')

^{−1}or [3(μ

*' + μ*

_{s}*)]*

_{a}^{−1}[11,13,18]. Additionally, the way in which the probe light source is modeled impacts the form of Eq. (3), as outlined by Farrell

*et al*. [11]. Thus, if it is modeled as a buried point source, the diffusion equation takes the form in Eq. (3) but without the reduced albedo,

*a*' = µ

*'/(µ*

_{s}*' + µ*

_{s}*). If it is modeled as an exponentially decaying line source extending into the tissue, then Eq. (3) includes the*

_{a}*a*' term. This will be discussed further below, where we empirically show that

*D*= (3µ

*')*

_{s}^{−1}and the exponential line source model is most suitable for this fiberoptic geometry. Note that for μ

*= 0 this issue does not matter, since then*

_{a}*D*= (3μ

*')*

_{s}^{−1}and

*a*' = 1 in all cases.

#### 2.2 Upper and lower bounds of validity of the diffusion theory model

In practice, it is not a simple matter to apply the above method, since, for a given *r* value, there is a range of μ* _{a}* and μ

*' values over which Eq. (3) can be accurately applied to solve the inverse problem to derive the optical properties. Hence, in order to increase the overall range, we have used three source-collector distances (*

_{s}*r*= 260, 520 and 780 μm). For a fixed

*r*, the diffuse reflectance does not monotonically increase with increasing μ

*',*

_{s}*i*.

*e*. there is a peak reflectance (Fig. 2 ). Hence, to avoid ambiguity correlating μ

*' to reflectance, the range of μ*

_{s}*' must be restricted to either the monotonically increasing or monotonically decreasing part of the curve. The absorption coefficient does not have this problem, since increasing μ*

_{s}*always reduces the reflectance signal. Here, we have used the monotonically increasing part since the reflectance is far more sensitive to changes in μ*

_{a}*' over this region, as evidenced by the steeper slope in Fig. 2. The peak reflectance then represents an upper bound for estimating μ*

_{s}*'. As shown in Table 1 , this upper bound decreases with μ*

_{s}*, so it should be taken as the largest expected μ*

_{a}*value (here, 10 cm*

_{a}^{−1}). In practice, this was set as the value of μ

*' at 90% peak reflectance, to provide an additional safety margin. From Table 1, the upper bounds for*

_{s}*r*= 260, 520 and 780 μm were then μ

*' = 52.9, 26.1 and 17.1 cm*

_{s}^{−1}, respectively. Note that the existence of the upper bound has nothing to do with diffusion model accuracy in the monotonically decreasing part of the reflectance-μ

*' curve; rather, the upper bound is placed to ensure that only the monotonically increasing part of the curve is used (for the reasons stated above) to solve the inverse problem to extract μ*

_{s}*and μ*

_{a}*', as detailed later on in Section 3.3.*

_{s}There is also a lower bound of validity for the diffusion model. At small μ* _{s}*', single scattering dominates, invalidating the Similarity Principle defining μ

*' = μ*

_{s}*(1-*

_{s}*g*), where

*g*is the scattering anisotropy, and thus also invalidating the diffusion approximation. This condition was investigated using Monte Carlo modeling for a pencil light beam incident on an optically semi-infinite turbid medium, using the on-line C-code implementation developed by S. Jacques and colleagues [19]. The roulette technique was employed to increase computational speed. The epoxy-packed space around the fibers was considered to be a perfect absorber and index-matched to the tissue. This Monte Carlo model was used to create 3-D look-up tables for each

*r*value and for µ

*' = 2-20 cm*

_{s}^{−1}, µ

*= 0-1 cm*

_{a}^{−1}and

*g*= 0.6-0.95. This range of

*g*is typical for tissues as determined

*ex vivo*in the breast [20], brain [5] and bone [21], as well as the phantom material Intralipid [22]. A Matlab (MathWorks, Natick, MA, USA) 3-D interpolation algorithm was applied to the data in the µ

*', µ*

_{s}*and*

_{a}*g*dimensions. Examples of the resulting Monte Carlo reflectance

*versus*µ

*' curves are shown in Fig. 3a , together with the diffusion theory model graphs. The root-mean-square (RMS) error,*

_{s}*ε*, of the Monte Carlo data compared to the diffusion theory model was used as a measure of the goodness of fit, as illustrated in Fig. 3b. We applied an

_{RMS}*ε*cut-off of 10% to determine the ranges of validity of the diffusion theory model. As would be expected, as

_{RMS}*r*decreased the corresponding µ

*' value at*

_{s}*ε*increased, since single scattering effects are amplified with smaller inter-fiber distances. An interesting artifact is evident in Fig. 3b, where the

_{RMS}*ε*increases slightly with increasing µ

_{RMS}*' after reaching a minimum. This is likely due to loss of photons outside the finite (2 cm dia.) tissue volume used in the Monte Carlo simulations to reduce the computation time.*

_{s}Table 2
shows the µ* _{s}*' values at the 10%

*ε*crossover points, for the three

_{RMS}*r*distances and for µ

*= 0.1, 0.5 and 1 cm*

_{a}^{−1}. Based on the above, Eq. (3) is valid in the ranges µ

*' >16.4, 10.1 and 5.8 cm*

_{s}^{−1}for

*r*= 260, 520 and 780 µm, respectively. Figure 4 then displays the corresponding ranges of validity of the diffusion theory model. Recall that the motivation for using three source-collector distances was to provide overlapping regions of validity to increase the overall dynamic range of the optical coefficient measurements. The resulting dynamic range then has µ

*' ranges of 5.8-52.9 cm*

_{s}^{−1}for a µ

*range of 0-10 cm*

_{a}^{−1}. Note that if the cut-off is >10%, then the lower limit of µ

*' for all fiber distances will be decreased.*

_{s}Figure 4 also displays the calculated ranges for *r* = 1 and 2 mm to further demonstrate how increasingly larger fiberoptic separations have a desirable decrease in the lower limit, but also an undesirable corresponding decrease in the overall range. This trade-off between the lower limit and overall range is important to inform the design of fiberoptic reflectance probes using this technique.

## 3. Materials and methods

#### 3.1 Instrumentation

A schematic and photographs of the fiberoptic probe are shown in Fig. 5
. An optical multiplexer (Model MPM-2000, Ocean Optics, Dunedin, FL, USA) was used to control the flow of input and output optical signals. A tungsten-halogen white light source (Model LS-1, Ocean Optics) was connected to one of the ports of the optical multiplexer such that the light could be selected to enter the probe fibers as well as be fed back into the spectrometer (Model S2000, Ocean Optics). The four fiberoptic leads were connected to the multiplexer so that the detected light could be measured sequentially by the spectrometer. A measurement consists of the following steps: 1) a background measurement is taken, 2) a reference spectrum is measured of the white light source to compensate for variations in lamp output, 3) the white light is sequentially channeled to each of the 3 source fibers, with spectrometer measurements taken from a collector fiber, thereby generating diffuse reflectance spectra for each *r*. The total time for each measurement was approximately 3 sec.

The day-to-day standard deviation of the reflectance measurements was calculated from 6 measurements of a standard Intralipid phantom taken on different days spanning a period of 4 months. The instrument demonstrated excellent repeatability, with the reflectance measurements (over all *r* distances) having a standard deviation of only 0.37 cm^{−2}, or 3.6%.

#### 3.2 Calibration

Calibration of the measured reflectance spectrum to the diffusion theory model is critical to retrieve μ* _{a}* and μ

*'. Fortuitously, the diffuse reflectance curve with respect to μ*

_{s}*' has a characteristic peaked shape (Fig. 2) that may be exploited. The approach was to measure the reflectance in a scattering fluid that was diluted over a range such that the reflectance*

_{s}*versus*μ

*' curve (of all dilutions) captures the peak reflectance over all wavelengths. The aliquot fraction of scattering fluid was a 3% concentration of Intralipid-20% (Fresenius Kabi, Uppsala, Sweden) in distilled water. The dilutions were then 3, 6, 9, ..., 48%. Figure 6a displays the uncalibrated reflectance at 600 nm, with both the*

_{s}*x*- and

*y*-axis needing calibration.

The reflectance measurements were then polynomial-fitted to obtain a smooth curve as a function of dilution. The peak of the reflectance *versus* μ* _{s}*' curve was then used to fit the reflectance measurements to the diffusion theory model with the appropriate

*x*- and

*y*-axis scaling. The

*y*-axis scale at each wavelength and source-collector distance is calculated as

*V*is the subset of μ

*' for which the diffusion theory model is valid (see Fig. 6b for properly scaled, calibrated data). This scale factor (Eq. (4)) was used to scale the (relative) reflectance measurements with the probe in order to yield the reflectance in absolute units (*

_{s}*i.e.*cm

^{−2}) so that the inverse problem may be solved to extract µ

*and µ*

_{a}*'.*

_{s}#### 3.3 Inverse algorithm to recover optical properties

For each acquisition, three reflectance spectra are taken in sequence, one for each *r* (260, 520 and 780 μm). To recover μ* _{a}*(

*λ*) and μ

*'(*

_{s}*λ*), a Levenberg-Marquardt non-linear least squares algorithm was applied to Eqs. (1)-(3) over the spectral range

*λ*= 450-850 nm. This minimized the variance between the diffusion theory reflectance equation,

*R*(

_{DT}*λ*), and the reflectance measurement,

*R*(

_{meas}*λ*), with

*f*,

_{Hb}*StO*

_{2},

*A*and

*k*as the free parameters. The optical properties spectra, μ

*(*

_{a}*λ*) and μ

*'(*

_{s}*λ*), can then be derived. Only one reflectance spectrum is required to estimate μ

*(*

_{a}*λ*) and μ

*'(*

_{s}*λ*). The selection of this reflectance spectrum is based on the optical properties range for each

*r*, as shown in Fig. 4. The following inversion algorithm was found to be suitable, with values for the μ

*' boundaries taken from Fig. 4.*

_{s}- i) Perform inversion with
*r*= 260 μm. If μ'>16.4 cm_{s}^{−1}for >50% of the spectral range (*i.e.*450-850 nm), output μ(_{a}*λ*) and μ'(_{s}*λ*) and end; else go to ii). - ii) Perform inversion with
*r*= 520 μm. If μ' > 10.1 cm_{s}^{−1}for > 50% of the spectral range, output μ(_{a}*λ*) and μ'(_{s}*λ*) and end; else go to iii). - iii) Perform inversion with
*r*= 780 μm. Output μ(_{a}*λ*) and μ'(_{s}*λ*).

#### 3.4 Phantom measurements for diffusion theory model validation

Phantom measurements were used to validate and optimize the diffusion model for the probe geometry. Although this type of analysis has been reported by others [11,18], the fiber separation distances are shorter (*i.e.* <1 mm) relative to these previous studies, necessitating confirmation of the model. As mentioned in the Theory section, there are different expressions for the diffusion coefficient, *D*. The general form is *D* = [3(µ* _{s}*' + αµ

*)]*

_{a}^{−1}, with α variously cited as 0, 1 or some function of the optical properties [11,18]. As also discussed above, there may be two different forms of Eq. (3), depending on how the light source in tissue is modeled (buried point or exponential line source). Hence, phantom measurements were used to determine the optimal form of

*D*and light source model for our probe geometry.

Naphthol Green (NG) dye and Intralipid were used as the absorber and scattering medium, respectively. Reflectance measurements at 715 nm (the absorption peak of Naphthol Green) were used for the following analysis. Nine phantoms were formulated, with μ* _{a}* = 1, 5 or 10 cm

^{−1}and μ

*' = 7, 14 or 21, prepared in all combinations. Since there are three*

_{s}*r*distances, this results in 27 total data points. This range of μ

*and μ*

_{a}*' spans the majority of tabulated*

_{s}*in vivo*optical properties in the review by Kim and Wilson for

*λ*= 450-850 nm [4]. Note that this phantom experiment does not include cases where μ

*is close to zero, since both forms of*

_{a}*D*and both forms of the light source model converge to the same model at μ

*= 0.*

_{a}The reduced scattering coefficient was verified using the single integrating sphere technique (see next section). The absorption coefficient of NG was measured in a standard spectrophotometer. Probe measurements were taken of the nine phantom solutions. The probe reflectance measurements were then compared to the diffusion model with both forms of the diffusion constant and the light source model. Note that probe calibration (with the varying dilutions of Intralipid) is not affected by the different forms of Eq. (3), since μ* _{a}* of Intralipid alone is assumed to be negligible for

*λ*= 450-850 nm.

#### 3.5 Phantom measurements to determine probe accuracy

A set of phantom data was prepared to obtain accuracy statistics for the probe-derived optical property values, using the NG absorption curve instead of that of hemoglobin in Eq. (1). The probe estimates of optical properties were compared with measurements made using the single integrating sphere technique. The latter has been reported in detail previously [23]. Briefly, the samples were placed in a custom sample holder consisting of two quartz disks separated by a 1 mm thick ring spacer. A tungsten-halogen lamp (L.O.T. Oriel: Darmstadt, Germany) and collimating optics were used to illuminate the sample with a 5 mm diameter white-light beam. The total diffuse transmittance, *T _{d}*(λ), and diffuse reflectance,

*R*(λ), spectra were measured with a 15 cm diameter integrating sphere (SphereOptics: Contoocook, NH, USA) coupled to a spectrometer (S2000, Ocean Optics). A Monte Carlo simulation was used to calculate the expected

_{d}*R*and

_{d}*T*for µ

_{d}*and µ*

_{a}*' values ranging from 0 to 100 cm*

_{s}^{−1}and 0-100 cm

^{−1}, respectively. The tissue optical properties were then calculated from

*T*(λ) and

_{d}*R*(λ) using an inverse interpolation algorithm. For this set of measurements, the phantoms were formulated first with μ

_{d}*= 21.7 and μ*

_{a}*' = 24.7cm*

_{s}^{−1}(at 715 nm) and then diluted into serial fractions of 95 to 5%, such that μ

*and μ*

_{a}*' scale linearly with concentration. Each of the 20 phantoms was measured by both the probe and the integrating sphere.*

_{s}*3.6* In vivo *measurements to demonstrate utility*

In order to test the fiberoptic probe *in vivo*, as well as to obtain brain optical properties for separate studies on optical diagnostics during brain resection surgery, female Lewis rats (Charles River, QC, Canada) were used, under institutional ethics approval (University Health Network, Toronto). The animals were brought under general anesthesia with 4% isofluorane (oxygen flow at 2 L/min) and sustained by an injection of ketamine/xylazine (80/13 mg/kg, i.p.), and the eyes lubricated with tear gel. The scalp was reflected and a 1 cm dia. craniotomy was performed using a 1 mm drill bit, exposing both hemispheres. The dura was cut with microscissors, exposing the cortical surface. The fiberoptic probe was placed in gentle contact with the brain tissue and measurements taken. As well, measurements were obtained from exposed facial muscle adjacent to the craniotomy site. After measurements were taken, 120 mg/kg bodyweight of Euthanyl under heavy anesthesia (2.5% isofluorane with 1 L/min oxygen) was used for euthanasia. In this article, representative data from these measurements are presented to demonstrate the utility of the probe in an *in vivo* application, although 5 animals were used in total. All studies were carried out under institutional animal-care approval (University Health Network, Toronto, Canada).

## 4. Results

#### 4.1 Phantom measurements for diffusion theory model validation

The errors between the phantom reflectance measurements and diffusion theory reflectance values were calculated for all *r*, μ* _{a}* and μ

*' (with the exception of the data with μ*

_{s}*' values lower than the lower bounds on the diffusion theory model, as defined previously). The fit to the variations in the diffusion theory model was quantified using the coefficient of determination (*

_{s}*R*

^{2}) and the normalized root-mean-square error (NRMSE). This was done for all combinations of the two diffusion constant variations, and the two light source models (buried point source and exponential line source). The statistics for these four cases are shown in Table 3 . Based on this analysis, the exponential line source model and a diffusion constant of

*D*= (3µ

*')*

_{s}^{−1}were found to be optimal for the probe geometry. The measured reflectance values (from the probe) were then plotted against the modeled reflectance values (based on μ

*and μ*

_{a}*' measurements from the integrating sphere), again with the proviso that, for a given μ*

_{s}*', the values were within the range of validity as discussed in the Theory section (Fig. 7 ).*

_{s}#### 4.2 Phantom measurements to determine probe accuracy

For the 20 phantom measurements, the RMS deviation between the probe and integrating sphere measurements were 5.4% and 4.3% for μ* _{a}* and μ

*', respectively. Figure 8a shows corresponding optical spectra for one measurement from this data set, demonstrating good correlation between the probe and integrating sphere measurements. Figure 8b then also shows good agreement between the diffuse reflectance measurement and the fit to the diffusion theory model.*

_{s}*4.3* In vivo *measurements to demonstrate feasibility*

Representative *in vivo* optical properties data measured at the rat brain cortical surface and facial muscle are shown in Figs. 9a
and 9c. The estimated free parameters from the brain measurement are *f _{Hb}* = 4.89 g/L,

*StO*

_{2}= 40.5%,

*A*= 125.46 and

*k*= 0.2576. From the muscle data set, the values are

*f*= 2.66 g/L,

_{Hb}*StO*

_{2}= 68.5%,

*A*= 12.49 and

*k*= 0.0934. The diffusion theory model of reflectance fits very well to both data sets, with

*R*

^{2}= 0.963 and 0.971 for brain and muscle, respectively (Fig. 9b and 9d).

It would be helpful to have a useful comparison between the *in vivo* probe data and the *in vitro* integrating sphere technique; however, this was not technically feasible due to significant distortions moving from the *in vivo* to the *in vitro* situation. For example, we took probe measurements in cortical brain tissue *in vivo* and also after the brain was extracted post-sacrifice. We found on average that μ* _{a}* dropped by 44% at 500 nm from the

*in vivo*to the

*ex vivo*situation due to loss of blood perfusion upon death and/or brain extraction (

*n*= 5 animals). Since tissue needs to be further prepared by slicing for integrating sphere measurements (and possibly frozen to preserve the tissue prior to these measurements) this is highly likely to introduce further handling artifacts, as demonstrated by Chan

*et al*. [6]. For these reasons, the integrating sphere technique was applied with phantoms as in Section 4.2 for validation purposes.

Note that these single measurements in tissue are to demonstrate the utility of the probe *in vivo* as well as to demonstrate algorithm convergence in tissue—estimates of the inter-animal variation on the derived optical properties in larger sets of animals are planned and will be reported in future work, with the focus of this current article on instrument/algorithm development and validation.

## 5. Discussion

One of the main issues addressed with this technique is that there is a limited range of μ* _{a}* and μ

*' values that can be determined from the reflectance measurements for any given source-collector distance,*

_{s}*r*. Using multiple inter-fiber distances to overlap these ranges increases the overall dynamic range of optical properties that can be accurately measured. The lower limit of μ

*' is the major issue in utilizing the technique described here. One potential improvement is to add reflectance measurements at*

_{s}*r*> 780 µm to reach μ

*'<5.8 cm*

_{s}^{−1}. There is a tradeoff, however: with increasing

*r*, the signal-to-noise decreases significantly and the probe head necessarily has to be larger (limiting the versatility of the technique). As well, the overall μ

*' range decreases with increasing*

_{s}*r*, as shown in Fig. 4. Restricting

*r*to small values also limits the effective tissue sampling depth of the measurements, which is advantageous for highly localized measurements and for application in small tissue structures. Figure 10 shows Monte Carlo modeling using the current probe geometry to determine the effective sampling depth (90% of the detected photons) for the different μ

*' and μ*

_{s}*values, with the deepest penetration occurring at low μ*

_{a}*' and low μ*

_{s}*.*

_{a}It is useful to compare our results with those in the literature concerning diffusion theory breakdown with fiberoptic separation distances <1 mm. Reif *et al.* implemented a probe with *r* = 250 µm, and used Monte Carlo simulations of the reflectance signal to determine the effect of varying *g* while holding μ* _{s}*' constant [14]. The variation was <15% for

*g*= 0.75-0.95 and μ

*' = 5, 10 and 20 cm*

_{s}^{−1}. We report very similar findings extrapolated from our Monte Carlo data, with the reflectance at

*r*= 260 μm reflectance having a variation <15% for μ

*' > 5.1 cm*

_{s}^{−1}for

*g*= 0.6-0.95. A similar diffusion theory analysis to that used here was performed by Sun

*et al.*[24], using Monte Carlo simulations to determine how errors in the diffusion theory model vary with

*r*: for μ

*= 2.5 cm*

_{a}^{−1}, μ

*' = 6.4 cm*

_{s}^{−1}and

*g*= 0.84, the diffusion theory errors were <20% for

*r*> 500 μm. Our results are comparable, with the

*r*= 520 μm fiber giving an error <10% for μ

*' > 9.3 cm*

_{s}^{−1}(μ

*= 1 cm*

_{a}^{−1},

*g*= 0.6-0.95) (see Fig. 3b).

We have used the form *D* = (3µ* _{s}*')

^{−1}as the diffusion constant, so that

*z*

_{0}= (µ

*')*

_{s}^{−1}and µ

*= (3µ*

_{eff}*µ*

_{a}*')*

_{s}^{1/2}. In practice, this simpler form of

*D*is not only more accurate, it also makes the inversion of Eq. (3) more robust than the alternative with

*D*= [3(µ

*+ µ*

_{a}*')]*

_{s}^{−1}(and, thereby,

*z*

_{0}= (µ

*+ µ*

_{a}*')*

_{s}^{−1}and µ

*= [3µ*

_{eff}*(µ*

_{a}*+ µ*

_{a}*')]*

_{s}^{1/2}): it was found that the nonlinear least squares algorithm used for inversion often did not converge in this alternative formulation. We note also that the exponential line source model is more accurate than the buried point source model under these experimental conditions, possibly because of the close source-collector separations used.

For skin studies, the presence of melanin and the shape of the μ* _{s}*' spectrum may cause difficulties with this technique, as outlined in Tseng

*et al*. [25]. The spectral shape of melanin is similar to the power law-dependence of μ

*', making it difficult to spectrally constrain the solution. As well, it has been shown in skin that a piece-wise power law function fits better to μ*

_{s}*', rather than a single power law function as used in this work. For these reasons, the technique presented in this work is likely not well-suited to skin in its present form, although it may be extended or modified to include skin. It is not obvious*

_{s}*a priori*that an algorithm based on homogeneous tissue will translate to layered turbid media such as skin.

An interesting consequence of the Similarity Principle breaking down at low μ* _{s}*' and low

*r*is that scattering phase function information is encoded in the reflectance signal. The Monte Carlo results in Fig. 3 suggest that, for μ

*' < 16.4 cm*

_{s}^{−1}at low absorption (μ

*= 0.1 cm*

_{a}^{−1}), the reflectance at

*r*= 260 µm will be sensitive to both μ

*' and*

_{s}*g*, whereas at

*r*= 780 µm it is sensitive only to μ

*'. The idea that anisotropy is encoded into the reflectance signal at very small*

_{s}*r*separations may be useful in some applications. Since the scattering phase function depends only on the ‘morphology’ of the scattering structures, while μ

*' depends on both morphology and abundance, a measure of both may provide additional biological information.*

_{s}In this study, we have chosen to apply diffusion theory to solve the inverse problem, rather than say, using Monte Carlo-generated look-up tables for reflectance versus μ* _{a}* and μ

*' and then applying a*

_{s}*post hoc*spectral constraint to determine the optical properties. The main advantage of using the diffusion theory approach is that it allows use of a simple, closed-form analytic equation that integrates the spectral constraint with the reflectance model, allowing the inverse problem to be solved in a straightforward manner using a Levenberg-Marquardt algorithm. At this time, it is not obvious which approach (diffusion theory or Monte Carlo) is better for this probe geometry and under what conditions. Relevant factors include the overall accuracy, ease of implementation, computational speed and robustness against, for example, tissue inhomogeneity, out-of-range μ

*and μ*

_{a}*' values and measurement noise. This would be an interesting subject for future studies.*

_{s}It is perhaps surprising that the diffusion modeling and the experimental data are in such good agreement given the number of transport mean free paths (1/μ* _{s}*') travelled by the detected photons is relatively small; Monte Carlo simulations show that this is in the range of roughly 2-5 transport mean free paths for μ

*' between 5 and 25 cm*

_{s}^{−1}. We speculate that this may be partially due to the fact that we are applying a spectral constraint to the model, which differs from the more typical situation for when diffusion theory is applied at single wavelengths, which might require a larger number of transport mean free paths to be robust. It may also be that the exponential line source model is more accurate than the point source model for close fiber separations.

## 6. Conclusions

We report a fiberoptic probe to recover accurate tissue optical absorption and reduced scattering coefficients using a diffusion theory model of the diffuse reflectance collected at the tissue surface. Multiple source-collector distances enabled spanning of a large range of μ* _{a}* and μ

*' values, beyond that of any single source-collector separation. The dynamic range is µ*

_{s}*' = 5.8-52.9 cm*

_{s}^{−1}for µ

*= 0-10 cm*

_{a}^{−1}for this geometry. Optical phantoms experiments demonstrated that the derived μ

*and μ*

_{a}*' values are accurate to 5.4% and 4.3%, respectively, when compared against integrating sphere estimates.*

_{s}The authors gratefully acknowledge support National Institutes of Health (grant # R01NS052274-01A1). Partial core support is provided by the Ontario Ministry of Health and Long-Term Care: the views expressed do not necessarily reflect those of the OMOHLTC. A. Kim is also supported in part by Natural Sciences and Engineering Research Council of Canada.

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