1. Introduction

Accurate measurement of tissue optical properties, specifically the absorption (μ_a) and transport (reduced) elastic scattering (μ_s') 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].

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 cm² [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_s' relies upon spectral constraints, i.e. applying a priori knowledge of the shapes of μ_a(λ) and μ_s'(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 mm³ and fast acquisition times (<1 sec to ~seconds).

 

Fig. 1 Source-collector pair for measuring the diffuse reflectance spectrum. A broadband white light source is used to illuminate the tissue. A collector fiber connected to a spectrometer records the diffuse reflectance spectrum.

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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_s' 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_a and μ_s' 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.

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]:

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

The diffusion constant, D, has been variously cited as (3μ_s')⁻¹ or [3(μ_s' + μ_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_s'/(µ_s' + µ_a). If it is modeled as an exponentially decaying line source extending into the tissue, then Eq. (3) includes the 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_a = 0 this issue does not matter, since then 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_s' 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 (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_s' 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_a always reduces the reflectance signal. Here, we have used the monotonically increasing part since the reflectance is far more sensitive to changes in μ_s' 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_a, so it should be taken as the largest expected μ_a value (here, 10 cm⁻¹). In practice, this was set as the value of μ_s' at 90% peak reflectance, to provide an additional safety margin. From Table 1, the upper bounds for r = 260, 520 and 780 μm were then μ_s' = 52.9, 26.1 and 17.1 cm⁻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_s' 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_a and μ_s', as detailed later on in Section 3.3.

 

Fig. 2 Diffuse reflectance varying with μs' as per the diffusion theory model equation (Eq. (3)), with r = 500 μm and μa = 0.01 cm1.

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Table 1. μ_s' values at the maximum of the diffuse reflectance curve and at 90% of the maximum, for each of the three source-collector separations

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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_s' = μ_s(1-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_s' = 2-20 cm⁻¹, µ_a = 0-1 cm⁻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_s', µ_a and g dimensions. Examples of the resulting Monte Carlo reflectance versus µ_s' curves are shown in Fig. 3a , together with the diffusion theory model graphs. The root-mean-square (RMS) error, ε_RMS, 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 ε_RMS cut-off of 10% to determine the ranges of validity of the diffusion theory model. As would be expected, as r decreased the corresponding µ_s' value at ε_RMS increased, since single scattering effects are amplified with smaller inter-fiber distances. An interesting artifact is evident in Fig. 3b, where the ε_RMS increases slightly with increasing µ_s' 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.

 

Fig. 3 a) Reflectance as a function of µ_s' for r = 520 µm, with µ_a = 0.1 cm⁻¹ and a range of g. The line-connected symbols are Monte Carlo values; the thick solid line is the diffusion theory graph. b) RMS errors of the Monte Carlo data compared to diffusion theory (g = 0.6-0.95) as a function of μ_s', for the three source-collector separations.

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Table 2 shows the µ_s' values at the 10% ε_RMS crossover points, for the three r distances and for µ_a = 0.1, 0.5 and 1 cm⁻¹. Based on the above, Eq. (3) is valid in the ranges µ_s' >16.4, 10.1 and 5.8 cm⁻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_s' ranges of 5.8-52.9 cm⁻¹ for a µ_a range of 0-10 cm⁻¹. Note that if the cut-off is >10%, then the lower limit of µ_s' for all fiber distances will be decreased.

Table 2. μ_s' values (cm⁻¹) below which there is >10% RMS error between Monte Carlo and diffusion theory (g = 0.6-0.95) for the three r distances

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Fig. 4 Range of μ_s' within which the diffusion theory model can be used to solve for μ_a and μ_s' using spectral constraint, for each r. The bars shaded in gray are for the r values that are used in the fiberoptic probe; the white bars are for larger r distances than currently used in the probe to graphically display the trend with increasing r.

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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.

 

Fig. 5 (a) Geometry of the probe tip, (b) the reflectance probe, (c) complete system.

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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_s'. 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 versus μ_s' 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 x- and y-axis needing calibration.

 

Fig. 6 (a) Experimental diffuse reflectance measurements (□’s) of diluted Intralipid concentrations measured at 600 nm and r = 520µm. Here, the absolute value of μ_s' for each dilution is unknown; however, the relative scatterer concentration of each dilution is known, and the uncalibrated result is graphed with the diffusion theory curve. (b) The μ_s' value at the peak reflectance is used to scale the data to match that of diffusion theory curve. Although the data shown are at 600 nm, the calibration was done at every wavelength and source-collector distance.

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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

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_s' boundaries taken from Fig. 4.

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_s' = 7, 14 or 21, prepared in all combinations. Since there are three r distances, this results in 27 total data points. This range of μ_a and μ_s' spans the majority of tabulated 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_a is close to zero, since both forms of D and both forms of the light source model converge to the same model at μ_a = 0.

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_d(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 R_d and T_d for µ_a and µ_s' values ranging from 0 to 100 cm⁻¹ and 0-100 cm⁻¹, respectively. The tissue optical properties were then calculated from T_d(λ) and R_d(λ) using an inverse interpolation algorithm. For this set of measurements, the phantoms were formulated first with μ_a = 21.7 and μ_s' = 24.7cm⁻¹ (at 715 nm) and then diluted into serial fractions of 95 to 5%, such that μ_a and μ_s' scale linearly with concentration. Each of the 20 phantoms was measured by both the probe and the integrating sphere.

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_s' (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 (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_a and μ_s' 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 ).

Table 3. R² and NRMS error quantifying the goodness-of-fit of the four different forms of the diffusion theory model compared with phantom measurements

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Fig. 7 Diffuse reflectance derived from the diffusion theory model versus measured reflectance values using the probe in the nine Intralipid-NG phantoms for the three r values (providing that the reflectances measured are within the range of validity). Here, the form of the diffusion theory model shown is with the exponential line source and D = (3μ_s')⁻¹, with R ² = 0.984. Representative error bars at the maximum value and in the mid-range are displayed, with y-axis error bars representing the uncertainty in the reflectance model due to uncertainty in the μ_s' value as measured by the integrating sphere; x-axis error bars represent standard deviation in the reflectance measurements.

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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_s', 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.

 

Fig. 8 (a) Absorption and reduced scattering coefficient spectra as measured by the probe and integrating sphere, for 50% dilution of the Intralipid-NG phantom. (b) Probe reflectance measurement (r = 520 µm) compared to the diffusion theory model fit, with R ² = 0.992.

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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_Hb = 2.66 g/L, 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).

 

Fig. 9 Representative in vivo absorption and scattering spectra as measured by the probe in (a) brain cortical tissue and (c) muscle. The corresponding reflectance measurements and fits to the diffusion theory model are shown in (b) and (d), with R ² values of 0.963 and 0.971, respectively.

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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_s' values that can be determined from the reflectance measurements for any given source-collector distance, 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_s' is the major issue in utilizing the technique described here. One potential improvement is to add reflectance measurements at r > 780 µm to reach μ_s'<5.8 cm⁻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_s' range decreases with increasing 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_s' and μ_a values, with the deepest penetration occurring at low μ_s' and low μ_a.

 

Fig. 10 Effective sampling depth of the probe as a function of μ_a and μ_s'. The r distance used for these depth calculations depends on the range of validity for each µ_s', i.e. for the µ_s' = 5 and 10 cm⁻¹ data points, r = 780µm; for µ_s' = 15 cm⁻¹, r = 520 µm; for µ_s' = 20 cm⁻¹, r = 260 µm. Representative error bars are shown at the highest and lowest data points. Error bars represent standard deviation in the Monte Carlo simulations with five repetitions of the simulations.

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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_s' = 5, 10 and 20 cm⁻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_s' > 5.1 cm⁻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_a = 2.5 cm⁻¹, μ_s' = 6.4 cm⁻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_s' > 9.3 cm⁻¹ (μ_a = 1 cm⁻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_eff = (3µ_aµ_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_a + µ_s')]⁻¹ (and, thereby, z ₀ = (µ_a + µ_s')⁻¹ and µ_eff = [3µ_a(µ_a + µ_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_s', 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 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_s' < 16.4 cm⁻¹ at low absorption (μ_a = 0.1 cm⁻¹), the reflectance at r = 260 µm will be sensitive to both μ_s' and g, whereas at r = 780 µm it is sensitive only to μ_s'. The idea that anisotropy is encoded into the reflectance signal at very small r separations may be useful in some applications. Since the scattering phase function depends only on the ‘morphology’ of the scattering structures, while μ_s' depends on both morphology and abundance, a measure of both may provide additional biological information.

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_s' and then applying a 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_a and μ_s' values and measurement noise. This would be an interesting subject for future studies.

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_s' between 5 and 25 cm⁻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_s' values, beyond that of any single source-collector separation. The dynamic range is µ_s' = 5.8-52.9 cm⁻¹ for µ_a = 0-10 cm⁻¹ for this geometry. Optical phantoms experiments demonstrated that the derived μ_a and μ_s' values are accurate to 5.4% and 4.3%, respectively, when compared against integrating sphere estimates.

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.