1. Introduction

Quantification of the tissue optical properties has long been an area of extensive research in biomedicine. Specifically, the in vivo determination of the optical properties can be used to calculate the fluence distribution of light for both pretreatment planning and online optimization during photodynamic therapy or thermal therapy [1–3]. Besides, the determination of the optical properties also can be applied for the early cancer detection of tissue according to the differences in the optical properties between cancerous and normal tissues, caused by the variations in extracellular and nuclear structures and chromophore compositions [4, 5].

The continuous wave (CW) techniques are commonly employed to determinate the optical properties due to relatively low cost and simple equipment. Numerous studies have been conducted for noninvasively obtaining the optical properties of tissue such as skin from the spatially resolved surface reflectance measurements [6, 7]. However, applications of such methods are limited by shallow light penetration and inappropriate for diagnosis of internal organs such as the prostate. In such cases, the minimally invasive measurements are desirable and there has been a good deal of work done on this topic. One part of groups employed the interstitial spectroscopy techniques by a single encapsulated probe for determining the optical properties. Bargo et al. implemented a spectrally resolved CW diffuse reflectance method where only two fibers (one source and one detector) spaced 2.5 mm apart are used for the determination of the optical properties interstitially [8]. However, such technique requires extensive calibration with a large library of optical phantoms [9]. Baran et al. developed a custom fiber optic probe which consisted of six side-firing fibers axially separated by 2 mm to interstitially obtain spatially and spectrally resolved reflectance data from turbid media [9]. The optical properties were extracted by a Monte Carlo (MC)-based fitting algorithm. However, the relatively small number of source detector separations (SDS) available by this probe places potential limits on the accuracy of the optical property determination [10]. Another part of groups inserted the source and detector fibers into the tissue at a prescribed spacing using guidance templates. Zhu et al. measured the ratio of light fluence rate to source power along a linear channel at a fixed distance of 5 mm and the diffusion approximation (DA) was used for fitting the measured data to extract the optical properties [11, 12]. However, an integrating sphere which is difficult to be implemented in vivo is required for calibrating an isotropic detector and a light source power [13].

Recently, a CW radiance technique measuring light from multiple directions has been proposed as another minimally invasive method for determining the optical properties under an interstitial probe arrangement, since such method can minimize the required number of SDSs [13–17]. Dickey et al. first demonstrated the capability to obtain the optical properties of liquid phantoms from radiance measurement data at a single SDS by using the P3 approximation for fitting [14]. However, their research was limited to a single optical property pair and a set of experimental parameters. Chin et al. had further investigated the determination of the optical properties using the P3 approximation for fitting from the MC simulated and experimentally measured radiance caused by an isotropic source [15, 16]. However, the shape-based fitting algorithm disclosed a significant breakdown in estimating the optical properties for the transport albedos, $a^{\prime }$ defined as $a^{\prime }={{\mu }^{\prime }}_s/({{\mu }^{\prime }}_s+{\mu }_a)$, less than ~0.9 due to the lack of inherent uniqueness in fitting algorithms, where ${\mu }_a$ and ${{\mu }^{\prime }}_s$ are the absorption coefficient and the reduced scattering coefficient, respectively. Grabtchak et al. derived two analytical expressions based on the DA to extract the effective attenuation coefficient, ${\mu }_{eff}$, and the diffusion coefficient, D, directly from radiance measurements caused by an isotropic source [13, 17]. Although such an analytical method could avoid the problems of lack of inherent uniqueness in fitting algorithms, the optical properties can be recovered finely just under the condition of ${{\mu }^{\prime }}_s>>{\mu }_a$ for which the DA is valid.

For the tissue with high absorption or low scattering that may be encountered in the clinical environment [18, 19], a higher order approximation for radiance is required for recovering the optical properties analytically. The P3 approximation, the P3 for short, is naturally chosen as a light propagation model to recover the optical properties. The P3 has a solution consisting of two parts; one is the transient part which contributes significantly only for the SDSs of less than two transport mean free paths, $2{l^{\prime }}_t$, where ${l^{\prime }}_t=1/({{\mu }^{\prime }}_s+{\mu }_a)$, and the other is the asymptotic part which dominates for the large SDSs [20]. However, analytical expressions for recovering the optical properties cannot be constructed based on the P3. Besides, small SDSs less than $2{l^{\prime }}_t$ are not used in interstitial radiance measurements, generally. Thus, in this paper, we have proposed an incomplete P3 approximation for radiance, the P3_in for short, which consists of only the asymptotic part of the solution for the P3. The proposed P3_in has a simpler expression than the complete solution for the P3, but provides a higher accuracy than the DA for modeling light propagation in the turbid media with high absorption or low scattering. Based on the P3_in, we have developed analytical formulae for extracting the optical properties directly by using simple analytical expressions constructed from CW radiance measurements at only two SDSs and four angles.

2. Methodology

2.1 The P3_in approximation for radiance

When the PN method is applied to solve the radiative transfer equation (RTE), the angular quantities are expanded in spherical harmonics, and for the case of spherical symmetry the radiance is given by Eq. (1) in a spherical coordinate [21]:

In the Eq. (3), the terms associated with $h_l({\nu }^{+})Q_l(-{\nu }^{+}r)$ represent the transient part and the terms associated with $h_l({\nu }^{-})Q_l(-{\nu }^{-}r)$ represent the asymptotic part. As $Q_l(-{\nu }^{+}r)$ $\text{(}l=0,\mathrm{...},3\text{)}$ decay rapidly to be zero with the increase in r due to the relatively large value of ${\nu }^{+}$, the transient part mainly contributes to the total radiance only for the SDSs less than $2{l^{\prime }}_t$. The value of ${\nu }^{-}$ is relatively small, and $Q_l(-{\nu }^{-}r)$ $\text{(}l=0,\mathrm{...},3\text{)}$ decay slowly, thus the asymptotic part dominates for large SDSs.

Conventionally, the P3 approximation has been used for recovering the optical properties analytically from the radiance measurements. For each $l=0,\mathrm{...},3$, the values of $S_0[C^{\prime }{h}_l({\nu }^{-})Q_l(-{\nu }^{-}r)+D^{\prime }{h}_l({\nu }^{+})Q_l(-{\nu }^{+}r)]$ in Eq. (3) can be obtained by measuring the radiances at different angles with a fixed r to determine the values of $P_l(cos\theta )$. However, the values of $S_0{C}^{\prime }{h}_l({\nu }^{-})$ and $S_0{D}^{\prime }{h}_l({\nu }^{+})$ cannot be separately obtained by measuring the radiances at different SDSs to determine the values of $Q_l(-{\nu }^{\pm }r)$ because ${\nu }^{\pm }$ appearing in $Q_l(-{\nu }^{\pm }r)$ are unknown. Thus $h_l({\nu }^{-})$ and $h_l({\nu }^{+})$ cannot be obtained analytically, and from Eq. (5) it is deduced that the optical properties cannot be recovered in the form of analytical expressions.

Analytical expressions for recovering the optical properties cannot be constructed based on the P3, but small SDSs less than $2{l^{\prime }}_t$ are usually not used for interstitial radiance measurements. Thus, in this paper, we propose an incomplete P3 approximation for the radiance, the P3_in for short, which is the asymptotic part of the solution for the P3. Hereafter, ${\nu }^{-}$ is replaced by $\nu$ for simplicity, and the radiance by the P3_in is given as:

2.2 Accuracy of the P3_in approximation

In order to evaluate the differences between the P3_in and the P3, we examined the influence of the abandoned transient part of the solution in the P3 by the relative differences, $\text{ε}=|L(r,\theta )-L_{in}(r,\theta )|/L(r,\theta )$, at different SDSs and angles for different optical properties. In the analysis, the optical properties of prostate with $0.01{\text{mm}}^{-1}\le {\mu }_a\le 1.0{\text{mm}}^{-1}$ and $0.5{\text{mm}}^{-1}\le {{\mu }^{\prime }}_s\le 4.5{\text{mm}}^{-1}$ were considered [18, 19, 23]. The SDSs were chosen from 5 mm to 10 mm which are typical SDSs during interstitial laser therapies [16, 23].

Figure 1 plots the maximum values of $\text{ε}$, ${\text{ε}}_{\text{max}}$, in the optical property range mentioned above at SDSs from 5 mm to 10 mm and angles from $0°$ to $180°$. The results show that the ${\text{ε}}_{\text{max}}$ values are very small for angles from $0°$ to $160°$ and the ${\text{ε}}_{\text{max}}$ values fluctuate strongly for angles from $160°$ to $180°$. It also can be seen that the ${\text{ε}}_{\text{max}}$ values decrease with the increase in r. Beside, Fig. 1 also shows that the sharp drops appear at the angls of about $40°$, $115°$ and $145°$, since the P3_in and the P3 have almost the same values at such angles. The ${\text{ε}}_{\text{max}}$ values for angles from $0°$ to $160°$ are within 0.48%, 1.94 × 10^-2%, 6.48 × 10^-4%, 4.78 × 10^-5%, 5.67 × 10^-6% and 6.76 × 10^-7% for the SDSs of 5, 6, 7, 8, 9 and 10 mm, respectively. Whereas, the ${\text{ε}}_{\text{max}}$ values for angles from $160°$ to $180°$ are within 17.11%, 2.46%, 0.18%, 2.12 × 10^-3%, 7.71 × 10^-5% and 2.20 × 10^-6% for the SDSs of 5, 6, 7, 8, 9 and 10 mm, respectively. In summary, the relative differences between the P3_in and the P3 are within 0.48% for angles from $0°$ to $160°$ and within 17.11% for angles from $160°$ to $180°$for the optical property range mentioned above at SDSs from 5 mm to 10 mm. It indicates that the P3_in is acceptable for an alternative to the P3 as a light propagation model to recover the optical properties for angles from $0°$ to $160°$.

 

Fig. 1 The maximum relative differences between the P3_in and the P3, ${\text{ε}}_{\text{max}}$, in the optical property range of $0.01{\text{mm}}^{-1}\le {\mu }_a\le 1.0{\text{mm}}^{-1}$ and $0.5{\text{mm}}^{-1}\le {{\mu }^{\prime }}_s\le 4.5{\text{mm}}^{-1}$ at SDSs from 5 mm to 10 mm and angles from $0°$ to $180°$.

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2.3 Recovery of the optical properties based on the P3_in approximation

Based on the P3_in expressed by Eq. (6), this study formulates an analytical method to recover the optical properties from the radiance measurements, $L_m(r,\theta )$, at several pairs of $(r,\theta )$. From Eq. (5), it is deduced that the optical properties of the medium can be determined analytically if $\nu$, $h_1(\nu )$ and $h_2(\nu )$ were obtained. As the first step for obtaining $\nu$, $h_1(\nu )$ and $h_2(\nu )$, the terms of $S_0{C}^{\prime }{h}_l(\nu )Q_l(-\nu r)$ $(l=0,1,2)$should be determined. As the terms of $S_0{C}^{\prime }{h}_l(\nu )Q_l(-\nu r)$ with four different numbers of l in Eq. (6) are simply added, they can be separated from the radiance measurement data of $L_m(r,{\theta }_i)$ $(i=0,\mathrm{...},3)$ at four angles of ${\theta }_0$, ${\theta }_1$, ${\theta }_2$,${\theta }_3$with a fixed r, as shown in Fig. 2, by a set of linear equations given as:

 

Fig. 2 Geometry for measurements of the radiances, $L_m(r,\theta )$, at SDSs of r and $r^{\prime }$ and angles of ${\theta }_0$, ${\theta }_1$, ${\theta }_2$ and ${\theta }_3$.

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As the second step for obtaining $\nu$, $h_1(\nu )$ and $h_2(\nu )$, the ratio of $f_0(r,{\theta }_0,{\theta }_1,{\theta }_2,{\theta }_3)$ to $f_0(r^{\prime },{\theta }_0,{\theta }_1,{\theta }_2,{\theta }_3)$ is taken from the radiance measurements at two SDSs, r and $r^{\prime }$, as shown in Fig. 2, to delete $S_0$ for avoiding the measurements of the absolute radiances:

Now $\nu$, $h_1(\nu )$ and $h_2(\nu )$ are obtained, and the optical properties can be determined from Eq. (5) as the following:

The advantages of this recovery algorithm of the optical properties are; (i) data fitting, a mass of measurement data and the absolute radiance measurements are not required, and (ii) both ${\mu }_a$ and ${{\mu }^{\prime }}_s$ are determined directly from the radiance measurements at two SDSs and four angles using the analytical formulas from Eq. (8) to Eq. (14).

3. Results

3.1 Simulation evaluation using the P25 approximation

The developed algorithm of recovering the optical properties based on the P3_in was evaluated using simulated radiance data calculated by the P25 approximation for the radiance. The P25 approximation solution of the RTE produced by an isotropic source placed inside an infinitely extended turbid medium was used as the light propagation model to generate the simulated data because the excellent agreement between the P25 approximation solution and the results of a MC method has been verified [21, 24]. In this analysis, the values of 1 mm⁻¹ and 0.5 mm⁻¹ were chosen for ${{\mu }^{\prime }}_s$ as typical and low scattering cases, respectively, while ${\mu }_a$ ranged from 0.01 mm⁻¹ to 0.5 mm⁻¹, so that the albedos ranged from 0.5 to 0.99. The chosen optical property sets roughly covered the range of the albedos expected in interstitial therapies [18, 19, 23, 25]. The radiance measurement data were generated at four angles of $40°$, $80°$, $120°$ and $160°$. This set of angles avoided the radiances at angles larger than $160°$ where the differences between the P3_in and the P3 are relative remarkable. The SDS of $r^{\prime }$ was fixed at 5 mm, while 6 mm and 10 mm were chosen for the SDSs of r. At the same time, the DA was also employed to extract the optical properties from the same simulated radiance data in order to compare with the P3_in [13]. The results are presented for the recovery algorithms based on both the DA and the P3_in.

We evaluated the performance of the developed algorithm based on the P3_in in the case of ${{\mu }^{\prime }}_s=1.0{\text{mm}}^{-1}$ first. In Figs. 3(a) and 3(b), the results from the algorithm based on the DA show that the accuracy of the recovered optical properties decreases dramatically with the increase in the true ${\mu }_a$ and that the accuracy is improved only a little with the increase in r. Just in the case of ${\mu }_a\le 0.1{\text{mm}}^{-1}$, i.e., $a^{\prime }\ge 0.91$, both ${\mu }_a$ and ${{\mu }^{\prime }}_s$ are recovered better with the relative errors (REs) less than 14.75% where the REs are defined as $|{\mu }_{a\_r}-{\mu }_{a\_t}|/{\mu }_{a\_t}$, $|{{\mu }^{\prime }}_{s\_r}-{{\mu }^{\prime }}_{s\_t}|/{{\mu }^{\prime }}_{s\_t}$ for ${\mu }_a$ and ${{\mu }^{\prime }}_s$, respectively, with ${\mu }_{a\_r}$, ${\mu }_{a\_t}$, ${{\mu }^{\prime }}_{s\_r}$ and ${{\mu }^{\prime }}_{s\_t}$ indicating the recovered ${\mu }_a$, the true ${\mu }_a$, the recovered ${{\mu }^{\prime }}_s$ and the true ${{\mu }^{\prime }}_s$. The likely explanation is that the DA provides a poor description of the radiance in the case of a low albedo even at large SDSs [15]. In contrast to the DA, Figs. 3(a) and 3(b) demonstrate that the developed recovery algorithm based on the P3_in is able to recover both ${\mu }_a$ and ${{\mu }^{\prime }}_s$ accurately. A small improvement in the accuracy of the recovered optical properties has also been observed with the increase in r similarly to the DA. The REs of the recovered ${\mu }_a$ and ${{\mu }^{\prime }}_s$ based on the P3_in are less than 1.30% and 0.96%, respectively, for the albedos ranging from 0.67 to 0.99.

 

Fig. 3 Recovered optical properties based on the DA and the P3_in from the simulated radiance data calculated by the P25 in the case of typical scattering. (a) is for ${\mu }_a$ and (b) is for ${{\mu }^{\prime }}_s$. The insets show the relative errors (REs) of the recovered optical properties.

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We also evaluated the performance of the developed algorithm for the case of ${{\mu }^{\prime }}_s=0.5{\text{mm}}^{-1}$ which may be encountered during interstitial laser therapies [18, 19]. Figures 4(a) and 4(b) show that the recovered ${\mu }_a$ and ${{\mu }^{\prime }}_s$ based on the P3_in are also very close to true values, whereas the recovered ${\mu }_a$ and ${{\mu }^{\prime }}_s$ based on the DA exhibit significant errors. The recovery algorithm based on the DA disclosed a significant breakdown in recovering ${\mu }_a$ with REs larger than 20.72%, although ${{\mu }^{\prime }}_s$ is recovered with the REs within 9.54% for the true ${\mu }_a$ less than 0.05 mm⁻¹. In contrast to the DA, the REs of the recovered ${\mu }_a$ and ${{\mu }^{\prime }}_s$ based on the P3_in are less than 5.28% and 3.86%, respectively, for the albedos ranging from 0.5 to 0.98. Thus the results show that the recovery algorithm based on the P3_in is able to recover the optical properties accurately in the cases of both typical and low scattering.

 

Fig. 4 Recovered optical properties based on the DA and the P3_in from the simulated radiance data calculated by the P25 in the case of low scattering. (a) is for ${\mu }_a$ and (b) is for ${{\mu }^{\prime }}_s$. The insets show the REs of the recovered optical properties.

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3.2. Experimental evaluation by liquid phantoms

Figure 5 presents a schematic of the relevant experimental setup. An $850\text{μm}$ diameter isotropic spherical diffuser (Medlight, Switzerland) was coupled to a 675nm laser (Thorlabs, USA) with an optical attenuator (EXFO, Canada) to illuminate a liquid phantom interstitially. The source fiber was attached to a holder mounted on a two-dimensional precision translation stage that allowed changing the source fiber position relative to the radiance detector probe. The radiance detector probe was constructed similarly to that of Barajas et al. by attaching a $\text{500}\text{μm}$ right-angle prism to a cleaved plane cut $475\text{μm}$ fiber with a numerical aperture of 0.22 [26]. The detector fiber was mounted on a motorized rotation stage that allowed complete $360°$ rotation. The detector fiber was connected to a photomultiplier tube (PMT) (Hamamatsu, Japan) for data acquisition.

 

Fig. 5 A schematic of measuring the radiances from an isotropic point source at SDS of r and angle of $\theta$.

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Similarly to the conditions in the evaluation using the simulated radiances, the radiance measurements were made at four angles of $40°$, $80°$, $120°$ and $160°$. The SDS of $r^{\prime }$ was fixed at 5 mm, while 6 mm and 10 mm were chosen for the SDSs of r. A gate time of 0.5 s and the average measurements of 10 times were chosen for the PMT. All measurements were background subtracted and were conducted in the center region of a 10 × 10 × 10 cm³ black acrylic box which minimized reflections from the box boundary. Tissue simulating phantoms were prepared using a combination of 20% Intralipid and India ink for ${{\mu }^{\prime }}_s$ of 1 mm⁻¹ and 0.5 mm⁻¹ and ${\mu }_a$ ranging from 0.01 mm⁻¹ to 0.5 mm⁻¹. The concentrations of ink were determined according to the absorbance measured by a spectrometer while the concentrations of Intralipid were determined according to the equations derived by van Staveren et al [27].

The developed recovery algorithm based on the P3_in was evaluated by the experimental radiance data in the case of typical scattering of ${{\mu }^{\prime }}_s=1.0{\text{mm}}^{-1}$ in Figs. 6(a) and 6(b). The trends obtained by the phantom measurements were similar to those shown in Figs. 3(a) and 3(b) by the corresponding simulation. The results show that the REs of the recovered optical properties based on the DA increase dramatically with the increase in the true ${\mu }_a$ and that only for ${\mu }_a\le 0.1{\text{mm}}^{-1}$, i.e., $a^{\prime }\ge 0.91$, both ${\mu }_a$ and ${{\mu }^{\prime }}_s$ were recovered better with the REs within 16.12%. The results using the recovery algorithm based on the P3_in show that significant improvements in the accuracy of the recovered optical properties have been obtained when compared with those based on the DA. Although the REs of the recovered optical properties from the phantom experiments are slightly larger than those from the simulations due to the systematic errors in measurements, the REs of the recovered ${\mu }_a$ and ${{\mu }^{\prime }}_s$ based on the P3_in are less than 10.39% and 9.86%, respectively, for the albedos ranging from 0.67 to 0.99.

 

Fig. 6 Recovered optical properties based on the DA and the P3_in from the phantom measurements in the case of typical scattering. (a) is for ${\mu }_a$ and (b) is for ${{\mu }^{\prime }}_s$. The insets show the REs of the recovered optical properties.

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The developed recovery algorithm based on the P3_in was also evaluated by the experimental radiance data in the case of low scattering of ${{\mu }^{\prime }}_s=0.5{\text{mm}}^{-1}$ in Figs. 7(a) and 7(b). It is observed that the recovery algorithm based on the DA disclosed a breakdown in recovering ${\mu }_a$ with the REs larger than 17.07%. ${{\mu }^{\prime }}_s$ were recovered finely only for the true ${\mu }_a$ less than $\text{0}\text{.05}{\text{mm}}^{-1}$ with the REs within 8.15%. In contrast to the DA, the REs of the recovered ${\mu }_a$ and ${{\mu }^{\prime }}_s$ based on the P3_in are less than 10.82% and 10.67%, respectively, for the albedos ranging from 0.5 to 0.98.

 

Fig. 7 Recovered optical properties obtained from the phantom measurements in the case of low scattering. (a) is for ${\mu }_a$ and (b) is for ${{\mu }^{\prime }}_s$. The insets show the REs of the recovered optical properties.

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In summary, the recovery algorithm based on the DA can recover the optical properties from simulated and experimental data with the REs within 16.12% only for $a^{\prime }\ge 0.91$, whereas the recovery algorithm based on the P3_in can recover the optical properties with REs within 10.82% over a wide range of albedos from 0.5 to 0.99.

4. Discussion

Although the recovery algorithm based the P3_in greatly improved the accuracy of the recovered optical properties from simulated and experimental radiances when compared with the DA, the P5_in approximation as a light propagation model to recover the optical properties was also explored to examine whether the accuracy of the recovered optical properties could be further improved. Similarly to the P3_in, the solution of the P5_in is the asymptotic part of the solution for the radiance by the P5 approximation. The recovery algorithm based on the P5_in was developed and the optical properties were extracted using the analytical expressions constructed from the radiance measurements at two SDSs and six angles.

The recovery algorithms based on the P3_in and the P5_in were evaluated by the simulated radiance data calculated by the P25 for the optical property range of $0.01{\text{mm}}^{-1}\le {\mu }_a\le 1.0{\text{mm}}^{-1}$ and $0.5{\text{mm}}^{-1}\le {{\mu }^{\prime }}_s\le 4.5{\text{mm}}^{-1}$. $r^{\prime }$ and r were chosen as 5 mm and 6 mm, respectively. Four angles of $40°$, $80°$, $120°$ and $160°$ for the P3_in and six angles of $40°$, $70°$, $90°$, $120°$, $140°$ and $160°$ for the P5_in were chosen.

Figures 8(a)-8(d) plot the REs of the recovered optical properties based on the P3_in and the P5_in, respectively. The REs of the recovered ${\mu }_a$ and ${{\mu }^{\prime }}_s$ based on the P3_in are less than 6.59% and 7.49%, respectively, whereas those based on the P5_in are less than 4.19% and 4.62%, respectively. Although more radiance measurements are required for optical property recovery based on the P5_in, minor improvement in the accuracy of the recovered optical properties by the P5_in was obtained when compared with the P3_in.

 

Fig. 8 REs of the recovered optical properties for $0.01{\text{mm}}^{-1}\le {\mu }_a\le 1.0{\text{mm}}^{-1}$ and $0.5{\text{mm}}^{-1}\le {{\mu }^{\prime }}_s\le 4.5{\text{mm}}^{-1}$. (a) and (b) are based on the P3_in. (c) and (d) are based on the P5_in. (a) and (c) are for ${\mu }_a$. (b) and (d) are for ${{\mu }^{\prime }}_s$.

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Here, we present a recovery algorithm based on the P3_in that is capable of recovering ${\mu }_a$ and ${{\mu }^{\prime }}_s$ with the REs within 10.82% and 10.67%, respectively, over a wide range of albedos from 0.5 to 0.99. The REs of the recovered optical properties based on the P3_in were comparable to the errors found in previous studies of interstitial optical property recovery. Zhu et al. demonstrated the recovery of ${\mu }_a$ and ${{\mu }^{\prime }}_s$ with standard (and maximum) deviations of 8% (15%) and 18% (32%), respectively [12]. The method developed by Chin et al. resulted in the recovery of the optical properties to within ~20% for the albedos larger than ~0.9 from relative radiance measurements [9, 16]. Baran et al. demonstrated that ${\mu }_a$ and ${{\mu }^{\prime }}_s$ were recovered with mean errors of 9% and 19%, respectively, for the albedos larger than 0.95 [9]. Grabtchak et al. showed that they were able to recover ${\mu }_{eff}$ and D within 2% and 40%, respectively, using a liquid phantom of Intralipid-1%, but they did not comment on the accuracy of the recovered ${\mu }_a$ and ${{\mu }^{\prime }}_s$ directly [13]. It indicates that the developed algorithm based on the P3_in in this paper can recover the optical properties of turbid media with albedos as low as 0.5 accurately when compared with the results found in previous studies.

The developed P3_in-based radiance technique has three advantages over other CW techniques of interstitial optical property determination [11, 12, 16]. Firstly, for the P3_in-based radiance technique, neither absolute measurement nor a priori information is required, making the technique easier to be implemented in interstitial probe arrangements for clinical applications. Secondly, the developed method that only single wavelength measurements are employed, and resultantly both characterization and treatment of target tissues can be conducted by one laser. Thirdly, the shape-based fitting algorithms require large number of measurement data, whereas the developed recovery algorithm just requires the radiance measurements at two SDSs and four angles.

In theory, CW radiance measurements at two SDSs and four angles are sufficient to determine the optical properties of the homogeneous medium using the developed recovery algorithm based on the P3_in. However, for the inhomogeneous tissue sample in the clinical environments, more radiance measurement data are required to obtain a reliable result of optical property recovery. Similar to the arrangement of PDT for prostate [23], the minimally invasive measurements for tissue sample in clinical application can be conducted by a source catheter and two detector catheters which can be inserted into the tissue at a prescribed spacing using guidance templates. In order to minimize the invasiveness during the interstitial measurements, radiance measurements are still carried out at two fixed SDSs. Instead, radiance measurements at multiple angles can be obtained by rotating the detector fibers. The detecting angles can be chosen from $0°$ to $160°$by the step of $5°$. For optical property recovery, CW radiance measurements at two SDSs and at four angles are employed to extract a set of optical properties where the four angles are randomly selected from four angle ranges of $0°~40°$, $41°~80°$, $81°~120°$ and $121°~160°$, separately. The process of optical property recovery can be repeated many times for statistical analysis.

5. Conclusion

In this paper, we have proposed the incomplete P3 approximation, the P3_in for short, which is the asymptotic part of the solution for the radiance by the P3 approximation. The proposed P3_in provides the simpler formulation than the P3 for recovering the optical properties and the higher degree approximation than the DA for modeling light propagation in the medium with a low albedo. Based on the P3_in, we have developed an analytical formulation method for extracting the optical properties directly from the radiance measurements at only two SDSs and four angles. The developed algorithm can recover ${\mu }_a$ and ${{\mu }^{\prime }}_s$ of the turbid media with albedos as low as 0.5 accurately with the REs within 10.82% and 10.67%, respectively, which demonstrates the effectiveness of the developed recovery algorithm based on the P3_in for recovering the optical properties of the turbid medium with a high absorption or a low scattering.

The ultimate goal of our work is to interstitially determine the optical properties of internal organs such as the prostate. Patient invasiveness limits the clinically available number of sources and detectors employed during interstitial laser treatments. Hence, it is interesting to explore whether ${\mu }_a$ and ${{\mu }^{\prime }}_s$ can be extracted, separately from relative radiance measurements at a single small SDS in the future. Thus, a source fiber and a detector fiber can be encapsulated in a probe. By replacing the laser with a broadband white light source and replacing the PMT with a spectrometer, CW interstitial radiance spectroscopy measurements can be carried out using the probe to determine the chromophore concentrations and the reduced scattering coefficient spectrum of tissue.