Abstract

Interstitial determination of the tissue optical properties is important in biomedicine, especially for interstitial laser therapies. Continuous wave (CW) radiance techniques which examine light from multiple directions have been proposed as minimally invasive methods for determining the optical properties under an interstitial probe arrangement. However, both the fitting algorithm based on the P3 approximation and the analytical method based on the diffusion approximation (DA), which are currently used recovery algorithms, cannot extract the optical properties of tissue with low transport albedos accurately from radiance measurements. In this paper, we proposed an incomplete P3 approximation for the radiance, the P3in for short, which is the asymptotic part of the solution for the P3 approximation. The relative differences between the P3in and the P3 were within 0.48% over a wide range of clinically relevant optical properties for measurements at source detector separations (SDS) from 5 mm to 10 mm and angles from 0° to 160°. Based on the P3in, we developed an analytical method for extracting the optical properties directly using simple expressions constructed from the radiance measurements at only two SDSs and four angles. The developed recovery algorithm was verified by simulated and experimental radiance data. The results show that both the absorption and reduced scattering coefficients were recovered accurately with relative errors within 5.28% and 3.86%, respectively, from the simulated data and with relative errors within 10.82% and 10.67%, respectively, from the experimental data over a wide range of albedos from 0.5 to 0.99. Since the developed P3in-based radiance technique can obtain the optical properties rapidly from the measurements at only two SDSs and four angles, it is expected to be used for in vivo and in situ determination of the optical properties in online treatment planning during laser therapies.

© 2017 Optical Society of America

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 defined as a=μs/(μs+μa), less than ~0.9 due to the lack of inherent uniqueness in fitting algorithms, where μa and μs 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, μeff, 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 μs>>μa 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, 2lt, where lt=1/(μs+μa), 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 2lt are not used in interstitial radiance measurements, generally. Thus, in this paper, we have proposed an incomplete P3 approximation for radiance, the P3in for short, which consists of only the asymptotic part of the solution for the P3. The proposed P3in 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 P3in, 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 P3in 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]:

L(r,θ)=S04πl=0N(2l+1)φl(r)Pl(cosθ),
where L is the radiance, r is the position, θ is the angle between the direction of propagation and the direction of the position r from the origin, S0 is the source power, φl(r) is the l-th order Legendre moment, and Pl is Legendre polynomial of l-th degree. In the case of the P3, the Legendre moments are given by the following set of the ordinary differential equations for l=0,...,3 [22]:
l+12l+1[ddr+l+2r]φl+1(r)+l2l+1[ddrl1r]φl1(r)+σlφl(r)=S0δl0,l=0,...,3,
where σl=μa+μs(1gl)(l=0,...,3) are l-th order absorption coefficients with g and μs indicating the anisotropy factor and scattering coefficient, respectively. By setting φ4(r)=0 in Eq. (2), the solution of the radiance in the P3 for an isotropic point source in an infinite medium is given as [20]:
L(r,θ)=S04πl=03(2l+1)[Chl(ν)Ql(νr)+Dhl(ν+)Ql(ν+r)]Pl(cosθ),
where ν and ν+ are the asymptotic and transient attenuation coefficients, respectively, given as the positive eigenvalues resulting from the third-order expansion of the RTE, Q/(x) are the modified spherical Bessel function of the second kind given by the recurrence relation Ql(x)=Ql2(x)Ql1(x)(2l1)/x for l=2,3with Q0(x)=x1ex and Q1(x)=(x1x2)ex, and hl(ν±)(l=0,...,3) are functions of μa, μs, g and either ν or ν+, and C and D derived from the boundary conditions for infinite media are given by the following equations:
C=(ν)512πμa2σ1(3μaσ1(ν+)2)((ν)2(ν+)2),D=(ν+)512πμa2σ1(3μaσ1(ν)2)((ν+)2(ν)2).
The optical properties of the medium are related to the parameters, hl(ν±), appearing in Eq. (3) as:

h0=1,h1(ν±)=μaν±,h2(ν±)=12+3μaσ12(ν±)2,h3(ν±)=9μaσ114σ3ν±3ν±14σ3.

In the Eq. (3), the terms associated with hl(ν+)Ql(ν+r) represent the transient part and the terms associated with hl(ν)Ql(νr) represent the asymptotic part. As Ql(ν+r) (l=0,...,3) decay rapidly to be zero with the increase in r due to the relatively large value of ν+, the transient part mainly contributes to the total radiance only for the SDSs less than 2lt. The value of ν is relatively small, and Ql(νr) (l=0,...,3) 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,...,3, the values of S0[Chl(ν)Ql(νr)+Dhl(ν+)Ql(ν+r)] in Eq. (3) can be obtained by measuring the radiances at different angles with a fixed r to determine the values of Pl(cosθ). However, the values of S0Chl(ν) and S0Dhl(ν+) cannot be separately obtained by measuring the radiances at different SDSs to determine the values of Ql(ν±r) because ν± appearing in Ql(ν±r) are unknown. Thus hl(ν) and hl(ν+) 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 2lt are usually not used for interstitial radiance measurements. Thus, in this paper, we propose an incomplete P3 approximation for the radiance, the P3in for short, which is the asymptotic part of the solution for the P3. Hereafter, ν is replaced by ν for simplicity, and the radiance by the P3in is given as:

Lin(r,θ)=S0C4πl=03(2l+1)hl(ν)Ql(νr)Pl(cosθ).

2.2 Accuracy of the P3in approximation

In order to evaluate the differences between the P3in and the P3, we examined the influence of the abandoned transient part of the solution in the P3 by the relative differences, ε=|L(r,θ)Lin(r,θ)|/L(r,θ), at different SDSs and angles for different optical properties. In the analysis, the optical properties of prostate with 0.01mm1μa1.0mm1 and 0.5mm1μs4.5mm1 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 ε, εmax, 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 εmax values are very small for angles from 0° to 160° and the εmax values fluctuate strongly for angles from 160° to 180°. It also can be seen that the εmax 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 P3in and the P3 have almost the same values at such angles. The εmax 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 εmax 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 P3in 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 P3in is acceptable for an alternative to the P3 as a light propagation model to recover the optical properties for angles from 0° to 160°.

 figure: Fig. 1

Fig. 1 The maximum relative differences between the P3in and the P3, εmax, in the optical property range of 0.01mm1μa1.0mm1 and 0.5mm1μs4.5mm1 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 P3in approximation

Based on the P3in expressed by Eq. (6), this study formulates an analytical method to recover the optical properties from the radiance measurements, Lm(r,θ), at several pairs of (r,θ). From Eq. (5), it is deduced that the optical properties of the medium can be determined analytically if ν, h1(ν) and h2(ν) were obtained. As the first step for obtaining ν, h1(ν) and h2(ν), the terms of S0Chl(ν)Ql(νr) (l=0,1,2)should be determined. As the terms of S0Chl(ν)Ql(νr) with four different numbers of l in Eq. (6) are simply added, they can be separated from the radiance measurement data of Lm(r,θi) (i=0,...,3) at four angles of θ0, θ1, θ2,θ3with a fixed r, as shown in Fig. 2, by a set of linear equations given as:

l=03S0C4π(2l+1)hl(ν)Ql(νr)Pl(cosθi)=Lm(r,θi),i=0,...,3.
The terms of S0Chl(ν)Ql(νr) (l=0,1,2) are determined as:
S0Ch0(ν)Q0(νr)=4π(α3(r,θ0,θ1,θ2)β13(θ0,θ1,θ3)α3(r,θ0,θ1,θ3)β13(θ0,θ1,θ2))(β03(θ0,θ1,θ2)β13(θ0,θ1,θ3)β03(θ0,θ1,θ3)β13(θ0,θ1,θ2)),
S0Ch1(ν)Q1(νr)=4π(α3(r,θ0,θ1,θ2)β03(θ0,θ1,θ3)α3(r,θ0,θ1,θ3)β03(θ0,θ1,θ2))3(β13(θ0,θ1,θ2)β03(θ0,θ1,θ3)β13(θ0,θ1,θ3)β03(θ0,θ1,θ2)),
S0Ch2(ν)Q2(νr)=4π(α1(r,θ0,θ1,θ2)β31(θ0,θ1,θ3)α1(r,θ0,θ1,θ3)β31(θ0,θ1,θ2))5(β21(θ0,θ1,θ2)β31(θ0,θ1,θ3)β21(θ0,θ1,θ3)β31(θ0,θ1,θ2)),
where αi(r,θ0,θ1,θ2) and βij(θ0,θ1,θ2) are defined as:
αi(r,θ0,θ1,θ2)=(Lm(r,θ0)Pi(cosθ1)Lm(r,θ1)Pi(cosθ0))(Pi1(cosθ0)Pi(cosθ2)Pi1(cosθ2)Pi(cosθ0)),(Lm(r,θ0)Pi(cosθ2)Lm(r,θ2)Pi(cosθ0))(Pi1(cosθ0)Pi(cosθ1)Pi1(cosθ1)Pi(cosθ0))
βij(θ0,θ1,θ2)=(Pi(cosθ0)Pj(cosθ1)Pi(cosθ1)Pj(cosθ0))(Pj1(cosθ0)Pj(cosθ1)Pj1(cosθ1)Pj(cosθ0)).(Pi(cosθ0)Pj(cosθ2)Pi(cosθ2)Pj(cosθ0))(Pj1(cosθ0)Pj(cosθ2)Pj1(cosθ2)Pj(cosθ0))
Replacing θ2 by θ3 in the αi(r,θ0,θ1,θ2) and βij(θ0,θ1,θ2) provides the equations for αi(r,θ0,θ1,θ3) and βij(θ0,θ1,θ3). For simplicity of expression, we write the right hand sides of Eqs. (8a), (8b) and (8c) as f0(r,θ0,θ1,θ2,θ3), f1(r,θ0,θ1,θ2,θ3), and f2(r,θ0,θ1,θ2,θ3), respectively.

 figure: Fig. 2

Fig. 2 Geometry for measurements of the radiances, Lm(r,θ), at SDSs of r and r and angles of θ0, θ1, θ2 and θ3.

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As the second step for obtaining ν, h1(ν) and h2(ν), the ratio of f0(r,θ0,θ1,θ2,θ3) to f0(r,θ0,θ1,θ2,θ3) is taken from the radiance measurements at two SDSs, r and r, as shown in Fig. 2, to delete S0 for avoiding the measurements of the absolute radiances:

h0(ν)Q0(νr)h0(ν)Q0(νr)=f0(r,θ0,θ1,θ2,θ3)f0(r,θ0,θ1,θ2,θ3).
Then, by substituting the definitions of h0(ν) and Q0(νr) into Eq. (10) and by simplifying the expression, ν is obtained as:
ν=ln[f0(r,θ0,θ1,θ2,θ3)f0(r,θ0,θ1,θ2,θ3)rr]/(rr).
For obtaining h1(ν) and h2(ν), the ratios of fl(r,θ0,θ1,θ2,θ3) (l=1,2) and f0(r,θ0,θ1,θ2,θ3) are taken using Eqs. (8a), (8b) and (8c) as:
hl(ν)Ql(νr)h0(ν)Q0(νr)=fl(r,θ0,θ1,θ2,θ3)f0(r,θ0,θ1,θ2,θ3),l=1,2.
The values of Ql(νr) (l=0,1,2) in Eq. (12) are known since ν has been obtained by Eq. (11). Thus h1(ν) and h2(ν) are determined as the following:

hl(ν)=Q0(νr)Ql(νr)fl(r,θ0,θ1,θ2,θ3)f0(r,θ0,θ1,θ2,θ3),l=1,2.

Now ν, h1(ν) and h2(ν) are obtained, and the optical properties can be determined from Eq. (5) as the following:

μa=h1(ν)ν,μs=(2h2(ν)+1)ν3h2(ν)h1(ν)ν.

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 μa and μs 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 P3in 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 μs as typical and low scattering cases, respectively, while μa 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 P3in and the P3 are relative remarkable. The SDS of r 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 P3in [13]. The results are presented for the recovery algorithms based on both the DA and the P3in.

We evaluated the performance of the developed algorithm based on the P3in in the case of μs=1.0mm1 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 μa and that the accuracy is improved only a little with the increase in r. Just in the case of μa0.1mm1, i.e., a0.91, both μa and μs are recovered better with the relative errors (REs) less than 14.75% where the REs are defined as |μa_rμa_t|/μa_t, |μs_rμs_t|/μs_t for μa and μs, respectively, with μa_r, μa_t, μs_r and μs_t indicating the recovered μa, the true μa, the recovered μs and the true μs. 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 P3in is able to recover both μa and μs 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 μa and μs based on the P3in are less than 1.30% and 0.96%, respectively, for the albedos ranging from 0.67 to 0.99.

 figure: Fig. 3

Fig. 3 Recovered optical properties based on the DA and the P3in from the simulated radiance data calculated by the P25 in the case of typical scattering. (a) is for μa and (b) is for μs. 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 μs=0.5mm1 which may be encountered during interstitial laser therapies [18, 19]. Figures 4(a) and 4(b) show that the recovered μa and μs based on the P3in are also very close to true values, whereas the recovered μa and μs based on the DA exhibit significant errors. The recovery algorithm based on the DA disclosed a significant breakdown in recovering μa with REs larger than 20.72%, although μs is recovered with the REs within 9.54% for the true μa less than 0.05 mm−1. In contrast to the DA, the REs of the recovered μa and μs based on the P3in 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 P3in is able to recover the optical properties accurately in the cases of both typical and low scattering.

 figure: Fig. 4

Fig. 4 Recovered optical properties based on the DA and the P3in from the simulated radiance data calculated by the P25 in the case of low scattering. (a) is for μa and (b) is for μs. 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 850μm 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 500μm right-angle prism to a cleaved plane cut 475μm 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.

 figure: Fig. 5

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

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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 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 μs of 1 mm−1 and 0.5 mm−1 and μa 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 P3in was evaluated by the experimental radiance data in the case of typical scattering of μs=1.0mm1 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 μa and that only for μa0.1mm1, i.e., a0.91, both μa and μs were recovered better with the REs within 16.12%. The results using the recovery algorithm based on the P3in 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 μa and μs based on the P3in are less than 10.39% and 9.86%, respectively, for the albedos ranging from 0.67 to 0.99.

 figure: Fig. 6

Fig. 6 Recovered optical properties based on the DA and the P3in from the phantom measurements in the case of typical scattering. (a) is for μa and (b) is for μs. The insets show the REs of the recovered optical properties.

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The developed recovery algorithm based on the P3in was also evaluated by the experimental radiance data in the case of low scattering of μs=0.5mm1 in Figs. 7(a) and 7(b). It is observed that the recovery algorithm based on the DA disclosed a breakdown in recovering μa with the REs larger than 17.07%. μs were recovered finely only for the true μa less than 0.05mm1 with the REs within 8.15%. In contrast to the DA, the REs of the recovered μa and μs based on the P3in are less than 10.82% and 10.67%, respectively, for the albedos ranging from 0.5 to 0.98.

 figure: Fig. 7

Fig. 7 Recovered optical properties obtained from the phantom measurements in the case of low scattering. (a) is for μa and (b) is for μs. 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 a0.91, whereas the recovery algorithm based on the P3in 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 P3in greatly improved the accuracy of the recovered optical properties from simulated and experimental radiances when compared with the DA, the P5in 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 P3in, the solution of the P5in is the asymptotic part of the solution for the radiance by the P5 approximation. The recovery algorithm based on the P5in 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 P3in and the P5in were evaluated by the simulated radiance data calculated by the P25 for the optical property range of 0.01mm1μa1.0mm1 and 0.5mm1μs4.5mm1. r and r were chosen as 5 mm and 6 mm, respectively. Four angles of 40°, 80°, 120° and 160° for the P3in and six angles of 40°, 70°, 90°, 120°, 140° and 160° for the P5in were chosen.

Figures 8(a)-8(d) plot the REs of the recovered optical properties based on the P3in and the P5in, respectively. The REs of the recovered μa and μs based on the P3in are less than 6.59% and 7.49%, respectively, whereas those based on the P5in are less than 4.19% and 4.62%, respectively. Although more radiance measurements are required for optical property recovery based on the P5in, minor improvement in the accuracy of the recovered optical properties by the P5in was obtained when compared with the P3in.

 figure: Fig. 8

Fig. 8 REs of the recovered optical properties for 0.01mm1μa1.0mm1 and 0.5mm1μs4.5mm1. (a) and (b) are based on the P3in. (c) and (d) are based on the P5in. (a) and (c) are for μa. (b) and (d) are for μs.

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Here, we present a recovery algorithm based on the P3in that is capable of recovering μa and μs 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 P3in were comparable to the errors found in previous studies of interstitial optical property recovery. Zhu et al. demonstrated the recovery of μa and μs 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 μa and μs 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 μeff 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 μa and μs directly [13]. It indicates that the developed algorithm based on the P3in 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 P3in-based radiance technique has three advantages over other CW techniques of interstitial optical property determination [11, 12, 16]. Firstly, for the P3in-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 P3in. 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 P3in for short, which is the asymptotic part of the solution for the radiance by the P3 approximation. The proposed P3in 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 P3in, 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 μa and μs 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 P3in 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 μa and μs 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.

Funding

The authors acknowledge the funding supports from the National Natural Science Foundation of China (81401453, 81371602, 61475115, 61475116, 61575140, 81571723, 81671728); Tianjin Municipal Government of China (15JCZDJC31800, 16JCZDJC31200, 17JCZDJC32700,17JCQNJC12700); the 111 Project (B07014).

Acknowledgments

The authors acknowledge the insightful suggestions and revisions given by Prof. Yukio Yamada from University of Electro-Communications, Japan.

References and links

1. G. Shafirstein, D. Bellnier, E. Oakley, S. Hamilton, M. Potasek, K. Beeson, and E. Parilov, “Interstitial photodynamic therapy-a focused review,” Cancers (Basel) 9(2), 12 (2017). [PubMed]  

2. T. M. Baran, “Recovery of optical properties using interstitial cylindrical diffusers as source and detector fibers,” J. Biomed. Opt. 21(7), 77001 (2016). [PubMed]  

3. V. K. Nagarajan and B. Yu, “Monitoring of tissue optical properties during thermal coagulation of ex vivo tissues,” Lasers Surg. Med. 48(7), 686–694 (2016). [PubMed]  

4. R. Nachabé, D. J. Evers, B. H. Hendriks, G. W. Lucassen, M. van der Voort, J. Wesseling, and T. J. Ruers, “Effect of bile absorption coefficients on the estimation of liver tissue optical properties and related implications in discriminating healthy and tumorous samples,” Biomed. Opt. Express 2(3), 600–614 (2011). [PubMed]  

5. Y. Pu, W. Wang, M. Al-Rubaiee, S. K. Gayen, and M. Xu, “Determination of optical coefficients and fractal dimensional parameters of cancerous and normal prostate tissues,” Appl. Spectrosc. 66(7), 828–834 (2012). [PubMed]  

6. R. Watté, B. Aernouts, R. Van Beers, and W. Saeys, “Robust metamodel-based inverse estimation of bulk optical properties of turbid media from spatially resolved diffuse reflectance measurements,” Opt. Express 23(21), 27880–27898 (2015). [PubMed]  

7. S.-Y. Tzeng, J.-Y. Guo, C.-C. Yang, C.-K. Hsu, H.-J. Huang, S.-J. Chou, C.-H. Hwang, and S.-H. Tseng, “Portable handheld diffuse reflectance spectroscopy system for clinical evaluation of skin: a pilot study in psoriasis patients,” Biomed. Opt. Express 7(2), 616–628 (2016). [PubMed]  

8. P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and an empirical light transport model during endoscopy,” J. Biomed. Opt. 10(3), 034018 (2005). [PubMed]  

9. T. M. Baran, M. C. Fenn, and T. H. Foster, “Determination of optical properties by interstitial white light spectroscopy using a custom fiber optic probe,” J. Biomed. Opt. 18(10), 107007 (2013). [PubMed]  

10. T. M. Baran, J. D. Wilson, S. Mitra, J. L. Yao, E. M. Messing, D. L. Waldman, and T. H. Foster, “Optical property measurements establish the feasibility of photodynamic therapy as a minimally invasive intervention for tumors of the kidney,” J. Biomed. Opt. 17(9), 98002 (2012). [PubMed]  

11. A. Dimofte, J. C. Finlay, and T. C. Zhu, “A method for determination of the absorption and scattering properties interstitially in turbid media,” Phys. Med. Biol. 50(10), 2291–2311 (2005). [PubMed]  

12. T. C. Zhu, J. C. Finlay, and S. M. Hahn, “Determination of the distribution of light, optical properties, drug concentration, and tissue oxygenation in-vivo in human prostate during motexafin lutetium-mediated photodynamic therapy,” J. Photochem. Photobiol. B 79(3), 231–241 (2005). [PubMed]  

13. S. Grabtchak and W. M. Whelan, “Separation of absorption and scattering properties of turbid media using relative spectrally resolved cw radiance measurements,” Biomed. Opt. Express 3(10), 2371–2380 (2012). [PubMed]  

14. D. J. Dickey, R. B. Moore, D. C. Rayner, and J. Tulip, “Light dosimetry using the P3 approximation,” Phys. Med. Biol. 46(9), 2359–2370 (2001). [PubMed]  

15. L. C. L. Chin, W. M. Whelan, and I. A. Vitkin, “Information content of point radiance measurements in turbid media: implications for interstitial optical property quantification,” Appl. Opt. 45(9), 2101–2114 (2006). [PubMed]  

16. L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation, and sensitivity analysis,” J. Biomed. Opt. 12(6), 064027 (2007). [PubMed]  

17. S. Grabtchak, L. G. Montgomery, and W. M. Whelan, “Feasibility of interstitial near-infrared radiance spectroscopy platform for ex vivo canine prostate studies: optical properties extraction, hemoglobin and water concentration, and gold nanoparticles detection,” J. Biomed. Opt. 19(5), 057003 (2014). [PubMed]  

18. D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE. J. Sel. Top. Quant. 16(4), 715–729 (2010).

19. T. Svensson, E. Alerstam, M. Einarsdóttír, K. Svanberg, and S. Andersson-Engels, “Towards accurate in vivo spectroscopy of the human prostate,” J. Biophotonics 1(3), 200–203 (2008). [PubMed]  

20. E. L. Hull and T. H. Foster, “Steady-state reflectance spectroscopy in the P3 approximation,” J. Opt. Soc. Am. A 18(3), 584–599 (2001).

21. A. Liemert and A. Kienle, “Analytical Green’s function of the radiative transfer radiance for the infinite medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(3 Pt 2), 036605 (2011). [PubMed]  

22. T. Khan and A. Thomas, “Comparison of PN or spherical harmonics approximation for scattering media with spatially varying and spatially constant refractive indices,” Opt. Commun. 255(1), 130–166 (2005).

23. J. Li and T. C. Zhu, “Determination of in vivo light fluence distribution in a heterogeneous prostate during photodynamic therapy,” Phys. Med. Biol. 53(8), 2103–2114 (2008). [PubMed]  

24. S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt. 17(6), 067007 (2012). [PubMed]  

25. H. Xu and M. S. Patterson, “Determination of the optical properties of tissue-simulating phantoms from interstitial frequency domain measurements of relative fluence and phase difference,” Opt. Express 14(14), 6485–6501 (2006). [PubMed]  

26. O. Barajas, Å. M. Ballangrud, G. G. Miller, R. B. Moore, and J. Tulip, “Monte Carlo modelling of angular radiance in tissue phantoms and human prostate: PDT light dosimetry,” Phys. Med. Biol. 42(9), 1675–1687 (1997). [PubMed]  

27. H. J. van Staveren, C. J. Moes, J. van Marie, S. A. Prahl, and M. J. van Gemert, “Light scattering in Intralipid-10% in the wavelength range of 400-1100 nm,” Appl. Opt. 30(31), 4507–4514 (1991). [PubMed]  

References

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  1. G. Shafirstein, D. Bellnier, E. Oakley, S. Hamilton, M. Potasek, K. Beeson, and E. Parilov, “Interstitial photodynamic therapy-a focused review,” Cancers (Basel) 9(2), 12 (2017).
    [PubMed]
  2. T. M. Baran, “Recovery of optical properties using interstitial cylindrical diffusers as source and detector fibers,” J. Biomed. Opt. 21(7), 77001 (2016).
    [PubMed]
  3. V. K. Nagarajan and B. Yu, “Monitoring of tissue optical properties during thermal coagulation of ex vivo tissues,” Lasers Surg. Med. 48(7), 686–694 (2016).
    [PubMed]
  4. R. Nachabé, D. J. Evers, B. H. Hendriks, G. W. Lucassen, M. van der Voort, J. Wesseling, and T. J. Ruers, “Effect of bile absorption coefficients on the estimation of liver tissue optical properties and related implications in discriminating healthy and tumorous samples,” Biomed. Opt. Express 2(3), 600–614 (2011).
    [PubMed]
  5. Y. Pu, W. Wang, M. Al-Rubaiee, S. K. Gayen, and M. Xu, “Determination of optical coefficients and fractal dimensional parameters of cancerous and normal prostate tissues,” Appl. Spectrosc. 66(7), 828–834 (2012).
    [PubMed]
  6. R. Watté, B. Aernouts, R. Van Beers, and W. Saeys, “Robust metamodel-based inverse estimation of bulk optical properties of turbid media from spatially resolved diffuse reflectance measurements,” Opt. Express 23(21), 27880–27898 (2015).
    [PubMed]
  7. S.-Y. Tzeng, J.-Y. Guo, C.-C. Yang, C.-K. Hsu, H.-J. Huang, S.-J. Chou, C.-H. Hwang, and S.-H. Tseng, “Portable handheld diffuse reflectance spectroscopy system for clinical evaluation of skin: a pilot study in psoriasis patients,” Biomed. Opt. Express 7(2), 616–628 (2016).
    [PubMed]
  8. P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and an empirical light transport model during endoscopy,” J. Biomed. Opt. 10(3), 034018 (2005).
    [PubMed]
  9. T. M. Baran, M. C. Fenn, and T. H. Foster, “Determination of optical properties by interstitial white light spectroscopy using a custom fiber optic probe,” J. Biomed. Opt. 18(10), 107007 (2013).
    [PubMed]
  10. T. M. Baran, J. D. Wilson, S. Mitra, J. L. Yao, E. M. Messing, D. L. Waldman, and T. H. Foster, “Optical property measurements establish the feasibility of photodynamic therapy as a minimally invasive intervention for tumors of the kidney,” J. Biomed. Opt. 17(9), 98002 (2012).
    [PubMed]
  11. A. Dimofte, J. C. Finlay, and T. C. Zhu, “A method for determination of the absorption and scattering properties interstitially in turbid media,” Phys. Med. Biol. 50(10), 2291–2311 (2005).
    [PubMed]
  12. T. C. Zhu, J. C. Finlay, and S. M. Hahn, “Determination of the distribution of light, optical properties, drug concentration, and tissue oxygenation in-vivo in human prostate during motexafin lutetium-mediated photodynamic therapy,” J. Photochem. Photobiol. B 79(3), 231–241 (2005).
    [PubMed]
  13. S. Grabtchak and W. M. Whelan, “Separation of absorption and scattering properties of turbid media using relative spectrally resolved cw radiance measurements,” Biomed. Opt. Express 3(10), 2371–2380 (2012).
    [PubMed]
  14. D. J. Dickey, R. B. Moore, D. C. Rayner, and J. Tulip, “Light dosimetry using the P3 approximation,” Phys. Med. Biol. 46(9), 2359–2370 (2001).
    [PubMed]
  15. L. C. L. Chin, W. M. Whelan, and I. A. Vitkin, “Information content of point radiance measurements in turbid media: implications for interstitial optical property quantification,” Appl. Opt. 45(9), 2101–2114 (2006).
    [PubMed]
  16. L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation, and sensitivity analysis,” J. Biomed. Opt. 12(6), 064027 (2007).
    [PubMed]
  17. S. Grabtchak, L. G. Montgomery, and W. M. Whelan, “Feasibility of interstitial near-infrared radiance spectroscopy platform for ex vivo canine prostate studies: optical properties extraction, hemoglobin and water concentration, and gold nanoparticles detection,” J. Biomed. Opt. 19(5), 057003 (2014).
    [PubMed]
  18. D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE. J. Sel. Top. Quant. 16(4), 715–729 (2010).
  19. T. Svensson, E. Alerstam, M. Einarsdóttír, K. Svanberg, and S. Andersson-Engels, “Towards accurate in vivo spectroscopy of the human prostate,” J. Biophotonics 1(3), 200–203 (2008).
    [PubMed]
  20. E. L. Hull and T. H. Foster, “Steady-state reflectance spectroscopy in the P3 approximation,” J. Opt. Soc. Am. A 18(3), 584–599 (2001).
  21. A. Liemert and A. Kienle, “Analytical Green’s function of the radiative transfer radiance for the infinite medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(3 Pt 2), 036605 (2011).
    [PubMed]
  22. T. Khan and A. Thomas, “Comparison of PN or spherical harmonics approximation for scattering media with spatially varying and spatially constant refractive indices,” Opt. Commun. 255(1), 130–166 (2005).
  23. J. Li and T. C. Zhu, “Determination of in vivo light fluence distribution in a heterogeneous prostate during photodynamic therapy,” Phys. Med. Biol. 53(8), 2103–2114 (2008).
    [PubMed]
  24. S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt. 17(6), 067007 (2012).
    [PubMed]
  25. H. Xu and M. S. Patterson, “Determination of the optical properties of tissue-simulating phantoms from interstitial frequency domain measurements of relative fluence and phase difference,” Opt. Express 14(14), 6485–6501 (2006).
    [PubMed]
  26. O. Barajas, Å. M. Ballangrud, G. G. Miller, R. B. Moore, and J. Tulip, “Monte Carlo modelling of angular radiance in tissue phantoms and human prostate: PDT light dosimetry,” Phys. Med. Biol. 42(9), 1675–1687 (1997).
    [PubMed]
  27. H. J. van Staveren, C. J. Moes, J. van Marie, S. A. Prahl, and M. J. van Gemert, “Light scattering in Intralipid-10% in the wavelength range of 400-1100 nm,” Appl. Opt. 30(31), 4507–4514 (1991).
    [PubMed]

2017 (1)

G. Shafirstein, D. Bellnier, E. Oakley, S. Hamilton, M. Potasek, K. Beeson, and E. Parilov, “Interstitial photodynamic therapy-a focused review,” Cancers (Basel) 9(2), 12 (2017).
[PubMed]

2016 (3)

T. M. Baran, “Recovery of optical properties using interstitial cylindrical diffusers as source and detector fibers,” J. Biomed. Opt. 21(7), 77001 (2016).
[PubMed]

V. K. Nagarajan and B. Yu, “Monitoring of tissue optical properties during thermal coagulation of ex vivo tissues,” Lasers Surg. Med. 48(7), 686–694 (2016).
[PubMed]

S.-Y. Tzeng, J.-Y. Guo, C.-C. Yang, C.-K. Hsu, H.-J. Huang, S.-J. Chou, C.-H. Hwang, and S.-H. Tseng, “Portable handheld diffuse reflectance spectroscopy system for clinical evaluation of skin: a pilot study in psoriasis patients,” Biomed. Opt. Express 7(2), 616–628 (2016).
[PubMed]

2015 (1)

2014 (1)

S. Grabtchak, L. G. Montgomery, and W. M. Whelan, “Feasibility of interstitial near-infrared radiance spectroscopy platform for ex vivo canine prostate studies: optical properties extraction, hemoglobin and water concentration, and gold nanoparticles detection,” J. Biomed. Opt. 19(5), 057003 (2014).
[PubMed]

2013 (1)

T. M. Baran, M. C. Fenn, and T. H. Foster, “Determination of optical properties by interstitial white light spectroscopy using a custom fiber optic probe,” J. Biomed. Opt. 18(10), 107007 (2013).
[PubMed]

2012 (4)

T. M. Baran, J. D. Wilson, S. Mitra, J. L. Yao, E. M. Messing, D. L. Waldman, and T. H. Foster, “Optical property measurements establish the feasibility of photodynamic therapy as a minimally invasive intervention for tumors of the kidney,” J. Biomed. Opt. 17(9), 98002 (2012).
[PubMed]

S. Grabtchak and W. M. Whelan, “Separation of absorption and scattering properties of turbid media using relative spectrally resolved cw radiance measurements,” Biomed. Opt. Express 3(10), 2371–2380 (2012).
[PubMed]

Y. Pu, W. Wang, M. Al-Rubaiee, S. K. Gayen, and M. Xu, “Determination of optical coefficients and fractal dimensional parameters of cancerous and normal prostate tissues,” Appl. Spectrosc. 66(7), 828–834 (2012).
[PubMed]

S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt. 17(6), 067007 (2012).
[PubMed]

2011 (2)

2010 (1)

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE. J. Sel. Top. Quant. 16(4), 715–729 (2010).

2008 (2)

T. Svensson, E. Alerstam, M. Einarsdóttír, K. Svanberg, and S. Andersson-Engels, “Towards accurate in vivo spectroscopy of the human prostate,” J. Biophotonics 1(3), 200–203 (2008).
[PubMed]

J. Li and T. C. Zhu, “Determination of in vivo light fluence distribution in a heterogeneous prostate during photodynamic therapy,” Phys. Med. Biol. 53(8), 2103–2114 (2008).
[PubMed]

2007 (1)

L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation, and sensitivity analysis,” J. Biomed. Opt. 12(6), 064027 (2007).
[PubMed]

2006 (2)

2005 (4)

T. Khan and A. Thomas, “Comparison of PN or spherical harmonics approximation for scattering media with spatially varying and spatially constant refractive indices,” Opt. Commun. 255(1), 130–166 (2005).

A. Dimofte, J. C. Finlay, and T. C. Zhu, “A method for determination of the absorption and scattering properties interstitially in turbid media,” Phys. Med. Biol. 50(10), 2291–2311 (2005).
[PubMed]

T. C. Zhu, J. C. Finlay, and S. M. Hahn, “Determination of the distribution of light, optical properties, drug concentration, and tissue oxygenation in-vivo in human prostate during motexafin lutetium-mediated photodynamic therapy,” J. Photochem. Photobiol. B 79(3), 231–241 (2005).
[PubMed]

P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and an empirical light transport model during endoscopy,” J. Biomed. Opt. 10(3), 034018 (2005).
[PubMed]

2001 (2)

D. J. Dickey, R. B. Moore, D. C. Rayner, and J. Tulip, “Light dosimetry using the P3 approximation,” Phys. Med. Biol. 46(9), 2359–2370 (2001).
[PubMed]

E. L. Hull and T. H. Foster, “Steady-state reflectance spectroscopy in the P3 approximation,” J. Opt. Soc. Am. A 18(3), 584–599 (2001).

1997 (1)

O. Barajas, Å. M. Ballangrud, G. G. Miller, R. B. Moore, and J. Tulip, “Monte Carlo modelling of angular radiance in tissue phantoms and human prostate: PDT light dosimetry,” Phys. Med. Biol. 42(9), 1675–1687 (1997).
[PubMed]

1991 (1)

Aernouts, B.

Alerstam, E.

T. Svensson, E. Alerstam, M. Einarsdóttír, K. Svanberg, and S. Andersson-Engels, “Towards accurate in vivo spectroscopy of the human prostate,” J. Biophotonics 1(3), 200–203 (2008).
[PubMed]

Al-Rubaiee, M.

Andersson-Engels, S.

T. Svensson, E. Alerstam, M. Einarsdóttír, K. Svanberg, and S. Andersson-Engels, “Towards accurate in vivo spectroscopy of the human prostate,” J. Biophotonics 1(3), 200–203 (2008).
[PubMed]

Ballangrud, Å. M.

O. Barajas, Å. M. Ballangrud, G. G. Miller, R. B. Moore, and J. Tulip, “Monte Carlo modelling of angular radiance in tissue phantoms and human prostate: PDT light dosimetry,” Phys. Med. Biol. 42(9), 1675–1687 (1997).
[PubMed]

Barajas, O.

O. Barajas, Å. M. Ballangrud, G. G. Miller, R. B. Moore, and J. Tulip, “Monte Carlo modelling of angular radiance in tissue phantoms and human prostate: PDT light dosimetry,” Phys. Med. Biol. 42(9), 1675–1687 (1997).
[PubMed]

Baran, T. M.

T. M. Baran, “Recovery of optical properties using interstitial cylindrical diffusers as source and detector fibers,” J. Biomed. Opt. 21(7), 77001 (2016).
[PubMed]

T. M. Baran, M. C. Fenn, and T. H. Foster, “Determination of optical properties by interstitial white light spectroscopy using a custom fiber optic probe,” J. Biomed. Opt. 18(10), 107007 (2013).
[PubMed]

T. M. Baran, J. D. Wilson, S. Mitra, J. L. Yao, E. M. Messing, D. L. Waldman, and T. H. Foster, “Optical property measurements establish the feasibility of photodynamic therapy as a minimally invasive intervention for tumors of the kidney,” J. Biomed. Opt. 17(9), 98002 (2012).
[PubMed]

Bargo, P. R.

P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and an empirical light transport model during endoscopy,” J. Biomed. Opt. 10(3), 034018 (2005).
[PubMed]

Bartels, K. E.

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE. J. Sel. Top. Quant. 16(4), 715–729 (2010).

Beeson, K.

G. Shafirstein, D. Bellnier, E. Oakley, S. Hamilton, M. Potasek, K. Beeson, and E. Parilov, “Interstitial photodynamic therapy-a focused review,” Cancers (Basel) 9(2), 12 (2017).
[PubMed]

Bellnier, D.

G. Shafirstein, D. Bellnier, E. Oakley, S. Hamilton, M. Potasek, K. Beeson, and E. Parilov, “Interstitial photodynamic therapy-a focused review,” Cancers (Basel) 9(2), 12 (2017).
[PubMed]

Blair, G.

P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and an empirical light transport model during endoscopy,” J. Biomed. Opt. 10(3), 034018 (2005).
[PubMed]

Bunting, C. F.

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE. J. Sel. Top. Quant. 16(4), 715–729 (2010).

Chin, L. C. L.

L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation, and sensitivity analysis,” J. Biomed. Opt. 12(6), 064027 (2007).
[PubMed]

L. C. L. Chin, W. M. Whelan, and I. A. Vitkin, “Information content of point radiance measurements in turbid media: implications for interstitial optical property quantification,” Appl. Opt. 45(9), 2101–2114 (2006).
[PubMed]

Chou, S.-J.

Dickey, D. J.

D. J. Dickey, R. B. Moore, D. C. Rayner, and J. Tulip, “Light dosimetry using the P3 approximation,” Phys. Med. Biol. 46(9), 2359–2370 (2001).
[PubMed]

Dimofte, A.

A. Dimofte, J. C. Finlay, and T. C. Zhu, “A method for determination of the absorption and scattering properties interstitially in turbid media,” Phys. Med. Biol. 50(10), 2291–2311 (2005).
[PubMed]

Einarsdóttír, M.

T. Svensson, E. Alerstam, M. Einarsdóttír, K. Svanberg, and S. Andersson-Engels, “Towards accurate in vivo spectroscopy of the human prostate,” J. Biophotonics 1(3), 200–203 (2008).
[PubMed]

Evers, D. J.

Fenn, M. C.

T. M. Baran, M. C. Fenn, and T. H. Foster, “Determination of optical properties by interstitial white light spectroscopy using a custom fiber optic probe,” J. Biomed. Opt. 18(10), 107007 (2013).
[PubMed]

Finlay, J. C.

A. Dimofte, J. C. Finlay, and T. C. Zhu, “A method for determination of the absorption and scattering properties interstitially in turbid media,” Phys. Med. Biol. 50(10), 2291–2311 (2005).
[PubMed]

T. C. Zhu, J. C. Finlay, and S. M. Hahn, “Determination of the distribution of light, optical properties, drug concentration, and tissue oxygenation in-vivo in human prostate during motexafin lutetium-mediated photodynamic therapy,” J. Photochem. Photobiol. B 79(3), 231–241 (2005).
[PubMed]

Foschum, F.

S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt. 17(6), 067007 (2012).
[PubMed]

Foster, T. H.

T. M. Baran, M. C. Fenn, and T. H. Foster, “Determination of optical properties by interstitial white light spectroscopy using a custom fiber optic probe,” J. Biomed. Opt. 18(10), 107007 (2013).
[PubMed]

T. M. Baran, J. D. Wilson, S. Mitra, J. L. Yao, E. M. Messing, D. L. Waldman, and T. H. Foster, “Optical property measurements establish the feasibility of photodynamic therapy as a minimally invasive intervention for tumors of the kidney,” J. Biomed. Opt. 17(9), 98002 (2012).
[PubMed]

E. L. Hull and T. H. Foster, “Steady-state reflectance spectroscopy in the P3 approximation,” J. Opt. Soc. Am. A 18(3), 584–599 (2001).

Gayen, S. K.

Goodell, T. T.

P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and an empirical light transport model during endoscopy,” J. Biomed. Opt. 10(3), 034018 (2005).
[PubMed]

Grabtchak, S.

S. Grabtchak, L. G. Montgomery, and W. M. Whelan, “Feasibility of interstitial near-infrared radiance spectroscopy platform for ex vivo canine prostate studies: optical properties extraction, hemoglobin and water concentration, and gold nanoparticles detection,” J. Biomed. Opt. 19(5), 057003 (2014).
[PubMed]

S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt. 17(6), 067007 (2012).
[PubMed]

S. Grabtchak and W. M. Whelan, “Separation of absorption and scattering properties of turbid media using relative spectrally resolved cw radiance measurements,” Biomed. Opt. Express 3(10), 2371–2380 (2012).
[PubMed]

Guo, J.-Y.

Hahn, S. M.

T. C. Zhu, J. C. Finlay, and S. M. Hahn, “Determination of the distribution of light, optical properties, drug concentration, and tissue oxygenation in-vivo in human prostate during motexafin lutetium-mediated photodynamic therapy,” J. Photochem. Photobiol. B 79(3), 231–241 (2005).
[PubMed]

Hamilton, S.

G. Shafirstein, D. Bellnier, E. Oakley, S. Hamilton, M. Potasek, K. Beeson, and E. Parilov, “Interstitial photodynamic therapy-a focused review,” Cancers (Basel) 9(2), 12 (2017).
[PubMed]

Hendriks, B. H.

Holyoak, G. R.

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE. J. Sel. Top. Quant. 16(4), 715–729 (2010).

Hsu, C.-K.

Huang, H.-J.

Hull, E. L.

Hwang, C.-H.

Jacques, S. L.

P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and an empirical light transport model during endoscopy,” J. Biomed. Opt. 10(3), 034018 (2005).
[PubMed]

Jiang, Z.

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE. J. Sel. Top. Quant. 16(4), 715–729 (2010).

Khan, T.

T. Khan and A. Thomas, “Comparison of PN or spherical harmonics approximation for scattering media with spatially varying and spatially constant refractive indices,” Opt. Commun. 255(1), 130–166 (2005).

Kienle, A.

S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt. 17(6), 067007 (2012).
[PubMed]

A. Liemert and A. Kienle, “Analytical Green’s function of the radiative transfer radiance for the infinite medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(3 Pt 2), 036605 (2011).
[PubMed]

Koval, G.

P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and an empirical light transport model during endoscopy,” J. Biomed. Opt. 10(3), 034018 (2005).
[PubMed]

Li, J.

J. Li and T. C. Zhu, “Determination of in vivo light fluence distribution in a heterogeneous prostate during photodynamic therapy,” Phys. Med. Biol. 53(8), 2103–2114 (2008).
[PubMed]

Liemert, A.

S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt. 17(6), 067007 (2012).
[PubMed]

A. Liemert and A. Kienle, “Analytical Green’s function of the radiative transfer radiance for the infinite medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(3 Pt 2), 036605 (2011).
[PubMed]

Lucassen, G. W.

Messing, E. M.

T. M. Baran, J. D. Wilson, S. Mitra, J. L. Yao, E. M. Messing, D. L. Waldman, and T. H. Foster, “Optical property measurements establish the feasibility of photodynamic therapy as a minimally invasive intervention for tumors of the kidney,” J. Biomed. Opt. 17(9), 98002 (2012).
[PubMed]

Miller, G. G.

O. Barajas, Å. M. Ballangrud, G. G. Miller, R. B. Moore, and J. Tulip, “Monte Carlo modelling of angular radiance in tissue phantoms and human prostate: PDT light dosimetry,” Phys. Med. Biol. 42(9), 1675–1687 (1997).
[PubMed]

Mitra, S.

T. M. Baran, J. D. Wilson, S. Mitra, J. L. Yao, E. M. Messing, D. L. Waldman, and T. H. Foster, “Optical property measurements establish the feasibility of photodynamic therapy as a minimally invasive intervention for tumors of the kidney,” J. Biomed. Opt. 17(9), 98002 (2012).
[PubMed]

Moes, C. J.

Montgomery, L. G.

S. Grabtchak, L. G. Montgomery, and W. M. Whelan, “Feasibility of interstitial near-infrared radiance spectroscopy platform for ex vivo canine prostate studies: optical properties extraction, hemoglobin and water concentration, and gold nanoparticles detection,” J. Biomed. Opt. 19(5), 057003 (2014).
[PubMed]

Moore, R. B.

D. J. Dickey, R. B. Moore, D. C. Rayner, and J. Tulip, “Light dosimetry using the P3 approximation,” Phys. Med. Biol. 46(9), 2359–2370 (2001).
[PubMed]

O. Barajas, Å. M. Ballangrud, G. G. Miller, R. B. Moore, and J. Tulip, “Monte Carlo modelling of angular radiance in tissue phantoms and human prostate: PDT light dosimetry,” Phys. Med. Biol. 42(9), 1675–1687 (1997).
[PubMed]

Nachabé, R.

Nagarajan, V. K.

V. K. Nagarajan and B. Yu, “Monitoring of tissue optical properties during thermal coagulation of ex vivo tissues,” Lasers Surg. Med. 48(7), 686–694 (2016).
[PubMed]

Oakley, E.

G. Shafirstein, D. Bellnier, E. Oakley, S. Hamilton, M. Potasek, K. Beeson, and E. Parilov, “Interstitial photodynamic therapy-a focused review,” Cancers (Basel) 9(2), 12 (2017).
[PubMed]

Palmer, T. J.

S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt. 17(6), 067007 (2012).
[PubMed]

Parilov, E.

G. Shafirstein, D. Bellnier, E. Oakley, S. Hamilton, M. Potasek, K. Beeson, and E. Parilov, “Interstitial photodynamic therapy-a focused review,” Cancers (Basel) 9(2), 12 (2017).
[PubMed]

Patterson, M. S.

Piao, D.

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE. J. Sel. Top. Quant. 16(4), 715–729 (2010).

Potasek, M.

G. Shafirstein, D. Bellnier, E. Oakley, S. Hamilton, M. Potasek, K. Beeson, and E. Parilov, “Interstitial photodynamic therapy-a focused review,” Cancers (Basel) 9(2), 12 (2017).
[PubMed]

Prahl, S. A.

P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and an empirical light transport model during endoscopy,” J. Biomed. Opt. 10(3), 034018 (2005).
[PubMed]

H. J. van Staveren, C. J. Moes, J. van Marie, S. A. Prahl, and M. J. van Gemert, “Light scattering in Intralipid-10% in the wavelength range of 400-1100 nm,” Appl. Opt. 30(31), 4507–4514 (1991).
[PubMed]

Pu, Y.

Rayner, D. C.

D. J. Dickey, R. B. Moore, D. C. Rayner, and J. Tulip, “Light dosimetry using the P3 approximation,” Phys. Med. Biol. 46(9), 2359–2370 (2001).
[PubMed]

Ritchey, J. W.

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE. J. Sel. Top. Quant. 16(4), 715–729 (2010).

Ruers, T. J.

Saeys, W.

Shafirstein, G.

G. Shafirstein, D. Bellnier, E. Oakley, S. Hamilton, M. Potasek, K. Beeson, and E. Parilov, “Interstitial photodynamic therapy-a focused review,” Cancers (Basel) 9(2), 12 (2017).
[PubMed]

Sleven, R. A.

P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and an empirical light transport model during endoscopy,” J. Biomed. Opt. 10(3), 034018 (2005).
[PubMed]

Slobodov, G.

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE. J. Sel. Top. Quant. 16(4), 715–729 (2010).

Svanberg, K.

T. Svensson, E. Alerstam, M. Einarsdóttír, K. Svanberg, and S. Andersson-Engels, “Towards accurate in vivo spectroscopy of the human prostate,” J. Biophotonics 1(3), 200–203 (2008).
[PubMed]

Svensson, T.

T. Svensson, E. Alerstam, M. Einarsdóttír, K. Svanberg, and S. Andersson-Engels, “Towards accurate in vivo spectroscopy of the human prostate,” J. Biophotonics 1(3), 200–203 (2008).
[PubMed]

Thomas, A.

T. Khan and A. Thomas, “Comparison of PN or spherical harmonics approximation for scattering media with spatially varying and spatially constant refractive indices,” Opt. Commun. 255(1), 130–166 (2005).

Tseng, S.-H.

Tulip, J.

D. J. Dickey, R. B. Moore, D. C. Rayner, and J. Tulip, “Light dosimetry using the P3 approximation,” Phys. Med. Biol. 46(9), 2359–2370 (2001).
[PubMed]

O. Barajas, Å. M. Ballangrud, G. G. Miller, R. B. Moore, and J. Tulip, “Monte Carlo modelling of angular radiance in tissue phantoms and human prostate: PDT light dosimetry,” Phys. Med. Biol. 42(9), 1675–1687 (1997).
[PubMed]

Tzeng, S.-Y.

Van Beers, R.

van der Voort, M.

van Gemert, M. J.

van Marie, J.

van Staveren, H. J.

Vitkin, I. A.

L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation, and sensitivity analysis,” J. Biomed. Opt. 12(6), 064027 (2007).
[PubMed]

L. C. L. Chin, W. M. Whelan, and I. A. Vitkin, “Information content of point radiance measurements in turbid media: implications for interstitial optical property quantification,” Appl. Opt. 45(9), 2101–2114 (2006).
[PubMed]

Waldman, D. L.

T. M. Baran, J. D. Wilson, S. Mitra, J. L. Yao, E. M. Messing, D. L. Waldman, and T. H. Foster, “Optical property measurements establish the feasibility of photodynamic therapy as a minimally invasive intervention for tumors of the kidney,” J. Biomed. Opt. 17(9), 98002 (2012).
[PubMed]

Wang, W.

Watté, R.

Wesseling, J.

Whelan, W. M.

S. Grabtchak, L. G. Montgomery, and W. M. Whelan, “Feasibility of interstitial near-infrared radiance spectroscopy platform for ex vivo canine prostate studies: optical properties extraction, hemoglobin and water concentration, and gold nanoparticles detection,” J. Biomed. Opt. 19(5), 057003 (2014).
[PubMed]

S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt. 17(6), 067007 (2012).
[PubMed]

S. Grabtchak and W. M. Whelan, “Separation of absorption and scattering properties of turbid media using relative spectrally resolved cw radiance measurements,” Biomed. Opt. Express 3(10), 2371–2380 (2012).
[PubMed]

L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation, and sensitivity analysis,” J. Biomed. Opt. 12(6), 064027 (2007).
[PubMed]

L. C. L. Chin, W. M. Whelan, and I. A. Vitkin, “Information content of point radiance measurements in turbid media: implications for interstitial optical property quantification,” Appl. Opt. 45(9), 2101–2114 (2006).
[PubMed]

Wilson, J. D.

T. M. Baran, J. D. Wilson, S. Mitra, J. L. Yao, E. M. Messing, D. L. Waldman, and T. H. Foster, “Optical property measurements establish the feasibility of photodynamic therapy as a minimally invasive intervention for tumors of the kidney,” J. Biomed. Opt. 17(9), 98002 (2012).
[PubMed]

Worthington, A. E.

L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation, and sensitivity analysis,” J. Biomed. Opt. 12(6), 064027 (2007).
[PubMed]

Xu, G.

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE. J. Sel. Top. Quant. 16(4), 715–729 (2010).

Xu, H.

Xu, M.

Yang, C.-C.

Yao, J. L.

T. M. Baran, J. D. Wilson, S. Mitra, J. L. Yao, E. M. Messing, D. L. Waldman, and T. H. Foster, “Optical property measurements establish the feasibility of photodynamic therapy as a minimally invasive intervention for tumors of the kidney,” J. Biomed. Opt. 17(9), 98002 (2012).
[PubMed]

Yu, B.

V. K. Nagarajan and B. Yu, “Monitoring of tissue optical properties during thermal coagulation of ex vivo tissues,” Lasers Surg. Med. 48(7), 686–694 (2016).
[PubMed]

Zhu, T. C.

J. Li and T. C. Zhu, “Determination of in vivo light fluence distribution in a heterogeneous prostate during photodynamic therapy,” Phys. Med. Biol. 53(8), 2103–2114 (2008).
[PubMed]

A. Dimofte, J. C. Finlay, and T. C. Zhu, “A method for determination of the absorption and scattering properties interstitially in turbid media,” Phys. Med. Biol. 50(10), 2291–2311 (2005).
[PubMed]

T. C. Zhu, J. C. Finlay, and S. M. Hahn, “Determination of the distribution of light, optical properties, drug concentration, and tissue oxygenation in-vivo in human prostate during motexafin lutetium-mediated photodynamic therapy,” J. Photochem. Photobiol. B 79(3), 231–241 (2005).
[PubMed]

Appl. Opt. (2)

Appl. Spectrosc. (1)

Biomed. Opt. Express (3)

Cancers (Basel) (1)

G. Shafirstein, D. Bellnier, E. Oakley, S. Hamilton, M. Potasek, K. Beeson, and E. Parilov, “Interstitial photodynamic therapy-a focused review,” Cancers (Basel) 9(2), 12 (2017).
[PubMed]

IEEE. J. Sel. Top. Quant. (1)

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE. J. Sel. Top. Quant. 16(4), 715–729 (2010).

J. Biomed. Opt. (7)

T. M. Baran, “Recovery of optical properties using interstitial cylindrical diffusers as source and detector fibers,” J. Biomed. Opt. 21(7), 77001 (2016).
[PubMed]

P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and an empirical light transport model during endoscopy,” J. Biomed. Opt. 10(3), 034018 (2005).
[PubMed]

T. M. Baran, M. C. Fenn, and T. H. Foster, “Determination of optical properties by interstitial white light spectroscopy using a custom fiber optic probe,” J. Biomed. Opt. 18(10), 107007 (2013).
[PubMed]

T. M. Baran, J. D. Wilson, S. Mitra, J. L. Yao, E. M. Messing, D. L. Waldman, and T. H. Foster, “Optical property measurements establish the feasibility of photodynamic therapy as a minimally invasive intervention for tumors of the kidney,” J. Biomed. Opt. 17(9), 98002 (2012).
[PubMed]

S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt. 17(6), 067007 (2012).
[PubMed]

L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation, and sensitivity analysis,” J. Biomed. Opt. 12(6), 064027 (2007).
[PubMed]

S. Grabtchak, L. G. Montgomery, and W. M. Whelan, “Feasibility of interstitial near-infrared radiance spectroscopy platform for ex vivo canine prostate studies: optical properties extraction, hemoglobin and water concentration, and gold nanoparticles detection,” J. Biomed. Opt. 19(5), 057003 (2014).
[PubMed]

J. Biophotonics (1)

T. Svensson, E. Alerstam, M. Einarsdóttír, K. Svanberg, and S. Andersson-Engels, “Towards accurate in vivo spectroscopy of the human prostate,” J. Biophotonics 1(3), 200–203 (2008).
[PubMed]

J. Opt. Soc. Am. A (1)

J. Photochem. Photobiol. B (1)

T. C. Zhu, J. C. Finlay, and S. M. Hahn, “Determination of the distribution of light, optical properties, drug concentration, and tissue oxygenation in-vivo in human prostate during motexafin lutetium-mediated photodynamic therapy,” J. Photochem. Photobiol. B 79(3), 231–241 (2005).
[PubMed]

Lasers Surg. Med. (1)

V. K. Nagarajan and B. Yu, “Monitoring of tissue optical properties during thermal coagulation of ex vivo tissues,” Lasers Surg. Med. 48(7), 686–694 (2016).
[PubMed]

Opt. Commun. (1)

T. Khan and A. Thomas, “Comparison of PN or spherical harmonics approximation for scattering media with spatially varying and spatially constant refractive indices,” Opt. Commun. 255(1), 130–166 (2005).

Opt. Express (2)

Phys. Med. Biol. (4)

A. Dimofte, J. C. Finlay, and T. C. Zhu, “A method for determination of the absorption and scattering properties interstitially in turbid media,” Phys. Med. Biol. 50(10), 2291–2311 (2005).
[PubMed]

D. J. Dickey, R. B. Moore, D. C. Rayner, and J. Tulip, “Light dosimetry using the P3 approximation,” Phys. Med. Biol. 46(9), 2359–2370 (2001).
[PubMed]

O. Barajas, Å. M. Ballangrud, G. G. Miller, R. B. Moore, and J. Tulip, “Monte Carlo modelling of angular radiance in tissue phantoms and human prostate: PDT light dosimetry,” Phys. Med. Biol. 42(9), 1675–1687 (1997).
[PubMed]

J. Li and T. C. Zhu, “Determination of in vivo light fluence distribution in a heterogeneous prostate during photodynamic therapy,” Phys. Med. Biol. 53(8), 2103–2114 (2008).
[PubMed]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

A. Liemert and A. Kienle, “Analytical Green’s function of the radiative transfer radiance for the infinite medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(3 Pt 2), 036605 (2011).
[PubMed]

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Figures (8)

Fig. 1
Fig. 1 The maximum relative differences between the P3in and the P3, ε max , in the optical property range of 0.01 mm 1 μ a 1.0 mm 1 and 0.5 mm 1 μ s 4.5 mm 1 at SDSs from 5 mm to 10 mm and angles from 0 ° to 180 ° .
Fig. 2
Fig. 2 Geometry for measurements of the radiances, L m ( r , θ ) , at SDSs of r and r and angles of θ 0 , θ 1 , θ 2 and θ 3 .
Fig. 3
Fig. 3 Recovered optical properties based on the DA and the P3in from the simulated radiance data calculated by the P25 in the case of typical scattering. (a) is for μ a and (b) is for μ s . The insets show the relative errors (REs) of the recovered optical properties.
Fig. 4
Fig. 4 Recovered optical properties based on the DA and the P3in from the simulated radiance data calculated by the P25 in the case of low scattering. (a) is for μ a and (b) is for μ s . The insets show the REs of the recovered optical properties.
Fig. 5
Fig. 5 A schematic of measuring the radiances from an isotropic point source at SDS of r and angle of θ .
Fig. 6
Fig. 6 Recovered optical properties based on the DA and the P3in from the phantom measurements in the case of typical scattering. (a) is for μ a and (b) is for μ s . The insets show the REs of the recovered optical properties.
Fig. 7
Fig. 7 Recovered optical properties obtained from the phantom measurements in the case of low scattering. (a) is for μ a and (b) is for μ s . The insets show the REs of the recovered optical properties.
Fig. 8
Fig. 8 REs of the recovered optical properties for 0.01 mm 1 μ a 1.0 mm 1 and 0.5 mm 1 μ s 4.5 mm 1 . (a) and (b) are based on the P3in. (c) and (d) are based on the P5in. (a) and (c) are for μ a . (b) and (d) are for μ s .

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

L ( r , θ ) = S 0 4 π l = 0 N ( 2 l + 1 ) φ l ( r ) P l ( cos θ ) ,
l + 1 2 l + 1 [ d d r + l + 2 r ] φ l + 1 ( r ) + l 2 l + 1 [ d d r l 1 r ] φ l 1 ( r ) + σ l φ l ( r ) = S 0 δ l 0 , l = 0 , ... , 3 ,
L ( r , θ ) = S 0 4 π l = 0 3 ( 2 l + 1 ) [ C h l ( ν ) Q l ( ν r ) + D h l ( ν + ) Q l ( ν + r ) ] P l ( cos θ ) ,
C = ( ν ) 5 12 π μ a 2 σ 1 ( 3 μ a σ 1 ( ν + ) 2 ) ( ( ν ) 2 ( ν + ) 2 ) , D = ( ν + ) 5 12 π μ a 2 σ 1 ( 3 μ a σ 1 ( ν ) 2 ) ( ( ν + ) 2 ( ν ) 2 ) .
h 0 = 1 , h 1 ( ν ± ) = μ a ν ± , h 2 ( ν ± ) = 1 2 + 3 μ a σ 1 2 ( ν ± ) 2 , h 3 ( ν ± ) = 9 μ a σ 1 14 σ 3 ν ± 3 ν ± 14 σ 3 .
L i n ( r , θ ) = S 0 C 4 π l = 0 3 ( 2 l + 1 ) h l ( ν ) Q l ( ν r ) P l ( cos θ ) .
l = 0 3 S 0 C 4 π ( 2 l + 1 ) h l ( ν ) Q l ( ν r ) P l ( cos θ i ) = L m ( r , θ i ) , i = 0 , ... , 3.
S 0 C h 0 ( ν ) Q 0 ( ν r ) = 4 π ( α 3 ( r , θ 0 , θ 1 , θ 2 ) β 13 ( θ 0 , θ 1 , θ 3 ) α 3 ( r , θ 0 , θ 1 , θ 3 ) β 13 ( θ 0 , θ 1 , θ 2 ) ) ( β 03 ( θ 0 , θ 1 , θ 2 ) β 13 ( θ 0 , θ 1 , θ 3 ) β 03 ( θ 0 , θ 1 , θ 3 ) β 13 ( θ 0 , θ 1 , θ 2 ) ) ,
S 0 C h 1 ( ν ) Q 1 ( ν r ) = 4 π ( α 3 ( r , θ 0 , θ 1 , θ 2 ) β 03 ( θ 0 , θ 1 , θ 3 ) α 3 ( r , θ 0 , θ 1 , θ 3 ) β 03 ( θ 0 , θ 1 , θ 2 ) ) 3 ( β 13 ( θ 0 , θ 1 , θ 2 ) β 03 ( θ 0 , θ 1 , θ 3 ) β 13 ( θ 0 , θ 1 , θ 3 ) β 03 ( θ 0 , θ 1 , θ 2 ) ) ,
S 0 C h 2 ( ν ) Q 2 ( ν r ) = 4 π ( α 1 ( r , θ 0 , θ 1 , θ 2 ) β 31 ( θ 0 , θ 1 , θ 3 ) α 1 ( r , θ 0 , θ 1 , θ 3 ) β 31 ( θ 0 , θ 1 , θ 2 ) ) 5 ( β 21 ( θ 0 , θ 1 , θ 2 ) β 31 ( θ 0 , θ 1 , θ 3 ) β 21 ( θ 0 , θ 1 , θ 3 ) β 31 ( θ 0 , θ 1 , θ 2 ) ) ,
α i ( r , θ 0 , θ 1 , θ 2 ) = ( L m ( r , θ 0 ) P i ( cos θ 1 ) L m ( r , θ 1 ) P i ( cos θ 0 ) ) ( P i 1 ( cos θ 0 ) P i ( cos θ 2 ) P i 1 ( cos θ 2 ) P i ( cos θ 0 ) ) , ( L m ( r , θ 0 ) P i ( cos θ 2 ) L m ( r , θ 2 ) P i ( cos θ 0 ) ) ( P i 1 ( cos θ 0 ) P i ( cos θ 1 ) P i 1 ( cos θ 1 ) P i ( cos θ 0 ) )
β i j ( θ 0 , θ 1 , θ 2 ) = ( P i ( cos θ 0 ) P j ( cos θ 1 ) P i ( cos θ 1 ) P j ( cos θ 0 ) ) ( P j 1 ( cos θ 0 ) P j ( cos θ 1 ) P j 1 ( cos θ 1 ) P j ( cos θ 0 ) ) . ( P i ( cos θ 0 ) P j ( cos θ 2 ) P i ( cos θ 2 ) P j ( cos θ 0 ) ) ( P j 1 ( cos θ 0 ) P j ( cos θ 2 ) P j 1 ( cos θ 2 ) P j ( cos θ 0 ) )
h 0 ( ν ) Q 0 ( ν r ) h 0 ( ν ) Q 0 ( ν r ) = f 0 ( r , θ 0 , θ 1 , θ 2 , θ 3 ) f 0 ( r , θ 0 , θ 1 , θ 2 , θ 3 ) .
ν = ln [ f 0 ( r , θ 0 , θ 1 , θ 2 , θ 3 ) f 0 ( r , θ 0 , θ 1 , θ 2 , θ 3 ) r r ] / ( r r ) .
h l ( ν ) Q l ( ν r ) h 0 ( ν ) Q 0 ( ν r ) = f l ( r , θ 0 , θ 1 , θ 2 , θ 3 ) f 0 ( r , θ 0 , θ 1 , θ 2 , θ 3 ) , l = 1 , 2.
h l ( ν ) = Q 0 ( ν r ) Q l ( ν r ) f l ( r , θ 0 , θ 1 , θ 2 , θ 3 ) f 0 ( r , θ 0 , θ 1 , θ 2 , θ 3 ) , l = 1 , 2.
μ a = h 1 ( ν ) ν , μ s = ( 2 h 2 ( ν ) + 1 ) ν 3 h 2 ( ν ) h 1 ( ν ) ν .

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