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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 14 — Jul. 10, 2006
  • pp: 6485–6501
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Determination of the optical properties of tissue-simulating phantoms from interstitial frequency domain measurements of relative fluence and phase difference

Heping Xu and Michael S. Patterson  »View Author Affiliations


Optics Express, Vol. 14, Issue 14, pp. 6485-6501 (2006)
http://dx.doi.org/10.1364/OE.14.006485


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Abstract

We estimated the absorption and reduced scattering coefficients of tissue-simulating phantoms from interstitial measurements of the phase difference and relative amplitude signals at two distances from a sinusoidally modulated isotropic source. It was found that absorption and reduced scattering coefficients can be recovered within 10% and slightly over 10% respectively, using either the data collected by two detectors 3mm apart or by two detectors 5mm apart with light collected by one detector attenuated by a neutral density filter. This accuracy was achieved over a wide range of optical properties, µa=0.008 to 0.17mm-1 and µ s ’=0.3 to 1.8mm-1. Additional factors affecting accuracy including source anisotropy, uncertainty in fiber placement, phase amplitude crosstalk, and the forward light propagation model (the combined isotropic similarity model and standard diffusion approximation versus the modified spherical harmonics method) were studied by Monte Carlo simulations (first two factors) and experiments (last two factors).

© 2006 Optical Society of America

1. Introduction

PDT has been demonstrated to be effective for the treatment of cancer at several sites, including skin, retina, lung, and reproductive and urological tissues and organs, and has been clinically approved for some of these sites in Canada, the United States, Europe, and Japan [5

5. Z. Huang, “A review of progress in clinical photodynamic therapy,” Technol. Cancer. Res. Treat. 4, 283–293 (2005). [PubMed]

]. The treatment of deep-seated tumors such as prostate cancer requires that optical fibers be implanted within the organ. In this interstitial geometry, a spatially resolved steady state technique has been developed to determine the OPs in phantoms and in-vivo, where light was delivered and collected by isotropic fibers [6

6. T. C. Zhu, S. M. Hahn, A. S. Kapatkin, A. Dimofte, C. E. Rodriguez, T. G. Vulcan, E. Glatstein, and R. A. His, “In-vivo optical properties of normal canine prostate at 732nm using motexafin lutetium mediated photodynamic therapy,” Photochem. Photobiol. 77, 81–88 (2003). [CrossRef] [PubMed]

8

8. 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, 2291–2311 (2005). [CrossRef] [PubMed]

]. As opposed to this steady state approach, in the frequency domain measurements can be undertaken at only two positions to recover OPs (the absorption coefficient, µa and the reduced scattering coefficient, µs’) since the additional information of phase can be obtained. Figure 1 shows the detection geometry used in this paper. Two isotropic detectors at different radial positions collected light emitted from an isotropic, sinusoidally-modulated source. All fibers were immersed in a phantom sufficiently large to mimic the clinical situation. Optical properties were determined from the heterodyne frequency domain measurements of the phase difference and the ratio of the amplitudes (fluences) at two known positions, using inverse calculation based on either the combined isotropic similarity model (ISM) and standard diffusion approximation (SDA), ISM/SDA for short, [9

9. H. Xu, T. J. Farrell, and M. S. Patterson, “Investigation of light propagation models to determine the optical properties of tissue from interstitial frequency domain fluence measurements,” J. Biomed. Opt. (to be published 2006). [CrossRef] [PubMed]

] or the modified spherical harmonics method [10

10. V. A. Markel, “Modified spherical harmonics method for solving the radiative transport equation,” Waves Random Media 14, L13–L19 (2004). [CrossRef]

11

11. G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A , 39, 115–137 (2006). [CrossRef]

].

Fig. 1. The geometry of the two-distance detection scheme. Two detector fibers are placed parallel to each other as well as to the source fiber. For these experiments, a single detector fiber was translated between positions.

2. Theoretical background

Re(φ)=μt'4πr[k02cexp(a0μt'r)+01g(c,ξ)exp(μt'rξ)dξξ2
(1)

where µ′t =µa +µs ′; c, the albedo is µs /µ′t ; the real parameter a 0 is the root of c=a 0/tanh-a 0; k 0 is imaginary and defined as k 0=ia 0 ; r is the distance from source to detector; and the functional form of g(c,ξ) is 1/[(1- tanh-1 ξ)2+(πcξ/2)2]. The imaginary part, Im(φ), is:

Im(φ)=12πr.ωv.12πi+k'k'2+1.exp[ik'(μa+μs')r][1(ck')·tan1k']2dk'
(2)

where v is the speed of light in tissue, ω is the angular frequency of the source modulation, and k′ is a dummy integration variable.

An alternative to the ISM/SDA is the modified spherical harmonics method (MSH) [10

10. V. A. Markel, “Modified spherical harmonics method for solving the radiative transport equation,” Waves Random Media 14, L13–L19 (2004). [CrossRef]

,11

11. G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A , 39, 115–137 (2006). [CrossRef]

]. It has been found that this method is fast and accurate, thus it is particularly suitable for the inverse calculation because fluence and phase can be expressed in an analytical form [12

12. H. Xu and M. S. Patterson, “Application of the modified spherical harmonics method to some problems in biomedical optics,” Phys. Med. Biol. 51, N247–N251 (2006). [CrossRef] [PubMed]

]. The essence of this method is that the angular part of the Fourier transform of quantities such as the radiance and the phase function in the radiative transfer equation (RTE, also known as the Boltzmann equation), can be expanded in the basis of spherical harmonic functions defined in a reference frame (called the rotated frame) whose z-axis is aligned with the direction of the Fourier wave vector. Because of this specific choice of the basis spherical harmonic functions in the rotated reference frame, the RTE is simplified, compared to the RTE in the laboratory frame. Solving the RTE in the rotated frame reduces to solving the eigenvalues and eigenvectors of a tridiagonal symmetric matrix W. The accuracy of this method depends on the chosen order of W. In the original paper [10], it was argued that for typical biological tissue, the accuracy needed can be satisfied with order less than 50. The computation of the W matrix of this order or less can be done within seconds using Matlab code on a personal computer. In general, MSH is an efficient and accurate numerical method for the RTE in infinite geometry. The formalisms relevant to our work can be found in Appendix A. Since we are interested in the value of the reduced scattering coefficient µs’ instead of the individual scattering coefficient µs and anisotropy factor g, all the calculations in this article used g=0.9 when MSH was employed. Use of the preset value g in our particular two-distance detection scheme is required because this scheme can recover only two independent optical parameters instead of the three parameters (which are µa, µs’ and g).

3. Experimental methods

3.1. The frequency domain system

Standard heterodyne detection was used in our frequency domain system. Two frequency generators (Marconi Instruments, Model 2022A) were used to provide modulation (typically 100MHz) of the laser source (Oriel Diode Lasers, Model no. 79401) and the high voltage (frequency was typically offset by 200Hz compared to the frequency of the laser modulation) applied to the photomultiplier tube (PMT) (Hamamatsu R928) with the housing (ISS model no. P016). The wavelength of the diode laser is 750nm with maximum power of 4.3mW and 100% modulation depth. Source and detector fibers (0.4mm in diameter, Photoglow) are uniform within ±5% on 88% (symmetrical to the fiber line) of their surfaces. The signal detected from the probe fiber was fed into a lock-in amplifier (Stanford Research Systems model SR850) via a signal preamplifier (Ithaco model 1642). In this analysis we refer to two detector fibers, but for these experiments a single fiber was translated between positions using a stage with 0.02mm resolution.

3.2. Data acquisition

Data collection in the frequency domain measurement was performed with LabView (National Instruments, Austin, TX) through the RS-232 serial port of the lock-in amplifier. For each set of fluence and phase measurements, nine runs were collected and subsequently averaged to reduce the noise and obtain the standard deviation. All measurements were performed with the room lights off. The RF amplifier was positioned about two meters from other instruments to reduce electromagnetic interference.

3.3. Monte Carlo simulation

3.4. Inverse algorithm

3.5. Accurate measurement of OPs of Intralipid and India ink

μa=(3.28±0.15)·(%c)+(0.0038±0.0008)(mm1)
(3)

The intercept indicates the absorption due to water (milliQ water), which is close to the published data: 0.00261mm-1 [19

19. R. C. Smith and K.S. Baker, “Optical properties of the clearest natural waters (200–800nm),” Appl. Opt. 20, 177–184 (1981). [CrossRef] [PubMed]

]. It has been shown that India ink may have non-negligible scattering which can result in underestimation of scattering or overestimation of absorption of ink-based phantoms [20

20. S. J. Madsen, M. S. Patterson, and B. C. Wilson, “The use of India ink as an optical absorber in tissue-simulating phantoms,” Phys. Med. Biol. 37, 985–993 (1992). [CrossRef] [PubMed]

]. This scattering is due primarily to large particle aggregates (~1µm), which can scatter light significantly. We ignored the scattering component due to ink and show below that this assumption was justified.

The added-absorber method was employed to determine the reduced scattering coefficient (µs’) of Intralipid and check the values obtained above for µa of ink. For broad beam irradiation of a turbid medium the fluence rate varies with depth z as:

ϕ(z)=A(μs',μa)·exp(μeffz)
(4)

where μeff=3μa(μa+μs') under conditions where standard diffusion theory applies. In order to obtain a satisfactory uniform beam, the tip of an isotropic fiber (which can be regarded as an isotropic point source) was located at the focal point of a lens with a diameter of 10cm. A second isotropic fiber was placed within the phantom and used to measure the depth dependence of the fluence rate. The phantom container used was 15×15×11cm3. Care must be exercised because the phantom volume cannot be regarded as (semi) infinite if 1/µeff is less than about 12 times its diameter, in which case light losses at the boundaries cause an overestimation of µeff [21

21. L. Lilge and B. C. Wilson, “The accuracy of interstitial measurements of absolute light fluence rate in the determination of tissue optical properties,” in Proceedings of Laser-tissue Interaction IV, S. L. Jacques and A. Katzir, eds., Proc SPIE1882, 291–304 (1993). [CrossRef]

]. At the same time, µa should be kept less than 5% of µs’ in order for diffusion theory to be valid. Assuming the absorption of Intralipid itself is negligible, the following equation results:

μeff2=3μs'μaink+3μs'μaw
(5)

where µaink =ξL(ξ is a constant in mm-1, L is the concentration of the India ink in percent) and µaw =0.00261 mm -1. By fitting the measured (µeff)2 to a range of concentrations of added ink (0.02% to 0.04%) for a phantom with fixed µs’, the intercept gives the reduced scattering of IL and the slope gives the absorption of ink. The results are: µs’=1.034mm-1 for 1% IL and ξ=3.09mm-1 at 750nm, which agrees with the ink formula obtained in the spectrometer measurements (Eq. 3). By varying the IL concentration (0.3% to 2.0%) with fixed ink absorption (0.01mm-1) and repeating the added absorber experiment, we determined that µs’ due to IL is:

μ's=(0.97±0.08)·(%c)(mm1)
(6)

at 750nm.

3.6. Phase amplitude crosstalk

There is limited discussion of phase amplitude crosstalk in the literature [22

22. N. Ramanujam, C. Du, H. Y. Ma, and B. Chance, “Sources of phase noise in homodyne phase modulation devices used for tissue oximetry studies,” Rev. Sci. Instrum. 69, 3042–3054 (1998). [CrossRef]

26

26. B. Chance, M. Cope, E. Gratton, N. Ramanujam, and B. Tromberg, “Phase measurement of light absorption and scatter in human tissue,” Rev. Sci. Instrum. 69, 3457–3481 (1998). [CrossRef]

]. Ramanujam et al [22

22. N. Ramanujam, C. Du, H. Y. Ma, and B. Chance, “Sources of phase noise in homodyne phase modulation devices used for tissue oximetry studies,” Rev. Sci. Instrum. 69, 3042–3054 (1998). [CrossRef]

] showed that this phenomenon occurs in both homodyne and heterodyne detection techniques using either PMT (R928) or avalanche photodiode (APD) detectors. According to the paper by Ramanujam et al [22

22. N. Ramanujam, C. Du, H. Y. Ma, and B. Chance, “Sources of phase noise in homodyne phase modulation devices used for tissue oximetry studies,” Rev. Sci. Instrum. 69, 3042–3054 (1998). [CrossRef]

], the crosstalk was 12.23o/O.D. (i.e. factor of 10 change in amplitude) using an R928 PMT (high voltage was 740V) and heterodyne technique (modulation frequency was 200MHz). The accepted explanation is that detector rise time changes with input light intensity [25

25. K. Alford and Y. Wickramasinghe, “Phase-amplitude crosstalk in intensity modulated near infrared spectroscopy,” Rev. Sci. Instrum. 71, 2191–2195 (2000). [CrossRef]

]. The methods proposed to rectify this problem were 1) maintenance of a fairly constant amplitude at the output of the detector using a dynode feedback loop, or 2) use of a correction curve as the crosstalk is an intrinsic and reproducible property of the detector [26

26. B. Chance, M. Cope, E. Gratton, N. Ramanujam, and B. Tromberg, “Phase measurement of light absorption and scatter in human tissue,” Rev. Sci. Instrum. 69, 3457–3481 (1998). [CrossRef]

]. However, we found that the measured crosstalk effect is dependent on the geometry of the light path and the method by which light is attenuated.

Fig. 2. Phase shift vs. amplitude at high voltage of 700V on PMT. The light was attenuated using a variable optical attenuator. Error bars show the standard deviations.

Fig. 3. Phase shift due to the attenuation through four neutral density filters (O.D.=0.38, 0.43, 0.92, and 0.99). Separation between the two collimating lenses was 3.0cm. The output current at the PMT was maintained at 0.35uA. Error bars show the standard deviations.
Fig. 4. Phase shift caused by the insertion of 0.92 OD neutral density filter at four different separations (0.0, 3.1, 5.5, and 8.0cm) between the two collimating lenses. Error bars show the standard deviations.

4. Results

4.1. MC simulation of anisotropic source

As stated earlier, light intensity on a spherical surface centered at the tip of the fibers used in our work is only uniform (within 5%) for the polar angle between 40° and 180° (see Fig. 5). The detailed light intensity profile inside the cone (between 0° and 40°) of the fiber is not provided by the manufacturer. This anisotropic source will cause the fluence rate to depend not only on the distance to the measurement point but also its angular coordinates. In highly scattering media this dependence will be less evident as the source-detector distance increases. This was demonstrated by Monte Carlo simulations for three sets of OPs, (µa, µs’)=(0.01, 1), (0.08, 0.4), and (0.2, 0.3) in mm-1, which cover µs’/µa from 1.5 to 100. In the Monte Carlo simulation, the source was a sphere of diameter 0.4mm. Photons were launched from random locations on the sphere surface, outside the 40° cone. No photon was launched inside that cone (see the insert in Fig. 5). Detector fibers were assumed to have isotropic response to simplify the MC code. The scattering anisotropy factor was set to 0.9. For the medium with (µa, µs’)=(0.01, 1.00)mm-1, Fig. 5 shows the typical variation in fluence rate versus polar angle at a distance of 3mm from a unit strength source. The insert in Fig. 5 shows the source orientation and definition of angle. The fluence rate was 0.042mm-2s-1 at zero degrees (i.e. a point “behind” the fiber) and steadily increased to 0.049mm-1 at 180°. In order to examine how this variation in the fluence rate affects the result of inverse calculation, three pairs of distances (2, 3), (2, 8), (4, 8)mm were chosen. The detectors were located as shown in Fig. 1. The choice of this geometry is based on the fact that in most clinical PDT, fibers are inserted with the guide of a brachytherapy template in which fibers are parallel to each other. The expected recovered OPs, (µa, µs’)=(0.08, 0.4) mm-1, are summarized in Table. 1. Similar results were observed for the other sets of OPs (data not shown). The pair (4, 8) using MSH model shows OPs can be recovered within 3%. For the same positions, ISM/SDA gives about 8% error in estimating µs’. For the other two position pairs, the errors in recovered OPs can be as large as 30% (using ISM/SDA) and 12% (using MSH). Hence, in order to minimize the effect of anisotropy of the source fiber, the detection positions should be at least 3mm away from the source.

Fig. 5. Fluence rate (mm-2s-1) versus polar angle defined in the insert. Fluence rate was calculated at 3mm from the source. In the insert, the bold line represents the fiber, and the angle denoted by θ varies from 0 (positive z direction) to 180 (negative z direction).

Table. 1. Recovered OPs using ISM/SDA and MSH for three pairs of positions. The expected OPs: µs’=0.4mm-1, µa=0.08mm-1

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4.2. Experimental comparison of MSH and ISM/SDA models under “extreme” conditions

Table. 2. Experimental comparison of MSH, and ISM/SDA. Detection position pair was (5, 8). The expected µs’=0.2mm-1

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4.3. MC study of inaccurate fiber placement

Needles inserted into tissue for optical fiber placement may bend, so that even if a template is used, the real positions of the needles and optical fibers may not be those expected. Hence an analysis of the resulting errors in estimated OPs due to the misplacement of fibers is desirable. First, we found that the recovered OPs can be off by at least 10% if one of the fibers is misplaced by 0.1mm and the other is accurately positioned. The Monte Carlo generated data for (µa, µs’) were OP1: (0.005, 1.00), OP2: (0.015, 0.9), and OP3: (0.2, 0.5) in mm-1. The recovered OPs were obtained by offsetting the probe at 7mm (the other probe is at 10mm) by 0.1, 0.2, 0.3, 0.4, 0.5mm away from the source and using those data in the inverse algorithm. The results are shown in Table 3 and can be understood as follows. It is known that the fluence decreases sharply with distance close to the source and more slowly beyond around 10mm (the actually transition distance depends on the OPs). Hence small changes in distance close to the source result in relatively large changes in fluence.

Table 3. Monte Carlo study of relative difference of recovered OPs using ISM/SDA model in µa (left) and µs’ (right).

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The probe at 7mm was offset by 0.1, 0.2, 0.3, 0.4, and 0.5mm away from the source. The other probe is fixed at 10.0mm. OP. 1, 2, and 3 are (1.00, 0.005), (0.90, 0.015), and (0.50, 0.20) in mm-1, where 1st (2nd) number refers to absorption (reduced scattering).

Second, we investigated the error in recovered OPs in the situation where the separation between the two fibers is fixed at 3mm but both fibers are displaced. The expected OPs were µa=0.01mm-1, µs’=1.00mm-1. The nominal positions are (3, 6), (5, 8) and (7, 10) mm. Table. 4. shows the recovered OPs for the nominal positions of (3, 6) when both detectors were shifted by 0.1, 0.3, 0.5 1.0 1.5mm. The upper row in µa and µs’ corresponds to the two detectors shifted away from the source. The lower row in µa and µs’ shows the two detectors shifted toward the source. Two points can be observed: 1) the same shift will give larger error closer to the source. The shift starts to give significant error (10%) if it is more than 0.3mm when close to the source, and more than 1.0mm far from the source; 2) the error in recovered OPs due to the simultaneous shift of the two detectors is far less than that due to the offset of one fiber (with the other in the correct position). Hence the two-distance technique has the advantage that precise measurement of the absolute position of the two fibers is not required, but the relative distance between the two fibers is important. It also shows that the fibers should be placed relatively far from the source (point (1) above-mentioned) to minimize the errors.

Table 4. Monte Carlo study of the effect of the shift of two detectors on recovered OPs using ISM/SDA model in µa and µs’.

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The upper row in µas’) corresponds to the two detectors shifted away from the source. The lower row in µa and (µs’) shows the two detectors shifted toward the source.

4.4. Choosing OPs of prostate tissue-simulating phantoms

The in vivo optical properties of prostate demonstrate fairly large inter- and intra- animal variability compared to other tissues such as brain. The absorption and reduced scattering coefficients vary between 0.003–0.058 and 0.1–2.0 mm-1 (human prostate) at 732nm according to Zhu et al [6

6. T. C. Zhu, S. M. Hahn, A. S. Kapatkin, A. Dimofte, C. E. Rodriguez, T. G. Vulcan, E. Glatstein, and R. A. His, “In-vivo optical properties of normal canine prostate at 732nm using motexafin lutetium mediated photodynamic therapy,” Photochem. Photobiol. 77, 81–88 (2003). [CrossRef] [PubMed]

], and between 0.002–0.053 and 0.4–2.0mm-1 (canine prostate) at 660nm as reported by Lilge et al [27

27. L. Lilge, N. Pomerleau-Dalcourt, A. Douplik, S. H. Selman, R. W. Keck, M. Szkudlarek, M. Pestka, and J. Jankun, “Transperineal in vivo fluence-rate dosimetry in the canine prostate during SnET2-mediated PDT,” Phys. Med. Biol. 49, 3209–3225 (2004). [CrossRef] [PubMed]

]. During PDT, the presence of photosensitizer contributes to the absorption as well. Some drugs such as Tookad have high molar extinction coefficient (~105 M-1cm-1) at the peak absorption wavelength [28

28. S. Gross, A. Gilead, A. Scherz, M. Neeman, and Y. Salomon, “Monitoring photodynamic therapy of solid tumors on-line by BOLD-contrast MRI,” Nat. Med. 9, 1327–1331 (2003). [CrossRef] [PubMed]

]. All these form the basis for our choice of OPs used for the phantoms.

Table 5. OPs for the phantoms used in Figs. 4, 5 (top two rows), and 6 (bottom two rows).

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Table 6. The average of root-mean-square percent errors and standard deviations (in bracket) in recovered OPs for nine phantoms (for OPs, see text) using ISM/SDA and MSH.

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4.4.1. Case where a neutral density filter was not used

We have chosen three pairs: location 1: (4, 7) mm, location 2: (7, 10) mm and location 3: (5, 10) mm. The first and the second number in the brackets are distances of the first and second probe locations respectively, measured from the source fiber. The OPs are chosen as follows (for reasons, see above): µa=0.008, 0.023, 0.17mm-1, and µs’=0.3, 1.05, 1.8mm-1, out of which nine possible combinations can be formed and Table 5. shows how they are numbered. The measurements were not conducted for phantom #9 at location 3 due to the weak signal at the farther detector. Figures 6 and 7 show the percent errors in the OPs recovered using ISM/SDA and MSH respectively at the three locations for nine phantoms. To better understand which location pair is best, we calculated all the root-mean-square (RMS) percent errors (for repeated runs on the same phantom) and their standard deviations for all nine phantoms at each location pair, (see Table 6). The standard deviations are measures of how the errors spread among nine phantoms for each pair location. The errors at location 2 are the smallest: 13% in µs’ and 9% in µa on average. Thus two detectors should be placed “far” from the source to minimize errors in the estimated OPs.

Fig. 6. Percent errors in estimated OPs using ISM/SDA model for nine phantoms at locations 1, 2, and 3 that are shown by sub-Fig (a), (b) and (c) respectively. Phantom identification numbers can be found in Table 5.

4.4.2. Case where a NDF was used for the detector closer to the source

In a relatively high absorption turbid medium, the phase shift between two detectors is small if their separation is not sufficient. This dependence of phase shift, θ, on µa, r1, r2 can be expressed analytically using the diffusion approximation:

θ=3r1r22vμs'μaω2π
(7)

On the other hand, large separation, which gives larger phase shift, entails much weaker signal at the second detector, especially in high absorption media. This in turn will introduce relatively large error in the amplitude ratio. One simple way to obtain large phase shift and reduce the dynamic range of the amplitude measurement is to use a calibrated NDF to attenuate light reaching the PMT when it is detecting light from the near fiber. The correction to phase shift due to the difference in the speed of light between the NDF and the air was taken into account. The OPs chosen for this experiment are listed in Table. 5: µa and µs’ range from 0.008mm-1 to 0.10 mm-1, and 0.3mm-1 to 1.6mm-1 respectively. The O.D. of the NDF was 0.90±0.02. The two detectors were located at 5 and 10mm from the source. Figure 8 shows the error in estimated OPs using ISM/SDA [sub-Fig. (a)] and MSH [sub-Fig. (b)] models for the nine phantoms. Again, it can be seen that errors in µa were less than those in µs’ in most cases. Using the ISM/SDA model, the errors in recovered OPs were greater than 10% for phantom 3, 5, 6, and 9. The error in µa for phantom 5 is slightly greater than 10%. The other three phantoms show larger errors because: 1) ISM/SDA breaks down for phantom 3 because µs’/µa<5; 2) typically, larger fluctuations in measured phase and amplitude were found in these experiments, and this resulted in larger errors. Using MSH, errors in µa and µs’ for phantoms 6 and 9 were larger than 10%, and this can be explained by reason #2 mentioned above. However, we still observe that errors in µs’ for phantom 3 and 4 were slightly larger than 10%.

Fig. 7. Percent errors in estimated OPs using MSH model for nine phantoms at locations 1, 2, and 3 that are shown by sub-Figs. (a), (b) and (c) respectively. Phantom identification numbers can be found in Table 5.
Fig. 8. Percent errors in estimated OPs using ISM/SDA [sub-Fig. (a)] and MSH [sub-Fig. (b)] models for nine phantoms at location (5, 10)mm. Phantom identification numbers can be found in Table 5.

Two points are worthy of note: 1) the relative error in µs’ is always greater than that in µa, and 2) their errors are higher than those found using MC generated data where input noise of about 10% was comparable to experimental noise [9

9. H. Xu, T. J. Farrell, and M. S. Patterson, “Investigation of light propagation models to determine the optical properties of tissue from interstitial frequency domain fluence measurements,” J. Biomed. Opt. (to be published 2006). [CrossRef] [PubMed]

]. It is helpful to understand which can be thought of as random errors and which are not. The OPs can be expressed in terms of modulation frequency, light speed in tissue, probe locations relative to the source and measured amplitudes and phase difference. In our two-distance detection scheme, it is generally assumed that the errors in modulation frequency and light speed are negligible. Hence, errors originate mainly from the measurements of distances, phase and amplitudes. Once two locations are chosen, they are fixed during the measurements of phase and amplitudes. Thus, errors in phase and amplitudes can be regarded as random, and the standard deviation can be calculated using several repeated measurements. However, errors due to distances should be regarded as systematic error in our scheme. This is different from the work done by most investigators, where multi-distance schemes were used and the effect on estimated OPs due to the inaccuracy in measured distances may be “averaged out” [29

29. M. Gerken and G. W. Faris, “High-precision frequency-domain measurements of the optical properties of turbid media,” Opt. Lett. 24, 930–932, (1999). [CrossRef]

,30

30. J. B. Fishkin, P. T. C. So, A.E. Cerussi, S. Fantini, M. A. Franceschini, and E. Gratton, “Frequency-domain method for measuring spectral properties in multiple-scattering media: methemoglobin absorption spectrum in a tissuelike phantom,” Appl. Opt. 34, 1143–1155 (1995). [CrossRef] [PubMed]

]. We can use the diffusion approximation (simple yet qualitatively correct) to calculate the random error in recovered µa and µs’ due to measured phase difference and fluences (error due to distance measurements will be treated separately and shown below). At the low frequency limit (ω/2π<150MHz), µa can be rewritten as follows:

μa=ω2v·ln(r2A1r1A2)θ
(8)

where θ, A, and r are the phase difference, fluence, and distance from the source respectively; subscript denotes the position numbers; ω is the angular modulation frequency and v is light speed in the phantom. In accordance with the law of random error propagation, the random error in µa is:

Δμaμa=Δθθ+(ΔA1A1+ΔA2A2)ln(A1A2)
(9)

Similarly, µs’ is given by:

μs=4v23d2ω2μaθ2

where d is the distance between the two detector positions. The random error in µs’ is:

Δμsμs=2Δθθ+Δμaμa
(11)

The fact that Δµ′s /µ′s is always larger than Δµaa is obvious from Eq. (11). Although a translation stage with 0.02mm precision was used in measuring the displacement, there are still factors which could potentially contribute to the errors, such as the difficulty of determining the absolute values of the distance between the probe and the source. Error due to these distance measurements cannot be calculated using the above formula because this error is not random. Tables 3 and 4 demonstrate that estimates of µs’ are affected more by systematic errors than estimates of µa.

It can be seen that both random and non-random errors contribute more to µs’ than µa. This is found to be generally true for determining OPs in infinite media using fluence based detection schemes in the steady state or in the frequency domain. Other sources of systematic errors include the distortion of photon propagation in the presence of the detector tip and the fiber which is connected to it, and non-uniform acceptance of light by the supposedly isotropic probe. In high absorption media, the benchmark values of OPs for India ink, (Eq. 3) and Intralipid, (Eq. 6) may become less accurate because large aggregates of ink particles may form as depicted in the paper by Madsen et al [20

20. S. J. Madsen, M. S. Patterson, and B. C. Wilson, “The use of India ink as an optical absorber in tissue-simulating phantoms,” Phys. Med. Biol. 37, 985–993 (1992). [CrossRef] [PubMed]

]. This may potentially contribute to systematic errors.

5. Discussion and conclusion

The work presented here is intended as a tool to predict the biological effect induced by PDT treatment. By measuring the OPs of the target tissue we are able to calculate light fluence and drug concentration. Knowledge of light fluence and photosensitizer concentration (called explicit dosimetry) is one possible method to predict the biological damage during PDT. Other alternatives exist, such as measuring phosphorescence of the photosensitizer or directly measuring the amount of singlet oxygen using the oxygen luminescence [31

31. M. Niedre, M. S. Patterson, and B. C. Wilson, “Direct near-infrared luminescence detection of singlet oxygen generated by photodynamic therapy in cells in vitro and tissues in vivo,” Photochem. Photobio. 75, 382–391 (2002). [CrossRef]

], which we will not discuss here.

The drug, Tookad is currently under investigation for treating prostate cancer and shows promising results [32

32. Q. Chen, Z. Huang, D. Luck, J. Beckers, P. Brun, B. C. Wilson, A. Scherz, Y. Salomon, and F. W. Hetzel, “Preclinical studies in normal canine prostate of a novel palladium-bacteriopheophorbide (WST09) photosensitizer for photodynamic therapy of prostate cancer,” Photochem. Photobio. 76, 438–445, (2002). [CrossRef]

, 33

33. Z. Huang, Q. Chen, N. Trncic, S. M. LaRue, P. Brun, B. C. Wilson, H. Shapiro, and F. W. Hetzel, “Effects of Pd-bacteriopheophorbide (Tookad) mediated photodynamic therapy on canine prostate pretreated with ionizing radiation,” Radiat. Res. 161, 723–731, (2004). [CrossRef] [PubMed]

]. It appears that there is very limited information about the in vivo values of absorption coefficient of Tookad. Weersink et al [34

34. R. A. Weersink, K. Diamond, B. C. Wilson, and M. S. Patterson, “Determination of the peak absorption wavelength and pharmacokinetics of WST11 in vivo using dynamic, spatially resolved diffuse reflectance spectroscopy in a rabbit model,” Technical report (2003).

, 35

35. R. A. Weersink, A. Bogaards, M Gertner, S. R. H. Davidson, K. Zhang, G. Netchev, J. Trachtenberg, and B. C. Wilson, “Techniques for delivery and monitoring of TOOKAD (WST09)-mediated photodynamic therapy of the prostate: clinical experience and practicalities,” J. Photochem. Photobiol B: Biology. 79. 211–222 (2005). [CrossRef] [PubMed]

] used a steady state diffuse reflectance method applied on skins of rabbits and pigs, and determined the absorption at its peak value could be as high as 0.2mm-1 for 5mg/kg injection dose of WST11 (a hydrophilic form of Tookad). Considering the reduced scattering of prostate can be as low as 0.3 mm-1 [6

6. T. C. Zhu, S. M. Hahn, A. S. Kapatkin, A. Dimofte, C. E. Rodriguez, T. G. Vulcan, E. Glatstein, and R. A. His, “In-vivo optical properties of normal canine prostate at 732nm using motexafin lutetium mediated photodynamic therapy,” Photochem. Photobiol. 77, 81–88 (2003). [CrossRef] [PubMed]

] in canine prostate (although the wavelengths in two cases are slightly different), diffusion theory cannot give satisfactory results. Our work has shown that ISM/SDA or MSH can successfully address this situation. It is well known that PDT effects are reduced in hypoxia, and that hypoxia can be caused by high light fluence rate that leads to rapid consumption of oxygen [36

36. T. H. Foster and L. Gao, “Dosimetry in photodynamic therapy: oxygen and the critical importance of capillary density,” Radiat. Res. 130, 379–383, (1992). [CrossRef] [PubMed]

]. If severe oxygen-depletion occurs during the treatment, a dose metric that uses only light fluence and photosensitizer concentration will not be a satisfactory indicator of true biological damage. Hence monitoring pO2 during the treatment is desired. Many methods, such as Eppendorf probes or phosphorescence or fluorescence of exogenous probe molecules can be used to monitor partial oxygen tension pO2. If we intend to add no more instrumentation to the current system, it may be possible to use the absorption spectrum to determine hemoglobin saturation and pO2 [37

37. 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 during motexafin lutetium mediated photodynamic therapy,” J. Photochem. Photobiol B: Biology. 79, 231–241 (2005). [CrossRef] [PubMed]

]. Such an instrument would require a radio-frequency modulated source at several wavelengths, or a combination of frequency domain and continuous wave measurements as suggested by Bevilacqua et al [38

38. F. Bevilacqua, A. J. Berger, A. E. Cerussi, D. Jakubowski, and B. J. Tromberg, “Broadband absorption spectroscopy in turbid media by combined frequency-domain and steady-state methods,” Appl. Opt. 39, 6498–6507 (2000). [CrossRef]

] for reflectance.

In conclusion, our phantom experiments showed that interstitial measurement at two positions in the frequency domain can recover absorption within 10% and reduced scattering to slightly over 10% using either the data collected from two detectors 3mm apart or using those from two detectors 5mm apart with light collected by one detector attenuated by a neutral density filter. This was true for a wide range of OPs that can possibly be encountered in clinical PDT of the prostate, µa=0.008 to 0.17mm-1 and µs’=0.3 to 1.8mm-1. The experimental results confirmed the theoretical prediction that ISM/SDA can recover OPs within 5–10% for µs’/µa>5. The MSH method can recover OPs within 10% for µs’/µa>1, but generally greater than 10% if µs’/µa<1. We believe this to be caused by the large relative uncertainty in measured phase in high absorption media. Monte Carlo simulation showed that detectors should be situated at least 3mm away from the source in order to avoid the effects of source anisotropy. The simultaneous shift of two detectors either toward or away from the source by an amount S is less important than the situation where one detector is in the right position and the other is off by the same amount S. This suggests that accurate measurement for the distance between two detectors is required. The optimum strategy is probably to design a probe using 2 detector fibers in a fixed geometry, or even to fix all 3 optical fibers (one source, two detectors) in a single probe. Phase amplitude crosstalk, according to our findings, was practically insignificant. It is anticipated that the method described in this article will be used in future PDT dosimetry.

Appendix A: List of formalism in the MSH model

The phase function can be expanded in terms of spherical harmonics (note, in the rotated frame), its coefficients are denoted by f1 . When the phase function is chosen to be of Henyey-Greenstein type, f1 is equal to gl where g is the anisotropy factor. In the frequency domain, the modulation frequency is f (angular frequency ω=2πf). µ˜a is defined as µa-iω/, where µa is the absorption, and v is the light speed in the tissue. µs denotes the scattering coefficient. Define the matrix R, whose elements are:

Rll=blδl,l1+bl+1δl,l+1
(A1)

where bl=1(2l+1)(2l1). Define the diagonal matrix S, whose elements are:

Sll=μlδll
(A2)

whereµl =µ˜a-µs (1-f1 ) l′s take non-negative integers, δij denotes the Kronecker symbol. Define the matrix W=S-1RS-1, whose eigenvectors and eigenvalues are denoted by yn and λn, respectively, and n denotes the index of the eigenvectors and eigenvalues associated with the W. The fluence due to an isotropic point source can be written as:

φ(r)=2μ0rn0|ynyn|0λn2exp(rλn)
(A3)

where the sum is over all possible positive eigenvalues. In practice, the series is truncated. Setting the order of matrix W at 30 will be satisfactory in all biological tissues. The measured fluence is the absolute value of φ(r⃑). The phase is the inverse tangent of the ratio of the imaginary part and real part of φ(r⃑).

Acknowledgments

This research was supported by the National Institutes of Health, P01-CA43892.

References and links

1.

R. Richards-Kortum and E. Sevick-Muraca, “Quantitative optical spectroscopy for tissue diagnosis,” Annu. Rev. Phys. Chem. 47, 555–606 (1996). [CrossRef] [PubMed]

2.

R. J. Hunter, M. S. Patterson, T. J. Farrell, and J. E. Hayward, “Haemoglobin oxygenation of a two-layer tissue-simulating phantom from time-resolved reflectance: effect of top layer thickness,” Phys Med Biol. 47, 193–208 (2002). [CrossRef] [PubMed]

3.

R. Weissleder and V. Ntziachristos, “Shedding light onto live molecule targets,” Nature. Med. 9, 123–128 (2003). [CrossRef] [PubMed]

4.

M. S. Patterson and B. C. Wilson, “Photodynamic therapy,” in The Modern Technology of Radiation Oncology, J. Van Dyk, eds. (Medical Physics Publishing, Madison, Wisconsin1999), Chap. 23.

5.

Z. Huang, “A review of progress in clinical photodynamic therapy,” Technol. Cancer. Res. Treat. 4, 283–293 (2005). [PubMed]

6.

T. C. Zhu, S. M. Hahn, A. S. Kapatkin, A. Dimofte, C. E. Rodriguez, T. G. Vulcan, E. Glatstein, and R. A. His, “In-vivo optical properties of normal canine prostate at 732nm using motexafin lutetium mediated photodynamic therapy,” Photochem. Photobiol. 77, 81–88 (2003). [CrossRef] [PubMed]

7.

T. C. Zhu, A. Dimofte, F. C. Finlay, D. Stripp, T. Bush, J. Miles, R. Whittington, S. B. Malkowicz, Z. Tochner, E. Glatstein, and S. M. Hahn, “Optical properties of human prostate at 732nm measured in-vivo during motexafin lutetium mediated photodynamic therapy,” Photochem. Photobiol. 81, 96–105 (2005). [CrossRef]

8.

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, 2291–2311 (2005). [CrossRef] [PubMed]

9.

H. Xu, T. J. Farrell, and M. S. Patterson, “Investigation of light propagation models to determine the optical properties of tissue from interstitial frequency domain fluence measurements,” J. Biomed. Opt. (to be published 2006). [CrossRef] [PubMed]

10.

V. A. Markel, “Modified spherical harmonics method for solving the radiative transport equation,” Waves Random Media 14, L13–L19 (2004). [CrossRef]

11.

G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A , 39, 115–137 (2006). [CrossRef]

12.

H. Xu and M. S. Patterson, “Application of the modified spherical harmonics method to some problems in biomedical optics,” Phys. Med. Biol. 51, N247–N251 (2006). [CrossRef] [PubMed]

13.

J. B. Fishkin and E. Gratton, “Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge,” J. Opt. Soc. Am. A. 10, 127–140 (1993). [CrossRef] [PubMed]

14.

S. L. Jacques and L. Wang, “Monte Carlo modeling of light transport in tissues,” in Optical-thermal response of laser-irradiated tissue, A. J. Welch and M. J. van Gemert, eds. (Plenum Press, New York1995), Chap. 4.

15.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83, (1941). [CrossRef]

16.

P. R. Bevington and D. K. Robinson, Data reduction and error analysis for the physical sciences (2nd ed.McGraw-Hill, Inc.1992).

17.

H. J. van Staveren, C. J. M. Moes, J. Van Marle, S. A. Prahl, and M. J. C. Van Gemert, “Light scattering in Intralipid-10% in the wavelength range of 400–1100 nm,” Appl. Opt. 30, 4507–4514 (1997). [CrossRef]

18.

S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, and M. J. C. Van Gemert, “Optical properties of Liposyn: a phantom medium for light propagation studies,” Lasers Surg. Med. 12, 510–519 (1992). [CrossRef] [PubMed]

19.

R. C. Smith and K.S. Baker, “Optical properties of the clearest natural waters (200–800nm),” Appl. Opt. 20, 177–184 (1981). [CrossRef] [PubMed]

20.

S. J. Madsen, M. S. Patterson, and B. C. Wilson, “The use of India ink as an optical absorber in tissue-simulating phantoms,” Phys. Med. Biol. 37, 985–993 (1992). [CrossRef] [PubMed]

21.

L. Lilge and B. C. Wilson, “The accuracy of interstitial measurements of absolute light fluence rate in the determination of tissue optical properties,” in Proceedings of Laser-tissue Interaction IV, S. L. Jacques and A. Katzir, eds., Proc SPIE1882, 291–304 (1993). [CrossRef]

22.

N. Ramanujam, C. Du, H. Y. Ma, and B. Chance, “Sources of phase noise in homodyne phase modulation devices used for tissue oximetry studies,” Rev. Sci. Instrum. 69, 3042–3054 (1998). [CrossRef]

23.

I. Nissila, K. Kotilahti, K. Fallstrom, and T. Katila, “Instrumentation for the accurate measurement of phase and amplitude in optical tomography,” Rev. Sci. Instrum. 73, 3306–3312 (2002). [CrossRef]

24.

S. Yokoyama and A. Okamoto, “Examination to eliminate undesirable phase delay of an avalanche photodiode (APD) for intensity-modulated light,” Rev. Sci. Instrum. 66, 5331–5336 (1995). [CrossRef]

25.

K. Alford and Y. Wickramasinghe, “Phase-amplitude crosstalk in intensity modulated near infrared spectroscopy,” Rev. Sci. Instrum. 71, 2191–2195 (2000). [CrossRef]

26.

B. Chance, M. Cope, E. Gratton, N. Ramanujam, and B. Tromberg, “Phase measurement of light absorption and scatter in human tissue,” Rev. Sci. Instrum. 69, 3457–3481 (1998). [CrossRef]

27.

L. Lilge, N. Pomerleau-Dalcourt, A. Douplik, S. H. Selman, R. W. Keck, M. Szkudlarek, M. Pestka, and J. Jankun, “Transperineal in vivo fluence-rate dosimetry in the canine prostate during SnET2-mediated PDT,” Phys. Med. Biol. 49, 3209–3225 (2004). [CrossRef] [PubMed]

28.

S. Gross, A. Gilead, A. Scherz, M. Neeman, and Y. Salomon, “Monitoring photodynamic therapy of solid tumors on-line by BOLD-contrast MRI,” Nat. Med. 9, 1327–1331 (2003). [CrossRef] [PubMed]

29.

M. Gerken and G. W. Faris, “High-precision frequency-domain measurements of the optical properties of turbid media,” Opt. Lett. 24, 930–932, (1999). [CrossRef]

30.

J. B. Fishkin, P. T. C. So, A.E. Cerussi, S. Fantini, M. A. Franceschini, and E. Gratton, “Frequency-domain method for measuring spectral properties in multiple-scattering media: methemoglobin absorption spectrum in a tissuelike phantom,” Appl. Opt. 34, 1143–1155 (1995). [CrossRef] [PubMed]

31.

M. Niedre, M. S. Patterson, and B. C. Wilson, “Direct near-infrared luminescence detection of singlet oxygen generated by photodynamic therapy in cells in vitro and tissues in vivo,” Photochem. Photobio. 75, 382–391 (2002). [CrossRef]

32.

Q. Chen, Z. Huang, D. Luck, J. Beckers, P. Brun, B. C. Wilson, A. Scherz, Y. Salomon, and F. W. Hetzel, “Preclinical studies in normal canine prostate of a novel palladium-bacteriopheophorbide (WST09) photosensitizer for photodynamic therapy of prostate cancer,” Photochem. Photobio. 76, 438–445, (2002). [CrossRef]

33.

Z. Huang, Q. Chen, N. Trncic, S. M. LaRue, P. Brun, B. C. Wilson, H. Shapiro, and F. W. Hetzel, “Effects of Pd-bacteriopheophorbide (Tookad) mediated photodynamic therapy on canine prostate pretreated with ionizing radiation,” Radiat. Res. 161, 723–731, (2004). [CrossRef] [PubMed]

34.

R. A. Weersink, K. Diamond, B. C. Wilson, and M. S. Patterson, “Determination of the peak absorption wavelength and pharmacokinetics of WST11 in vivo using dynamic, spatially resolved diffuse reflectance spectroscopy in a rabbit model,” Technical report (2003).

35.

R. A. Weersink, A. Bogaards, M Gertner, S. R. H. Davidson, K. Zhang, G. Netchev, J. Trachtenberg, and B. C. Wilson, “Techniques for delivery and monitoring of TOOKAD (WST09)-mediated photodynamic therapy of the prostate: clinical experience and practicalities,” J. Photochem. Photobiol B: Biology. 79. 211–222 (2005). [CrossRef] [PubMed]

36.

T. H. Foster and L. Gao, “Dosimetry in photodynamic therapy: oxygen and the critical importance of capillary density,” Radiat. Res. 130, 379–383, (1992). [CrossRef] [PubMed]

37.

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 during motexafin lutetium mediated photodynamic therapy,” J. Photochem. Photobiol B: Biology. 79, 231–241 (2005). [CrossRef] [PubMed]

38.

F. Bevilacqua, A. J. Berger, A. E. Cerussi, D. Jakubowski, and B. J. Tromberg, “Broadband absorption spectroscopy in turbid media by combined frequency-domain and steady-state methods,” Appl. Opt. 39, 6498–6507 (2000). [CrossRef]

OCIS Codes
(170.5180) Medical optics and biotechnology : Photodynamic therapy
(170.5270) Medical optics and biotechnology : Photon density waves
(170.7050) Medical optics and biotechnology : Turbid media

ToC Category:
Medical Optics and Biotechnology

History
Original Manuscript: May 11, 2006
Revised Manuscript: June 15, 2006
Manuscript Accepted: June 27, 2006
Published: July 10, 2006

Virtual Issues
Vol. 1, Iss. 8 Virtual Journal for Biomedical Optics

Citation
Heping Xu and Michael 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, 6485-6501 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-14-6485


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References

  1. R. Richards-Kortum and E. Sevick-Muraca, "Quantitative optical spectroscopy for tissue diagnosis," Annu. Rev. Phys. Chem. 47,555-606 (1996). [CrossRef] [PubMed]
  2. R. J. Hunter, M. S. Patterson, T. J. Farrell, and J. E. Hayward, "Haemoglobin oxygenation of a two-layer tissue-simulating phantom from time-resolved reflectance: effect of top layer thickness," Phys Med Biol. 47,193-208 (2002). [CrossRef] [PubMed]
  3. R. Weissleder and V. Ntziachristos, "Shedding light onto live molecule targets," Nature. Med. 9,123-128 (2003). [CrossRef] [PubMed]
  4. M. S. Patterson and B. C. Wilson, "Photodynamic therapy," in The Modern Technology of Radiation Oncology, J. Van Dyk, eds. (Medical Physics Publishing, Madison, Wisconsin 1999), Chap. 23.
  5. Z. Huang, "A review of progress in clinical photodynamic therapy," Technol. Cancer. Res. Treat. 4,283-293 (2005). [PubMed]
  6. T. C. Zhu, S. M. Hahn, A. S. Kapatkin, A. Dimofte, C. E. Rodriguez, T. G. Vulcan, E. Glatstein, and R. A. His, "In-vivo optical properties of normal canine prostate at 732nm using motexafin lutetium mediated photodynamic therapy," Photochem. Photobiol. 77,81-88 (2003). [CrossRef] [PubMed]
  7. T. C. Zhu, A. Dimofte, F. C. Finlay, D. Stripp, T. Bush, J. Miles, R. Whittington, S. B. Malkowicz, Z. Tochner, E. Glatstein, and S. M. Hahn, "Optical properties of human prostate at 732nm measured in-vivo during motexafin lutetium mediated photodynamic therapy," Photochem. Photobiol. 81,96-105 (2005). [CrossRef]
  8. 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. 50,2291-2311 (2005). [CrossRef] [PubMed]
  9. H. Xu, T. J. Farrell, and M. S. Patterson, "Investigation of light propagation models to determine the optical properties of tissue from interstitial frequency domain fluence measurements," J. Biomed. Opt.(to be published2006). [CrossRef] [PubMed]
  10. V. A. Markel, "Modified spherical harmonics method for solving the radiative transport equation," Waves Random Media 14, L13-L19 (2004). [CrossRef]
  11. G. Panasyuk, J. C. Schotland, and V. A. Markel, "Radiative transport equation in rotated reference frames," J. Phys. A,  39,115-137 (2006). [CrossRef]
  12. H. Xu and M. S. Patterson, "Application of the modified spherical harmonics method to some problems in biomedical optics," Phys. Med. Biol. 51, N247-N251 (2006). [CrossRef] [PubMed]
  13. J. B. Fishkin and E. Gratton, "Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge," J. Opt. Soc. Am. A. 10,127-140 (1993). [CrossRef] [PubMed]
  14. S. L. Jacques, and L. Wang, "Monte Carlo modeling of light transport in tissues," in Optical-thermal response of laser-irradiated tissue, A. J. Welch and M. J. van Gemert, eds. (Plenum Press, New York 1995), Chap. 4.
  15. L. G. Henyey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 93,70-83, (1941). [CrossRef]
  16. P. R. Bevington and D. K. Robinson, Data reduction and error analysis for the physical sciences (2nd ed. McGraw-Hill, Inc. 1992).
  17. H. J. van Staveren, C. J. M. Moes, J. Van Marle, S. A. Prahl, and M. J. C. Van Gemert, "Light scattering in Intralipid-10% in the wavelength range of 400-1100 nm," Appl. Opt. 30,4507-4514 (1997). [CrossRef]
  18. S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, and M. J. C. Van Gemert, "Optical properties of Liposyn: a phantom medium for light propagation studies," Lasers Surg. Med. 12,510-519 (1992). [CrossRef] [PubMed]
  19. R. C. Smith and K.S. Baker, "Optical properties of the clearest natural waters (200-800nm)," Appl. Opt. 20,177-184 (1981). [CrossRef] [PubMed]
  20. S. J. Madsen, M. S. Patterson, and B. C. Wilson, "The use of India ink as an optical absorber in tissue-simulating phantoms," Phys. Med. Biol. 37, 985-993 (1992). [CrossRef] [PubMed]
  21. L. Lilge and B. C. Wilson, "The accuracy of interstitial measurements of absolute light fluence rate in the determination of tissue optical properties," in Proceedings of Laser-tissue Interaction IV, S. L. Jacques and A. Katzir, eds., Proc SPIE 1882, 291-304 (1993). [CrossRef]
  22. N. Ramanujam, C. Du, H. Y. Ma, and B. Chance, "Sources of phase noise in homodyne phase modulation devices used for tissue oximetry studies," Rev. Sci. Instrum. 69,3042-3054 (1998). [CrossRef]
  23. I. Nissila, K. Kotilahti, K. Fallstrom, and T. Katila, "Instrumentation for the accurate measurement of phase and amplitude in optical tomography," Rev. Sci. Instrum. 73,3306-3312 (2002). [CrossRef]
  24. S. Yokoyama and A. Okamoto, "Examination to eliminate undesirable phase delay of an avalanche photodiode (APD) for intensity-modulated light," Rev. Sci. Instrum. 66,5331-5336 (1995). [CrossRef]
  25. K. Alford and Y. Wickramasinghe, "Phase-amplitude crosstalk in intensity modulated near infrared spectroscopy," Rev. Sci. Instrum. 71,2191-2195 (2000). [CrossRef]
  26. B. Chance, M. Cope, E. Gratton, N. Ramanujam, and B. Tromberg, "Phase measurement of light absorption and scatter in human tissue," Rev. Sci. Instrum. 69,3457-3481 (1998). [CrossRef]
  27. L. Lilge, N. Pomerleau-Dalcourt, A. Douplik, S. H. Selman, R. W. Keck, M. Szkudlarek, M. Pestka, and J. Jankun, "Transperineal in vivo fluence-rate dosimetry in the canine prostate during SnET2-mediated PDT," Phys. Med. Biol. 49,3209-3225 (2004). [CrossRef] [PubMed]
  28. S. Gross, A. Gilead, A. Scherz, M. Neeman, and Y. Salomon, "Monitoring photodynamic therapy of solid tumors on-line by BOLD-contrast MRI," Nat. Med. 9, 1327-1331 (2003). [CrossRef] [PubMed]
  29. M. Gerken and G. W. Faris, "High-precision frequency-domain measurements of the optical properties of turbid media," Opt. Lett. 24,930-932, (1999). [CrossRef]
  30. J. B. Fishkin, P. T. C. So, A.E. Cerussi, S. Fantini, M. A. Franceschini, and E. Gratton, "Frequency-domain method for measuring spectral properties in multiple-scattering media: methemoglobin absorption spectrum in a tissuelike phantom," Appl. Opt. 34,1143-1155 (1995). [CrossRef] [PubMed]
  31. M. Niedre, M. S. Patterson, and B. C. Wilson, "Direct near-infrared luminescence detection of singlet oxygen generated by photodynamic therapy in cells in vitro and tissues in vivo," Photochem. Photobio. 75,382-391 (2002). [CrossRef]
  32. Q. Chen, Z. Huang, D. Luck, J. Beckers, P. Brun, B. C. Wilson, A. Scherz, Y. Salomon, and F. W. Hetzel, "Preclinical studies in normal canine prostate of a novel palladium-bacteriopheophorbide (WST09) photosensitizer for photodynamic therapy of prostate cancer," Photochem. Photobio. 76,438-445, (2002). [CrossRef]
  33. Z. Huang, Q. Chen, N. Trncic, S. M. LaRue, P. Brun, B. C. Wilson, H. Shapiro, and F. W. Hetzel, "Effects of Pd-bacteriopheophorbide (Tookad) mediated photodynamic therapy on canine prostate pretreated with ionizing radiation," Radiat. Res. 161, 723-731, (2004). [CrossRef] [PubMed]
  34. R. A. Weersink, K. Diamond, B. C. Wilson, and M. S. Patterson, "Determination of the peak absorption wavelength and pharmacokinetics of WST11 in vivo using dynamic, spatially resolved diffuse reflectance spectroscopy in a rabbit model," Technical report (2003).
  35. R. A. Weersink, A. Bogaards, M, Gertner, S. R. H. Davidson, K. Zhang, G. Netchev, J. Trachtenberg, and B. C. Wilson, "Techniques for delivery and monitoring of TOOKAD (WST09)-mediated photodynamic therapy of the prostate: clinical experience and practicalities," J. Photochem. Photobiol B: Biology. 79. 211-222 (2005). [CrossRef] [PubMed]
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