OSA's Digital Library

Applied Optics

Applied Optics

APPLICATIONS-CENTERED RESEARCH IN OPTICS

  • Vol. 45, Iss. 5 — Feb. 10, 2006
  • pp: 1062–1071

Optics InfoBase > Applied Optics > Volume 45 > Issue 5 > Page 1062

« Show journal navigation

Monte Carlo-based inverse model for calculating tissue optical properties. Part I: Theory and validation on synthetic phantoms

Gregory M. Palmer and Nirmala Ramanujam  »View Author Affiliations


Applied Optics, Vol. 45, Issue 5, pp. 1062-1071 (2006)
http://dx.doi.org/10.1364/AO.45.001062


View Full Text Article

Acrobat PDF (163 KB)


Advanced Search

Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

A flexible and fast Monte Carlo-based model of diffuse reflectance has been developed for the extraction of the absorption and scattering properties of turbid media, such as human tissues. This method is valid for a wide range of optical properties and is easily adaptable to existing probe geometries, provided a single phantom calibration measurement is made. A condensed Monte Carlo method was used to speed up the forward simulations. This model was validated by use of two sets of liquid-tissue phantoms containing Nigrosin or hemoglobin as absorbers and polystyrene spheres as scatterers. The phantoms had a wide range of absorption ( 0 – 20   cm − 1 ) and reduced scattering coefficients ( 7 – 33   cm − 1 ) . Mie theory and a spectrophotometer were used to determine the absorption and reduced scattering coefficients of the phantoms. The diffuse reflectance spectra of the phantoms were measured over a wavelength range of 350 – 8 5 0   nm . It was found that optical properties could be extracted from the experimentally measured diffuse reflectance spectra with an average error of 3 % or less for phantoms containing hemoglobin and 12 % or less for phantoms containing Nigrosin.

© 2006 Optical Society of America

1. Introduction

Diffuse reflectance spectroscopy is sensitive to the absorption and scattering properties of tissue. The absorption coefficient is directly related to the concentration of the chromophores, and the scattering coefficient reflects the size and density of scattering structures in the tissue. However, the combined influence of the absorption and scattering events on diffusely reflected light in tissue makes it difficult to interpret the physically meaningful information present in diffuse reflectance measurements, since a change in measured diffuse reflectance could reflect a change in absorption, scattering, or both. The ability to separately quantify the independent absorption and scattering coefficients of tissue from a diffuse reflectance spectrum will allow the extraction of the physiological and structural properties of the sample that influence the diffuse reflectance. Knowledge of these sources of optical contrast can provide insight into the underlying mechanisms that permit diagnosis and lead to an improvement in diagnostic algorithms.

Several groups have developed analytical and numerical models to extract the absorption and scattering coefficients of tissue from fiber-optic-based measurements of diffuse reflectance spectra measured in the ultraviolet–visible (UV–VIS) part of the electromagnetic spectrum. In general, these methods can be categorized into those that use an analytical approximation of the transport equation such as the diffusion equation,[1

G. Zonios, L. T. Perelman, V. Backman, R. Manoharan, M. Fitzmaurice, J. Van-Dam, and M. S. Feld, “Diffuse reflectance spectroscopy of human adenomatous colon polyps in vivo, ” Appl. Opt. 38, 6628–6637 (1999). [CrossRef]

, 2

N. Ghosh, S. K. Mohanty, S. K. Majumder, and P. K. Gupta, “Measurement of optical transport properties of normal and malignant human breast tissue,” Appl. Opt. 40, 176–184 (2001). [CrossRef]

, 3

J. C. Finlay and T. H. Foster, “Hemoglobin oxygen saturations in phantoms and in vivo from measurements of steady-state diffuse reflectance at a single, short source–detector separation,” Med. Phys. 31, 1949–1959 (2004). [CrossRef] [PubMed]

] a modified form of the Monte Carlo model,[4

P. Thueler, I. Charvet, F. Bevilacqua, M. St. Ghislain, G. Ory, P. Marquet, P. Meda, B. Vermeulen, and C. Depeursinge, “ In vivo endoscopic tissue diagnostics based on spectroscopic absorption, scattering, and phase function properties,” J. Biomed. Opt. 8, 495–503 (2003). [CrossRef] [PubMed]

] and empirical methods (for example, neural networks).[5

T. J. Pfefer, L. S. Matchette, C. L. Bennett, J. A. Gall, J. N. Wilke, A. J. Durkin, and M. N. Ediger, “Reflectance-based determination of optical properties in highly attenuating tissue,” J. Biomed. Opt. 8, 206–215 (2003). [CrossRef] [PubMed]

, 6

A. Amelink, H. J. Sterenborg, M. P. Bard, and S. A. Burgers, “ In vivo measurement of the local optical properties of tissue by use of differential path-length spectroscopy,” Opt. Lett. 29, 1087–1089 (2004). [CrossRef] [PubMed]

] The diffusion equation is valid for low to moderate tissue absorption, i.e., for the case in which tissues are scattering dominant and for relatively large photon travel paths, which is the case at red and near-infrared (NIR) wavelengths. Monte Carlo modeling is a numerical technique that is valid for a wide range of absorption and scattering coefficients and photon paths and thus can be used to model light transport over the entire UV–VIS–NIR range. However, unlike the diffusion equation, this technique is computationally intensive. The previously developed models are briefly discussed below.

Zonios et al.[1

G. Zonios, L. T. Perelman, V. Backman, R. Manoharan, M. Fitzmaurice, J. Van-Dam, and M. S. Feld, “Diffuse reflectance spectroscopy of human adenomatous colon polyps in vivo, ” Appl. Opt. 38, 6628–6637 (1999). [CrossRef]

] developed a forward model of light transport based on the diffusion approximation, which expressed the diffuse reflectance spectra as functions of the wavelength-dependent reduced scattering coefficient and absorption coefficient, an empirically determined constant related to the probe geometry, and an empirically determined adjustment to the diffusion equation for use with high-absorption coefficients. The empirical correction factors were determined by use of tissue phantom studies. The model was constrained to assume that hemoglobin is the only absorber that contributes to the wavelength-dependent absorption coefficient, and the reduced scattering coefficient was constrained to be always decreasing with increasing wavelength. A nonlinear least-squares optimization routine was then employed to minimize the difference between the measured and the fitted diffuse reflectance spectrum. The extracted reduced scattering coefficient was fit to Mie theory to extract the scatterer size and scatterer density. This model was then used to extract the total hemoglobin concentration, scatterer size, and scatterer density from diffuse reflectance spectra of colon tissues measured over the wavelength range of 350–700  nm. Both total hemoglobin concentration and scatterer size increased while scatterer density decreased in adenomatous colon polyps compared with normal tissues.

Finlay and Foster[3

J. C. Finlay and T. H. Foster, “Hemoglobin oxygen saturations in phantoms and in vivo from measurements of steady-state diffuse reflectance at a single, short source–detector separation,” Med. Phys. 31, 1949–1959 (2004). [CrossRef] [PubMed]

] improved on the standard diffusion equation described above by using a more complex model of light transport. They used a P3 approximation of the radiative transport equation, which incorporates higher-order moments of anisotropy than the standard diffusion approximation. They found that this modification performed better than the standard diffusion equation at relatively short source–detector separations ( >1  mm) and high-absorption coefficients corresponding to those at UV–VIS wavelengths. Furthermore, this model was adaptable to complex fiber-optic probe geometries. However, systematic deviations in the absorber concentration were observed at source–detector separations of less than 1  mm and for high absorber concentrations.

Ghosh et al.[2

N. Ghosh, S. K. Mohanty, S. K. Majumder, and P. K. Gupta, “Measurement of optical transport properties of normal and malignant human breast tissue,” Appl. Opt. 40, 176–184 (2001). [CrossRef]

] used a model based on the standard diffusion approximation to extract the absorption and scattering coefficients from spatially resolved diffuse reflectance measurements. They measured the spatially resolved diffuse reflectance at 12 source–detector separations, ranging from 1.2 to 12  mm, over the wavelength range of 450–650  nm from human breast tissue samples. The absorption and scattering coefficients were determined from fits to the diffusion approximation of the spatially resolved diffuse reflectance at each individual wavelength. The results of their study indicated that malignant tissues are more absorbing and scattering than normal tissues are at all wavelengths.

Thueler et al.[4

P. Thueler, I. Charvet, F. Bevilacqua, M. St. Ghislain, G. Ory, P. Marquet, P. Meda, B. Vermeulen, and C. Depeursinge, “ In vivo endoscopic tissue diagnostics based on spectroscopic absorption, scattering, and phase function properties,” J. Biomed. Opt. 8, 495–503 (2003). [CrossRef] [PubMed]

] used a model based on Monte Carlo simulations to extract absorption and scattering coefficients from spatially resolved diffuse reflectance measurements. They used a scaling procedure to increase the efficiency of their Monte Carlo model. However, the scaling procedure was valid for only a limited range of absorption[7

F. Bevilacqua and C. Depeursinge, “Monte Carlo study of diffuse reflectance at source–detector separations close to one transport mean free path,” J. Opt. Soc. Am. A 16, 2935–2945 (1999). [CrossRef]

] ( albedo >0.9). They measured spatially resolved diffuse reflectance measurements of esophageal tissues, using ten source– detector separations ranging from 0.3 to 1.35  mm, over a wavelength range of 480–950  nm, and fit the measurements at each wavelength to the Monte Carlo model. Their results showed that there are significant differences in the absorption and scattering coefficients of the normal antrum and fundus of the stomach.

Pfefer et al.[5

T. J. Pfefer, L. S. Matchette, C. L. Bennett, J. A. Gall, J. N. Wilke, A. J. Durkin, and M. N. Ediger, “Reflectance-based determination of optical properties in highly attenuating tissue,” J. Biomed. Opt. 8, 206–215 (2003). [CrossRef] [PubMed]

] developed an empirical method for the extraction of absorption and scattering coefficients from spatially resolved diffuse reflectance measurements. They employed a neural-network algorithm, which was trained on phantoms, and then used this neural-network model to extract optical properties from another set of phantoms. The spatially resolved diffuse reflectance was measured at six source–detector separations between 0.23 and 2.46  mm at a wavelength of 675  mm from phantoms with absorption coefficients ranging from 1  to   25   cm −1 and reduced scattering coefficients ranging from 5  to   25   cm −1. The neural-network model was able to extract the absorption and scattering coefficients of the phantoms to within root-mean-square (rms) errors of ±2  and± 3⁢ ⁢cm −1, respectively.

Amelink et al.[6

A. Amelink, H. J. Sterenborg, M. P. Bard, and S. A. Burgers, “ In vivo measurement of the local optical properties of tissue by use of differential path-length spectroscopy,” Opt. Lett. 29, 1087–1089 (2004). [CrossRef] [PubMed]

] employed a method they referred to as differential path-length spectroscopy to extract the absorption and scattering coefficients from diffuse reflectance spectra of tissues measured with a specific probe geometry. By subtracting the reflectance measured from two adjacent fibers at a given wavelength, they found that the resulting output was sensitive to a superficial portion of the tissue, and they were able to express it as a simple function of the absorption and scattering coefficients. They then constrained absorption and scattering coefficients in the model to be functions of the hemoglobin absorption and Mie scattering, respectively. They fit the diffuse reflectance spectra measured over a wavelength range of 350–1000  nm from normal and malignant bronchial mucosa to this model. They found a statistically significant decrease in blood oxygenation in malignant tissue compared with normal tissues.

Table 1 summarizes the previously described methods for extracting optical properties including the light-transport model described in this paper, the absorption range for which these models are valid, the calibration requirement for the use of each method, and the probe-geometry requirements. High, moderate, and low absorptions correspond generally to tissue absorption in the UV, VIS, and NIR wavelength ranges, respectively. For multiple separation probe geometries, the range of source–detector separations used in the study is provided. Table 1 shows that the previously described methods are limited in that they place one or more of the following constraints on their applicability: They are limited to wavelengths corresponding to moderate to low absorption[2

N. Ghosh, S. K. Mohanty, S. K. Majumder, and P. K. Gupta, “Measurement of optical transport properties of normal and malignant human breast tissue,” Appl. Opt. 40, 176–184 (2001). [CrossRef]

, 4

P. Thueler, I. Charvet, F. Bevilacqua, M. St. Ghislain, G. Ory, P. Marquet, P. Meda, B. Vermeulen, and C. Depeursinge, “ In vivo endoscopic tissue diagnostics based on spectroscopic absorption, scattering, and phase function properties,” J. Biomed. Opt. 8, 495–503 (2003). [CrossRef] [PubMed]

] (VIS and NIR wavelengths), they impose constraints on the fiber-optic probe geometries used or require multiple illumination–collection fiber separations,[2

N. Ghosh, S. K. Mohanty, S. K. Majumder, and P. K. Gupta, “Measurement of optical transport properties of normal and malignant human breast tissue,” Appl. Opt. 40, 176–184 (2001). [CrossRef]

, 3

J. C. Finlay and T. H. Foster, “Hemoglobin oxygen saturations in phantoms and in vivo from measurements of steady-state diffuse reflectance at a single, short source–detector separation,” Med. Phys. 31, 1949–1959 (2004). [CrossRef] [PubMed]

, 4

P. Thueler, I. Charvet, F. Bevilacqua, M. St. Ghislain, G. Ory, P. Marquet, P. Meda, B. Vermeulen, and C. Depeursinge, “ In vivo endoscopic tissue diagnostics based on spectroscopic absorption, scattering, and phase function properties,” J. Biomed. Opt. 8, 495–503 (2003). [CrossRef] [PubMed]

, 5

T. J. Pfefer, L. S. Matchette, C. L. Bennett, J. A. Gall, J. N. Wilke, A. J. Durkin, and M. N. Ediger, “Reflectance-based determination of optical properties in highly attenuating tissue,” J. Biomed. Opt. 8, 206–215 (2003). [CrossRef] [PubMed]

, 6

A. Amelink, H. J. Sterenborg, M. P. Bard, and S. A. Burgers, “ In vivo measurement of the local optical properties of tissue by use of differential path-length spectroscopy,” Opt. Lett. 29, 1087–1089 (2004). [CrossRef] [PubMed]

] or they require extensive phantom studies for empirical calibration of the model.[1

G. Zonios, L. T. Perelman, V. Backman, R. Manoharan, M. Fitzmaurice, J. Van-Dam, and M. S. Feld, “Diffuse reflectance spectroscopy of human adenomatous colon polyps in vivo, ” Appl. Opt. 38, 6628–6637 (1999). [CrossRef]

, 5

T. J. Pfefer, L. S. Matchette, C. L. Bennett, J. A. Gall, J. N. Wilke, A. J. Durkin, and M. N. Ediger, “Reflectance-based determination of optical properties in highly attenuating tissue,” J. Biomed. Opt. 8, 206–215 (2003). [CrossRef] [PubMed]

] These limitations make these techniques difficult or impossible to implement on the large body of existing data that have been collected with a wide variety of probe geometries over the UV–VIS wavelength range. To address these concerns, this paper outlines a method for extracting optical properties, which is based on Monte Carlo modeling of light transport and is valid for a wide range of optical properties, including high absorption.[8

Q. Liu, C. Zhu, and N. Ramanujam, “Experimental validation of Monte Carlo modeling of fluorescence in tissues in the UV–visible spectrum,” J. Biomed. Opt. 8, 223–236 (2003). [CrossRef] [PubMed]

] The method presented here is validated for a specific probe geometry by use of phantom studies; however, it imposes no inherent constraints on the probe geometry used and can be easily adapted to any arbitrary probe geometry provided a single phantom calibration measurement is made.

2. Methods and Results

2A. Forward and Inverse Models

Forward model.

The forward model refers to the model that relates the physiological and structural properties of the tissue to its modeled diffuse reflectance. Figure 1(a) shows the forward Monte Carlo model of diffuse reflectance. The model has two sets of inputs, which are used to determine the absorption and scattering coefficients. The concentration ( Ci) of each chromophore (free parameter) and the corresponding wavelength-dependent extinction coefficient [ εi (λ)] (fixed parameter) are used to determine the wavelength-dependent absorption coefficient [ μa (λ)], according to the relationship μa (λ)=∑ln ( 10) εi (λ) Ci. Mie theory for spherical particles[9

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

] is used to model scattering. The scatterer size and density are free parameters, and the refractive index is fixed according to known values for phantoms. The scattering coefficient [ μs (λ)] and the anisotropy factor [g(λ)] are then calculated at a given wavelength.

The optical properties μa (λ), μs (λ),and  g (λ) of the medium can then be input into a Monte Carlo model of light transport to obtain the modeled diffuse reflectance for a given wavelength. The code used for this purpose was adapted from that of Wang et al.[10

L. Wang, S. L. Jacques, and L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995). [CrossRef] [PubMed]

] Note that a Monte Carlo simulation would be required for each unique set of optical properties, thus making this step of the forward model computationally prohibitive. To increase the efficiency of this step, a scaling approach previously described by Graaff et al.[11

R. Graaff, M. H. Koelink, F. F. M. de Mul, W. G. Zijlstra, A. C. M. Dassel, and J. G. Aarnoudse, “Condensed Monte Carlo simulations for the description of light transport,” Appl. Opt. 32, 426–434 (1993). [CrossRef] [PubMed]

] was incorporated such that a single Monte Carlo simulation can be run for a particular set of optical properties, the output of which can be scaled to any set of optical properties. The method consists of running a single simulation for a given set of absorption ( μ a, sim) and scattering ( μ s, sim) coefficients and recording the exit weight ( W exit , sim), net distance traveled ( r t, sim), and total number of interactions for each photon (N) that exits the tissue surface. The scaling method then uses these stored parameters to calculate the new exit weight ( W exit , new) [Eq. (1)] and net distance traveled ( r t, new) [Eq. (2)] for a given photon that had a different absorption ( μ a,new) and scattering coefficient ( μ s,new) used in the same simulation. The scaling relationships, taken from Graaff et al.[11

R. Graaff, M. H. Koelink, F. F. M. de Mul, W. G. Zijlstra, A. C. M. Dassel, and J. G. Aarnoudse, “Condensed Monte Carlo simulations for the description of light transport,” Appl. Opt. 32, 426–434 (1993). [CrossRef] [PubMed]

] are
W exit,new= W exit,sim ( μ s, new μ s, new+ μ a, new  μ s, sim+ μ a, sim μ s, sim)N,
(1)
r t,new= r t,sim ( μ s, sim+ μ a, sim μ s, new+ μ a, new).
(2)
To further simplify the scaling process, it was assumed that, for a given value of the reduced scattering coefficient, μ s′= μ s ( 1−g), the diffuse reflectance would be the same for any values of μs and g that generate the same μ s′. This has been shown to be valid over the range of g values present in human tissue, i.e., for g values greater than 0.8.[11

R. Graaff, M. H. Koelink, F. F. M. de Mul, W. G. Zijlstra, A. C. M. Dassel, and J. G. Aarnoudse, “Condensed Monte Carlo simulations for the description of light transport,” Appl. Opt. 32, 426–434 (1993). [CrossRef] [PubMed]

, 12

A. Kienle and M. S. Patterson, “Determination of the optical properties of turbid media from a single Monte Carlo simulation,” Phys. Med. Biol. 41, 2221–2227 (1996). [CrossRef] [PubMed]

] By use of this similarity relation and the scaling procedure outlined above, only a single Monte Carlo simulation needed to be run to determine the output of a Monte Carlo simulation for any set of optical properties. The Henyey–Greenstein phase function was used in the single Monte Carlo simulation. The parameters of the single Monte Carlo simulation were as follows: number of photons, 40× 106; μs,150  cm −1; μa,0  cm −1; g,0.8; model dimensions, 2  cm   ( radius)×2cm   ( depth); refractive indices, 1.33 (medium for phantoms) and 1.452 (fiber-optic probe above medium).

Convolution was used to integrate over the illumination and collection fibers to determine the probability that a photon, traveling a fixed distance, would be collected for a given probe geometry. This takes advantage of the spatial invariance and rotational symmetry present in a homogeneous medium. For a pair of illumination and collection fibers, the probability of collection of a photon traveling a net distance rt between the points of entering and exiting the medium is given by
1 π 2 r i 2 ∫ max ( − r i,s− r t− r c) min ( r i,s− r t+ r c) ( s−x) cos⁡ −1 [ s 2+ ( s−x) 2− r i    2 2 ( s−x)s]                 ×  cos⁡ −1 [ r t      2+ ( s−x) 2− r i      2 2 ( s−x) r t]dx,
(3)
where ri is the radius of the illumination fiber, rc is the radius of the collection fiber, s is the separation between the centers of the illumination and the collection fibers, and x is the spatial variable over which the integral is taken (see appendixA for derivation). This equation was numerically integrated. To adapt this to the fiber bundle used in this study (the geometry of the fiber bundle is described below), the common end of the fiber bundle was imaged, and the centers of each illumination and collection fiber in the bundle were determined. Then the probe geometry was integrated pairwise (for each illumination–collection fiber pair) to determine the total probability of collection. It was found that imaging the fiber bundle to obtain the exact location of each illumination and collection fiber was necessary. Approximating the fiber bundle as solid rings of illumination and collection fibers produced significant errors in the model, likely because of imperfect physical placement of the fibers within the bundle.

It was found that the scaling process and subsequent numerical integration for the probe geometry of a large number of photons required approximately 1 s to complete, which, although much faster than running an independent simulation, was still rather slow for performing an inversion procedure. Therefore the diffuse reflectance values for a range of optical properties ( μ s,5–500  cm −1; μa,0–200  cm −1;  g,0.8) were determined ahead of time to form a lookup table and cubic splines were used to interpolate between table values. The smallest increment used in the lookup table was 0.1  cm −1 for μa and 2.5  cm −1 for μs.

2A1B. Inverse model.

Figure 1(b) shows the inverse Monte Carlo model of diffuse reflectance. First, an initial set of free parameters is input into the Monte Carlo-based forward model. These include the absorber concentrations and the scatterer size and density. The fixed parameters are the extinction coefficients of the absorber and the refractive index of the scatterer and the surrounding medium for a given wavelength. These input parameters are used in the forward model to generate the predicted reflectance as a function of wavelength. Next, the sum of squares error between the predicted and measured reflectance is computed. The free parameters are then iteratively updated until the sum of squares error is minimized. A Gauss–Newton nonlinear least-squares algorithm provided in the MATLAB optimization toolbox (Mathworks, Natick, Massachusetts) was used as the optimization algorithm. To ensure convergence to a global minimum, this procedure was repeated several times for each sample, with a different, randomly chosen set of starting parameters.

2B. Tissue Phantom Models

The forward and inverse models were tested on homogeneous tissue phantoms with optical properties representative of human tissues in the UV–VIS spectral range.[13

W.-F. Cheong, “Appendix to chapter 8: Summary of optical properties,” in Optical-Thermal Response of Laser-Irradiated Tissue , A. J. Welch and M. J. C. v. Gemert, eds. (Plenum, 1995), pp. 275–303.

] Two sets of liquid homogeneous phantoms were created, and all measurements were made the day the phantoms were made. The first set (phantom set 1) consisted of five phantoms containing variable concentrations of the absorber, hemoglobin, and the scatterer, 1 μm diameter polystyrene spheres (07310-15, Polysciences, Inc., Warrington, Pennsylvania). The hemoglobin used is water soluble and was obtained by the supplier from human blood (H0267, Sigma Co., St. Louis, Missouri). First, a solution with a fixed volume density of polystyrene spheres suspended in water was made and four titrations of a fixed amount of hemoglobin were added, with a diffuse reflectance measurement made after each titration. This produced a set of five phantoms (including the phantom with no absorber) with similar scattering properties and a range of absorption properties. The second set (phantom set 2) consisted of 25 phantoms containing variable concentrations of the same scatterer and a different absorber, Nigrosin (N-4754, Sigma Co., St. Louis, Missouri), to characterize the model's effectiveness by using an absorber with different absorption characteristics. For phantom set 2, five base phantoms were created, each with a different volume density of polystyrene spheres. For each base phantom, four titrations of a fixed volume of a Nigrosin solution were added, and a diffuse reflectance measurement was made after each titration. This produced a total of 25 phantoms (including the phantoms with no absorber) with a range of scattering and absorption coefficients. The phantoms were filled up to a height of at least 3  cm in a 1.6 cm diameter plastic container. The wavelength-dependent extinction coefficients for hemoglobin and Nigrosin were measured with a spectrophotometer (Cary 300, Varian, Palo Alto, California). It was assumed that the oxygenation of hemoglobin was constant throughout the course of the experiment. The reduced scattering coefficient was determined from Mie theory[9

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

] by use of freely available software,[14

S. Prahl, “Mie scattering program,” Oregon Medical Laser Center (2005), available at http://omlc.ogi.edu/software/mie/index.html.

] given the known size ( 1 μm), density, and refractive index of the spheres (1.60) and the surrounding medium, water (1.33). The refractive indices of polystyrene spheres have been reported to be constant to within approximately 1% of this value over the wavelength range used.[15

X. Ma, J. Q. Lu, R. S. Brock, K. M. Jacobs, P. Yang, and X. H. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48, 4165–4172 (2003). [CrossRef]

] Tables 2 and 3 show the means and ranges of the reduced scattering coefficient ( μ s′) and absorption coefficient ( μa), respectively, for each of the phantoms in phantom sets 1 and 2 over the wavelength range of 350–850  nm.

2C. Measurement System and Measurement Parameters

All diffuse reflectance spectroscopy measurements were made with a spectrofluorometer (SkinSkan, JY Horiba, Edison, New Jersey). This instrument consists of a 150 W xenon lamp, double-grating excitation, emission monochromators, and a photomultiplier tube (PMT). The illumination and collection of light from the phantom were coupled through a fiber-optic probe, consisting of a central collection core with a diameter of 1.52  mm surrounded by an illumination ring with an outer diameter of 2.18  mm. The instrument is designed for making fluorescence and diffuse reflectance measurements through a fiber-optic probe, and the standard probe and coupling mechanism were used. Figure 2 shows a schematic of the probe geometry used in this study, with the gray central region containing the collection fibers and the surrounding white region containing the illumination fibers. Both the illumination ring and collection core are made up of 31 individual fibers, each with a core–cladding diameter of 200  and   245 μm. The numerical apertures of the illumination and collection fibers are 0.125 and 0.12, respectively. The adjustable parameters of the system are the wavelength range and increment and the signal-acquisition time. The fixed parameters of the system are the excitation and emission bandpasses, which are set at 5  nm, and the PMT high voltage, which is set at 950  V. The instrument was allowed to warm up for 20  min before the measurement. The diffuse reflectance was measured over a wavelength range of 350–850  nm at 5 nm increments. The diffuse reflectance spectrum measured from each sample was background subtracted (background was measured from distilled water).

2D. Experimental Validation of the Forward Model

First, we evaluated the accuracy of the forward model, using phantom set 1, by dividing the experimentally measured spectra from each tissue phantom by the modeled (Monte Carlo) spectra corresponding to the optical properties present in that phantom. A ratio of the experimentally measured and modeled spectra should ideally yield a ratio of unity at all wavelengths (assuming the model accurately describes the experimentally measured data). However, the experimentally measured spectrum is affected by the throughput and the wavelength-dependent response of the SkinSkan spectrofluorometer. In addition, there is a difference in magnitude since the Monte Carlo results are on an absolute scale, whereas the experimental measurements are relative to a calibration standard. To eliminate these effects, the experimentally measured diffuse reflectance spectrum of each tissue phantom was normalized to that of a reference phantom with predefined optical properties at each wavelength point. To make the modeled spectra equivalent to the experimentally measured spectra, we calibrated the modeled diffuse reflectance spectrum of each tissue phantom by that of a reference phantom with the same predefined optical properties at each wavelength point.

Figure 3 shows the ratio of the calibrated measured and modeled diffuse reflectance spectra for four phantoms from phantom set 1 containing hemoglobin as the absorber (the phantom with no absorber added was used as the reference phantom) for the case in which the Monte Carlo forward model [Fig. 3(a)] and the diffusion equation for a semi-infinite medium [Fig. 3(b)] were used to generate the modeled spectra. In Fig. 3(a), the ratio is close to unity and constant across the wavelength range for all of the phantoms. This indicates good agreement between the modeled and the measured diffuse reflectances for the range of optical properties present in these phantoms. For comparison, this ratio was also calculated for the case in which the modeled spectra were obtained with the standard diffusion equation for a semi-infinite medium.[16

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo ,” Med. Phys. 19, 879–888 (1992). [CrossRef] [PubMed]

] In Fig. 3(b), it can be seen that there are significant deviations in the ratio from unity, particularly at wavelengths below 600  nm. These results confirm the fact that the diffusion equation breaks down in the UV–VIS range where absorption is comparable with or greater than scattering.

2E. Experimental Validation of the Inverse Model

Next, the accuracy of the inverse model was evaluated. As in the evaluation of the forward model, calibration with a reference phantom was needed to correct for the system response. We made the calibration by normalizing the experimentally measured and modeled spectra of the unknown sample to that of a reference phantom with predefined optical properties at each wavelength point. For this analysis, each of the phantoms was used as a reference phantom to calibrate every other phantom (target phantom) within phantom sets 1 and 2. This was done to ensure that the choice of optical properties of the reference phantom did not bias the accuracy with which optical properties were extracted from the target phantoms. To evaluate the accuracy with which the optical properties can be extracted, the rms percent errors or the absorption and the reduced scattering coefficients (over the entire wavelength range of measurement) were determined for each target–reference phantom combination, and then averaged over all phantoms within a phantom set to produce the average rms percent error. Phantoms containing no absorber were excluded because the rms percent errors were significantly higher when these were used as either reference or target phantoms. This is likely because the polystyrene spheres in these phantoms have nonzero absorption, which is not accounted for in the inverse model.

Figure 4 shows scatter plots of the extracted versus expected absorption coefficients ( μa) [Fig. 4(a)] and reduced scattering coefficients ( μ s′) [Fig. 4(b)] for all wavelengths for all target–reference phantom combinations from phantom set 1. The solid line in each graph is the line of perfect agreement. The results indicate there is excellent agreement between the extracted and expected optical properties, with correlation coefficients of 0.999 and 0.98 for the absorption and the reduced scattering coefficients, respectively. It should also be pointed out that no trend was seen in the rms error as the target and reference phantom optical properties became more dissimilar, indicating that the choice of optical properties for the reference phantom does not influence the accuracy with which the target phantom optical properties are extracted. The mean rms percent errors for extraction of optical properties from all the target–reference phantom combinations in phantom set 1 were ( 2.5±0.7)% and ( 3.1±1.1)% for the reduced scattering and absorption coefficients, respectively. The corresponding mean rms percent errors for extraction of optical properties from all the target–reference phantom combinations in phantom set 2 were ( 12.1±5.4)% and ( 10.7±3.7)% for the reduced scattering and absorption coefficients, respectively.

2F. Sensitivity to Model Assumptions

Next, the robustness of the inverse model with respect to potential errors in the underlying assumptions and differences in the measurement methodology was tested. First, the accuracy with which optical properties can be extracted from the experimentally measured diffuse reflectance spectra was investigated over different wavelength ranges. We evaluated this for all target–reference phantom combinations in phantom set 1 by constraining the wavelength range available to the model during the fit from the full range of 350–850  nm to various wavelength ranges of interest (a wavelength increment of 5  nm was used in all cases). Table 4 shows the mean rms percent errors in the reduced scattering coefficients ( μ s′) and absorption coefficients ( μa) for all target–reference phantom combinations in phantom set 1 for each wavelength regime. It can be seen that errors are similar to those of the full wavelength range when UV–VIS wavelengths are used, whereas the error in the absorption coefficient increases as the wavelength range is restricted to just the red and the NIR.

Next, the effect of varying the refractive index mismatch between the scatterer and surrounding medium (fixed parameters in the Mie theory model) over a range of values extending beyond that which would be expected in tissue[17

A. Brunsting and P. F. Mullaney, “Differential light scattering from spherical mammalian cells,” Biophys. J. 14, 439–453 (1974). [CrossRef] [PubMed]

, 8

Q. Liu, C. Zhu, and N. Ramanujam, “Experimental validation of Monte Carlo modeling of fluorescence in tissues in the UV–visible spectrum,” J. Biomed. Opt. 8, 223–236 (2003). [CrossRef] [PubMed]

] was evaluated. We did this by repeating the extraction of optical properties from the diffuse reflectance spectra over the wavelength range of 400–600  nm (this is the wavelength range used in the analysis of breast tissue diffuse reflectance spectra in part II of this study[19

G. M. Palmer, C. Zhu, T. M. Breslin, F. Xu, K. W. Gilchrist, and N. Ramanujam, “Monte Carlo-based inverse model of diffuse reflectance. Part II: Application to breast cancer diagnosis,” Appl. Opt. 45, 1072–1078 (2006). [CrossRef] [PubMed]

]) of the target phantoms in phantom set 1 by using refractive indices of 1.5, 1.4, and 1.34 for the polystyrene spheres rather than the correct value of 1.6. Table 5 shows the mean rms percent errors in the reduced scattering coefficients ( μ s′) and absorption coefficients ( μa) for all target–reference phantom combinations from phantom set 1. It can be seen that, for a wide range of refractive index mismatches, extending beyond that which would be expected in biological samples,[17

A. Brunsting and P. F. Mullaney, “Differential light scattering from spherical mammalian cells,” Biophys. J. 14, 439–453 (1974). [CrossRef] [PubMed]

, 18

H. Liu, B. Beauvoit, M. Kimura, and B. Chance, “Dependence of tissue optical properties of solute-induced changes in refractive index and osmolarity,” J. Biomed. Opt. 1, 200–211 (1996). [CrossRef]

] there is a negligible degradation in the accuracy with which the optical properties are extracted.

Finally, the validity of the assumption of a fixed scatterer size in the Mie theory model was assessed by use of Monte Carlo simulations. First, scattering coefficients were determined from Mie theory for a uniform distribution of scatterer sizes ranging from 0.5  to   1 .0  μm in diameter, which was reported as the range of scatterer sizes in cells by Mourant et al.[20

J. R. Mourant, T. M. Johnson, and J. P. Freyer, “Characterizing mammalian cells and cell phantoms by polarized backscattering fiber-optic measurements,” Appl. Opt. 40, 5114–5123 (2001). [CrossRef]

] In addition, scattering coefficients were generated from Mie theory for a power-law distribution of scatterer sizes, ranging from 0.01  to   1 .5  μm in diameter, as reported by Backman et al.[21

V. Backman, V. Gopal, M. Kalashnikov, K. Badizadegan, R. Gurjar, A. Wax, I. Georgakoudi, M. Mueller, C. W. Boone, R. R. Dasari, and M. S. Feld, “Measuring cellular structure at submicrometer scale with light scattering spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 7, 887–893 (2001). [CrossRef]

] The overall scatterer density was set such that the reduced scattering coefficient had a mean of 20  cm −1 in each case and ranged from 18 to 22 and 16  to   26   cm −1 for the uniform and the power-law distributions, respectively, over a wavelength range of 400–600 nm. The absorption spectrum of oxygenated hemoglobin was used to generate the absorption coefficient. The concentration of oxygenated hemoglobin was set such that the absorption coefficient ranged from 0.12 to 20   cm −1 over a wavelength range of 400–600  nm. Next, independent Monte Carlo simulations were run with 106 photons used at each wavelength. The inversion procedure was performed with the Mie theory model for a single scatterer size described above [Fig. 1(b)]. For a uniform scatter size distribution, the mean rms percent errors were 5.0% for the reduced scattering coefficient and 1.4% for the absorption coefficient. For a power-law distribution of scatterer sizes, the mean rms percent error was 6.6% for the reduced scattering coefficient and 7.1% for the absorption coefficient. For comparison, this procedure was repeated for the case in which a single size scatterer of 1.0 μm diameter was used in the simulations, which produced errors of 4.9% and 2.1% for the reduced scattering and absorption coefficients, respectively. The errors for the distribution of scatterer sizes are comparable with the mean rms percent errors for phantoms containing spherical scatterers of a single size. These results indicate that the model is able to extract optical properties with reasonable accuracy from a sample with a distribution of scatterer sizes.

3. Discussion and Conclusions

A Monte Carlo-based forward and inverse model has thus been developed for the extraction of the absorption and scattering coefficients of turbid media such as human tissue from diffuse reflectance spectroscopy measurements. The accuracy of the Monte Carlo model for extracting optical properties from tissue phantom diffuse reflectance spectra was evaluated. It was found that this model was more effective in extracting optical properties from the phantoms containing hemoglobin and polystyrene spheres (phantom set 1) compared with phantoms containing Nigrosin and polystyrene spheres (phantom set 2). One reason for this is that the model's accuracy depends in part on the contrast in absorption coefficients present for a given absorber across the wavelength range used. The extinction coefficient has a maximum contrast ratio of approximately 530:1 for hemoglobin over the wavelengths measured (defined as the ratio of the maximum and minimum extinction coefficients), whereas Nigrosin has a maximum contrast of less than 4:1.

In addition, the robustness of this method to changes in the wavelength regime and potential errors in the underlying assumptions of the scattering model was established. For the case of phantom set 1 (hemoglobin–polystyrene phantoms), it was shown that the method is relatively insensitive to the wavelength range used for extracting optical properties in the UV–VIS spectrum. However, restricting the wavelength range to include only those wavelengths in the NIR resulted in an increase in the mean rms percent errors. Hemoglobin absorption is low at these wavelengths, and so the absorption of light by the polystyrene spheres, which was not accounted for, may have contributed to the increased errors. Furthermore, it was shown that the assumption of a single scatterer size and variations over a wide range of the refractive index mismatch between the scatterer and surrounding medium did not introduce significant errors in the extracted absorption and reduced scattering coefficients when tested over a biologically relevant range of scatterer size distributions and refractive indices. Thus this model is appropriate for use in tissues in which the dominant absorbers are known and tissue scattering can be reasonably approximated by use of Mie theory.

This work was funded by University of Wisconsin Radiological Sciences Training grant 5T32CA009206-27, sponsored by the National Institutes of Health, the Department of Health and Human Services, and the Public Health Service. Additional funding was provided by the National Institutes of Health through grant 1R01CA100559-01A1.

Appendices

Appendix A

Derivation of Eq. (3):We wish to derive the probability that a photon launched into a circular illumination fiber of radius ri, which travels a given net distance r t, will be collected by a separate circular fiber of radius rc at a fixed center-to-center distance s from the illumination fiber (see Fig. 5). This is derived for uniform fiber illumination and collection efficiencies. Both fibers are normal to the medium, which produces a circularly symmetric and translationally invariant system, assuming a homogeneous or homogeneous layered medium.

Let the illumination fiber be centered at the origin, and the collection fiber be centered at (s, 0). First, we take the case in which photons are launched only at the center of the illumination fiber. Because the system is circularly symmetric, the photon may exit the surface anywhere along the circle centered at the origin, with radius rt, with equal probability. The probability that the photon will exit within the region contained by the collection fiber, and thus be collected, is given by p= rtθ/ ( 2π rt), which corresponds to the arc length contained within the collection fiber, divided by the total circumference of the circle that defines all possible exit locations. This can be shown to be
p= 1 π   cos⁡ −1 ( r t 2+ s 2+ r c 2 2s r t),s− r c< r t<s+ r c,
p=0,otherwise.
(A1)
We can then extend this to a line source located at y=0, and − ri≤x≤ ri, by noting that a displacement in x in the source effectively changes the source–detector separation s and then integrating. This is normalized to the length of the source line to produce the average probability for all source locations from which the photon could originate. For the following derivations we assume that the mean probability of collection is nonzero, i.e., s− ri− rc< rt<s+ ri+ rc. In this case the probability of collection is given by
p= 1 2 r i 1 π ∫ l⁢b u⁢b  cos -1 [ r t    2+ ( s−x) 2− r c    2 2 ( s−x) r t]dx,
(A2)
where ub=min⁡ ( ri,s− rt+ rc) and lb=max⁡ ( − ri,s− rt− rc). These bounds correspond to the launch locations for which the probability of collection is nonzero.

Finally, we can extend this system to a fiber source by noting that the probability of collection is the same for any source location at a given distance from the center of the collection fiber. Thus each point in the integral given in Eq. (A2) is weighted by the arc length of all source locations occurring within the source fiber, equidistant to the collection fiber center (as indicated by the dashed line in Fig. 5). The integral is then normalized to the area of the source fiber to produce the average probability of collection for all possible source locations. This gives
p= 1 π⁢r i    2 1 π ∫ l⁢b u⁢b( s−x) cos⁡ −1 [ s 2+ ( s−x) 2− r i    2 2 ( s−x)s]  cos⁡ −1 [ r t    2+ ( s−x) 2− r c    2 2 ( s−x) r t]dx,
(A3)
with the bounds of the integral being the same as those given above.

References and links

1.

G. Zonios, L. T. Perelman, V. Backman, R. Manoharan, M. Fitzmaurice, J. Van-Dam, and M. S. Feld, “Diffuse reflectance spectroscopy of human adenomatous colon polyps in vivo, ” Appl. Opt. 38, 6628–6637 (1999). [CrossRef]

2.

N. Ghosh, S. K. Mohanty, S. K. Majumder, and P. K. Gupta, “Measurement of optical transport properties of normal and malignant human breast tissue,” Appl. Opt. 40, 176–184 (2001). [CrossRef]

3.

J. C. Finlay and T. H. Foster, “Hemoglobin oxygen saturations in phantoms and in vivo from measurements of steady-state diffuse reflectance at a single, short source–detector separation,” Med. Phys. 31, 1949–1959 (2004). [CrossRef] [PubMed]

4.

P. Thueler, I. Charvet, F. Bevilacqua, M. St. Ghislain, G. Ory, P. Marquet, P. Meda, B. Vermeulen, and C. Depeursinge, “ In vivo endoscopic tissue diagnostics based on spectroscopic absorption, scattering, and phase function properties,” J. Biomed. Opt. 8, 495–503 (2003). [CrossRef] [PubMed]

5.

T. J. Pfefer, L. S. Matchette, C. L. Bennett, J. A. Gall, J. N. Wilke, A. J. Durkin, and M. N. Ediger, “Reflectance-based determination of optical properties in highly attenuating tissue,” J. Biomed. Opt. 8, 206–215 (2003). [CrossRef] [PubMed]

6.

A. Amelink, H. J. Sterenborg, M. P. Bard, and S. A. Burgers, “ In vivo measurement of the local optical properties of tissue by use of differential path-length spectroscopy,” Opt. Lett. 29, 1087–1089 (2004). [CrossRef] [PubMed]

7.

F. Bevilacqua and C. Depeursinge, “Monte Carlo study of diffuse reflectance at source–detector separations close to one transport mean free path,” J. Opt. Soc. Am. A 16, 2935–2945 (1999). [CrossRef]

8.

Q. Liu, C. Zhu, and N. Ramanujam, “Experimental validation of Monte Carlo modeling of fluorescence in tissues in the UV–visible spectrum,” J. Biomed. Opt. 8, 223–236 (2003). [CrossRef] [PubMed]

9.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

10.

L. Wang, S. L. Jacques, and L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995). [CrossRef] [PubMed]

11.

R. Graaff, M. H. Koelink, F. F. M. de Mul, W. G. Zijlstra, A. C. M. Dassel, and J. G. Aarnoudse, “Condensed Monte Carlo simulations for the description of light transport,” Appl. Opt. 32, 426–434 (1993). [CrossRef] [PubMed]

12.

A. Kienle and M. S. Patterson, “Determination of the optical properties of turbid media from a single Monte Carlo simulation,” Phys. Med. Biol. 41, 2221–2227 (1996). [CrossRef] [PubMed]

13.

W.-F. Cheong, “Appendix to chapter 8: Summary of optical properties,” in Optical-Thermal Response of Laser-Irradiated Tissue , A. J. Welch and M. J. C. v. Gemert, eds. (Plenum, 1995), pp. 275–303.

14.

S. Prahl, “Mie scattering program,” Oregon Medical Laser Center (2005), available at http://omlc.ogi.edu/software/mie/index.html.

15.

X. Ma, J. Q. Lu, R. S. Brock, K. M. Jacobs, P. Yang, and X. H. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48, 4165–4172 (2003). [CrossRef]

16.

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo ,” Med. Phys. 19, 879–888 (1992). [CrossRef] [PubMed]

17.

A. Brunsting and P. F. Mullaney, “Differential light scattering from spherical mammalian cells,” Biophys. J. 14, 439–453 (1974). [CrossRef] [PubMed]

18.

H. Liu, B. Beauvoit, M. Kimura, and B. Chance, “Dependence of tissue optical properties of solute-induced changes in refractive index and osmolarity,” J. Biomed. Opt. 1, 200–211 (1996). [CrossRef]

19.

G. M. Palmer, C. Zhu, T. M. Breslin, F. Xu, K. W. Gilchrist, and N. Ramanujam, “Monte Carlo-based inverse model of diffuse reflectance. Part II: Application to breast cancer diagnosis,” Appl. Opt. 45, 1072–1078 (2006). [CrossRef] [PubMed]

20.

J. R. Mourant, T. M. Johnson, and J. P. Freyer, “Characterizing mammalian cells and cell phantoms by polarized backscattering fiber-optic measurements,” Appl. Opt. 40, 5114–5123 (2001). [CrossRef]

21.

V. Backman, V. Gopal, M. Kalashnikov, K. Badizadegan, R. Gurjar, A. Wax, I. Georgakoudi, M. Mueller, C. W. Boone, R. R. Dasari, and M. S. Feld, “Measuring cellular structure at submicrometer scale with light scattering spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 7, 887–893 (2001). [CrossRef]

Table 1  Summary of Methods for Extracting Optical Properties a
AuthorLight-Transport ModelAbsorption RangeCalibration RequirementProbe Geometry
Zonios et al. (Ref. [1

G. Zonios, L. T. Perelman, V. Backman, R. Manoharan, M. Fitzmaurice, J. Van-Dam, and M. S. Feld, “Diffuse reflectance spectroscopy of human adenomatous colon polyps in vivo, ” Appl. Opt. 38, 6628–6637 (1999). [CrossRef]

])		Modified diffusion equationHigh to low absorptionExtensive phantom studies for empirical calibrationSingle source–detector separation
Finlay and Foster (Ref. [3

J. C. Finlay and T. H. Foster, “Hemoglobin oxygen saturations in phantoms and in vivo from measurements of steady-state diffuse reflectance at a single, short source–detector separation,” Med. Phys. 31, 1949–1959 (2004). [CrossRef] [PubMed]

])		 P 3 approximation of transport equationHigh to low absorptionSpectra normalized to a specific wavelengthSingle source–detector separation at >1 mm
Ghosh et al. (Ref. [2

N. Ghosh, S. K. Mohanty, S. K. Majumder, and P. K. Gupta, “Measurement of optical transport properties of normal and malignant human breast tissue,” Appl. Opt. 40, 176–184 (2001). [CrossRef]

])		Spatially resolved diffusion equationModerate to low absorptionSpectra normalized to a specific source–detector separationMultiple source–detector separations: 1.2 to 12 mm
Thueler et al. (Ref. [4

P. Thueler, I. Charvet, F. Bevilacqua, M. St. Ghislain, G. Ory, P. Marquet, P. Meda, B. Vermeulen, and C. Depeursinge, “ In vivo endoscopic tissue diagnostics based on spectroscopic absorption, scattering, and phase function properties,” J. Biomed. Opt. 8, 495–503 (2003). [CrossRef] [PubMed]

])		Spatially resolved Monte Carlo approximationModerate to low absorptionMultiple phantoms required for calibrationMultiple source–detector separations: 0.3 to 1.35 mm
Pfefer et al. (Ref. [5

T. J. Pfefer, L. S. Matchette, C. L. Bennett, J. A. Gall, J. N. Wilke, A. J. Durkin, and M. N. Ediger, “Reflectance-based determination of optical properties in highly attenuating tissue,” J. Biomed. Opt. 8, 206–215 (2003). [CrossRef] [PubMed]

])		Spatially resolved empirical methodHigh to low absorptionExtensive phantom studies for empirical calibrationMultiple source–detector separations: 0.23 to 2.46 mm
Amelink et al. (Ref. [6

A. Amelink, H. J. Sterenborg, M. P. Bard, and S. A. Burgers, “ In vivo measurement of the local optical properties of tissue by use of differential path-length spectroscopy,” Opt. Lett. 29, 1087–1089 (2004). [CrossRef] [PubMed]

])		Empirical methodHigh to low absorptionInternal calibration by means of subtraction of signal from two fibersSpecialized probe geometry
Palmer and Ramanujam (this paper)Monte CarloHigh to low absorptionCalibration on a single phantomRequires only a single source–detector separation and adaptable to any probe geometry
a High, moderate, and low absorptions correspond generally to tissue absorption in the UV, VIS, and NIR wavelength ranges, respectively. For multiple separation probes, the range of source–detector separations used in the study is provided.
Table 2  Means and Ranges of the Reduced Scattering Coefficient (μ s ′) for Each of the Phantoms in Phantom Sets 1 and 2 over the Wavelength Range of 350–850 nm a
Phantom Setμ s ′ LevelMean μ s ′ (cm−1)μ s ′ Range (cm−1)
1Fixed13.310.9–16.4
2Lowest8.97.3–10.9
213.310.9–16.4
317.814.6–21.8
422.218.2–27.3
Highest26.721.8–32.7
a Note that these are given for the phantoms for which no absorber had been added. The addition of absorber dilutes the scatterer and thus reduces the scattering coefficient, by as much as 16% and 20% for the hemoglobin and Nigrosin phantoms, respectively.
Table 3  Means and Ranges of the Absorption Coefficient (μ a ) for Each of the Phantoms in Phantom Sets 1 and 2 over the Wavelength Range of 350–850 nm
Phantom Setμ a LevelMean μ a (cm−1)μ a Range (cm−1)
1100–0
20.90.02–7.7
31.30.02–11.2
41.60.03–14.4
52.00.03–17.5
2100–0
23.81.7–5.9
37.23.1–11.2
410.34.4–15.9
513.05.6–20.1
Table 4  Mean rms Percent Errors in the Reduced Scattering Coefficients (μs′) and Absorption Coefficients (μ a ) for All Target–Reference Phantom Combinations in Phantom Set 1 for Each Wavelength Regime
Wavelength Range (nm)350–850400–600400–500500–600550–850600–850
Mean rms error in μ s ′ (%)2.5 ± .082.0 ± 1.21.6 ± 0.82.2 ± 1.31.7 ± 0.81.5 ± 0.6
Mean rms error in μ a (%)3.1 ± 1.51.2 ± 1.31.0 ± 0.55.6 ± 5.53.6 ± 2.317.1 ± 17.3
Table 5  Mean rms Percent Errors in the Reduced Scattering Coefficients (μ s ′) and Absorption Coefficients (μ a ) for All Target–Reference Phantom Combinations in Phantom Set 1, Provided for a Range of Refractive Index Mismatches
Refractive Index (Scatterer–Medium)1.6–1.33 (True Value)1.5–1.331.4–1.331.34–1.33
Mean rms error in μ s ′ (%)2.0 ± 1.22.0 ± 0.92.7 ± 1.12.4 ± 0.8
Mean rms error in μ a (%)1.2 ± 1.31.2 ± 1.02.9 ± 2.22.0 ± 1.4
Fig. 1 (a) Forward and (b) inverse models of diffuse reflectance. Input boxes are gray, and output boxes are gray with a bold outline.
Fig. 2 Schematic of the probe geometry used in this study, with the gray central region containing the collection fibers and the surrounding white region containing the illumination fibers.
Fig. 3 Ratio of the calibrated measured and modeled diffuse reflectance spectra for four phantoms from phantom set 1 containing hemoglobin as the absorber (the phantom with no absorber added was used as the reference phantom) for the case in which (a) the Monte Carlo (MC) forward model and (b) the diffusion equation for a semi-infinite medium were used to generate the modeled spectra.
Fig. 4 Scatter plots of the extracted versus expected (a) absorption coefficients (μ a ) and (b) reduced scattering coefficients (μ s ′) for all wavelengths for all target–reference phantom combinations from phantom set 1. Note that each of the phantoms was used as a reference phantom to extract the optical properties of every other phantom.
Fig. 5 Schematic of the fiber configuration used in the derivation of Eq. (3), with the illumination fiber on the left and the collection fiber on the right.

OCIS Codes
(160.4760) Materials : Optical properties
(170.4580) Medical optics and biotechnology : Optical diagnostics for medicine
(170.6510) Medical optics and biotechnology : Spectroscopy, tissue diagnostics

ToC Category:
Medical Optics and Biotechnology

History
Original Manuscript: November 5, 2004
Revised Manuscript: March 29, 2005
Manuscript Accepted: May 8, 2005

Virtual Issues
Vol. 1, Iss. 3 Virtual Journal for Biomedical Optics

Citation
Gregory M. Palmer and Nirmala Ramanujam, "Monte Carlo-based inverse model for calculating tissue optical properties. Part I: Theory and validation on synthetic phantoms," Appl. Opt. 45, 1062-1071 (2006)
http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-45-5-1062


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. G. Zonios, L. T. Perelman, V. Backman, R. Manoharan, M. Fitzmaurice, J. Van-Dam, and M. S. Feld, "Diffuse reflectance spectroscopy of human adenomatous colon polyps in vivo," Appl. Opt. 38, 6628-6637 (1999). [CrossRef]
  2. N. Ghosh, S. K. Mohanty, S. K. Majumder, and P. K. Gupta, "Measurement of optical transport properties of normal and malignant human breast tissue," Appl. Opt. 40, 176-184 (2001). [CrossRef]
  3. J. C. Finlay and T. H. Foster, "Hemoglobin oxygen saturations in phantoms and in vivo from measurements of steady-state diffuse reflectance at a single, short source-detector separation," Med. Phys. 31, 1949-1959 (2004). [CrossRef] [PubMed]
  4. P. Thueler, I. Charvet, F. Bevilacqua, M. St. Ghislain, G. Ory, P. Marquet, P. Meda, B. Vermeulen, and C. Depeursinge, "In vivo endoscopic tissue diagnostics based on spectroscopic absorption, scattering, and phase function properties," J. Biomed. Opt. 8, 495-503 (2003). [CrossRef] [PubMed]
  5. T. J. Pfefer, L. S. Matchette, C. L. Bennett, J. A. Gall, J. N. Wilke, A. J. Durkin, and M. N. Ediger, "Reflectance-based determination of optical properties in highly attenuating tissue," J. Biomed. Opt. 8, 206-215 (2003). [CrossRef] [PubMed]
  6. A. Amelink, H. J. Sterenborg, M. P. Bard, and S. A. Burgers, "In vivo measurement of the local optical properties of tissue by use of differential path-length spectroscopy," Opt. Lett. 29, 1087-1089 (2004). [CrossRef] [PubMed]
  7. F. Bevilacqua and C. Depeursinge, "Monte Carlo study of diffuse reflectance at source-detector separations close to one transport mean free path," J. Opt. Soc. Am. A 16, 2935-2945 (1999). [CrossRef]
  8. Q. Liu, C. Zhu, and N. Ramanujam, "Experimental validation of Monte Carlo modeling of fluorescence in tissues in the UV-visible spectrum," J. Biomed. Opt. 8, 223-236 (2003). [CrossRef] [PubMed]
  9. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  10. L. Wang, S. L. Jacques, and L. Zheng, "MCML--Monte Carlo modeling of light transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995). [CrossRef] [PubMed]
  11. R. Graaff, M. H. Koelink, F. F. M. de Mul, W. G. Zijlstra, A. C. M. Dassel, and J. G. Aarnoudse, "Condensed Monte Carlo simulations for the description of light transport," Appl. Opt. 32, 426-434 (1993). [CrossRef] [PubMed]
  12. A. Kienle and M. S. Patterson, "Determination of the optical properties of turbid media from a single Monte Carlo simulation," Phys. Med. Biol. 41, 2221-2227 (1996). [CrossRef] [PubMed]
  13. W.-F. Cheong, "Appendix to chapter 8: Summary of optical properties," in Optical-Thermal Response of Laser-Irradiated Tissue, A.J.Welch and M.J. C. v.Gemert, eds. (Plenum, 1995), pp. 275-303.
  14. S. Prahl, "Mie scattering program," Oregon Medical Laser Center (2005), available at http://omlc.ogi.edu/software/mie/index.html.
  15. X. Ma, J. Q. Lu, R. S. Brock, K. M. Jacobs, P. Yang, and X. H. Hu, "Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm," Phys. Med. Biol. 48, 4165-4172 (2003). [CrossRef]
  16. T. J. Farrell, M. S. Patterson, and B. Wilson, "A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo," Med. Phys. 19, 879-888 (1992). [CrossRef] [PubMed]
  17. A. Brunsting and P. F. Mullaney, "Differential light scattering from spherical mammalian cells," Biophys. J. 14, 439-453 (1974). [CrossRef] [PubMed]
  18. H. Liu, B. Beauvoit, M. Kimura, and B. Chance, "Dependence of tissue optical properties of solute-induced changes in refractive index and osmolarity," J. Biomed. Opt. 1, 200-211 (1996). [CrossRef]
  19. G. M. Palmer, C. Zhu, T. M. Breslin, F. Xu, K. W. Gilchrist, and N. Ramanujam, "Monte Carlo-based inverse model of diffuse reflectance. Part II: Application to breast cancer diagnosis," Appl. Opt. 45, 1072-1078 (2006). [CrossRef] [PubMed]
  20. J. R. Mourant, T. M. Johnson, and J. P. Freyer, "Characterizing mammalian cells and cell phantoms by polarized backscattering fiber-optic measurements," Appl. Opt. 40, 5114-5123 (2001). [CrossRef]
  21. V. Backman, V. Gopal, M. Kalashnikov, K. Badizadegan, R. Gurjar, A. Wax, I. Georgakoudi, M. Mueller, C. W. Boone, R. R. Dasari, and M. S. Feld, "Measuring cellular structure at submicrometer scale with light scattering spectroscopy," IEEE J. Sel. Top. Quantum Electron. 7, 887-893 (2001). [CrossRef]

Cited By

 Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2 Fig. 3
 
Fig. 4 Fig. 5
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited