## Perturbation Monte Carlo methods for tissue structure alterations |

Biomedical Optics Express, Vol. 4, Issue 10, pp. 1946-1963 (2013)

http://dx.doi.org/10.1364/BOE.4.001946

Acrobat PDF (1319 KB)

### Abstract

This paper describes an extension of the perturbation Monte Carlo method to model light transport when the phase function is arbitrarily perturbed. Current perturbation Monte Carlo methods allow perturbation of both the scattering and absorption coefficients, however, the phase function can not be varied. The more complex method we develop and test here is not limited in this way. We derive a rigorous perturbation Monte Carlo extension that can be applied to a large family of important biomedical light transport problems and demonstrate its greater computational efficiency compared with using conventional Monte Carlo simulations to produce forward transport problem solutions. The gains of the perturbation method occur because only a single baseline Monte Carlo simulation is needed to obtain forward solutions to other closely related problems whose input is described by perturbing one or more parameters from the input of the baseline problem. The new perturbation Monte Carlo methods are tested using tissue light scattering parameters relevant to epithelia where many tumors originate. The tissue model has parameters for the number density and average size of three classes of scatterers; whole nuclei, organelles such as lysosomes and mitochondria, and small particles such as ribosomes or large protein complexes. When these parameters or the wavelength is varied the scattering coefficient and the phase function vary. Perturbation calculations give accurate results over variations of ∼15–25% of the scattering parameters.

© 2013 OSA

## 1. Introduction

1. L. Nieman, A. Myakov, J. Aaron, and K. Sokolov, “Optical sectioning using a fiber probe with an angled illumination-collection geometry: Evaluation in engineered tissue Phantoms,” Appl. Opt. **43**, 1308–1319 (2004). [CrossRef] [PubMed]

5. R. Reif, O. A‘Amar, and I. J. Bigio, “Analytical model of light reflectance for extraction of the optical properties in small volumes of turbid media,” Appl. Opt. **46**, 7317–7328 (2007). [CrossRef] [PubMed]

6. M. Canpolat and J. R. Mourant, “High-angle scattering events strongly affect light collection in clinically relevant measurement geometries for light transport through tissue,” Phys. Med. Biol. **45**, 1127–1140 (2000). [CrossRef] [PubMed]

*μ*m or less, the reflectance depends on the form of the phase function [6

6. M. Canpolat and J. R. Mourant, “High-angle scattering events strongly affect light collection in clinically relevant measurement geometries for light transport through tissue,” Phys. Med. Biol. **45**, 1127–1140 (2000). [CrossRef] [PubMed]

7. J. R. Mourant, J. Boyer, A. H. Hielscher, and I. J. Bigio, “Influence of the scattering phase function on light transport measurements in turbid media performed with small source detector separations,” Opt. Lett. **21**, 546–548 (1996). [CrossRef] [PubMed]

8. 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. **31**, 6628–6637 (1999). [CrossRef]

9. C. Lau, O. Šcepanović, J. Mirkovic, S. McGee, C.-C. Yu, S. Fulghum, M. Wallace, J. Tunnell, K. Bechtel, and M. Feld, “Re-evaluation of model-based light-scattering spectroscopy for tissue spectroscopy,” J. Biomed. Opt. **14**, 024031 (2009). [CrossRef] [PubMed]

10. R. Graaff, M. H. Koelink, F. F. M. de Mul, W. G. Zijistra, 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]

11. C Zhu and Q Liu, “Hybrid method for fast Monte Carlo simulation of diffuse reflectance from a multilayered tissue model with tumor-like heterogeneities.” J. Biomed. Opt. **17**010501 (2012). [CrossRef] [PubMed]

12. Q. Liu and N. Ramanujam, “Scaling method for fast Monte Carlo simulation of diffuse reflectance spectra from multilayered turbid media,” J. Opt. Soc. Am. A **24**, 1011–1025 (2007). [CrossRef]

13. Q. Wang, A. Agrawal, N. S. Wang, and T. J. Pfefer, “Condensed Monte Carlo modeling of reflectance from biological tissue with a single illuminationdetection fiber,” IEEE J. Sel. Top. Quantum Electron. **16**, 627–634 (2010). [CrossRef]

12. Q. Liu and N. Ramanujam, “Scaling method for fast Monte Carlo simulation of diffuse reflectance spectra from multilayered turbid media,” J. Opt. Soc. Am. A **24**, 1011–1025 (2007). [CrossRef]

14. C. Hayakawa, J. Spanier, F. Bevilacqua, A. Dunn, J. You, B. Tromberg, and V. Venugopalan, “Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues,” Opt. Lett. **26**, 1335–1337 (2001). [CrossRef]

15. I. Seo, J. You, C. Hayakawa, and V. Venugopalan, “Perturbation and differential Monte Carlo Methods for measurement of optical properties in a layered epithelial tissue model,” J. Biomed. Opt. **12**, 014030 (2007). [CrossRef] [PubMed]

16. P. Yalavarthy, K. Karlekar, H. Patel, R. Vasu, M. Pramanik, P. Mathias, B. Jain, and P. Gupta, “Experimental Investigation of Perturbation Monte-Carlo Based Derivative Estimation for Imaging Low-Scattering Tissue,” Opt. Express **13**, 985–997 (2005). [CrossRef] [PubMed]

17. J. Heiskala, M. Pollari, M. Metsäranta, and P. E. Grant, “Probabilistic atlas can improve reconstruction from optical imaging of the neonatal brain,” Opt. Express **17**, 14977–14992 (2009). [CrossRef] [PubMed]

18. J. Chen and X. Intes, “Time-gated perturbation Monte Carlo for whole body functional imaging in small animals,” Opt. Express **17**, 19566–19579 (2009). [CrossRef] [PubMed]

19. J. Chen and X. Intes, “Comparison of Monte Carlo methods for fluorescence molecular tomographycomputational efficiency,” Med. Phys. **38**, 5788–5798 (2011). [CrossRef] [PubMed]

12. Q. Liu and N. Ramanujam, “Scaling method for fast Monte Carlo simulation of diffuse reflectance spectra from multilayered turbid media,” J. Opt. Soc. Am. A **24**, 1011–1025 (2007). [CrossRef]

**24**, 1011–1025 (2007). [CrossRef]

**24**, 1011–1025 (2007). [CrossRef]

14. C. Hayakawa, J. Spanier, F. Bevilacqua, A. Dunn, J. You, B. Tromberg, and V. Venugopalan, “Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues,” Opt. Lett. **26**, 1335–1337 (2001). [CrossRef]

*θ*, is perturbed. In other words, the MC simulations are restricted to unpolarized scattering from spherical particles.

## 2. Theory and methods

### 2.1. Model of the scattering parameters of tissue

20. J. Schmitt and G. Kumar, “Optimal scattering properties of soft tissue: a discrete particle model,” Appl. Opt. **37**, 2788–2797 (1998). [CrossRef]

23. J. R. Mourant, T. M. Johnson, S. Carpenter, A. Guerra, T. Aida, and J. P. Freyer, “Polarized angular dependent spectroscopy of epithelial cells and epithelial cell nuclei to determine the size scale of scattering structures” J. Biomed. Opt. **7**, 378–387 (2002). [CrossRef] [PubMed]

24. J. Ramachandran, T. Powers, S. Carpenter, A. Garcia-Lopez, J. P. Freyer, and J. R. Mourant, “Light Scattering and microarchitectural differences between tumorigenic and non-tumorigenic cell models of tissue,” Opt. Express **15**, 4039–4053 (2007). [CrossRef] [PubMed]

*g*closer to that reported in the literature for bronchial epithelium [25

25. J. Qu, C. MacAulay, S. Lam, and B. Palcic, “Optical properties of normal and carcinomatous bronchial tissue,” Appl. Opt. **33**, 7397–7405 (1994). [CrossRef] [PubMed]

26. Y. N. Mirabal, S. K. Chang, E. N. Atkinson, A. Malpica, M. Follen, and R. Richards-Kortum, “Reflectance spectroscopy for in vivo detection of cervical precancer,” J. Biomed. Opt. **7**587–594 (2002). [CrossRef] [PubMed]

*n*to

_{scatterer}*n*is smaller for these particles. The third distribution represents the nuclei. The size was obtained from microscopy and the index of refraction values are taken from the literature [27

_{medium}27. A. Brunsting and P. F. Mullaney, “Differential light scattering from spherical mammalian cells,” Biophys. J. **14**, 439–453 (1974). [CrossRef] [PubMed]

*μ*= 126.3 cm

_{s}^{−1},

*g*= 0.954, and the reduced scattering coefficient

*μ′*= 5.84 cm

_{s}^{−1}at 620 nm.

*k*, according to Eqs. 1 and 2, where

*L*(

_{k}*x*) is the log normal distribution and

*C*is the cross section calculated from Mie theory for a particle of radius

_{sca}*x*[28].

*f*is the phase function for the

_{k}*k*th scattering type (or group),

*m*is the number of scatterer types or groups and has a value of 3 for the model used here, and

*μ*is the scattering coefficient of the

_{s,k}*k*th group of scatterers [29

29. B. Gélébart, E. Tinet, J. M. Tualle, and S. Avrillier, “Phase function simulation in tissue phantoms: a fractal approach,” Pure Appl. Opt. **5**, 377–388 (1996). [CrossRef]

### 2.2. Measurement geometry

*μ*m. The half angle for light delivery/collection is 21.7° for all source and collection fibers and the radius of the light cone at the sample surface is 240

*μ*m for each fiber. The collection fibers are tilted at an angle of 20° from normal towards the collection fiber. This tilting increases sampling of the clinically relevant surface epithelium and also increases the number of collected photons.

### 2.3. The connection between the RTE and Monte Carlo simulations

*area*/

*sr*),

*r*is position,

*ω*is a unit direction vector,

*μ*is the optical interaction coefficient,

_{t}*μ*is the optical scattering coefficient,

_{s}*f*is the single-scattering phase function that scatters photons from

*ω′*to

*ω*, and

*Q*is the volumetric source.

*P*= (

*r*,

*ω*) is a point in the phase space,. The kernel,

*K*, describes both the positional and directional changes involved in scattering and transporting photons at

*r′*with direction

*ω′*to

*r*with direction

*ω*. In the case of

*no absorption*, it is composed of the probability density for scattering from

*ω′*to

*ω*,

*f*(

*ω′*·

*ω*), and the transport kernel

*T*as shown in Eq. (7). The transport kernel,

*T*, describes transport of photons in the direction

*ω*from

*r′*to

*r*[31] with

*l*being the distance from

*r′*to

*r*, as in Eq. (8). Ψ(

*P*) is the collision density as shown in Eq. (9). Lastly,

*S*(

*P*) is the density of first collisions and Eq. (10) shows that the density of first collisions,

*S*(

*P*), is obtained by transporting each photon along its initial direction

*ω*from the physical source

*Q*to its collision location

*r*.

*d*(

*P*) is a “detector function” that describes the spatial locations and the unit direction vectors that characterize the physical detector, including its numerical aperture. Together, Eqs. 6 and 11 form the analytic model of the problem.

*ℬ*whose elements describe all of the possible photon biographies [31]. Their likelihoods are described by a probability measure

*M*on

*ℬ*(so that

*M*(

*ℬ*) =1). The simplest choice for

*M*is the analog measure

*M*that is induced when the starting location and direction of each biography is generated by sampling from a normalized version of the source function,

_{A}*S*, and the transport kernel is used to move the photon to the first collision point, and additional collision points are generated by using the kernel

*K*to change direction using

*f*, and then move the photon to a new location using the transport kernel,

*T*(

*r′*→

*r*,

*ω*). The final component of the probability model is an unbiased estimator

*ξ*on

*ℬ*that designates the contribution of every photon biography,

*C*∈

_{i}*ℬ*, to estimate the integral (11). The simplest example of such an unbiased estimator is the binomial estimator

*ξ*whose value on the photon biographies is: For these analog choices of the measure

*M*and unbiased estimator,

*ξ*, it is intuitively clear (and rigorously shown in [31]) that

### 2.4. Perturbation Monte Carlo

*M*and then define a new estimator that can be used to estimate collected light intensity

*using the same photon biographies*for different (perturbed) conditions. For the perturbed conditions, Eq. (14) becomes Eq. (15) where hats denote perturbed conditions.

*ξ̂*(

*C*) can be defined such that then The interpretation of (17) is that the expected value of

_{i}*ξ̂*with respect to the baseline measure

*M*is identical to the expected value of the original variable

*ξ*with respect to the modified measure

*M̂*.

*M̂*that is used to generate the biographies in the tumorogenic system is different from the measure

*M*used to generate the biographies in the original system. For this work we assume all tissue systems are homogeneous, although that assumption can be easily relaxed [14

14. C. Hayakawa, J. Spanier, F. Bevilacqua, A. Dunn, J. You, B. Tromberg, and V. Venugopalan, “Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues,” Opt. Lett. **26**, 1335–1337 (2001). [CrossRef]

*M*. Calculating the photon trajectories for each precancerous and cancerous condition could be a prohibitively costly process. Instead, with pMC a single set of biographies is generated using the baseline measure

_{α}*M*, and for each tumor condition the pMC estimator in Eq. (16) is used to estimate the collected intensity, Eq. (17).

*M*, is composed of the source term multiplied by a kernel term for every collision, i.e.

*S*(

*P*

_{0})

*T*(

*P*

_{0}→

*P*

_{1})

*K*(

*P*

_{1}→

*P*

_{2})

*K*(

*P*

_{2}→

*P*

_{3})... When the source does not change, Eq. (18) holds for photons that enter a scattering medium, where

*j*is the number of collisions undergone by the photon in the scattering medium.

*K*and

_{α}*K*are different for each scattering event,

*m*, because they depend on the scattering angle. This reweighting can be performed by a postprocessing algorithm that is quite inexpensive compared to the cost of generating different photon biographies for each set of tissue conditions. Explicit formulas will be given in the next section that make these ideas concrete.

### 2.5. Implementation of perturbation Monte Carlo

*K*and

*K̂*are: Using Eqs. (16) and (18) an estimator for the perturbed weight can be derived. In performing this derivation, the subtleties of the exit event must be considered. The exit or collection step of a photon transport Monte Carlo simulation is different than the others in that the photon does not go from one point to another point location, but rather must travel a distance at least as far as the distance to the collection surface. The probability of not reaching the detector is: where

*L*is the distance from the last (

_{s}*j*th) collision inside the tissue to the point of photon exit out the tissue surface. Therefore, the probability of reaching the detector is

*e*

^{−μsLs}. The estimator in Eq. (16) is then obtained using Eq. (18) and the expressions for

*T*, Eq. (8), and

*K*, Eq. (7). The entrance and collection events are separated out in the expression for the estimator in Eq. (22). where

*j*is the number of collisions inside the medium,

*l*is the length of step

_{m}*m*in the medium. The phase functions,

*f*and

*f̂*are functions of scattering angles

*θ*and

*ϕ*. Eq. (22) can be rewritten as where L is the total distance traveled inside the scattering medium.

*θ*values which is composed of the

*θ*angle through which the photon scattered at every collision. To perform the perturbation calculation, Eq. (23) is evaluated for each photon using the tissue scattering parameters and the stored photon parameters.

*x̄*, of a scatter size distribution and the number density,

_{k}*N*, of each scatter size distribution. For example, in the tissue model of this paper, the effects of an increase in the number of very small scatterers without any changes in the number or sizes of the nuclei and organelles could be determined by changing the number density of the smallest size distribution in Table 1.

_{k}## 3. Testing of the pMC method

*f*(

*θ*) and

*f̂*(

*θ*) for the full range of

*θ*values. These tables of 720 elements are used for rapid sampling of the phase function both in the cMC simulations and the pMC calculations. The pMC calculation uses unperturbed as well as perturbed scattering parameters. These parameters as well as the saved trajectories are used according to Eq. (23) to reweight the photons. A separate set of trajectories is used for each collection fiber. Therefore, there are effectively four replicates of each pMC calculation.

### 3.1. The simpler problem: one size of scatterers

*r*= 0.4475

*μm*, the number density,

*N*= 1.27 × 10

_{s}^{12}particles/cm

^{3}, the wavelength,

*λ*= 620 nm, the index of the medium was 1.332, the index of the scatterers was 1.390 and 20 million photons were incident through the delivery fiber. Particle concentration was perturbed by ±25% for the pMC calculations. Each cMC simulation took 10 min. and the pMC calculations took ∼1 min using computer 2. Consequently, the pMC results were obtained in 11 min., much shorter than the 130 min. needed for the cMC calculations. The agreement between cMC and pMC results is quite good as seen in Fig. 3. By varying the concentration, the sensitivity of the perturbed reflectance to the weight factors described in Eq. (23) are determined without the phase function contribution.

*r*, will vary the phase function along with

*μ*. Figure 4 shows the results of varying

_{s}*r*using the same baseline simulation used for Fig. 3. There is good agreement from 0.4175 to 0.4775

*μ*m, with some variation in the pMC results at

*r*= 0.4775

*μ*m.

*μ*, because the phase function and

_{s}*μ*depend on the size parameter which is a function of wavelength as shown in Eq. (24). In Fig. (5), pMC and cMC results are compared for varying values of wavelength and constant values for other parameters. The parameters for the baseline simulation are the same as for Figs. 3 and 4. The agreement between pMC and cMC is excellent over the range 580 nm to 650 nm. However, for wavelengths of 550 nm and below, the pMC calculations underestimate the reflectance. Interestingly, large standard errors of the mean are not found in all cases, e.g. the results for fiber 4 at 520 nm. At wavelengths above 650 nm pMC results for one fiber over estimate reflectance while pMC results for the other three collection fibers under estimate the reflectance. Nonetheless, in most cases the standard error of the mean overlaps with the cMC result. The cMC results are nearly identical for each fiber. (When plotted on the same graph, the symbols overlap.) The pMC results, however, show a different trend for each fiber in Fig. 5 as a function of wavelength. The different trends for each fiber are due to the fact the pMC calculations for each fiber use a different base set of trajectories.

_{s}*r*= 0.100

*μm*,

*N*= 1.83 × 10

_{s}^{10}particles/cm

^{3},

*λ*= 620 nm,

*n*= 1.332 and

_{medium}*n*= 1.390. Twenty million photons were incident for the baseline and cMC simulations.

_{scatterer}*N*, radius, and

_{s}*λ*of the 447.5 nm radius spheres, the parameters all overlap at the 0% point as in the top left graph for

*μ*.

_{s}*N*and radius were varied are quite similar as shown in Figs. 3, and 4. This similarity is because the scattering parameters for the two sets of simulations were nearly the same. The range of

_{s}*μ*values, shown in the top left panel of Fig. 7 is nearly the same. The anisotropy factor,

_{s}*g*, was 0.933 for the set of simulations in which

*N*was varied. For the set of simulations for which the radius was varied,

_{s}*g*ranged from 0.927 to 0.939. Given these similarities in scattering parameters it is not surprising that the results in Figs. 3 and 4 are similar.

*λ*is varied with

*r*= 447.5 nm, the range of scattering coefficients, 78–153 cm

^{−1}, is slightly larger than when radius is varied, 80–139 cm

^{−1}. The variation in

*μ*is nearly the same for the two sets of simulations when the wavelength range is restricted to 550–710 nm for simulations varying wavelength. However, Fig. 5 shows that pMC and cMC results are not the same over this range. Examination of the bottom right panel shows that

_{s}*g*is varying more when wavelength is varied than when radius is varied even when only the range 550 to 710 nm is considered. Not until the wavelength range is reduced to 580 to 670 nm is the variation in

*g*the same. This corresponds to the same range of wavelengths over which good agreement is found between the cMC and pMC results in Fig. 5. Clearly, the variation in

*g*reduces the accuracy of the pMC results when

*g*is large. However, if

*g*is smaller, a much bigger variation in

*g*can be tolerated as can be seen for the results using 100nm radius spheres in Fig. 6 and the bottom left panel of Fig. 7. Lastly, we note that variations in

*μ′*are not a good predictor of the accuracy of pMC for this geometry where delivery and collection fibers are close together. The top right panel shows that varying lambda resulted in the smallest variation in

_{s}*μ′*, while varying the concentration resulted in the largest variation of

_{s}*μ′*.

_{s}### 3.2. A more complex problem: three lognormal distributions of radii

*μ*m. The pMC and cMC results overlap for all radii and results are quite accurate at the smallest radius used,

*r*= 0.024

*μ*m. Figure 9(b) are results when the mean radius of the middle size distribution is varied from the baseline value of 0.045

*μ*m. The pMC and cMC results agree well from about 0.42

*μ*m to 0.49

*μ*m, but the pMC results differ greatly from the cMC results for smaller radii in one case. Similar results were obtained when the radius of the largest distribution was varied, Fig. 9(c).

*μ*varies, while

_{s}*g*is constant or nearly constant, 2)

*g*varies while

*μ*is nearly constant, 3) both

_{s}*g*and

*μ*vary for nearly constant

_{s}*μ′*, and 4)

_{s}*g*,

*μ*and

_{s}*μ′*all vary. For each of these classes, there are 2 or 3 relevant simulations. By examining the pMC results for these simulations we can determine the range of scattering parameters over which the pMC and cMC results agree to within 1% of the cMC results. The results of this analysis are shown in Table 2. When only

_{s}*μ*varied, pMC results are accurate over a range of ±15% the original value of

_{s}*μ*. When only

_{s}*g*varied, pMC results are accurate over a range of ±25% the original value of (1 −

*g*). When both

*g*and

*μ*varied, then the variation of

_{s}*μ*+ 0.5(1 −

_{s}*g*) can be ±20% when

*μ′*is constant and slightly less if

_{s}*μ′*varies.

_{s}## 4. Discussion

*θ*. Testing of the pMC method used homogeneous, realistic epithelial tissue properties and probe geometries identical or similar to that being used by many groups to develop optical methods of precancer and cancer detection. Absorption was set to 0 and not varied in order to focus on testing the new aspects of this method which relate to the scattering properties.

*μ*m source-detector separation when scattering coefficients were varied by about ±15% along with some changes in absorption using the scaling model [12

**24**, 1011–1025 (2007). [CrossRef]

*μ*m, 13.5% and 40%, respectively [12

**24**, 1011–1025 (2007). [CrossRef]

**26**, 1335–1337 (2001). [CrossRef]

3. U. Utzinger and R. R. Richards-Kortum, “Fiber optic probes for biomedical optical spectroscopy,” J. Biomed. Opt. **8**, 121–147 (2003). [CrossRef] [PubMed]

32. 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]

33. V. M. Turzhitsky, A. J. Gomes, Y. L. Kim, Y. Liu, A. Kromine, J. D. Rogers, M. Jameel, H. K. Roy, and V. Backman, “Measuring mucosal blood supply in vivo with a polarization-gating probe,” Appl. Opt , **47**, 6046–6057 (2008). [CrossRef] [PubMed]

## 5. Summary and conclusions

## Acknowledgments

## References and links

1. | L. Nieman, A. Myakov, J. Aaron, and K. Sokolov, “Optical sectioning using a fiber probe with an angled illumination-collection geometry: Evaluation in engineered tissue Phantoms,” Appl. Opt. |

2. | A. M. J. Wang, J. E. Bender, J. Pfefer, U. Utzinger, and R. A. Drezek, “Depth-sensitive reflectance measurements using obliquely oriented fiber probes,” J. Biomed. Opt. |

3. | U. Utzinger and R. R. Richards-Kortum, “Fiber optic probes for biomedical optical spectroscopy,” J. Biomed. Opt. |

4. | A. Amelink and H. Sterenborg, “Measurement of the local optical properties of turbid media by differential path-length spectroscopy,” Appl. Opt. |

5. | R. Reif, O. A‘Amar, and I. J. Bigio, “Analytical model of light reflectance for extraction of the optical properties in small volumes of turbid media,” Appl. Opt. |

6. | M. Canpolat and J. R. Mourant, “High-angle scattering events strongly affect light collection in clinically relevant measurement geometries for light transport through tissue,” Phys. Med. Biol. |

7. | J. R. Mourant, J. Boyer, A. H. Hielscher, and I. J. Bigio, “Influence of the scattering phase function on light transport measurements in turbid media performed with small source detector separations,” Opt. Lett. |

8. | 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. |

9. | C. Lau, O. Šcepanović, J. Mirkovic, S. McGee, C.-C. Yu, S. Fulghum, M. Wallace, J. Tunnell, K. Bechtel, and M. Feld, “Re-evaluation of model-based light-scattering spectroscopy for tissue spectroscopy,” J. Biomed. Opt. |

10. | R. Graaff, M. H. Koelink, F. F. M. de Mul, W. G. Zijistra, A. C. M. Dassel, and J. G. Aarnoudse, “Condensed Monte Carlo simulations for the description of light transport,” Appl. Opt. |

11. | C Zhu and Q Liu, “Hybrid method for fast Monte Carlo simulation of diffuse reflectance from a multilayered tissue model with tumor-like heterogeneities.” J. Biomed. Opt. |

12. | Q. Liu and N. Ramanujam, “Scaling method for fast Monte Carlo simulation of diffuse reflectance spectra from multilayered turbid media,” J. Opt. Soc. Am. A |

13. | Q. Wang, A. Agrawal, N. S. Wang, and T. J. Pfefer, “Condensed Monte Carlo modeling of reflectance from biological tissue with a single illuminationdetection fiber,” IEEE J. Sel. Top. Quantum Electron. |

14. | C. Hayakawa, J. Spanier, F. Bevilacqua, A. Dunn, J. You, B. Tromberg, and V. Venugopalan, “Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues,” Opt. Lett. |

15. | I. Seo, J. You, C. Hayakawa, and V. Venugopalan, “Perturbation and differential Monte Carlo Methods for measurement of optical properties in a layered epithelial tissue model,” J. Biomed. Opt. |

16. | P. Yalavarthy, K. Karlekar, H. Patel, R. Vasu, M. Pramanik, P. Mathias, B. Jain, and P. Gupta, “Experimental Investigation of Perturbation Monte-Carlo Based Derivative Estimation for Imaging Low-Scattering Tissue,” Opt. Express |

17. | J. Heiskala, M. Pollari, M. Metsäranta, and P. E. Grant, “Probabilistic atlas can improve reconstruction from optical imaging of the neonatal brain,” Opt. Express |

18. | J. Chen and X. Intes, “Time-gated perturbation Monte Carlo for whole body functional imaging in small animals,” Opt. Express |

19. | J. Chen and X. Intes, “Comparison of Monte Carlo methods for fluorescence molecular tomographycomputational efficiency,” Med. Phys. |

20. | J. Schmitt and G. Kumar, “Optimal scattering properties of soft tissue: a discrete particle model,” Appl. Opt. |

21. | M. Bartek, X. Wang, W. Wells, K. D. Paulsen, and P. W. Pogue, “Estimation of subcellular particle size histograms with electron microscopy for prediction of optical scattering in breast tissue,” J. Biomed. Opt. |

22. | J. D. Wilson and T. H. Foster, “Mie theory interpretations of light scattering from intact cells,” Opt. Lett. |

23. | J. R. Mourant, T. M. Johnson, S. Carpenter, A. Guerra, T. Aida, and J. P. Freyer, “Polarized angular dependent spectroscopy of epithelial cells and epithelial cell nuclei to determine the size scale of scattering structures” J. Biomed. Opt. |

24. | J. Ramachandran, T. Powers, S. Carpenter, A. Garcia-Lopez, J. P. Freyer, and J. R. Mourant, “Light Scattering and microarchitectural differences between tumorigenic and non-tumorigenic cell models of tissue,” Opt. Express |

25. | J. Qu, C. MacAulay, S. Lam, and B. Palcic, “Optical properties of normal and carcinomatous bronchial tissue,” Appl. Opt. |

26. | Y. N. Mirabal, S. K. Chang, E. N. Atkinson, A. Malpica, M. Follen, and R. Richards-Kortum, “Reflectance spectroscopy for in vivo detection of cervical precancer,” J. Biomed. Opt. |

27. | A. Brunsting and P. F. Mullaney, “Differential light scattering from spherical mammalian cells,” Biophys. J. |

28. | C. F. Bohren and D. R. Huffman, |

29. | B. Gélébart, E. Tinet, J. M. Tualle, and S. Avrillier, “Phase function simulation in tissue phantoms: a fractal approach,” Pure Appl. Opt. |

30. | G. Goertzel and M. K. Kalos, “Monte Carlo methods in transport problems,” Appendix 2 in |

31. | J. Spanier and E. Gelbard, |

32. | J. R. Mourant, T. M. Johnson, and J. P. Freyer, “Characterizing mammalian cells and cell phantoms by polarized backscattering fiber-optic measurements,” Appl. Opt. |

33. | V. M. Turzhitsky, A. J. Gomes, Y. L. Kim, Y. Liu, A. Kromine, J. D. Rogers, M. Jameel, H. K. Roy, and V. Backman, “Measuring mucosal blood supply in vivo with a polarization-gating probe,” Appl. Opt , |

**OCIS Codes**

(170.0170) Medical optics and biotechnology : Medical optics and biotechnology

(170.3660) Medical optics and biotechnology : Light propagation in tissues

(170.6510) Medical optics and biotechnology : Spectroscopy, tissue diagnostics

(170.6935) Medical optics and biotechnology : Tissue characterization

**ToC Category:**

Optics of Tissue and Turbid Media

**History**

Original Manuscript: June 19, 2013

Revised Manuscript: August 2, 2013

Manuscript Accepted: August 8, 2013

Published: September 4, 2013

**Citation**

Jennifer Nguyen, Carole K. Hayakawa, Judith R. Mourant, and Jerome Spanier, "Perturbation Monte Carlo methods for tissue structure alterations," Biomed. Opt. Express **4**, 1946-1963 (2013)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-4-10-1946

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