## An ideal-observer framework to investigate signal detectability in diffuse optical imaging |

Biomedical Optics Express, Vol. 4, Issue 10, pp. 2107-2123 (2013)

http://dx.doi.org/10.1364/BOE.4.002107

Acrobat PDF (1370 KB)

### Abstract

With the emergence of diffuse optical tomography (DOT) as a non-invasive imaging modality, there is a requirement to evaluate the performance of the developed DOT systems on clinically relevant tasks. One such important task is the detection of high-absorption signals in the tissue. To investigate signal detectability in DOT systems for system optimization, an appropriate approach is to use the Bayesian ideal observer, but this observer is computationally very intensive. It has been shown that the Fisher information can be used as a surrogate figure of merit (SFoM) that approximates the ideal observer performance. In this paper, we present a theoretical framework to use the Fisher information for investigating signal detectability in DOT systems. The usage of Fisher information requires evaluating the gradient of the photon distribution function with respect to the absorption coefficients. We derive the expressions to compute the gradient of the photon distribution function with respect to the scattering and absorption coefficients. We find that computing these gradients simply requires executing the radiative transport equation with a different source term. We then demonstrate the application of the SFoM to investigate signal detectability in DOT by performing various simulation studies, which help to validate the proposed framework and also present some insights on signal detectability in DOT.

© 2013 OSA

## 1. Introduction

1. A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. **50**, 1–43 (2005). [CrossRef]

4. H. Dehghani, S. Srinivasan, B. W. Pogue, and A. Gibson, “Numerical modelling and image reconstruction in diffuse optical tomography,” Phil. Trans. Royal Soc. A **367**, 3073–3093 (2009). [CrossRef]

5. H. Dehghani, B. W. Pogue, S. P. Poplack, and K. D. Paulsen, “Multiwavelength three-dimensional near-infrared tomography of the breast: initial simulation, phantom, and clinical results,” App. Optics **42**, 135–146 (2003). [CrossRef]

6. S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “In vivo hemoglobin and water concentrations, oxygen saturation, and scattering estimates from near-infrared breast tomography using spectral reconstruction,” Acad. Radiol. **13**, 195–202 (2006). [CrossRef] [PubMed]

7. T. Austin, A. P. Gibson, G. Branco, R. M. Yusof, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, “Three dimensional optical imaging of blood volume and oxygenation in the neonatal brain,” Neuroimage **31**, 1426–1433 (2006). [CrossRef] [PubMed]

8. B. W. Zeff, B. R. White, H. Dehghani, B. L. Schlaggar, and J. P. Culver, “Retinotopic mapping of adult human visual cortex with high-density diffuse optical tomography,” Proc. Nat. Acad. Sciences **104**, 12169–12174 (2007). [CrossRef]

9. A. H. Hielscher, A. D. Klose, A. K. Scheel, B. Moa-Anderson, M. Backhaus, U. Netz, and J. Beuthan, “Sagittal laser optical tomography for imaging of rheumatoid finger joints,” Phys. Med. Biol. **49**, 1147–1163 (2004). [CrossRef] [PubMed]

10. A. H. Hielscher, “Optical tomographic imaging of small animals,” Curr. Opinion in Biotech. **16**, 79–88 (2005). [CrossRef]

11. A. Li, E. L. Miller, M. E. Kilmer, T. J. Brukilacchio, T. Chaves, J. Stott, Q. Zhang, T. Wu, M. Chorlton, R. H. Moore, D. B. Kopans, and D. A. Boas, “Tomographic optical breast imaging guided by three-dimensional mammography,” Appl. Opt. **42**, 5181–5190 (2003). [CrossRef] [PubMed]

15. S. van de Ven, S. Elias, A. Wiethoff, M. van der Voort, A. Leproux, T. Nielsen, B. Brendel, L. Bakker, M. van der Mark, W. Mali, and P. Luijten, “Diffuse optical tomography of the breast: initial validation in benign cysts,” Mol. Imaging Biol. **11**, 64–70 (2009). [CrossRef]

16. B. W. Pogue, S. C. Davis, X. Song, B. A. Brooksby, H. Dehghani, and K. D. Paulsen, “Image analysis methods for diffuse optical tomography,” J. Biomed. Opt. **11**, 33001 (2006). [CrossRef] [PubMed]

17. V. C. Kavuri, Z. J. Lin, F. Tian, and H. Liu, “Sparsity enhanced spatial resolution and depth localization in diffuse optical tomography,” Biomed. Opt. Express **3**, 943–957 (2012). [CrossRef] [PubMed]

22. S. Morgan and K. Yong, “Controlling the phase response of a diffusive wave phased array system,” Opt. Express **7**, 540–546 (2000). [CrossRef] [PubMed]

23. J. P. Culver, A. M. Siegel, J. J. Stott, and D. A. Boas, “Volumetric diffuse optical tomography of brain activity,” Opt. Lett. **28**, 2061–2063 (2003). [CrossRef] [PubMed]

24. H. Niu, P. Guo, X. Song, and T. Jiang, “Improving depth resolution of diffuse optical tomography with an exponential adjustment method based on maximum singular value of layered sensitivity,” Chin. Opt. Lett. **6**, 886–888 (2008). [CrossRef]

26. F. Shen and E. Clarkson, “Using Fisher information to approximate ideal-observer performance on detection tasks for lumpy-background images,” J. Opt. Soc. Am. A **23**, 2406–2414 (2006). [CrossRef]

19. D. Kang and M. A. Kupinski, “Signal detectability in diffusive media using phased arrays in conjunction with detector arrays,” Opt. Express **19**, 12261–12274 (2011). [CrossRef] [PubMed]

26. F. Shen and E. Clarkson, “Using Fisher information to approximate ideal-observer performance on detection tasks for lumpy-background images,” J. Opt. Soc. Am. A **23**, 2406–2414 (2006). [CrossRef]

26. F. Shen and E. Clarkson, “Using Fisher information to approximate ideal-observer performance on detection tasks for lumpy-background images,” J. Opt. Soc. Am. A **23**, 2406–2414 (2006). [CrossRef]

27. E. Clarkson and F. Shen, “Fisher information and surrogate figures of merit for the task-based assessment of image quality,” J. Opt. Soc. Am. A **27**, 2313–2326 (2010). [CrossRef]

16. B. W. Pogue, S. C. Davis, X. Song, B. A. Brooksby, H. Dehghani, and K. D. Paulsen, “Image analysis methods for diffuse optical tomography,” J. Biomed. Opt. **11**, 33001 (2006). [CrossRef] [PubMed]

17. V. C. Kavuri, Z. J. Lin, F. Tian, and H. Liu, “Sparsity enhanced spatial resolution and depth localization in diffuse optical tomography,” Biomed. Opt. Express **3**, 943–957 (2012). [CrossRef] [PubMed]

18. H. Niu, Z. J. Lin, F. Tian, S. Dhamne, and H. Liu, “Comprehensive investigation of three-dimensional diffuse optical tomography with depth compensation algorithm,” J. Biomed Opt. **15**, 046005 (2010). [CrossRef] [PubMed]

24. H. Niu, P. Guo, X. Song, and T. Jiang, “Improving depth resolution of diffuse optical tomography with an exponential adjustment method based on maximum singular value of layered sensitivity,” Chin. Opt. Lett. **6**, 886–888 (2008). [CrossRef]

28. Y. Zhan, A. T. Eggebrecht, J. P. Culver, and H. Dehghani, “Image quality analysis of high-density diffuse optical tomography incorporating a subject-specific head model,” Front Neuroenergetics **4**, 6 (2012). [CrossRef] [PubMed]

*et al.*[29

29. R. Ziegler, B. Brendel, A. Schipper, R. Harbers, M. v. Beek, H. Rinneberg, and T. Nielsen, “Investigation of detection limits for diffuse optical tomography systems: I. Theory and experiment,” Phys. Med. Biol. **54**, 399–412 (2009). [CrossRef]

30. R. Ziegler, B. Brendel, H. Rinneberg, and T. Nielsen, “Investigation of detection limits for diffuse optical tomography systems: II. Analysis of slab and cup geometry for breast imaging,” Phys. Med. Biol. **54**, 413–431 (2009). [CrossRef]

*et al.*[20

20. S. P. Morgan, “Detection performance of a diffusive wave phased array,” Appl. Opt. **43**, 2071–2078 (2004). [CrossRef] [PubMed]

19. D. Kang and M. A. Kupinski, “Signal detectability in diffusive media using phased arrays in conjunction with detector arrays,” Opt. Express **19**, 12261–12274 (2011). [CrossRef] [PubMed]

32. A. K. Jha, M. A. Kupinski, T. Masumura, E. Clarkson, A. A. Maslov, and H. H. Barrett, “Simulating photon-transport in uniform media using the radiative transfer equation: A study using the Neumann-series approach,” J. Opt. Soc. Amer. A **29**, 1741–1757 (2012). [CrossRef]

33. A. K. Jha, M. A. Kupinski, H. H. Barrett, E. Clarkson, and J. H. Hartman, “Three-dimensional Neumann-series approach to model light transport in nonuniform media,” J. Opt. Soc. Am. A **29**, 1885–1899 (2012). [CrossRef]

## 2. Methods

### 2.1. Relation between signal detectability and Fisher information in DOT

*M*detector elements. Let the flux measured by the

*m*

^{th}detector element be denoted by

*g*, and the image acquired using the

_{m}*M*detector elements by the

*M*-dimensional vector

**g**. Let us also denote the signal-absent and the signal-present hypothesis by

*H*

_{0}and

*H*

_{1}, respectively. The objective in the detection task is to determine if

**g**is a sample from the signal-absent PDF, denoted by pr(

**g**|

*H*

_{0}), or the signal-present PDF, denoted by pr(

**g**|

*H*

_{1}). The Bayesian ideal observer performs this task by evaluating the likelihood ratio

*t*(

**g**) and comparing it to a threshold. The likelihood ratio

*t*(

**g**) is given by To use this observer in the DOT context, let us denote the absorption and scattering coefficients at location

**r**by

*μ*

_{a}(

**r**) and

*μ*

_{s}(

**r**), respectively. We discreteize this functions using a certain spatial basis with basis functions given by

*ϕ*(

_{n}**r**). The representation of the absorption and scattering functions in this basis is given by where

*μ*

_{a,n}and

*μ*

_{s,n}denote the absorption and scattering coefficients in the considered spatial basis. These basis functions could be the commonly used voxel basis functions. However, we are not placing any restriction on the spatial support of these basis functions. The basis function could in fact represent the support of different anatomical structures in the tissue, where a given anatomical structure is characterized by the same absorption and scattering coefficient. It must be pointed out that in general, the equality in Eqs. (2) and (3) holds in the limit of having an infinite number of basis functions, since we are approximating a continuous function by a finite basis set. Let us denote the

*N*-dimensional vector of the absorption coefficients by

**and the vector of scattering coefficients by**

*μ*_{a}**. For notational simplicity, let us denote**

*μ*_{s}**= {**

*μ***,**

*μ*_{a}**}. Also, again for notational simplicity, we define a general coefficient function**

*μ*_{s}*μ*(

**r**) that denotes the absorption/scattering coefficient at location

**r**. The representation of this general coefficient function in the spatial basis is given by where

*μ*denotes the absorption/scattering coefficients in the considered spatial basis.

_{n}### 2.2. Computing the Fisher information

19. D. Kang and M. A. Kupinski, “Signal detectability in diffusive media using phased arrays in conjunction with detector arrays,” Opt. Express **19**, 12261–12274 (2011). [CrossRef] [PubMed]

34. V. Toronov, E. D’Amico, D. Hueber, E. Gratton, B. Barbieri, and A. Webb, “Optimization of the signal-to-noise ratio of frequency-domain instrumentation for near-infrared spectro-imaging of the human brain,” Opt. Express **11**, 2717–2729 (2003). [CrossRef] [PubMed]

*ḡ*(

_{m}**) denote the noiseless mean image data as a function of the scattering and absorption coefficient vectors**

*μ***= {**

*μ***,**

*μ*_{s}**}. Since the Poisson noise in individual detector elements is independent of each other, the PDF for the acquired image data**

*μ*_{a}**g**in our DOT setup is given by Taking the logarithm on both sides, we get the log-likelihood of the image data, which we denote by ℒ(

**ḡ**|

**), as The (**

*μ**i*,

*j*)

^{th}element of the FIM

**F**(

**) is given by [25] where angled brackets denote statistical expectations, and where**

*μ**μ*denotes the

_{i}*i*

^{th}absorption/scattering coefficient as defined in Eq. (4). To obtain the elements of the FIM, we take the derivative of the log-likelihood (Eq. (9)) with respect to the coefficient

*μ*. This yields Further taking the derivative of the above expression with respect to

_{i}*μ*, we get Averaging the above expression over

_{j}**g**and using Eq. (10), we obtain since 〈

*g*〉

_{m}**|**

_{g}*=*

_{μ}*ḡ*(

_{m}**). We observe that to evaluate the elements of the FIM, we have to evaluate the gradient of the mean image data with respect to the absorption and scattering coefficients. This is the topic of the next section.**

*μ*### 2.3. Evaluating the gradient of the mean image data

*w*(

**r**,

**ŝ**, ℰ,

*t*). In terms of photons,

*w*(

**r**,

**ŝ**, ℰ,

*t*)Δ

*V*ΔΩΔℰ can be interpreted as the number of photons contained in volume Δ

*V*centered on the 3D position vector

**r**= (

*x,y,z*), traveling in a solid angle ΔΩ about direction

**ŝ**= (

*θ*,

*ϕ*), and having energies between ℰ and ℰ + Δℰ at time

*t*. Another radiometric quantity required to describe the RTE is the emission function Ξ(

**r**,

**ŝ**, ℰ,

*t*). Like the distribution function, Ξ(

**r**,

**ŝ**, ℰ,

*t*)Δ

*V*ΔΩΔℰ can be interpreted as the number of photons injected per second into volume Δ

*V*in energy range Δℰ, over solid angle ΔΩ, and at time

*t*. In the DOT implementation, we assume a mono-energetic time-independent emission source, so that the emission function can be written as Ξ(

**r**,

**ŝ**). Also, the dominant scattering mechanism in DOT is elastic scattering and the scattered photon does not lose any energy. Since there is no other energy-loss mechanism for the photons, we can completely drop the dependence of the distribution function on energy. Also, since we consider a time-independent emission source, the dependence of the distribution function on time is also dropped, and we write the distribution function as

*w*(

**r**,

**ŝ**).

**𝒦**and

**𝒳**denote the scattering and attenuation operators in integral form, which represent the effect of scattering, and the effect of attenuation and propagation of photons, respectively. The effect of the scattering operator

**𝒦**on the distribution function is given by [25] where

*c*

_{m}denote the speed of light in the medium,

**ŝ**and

**ŝ′**denote the direction of the outgoing and incoming photons, respectively, and

*p*(

**ŝ**,

**ŝ′**|

**r**) denotes the scattering phase function, which in biological tissue is typically given by the Henyey-Greenstein function [35

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

36. M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, “Light transport in biological tissue using three-dimensional frequency-domain simplified spherical harmonics equations,” Phys. Med. Biol. **54**, 2493–2509 (2009). [CrossRef] [PubMed]

*α*characterizes the angular distribution of scattering in the tissue. The attenuation operator

**𝒳**is the standard attenuated X-ray transform, and its effect on the distribution function is given by [25] where

*μ*

_{tot}(

**r**) =

*μ*

_{a}(

**r**) +

*μ*

_{s}(

**r**) is the total attenuation coefficient. In terms of the defined attenuation and scattering operators, for a mono-energetic time-independent source Ξ(

**r**,

**ŝ**), the RTE can be written as [25, 32

32. A. K. Jha, M. A. Kupinski, T. Masumura, E. Clarkson, A. A. Maslov, and H. H. Barrett, “Simulating photon-transport in uniform media using the radiative transfer equation: A study using the Neumann-series approach,” J. Opt. Soc. Amer. A **29**, 1741–1757 (2012). [CrossRef]

*n*

^{th}absorption/scattering coefficient

*μ*, we note that

_{n}*ḡ*(

_{m}**) is computed as where**

*μ**h*(

_{m}**r**,

**ŝ**) denotes the sensitivity of the

*m*

^{th}detector pixel to the distribution function. We can write the above equation in inner product notation as Taking the gradient of

*ḡ*(

_{m}**) with respect to**

*μ**μ*, we get Thus, to evaluate the gradient of the mean image data, we have to evaluate the gradient of the distribution function with respect to

_{n}*μ*. Using Eq. (17), we can write the expression for this gradient as Of the four terms in the above equation, the first two terms can be considered as a new operator

_{n}*μ*, where as mentioned earlier,

_{n}*μ*could denote the absorption/scattering coefficient, we get Defining the step function we can rewrite the integral over

_{n}*l″*in Eq. (24) as

*l″*varies from 0 to

**∞**, step(

*l″*) = 1, and that 1 − step(

*l″*− 1) exists only when

*l*>

*l″*. Splitting the exponential integral over

*l′*into two parts, we get

*l*. Simplifying this integral further by replacing

*l*−

*l″*by

*l̃*, we get

*l′*by replacing

*l′*−

*l″*by

*l̂*. We recognize that the last term is very similar to the attenuated X-ray transform. More precisely, using Eq. (16) Substituting this in Eq. (28), we get which, using Eq. (16), can be written in operator notation as This gives the first term in Eq. (21). The second term in Eq. (21) is very similar to the first term, and can be written as Using Eqs. (32) and (33) we get where in the second step, we have used Eq. (17).

*μ*

_{s}. Using Eq. (4), the gradient of the scattering operator is obtained as where

*β*= 0 when

*μ*denotes the absorption coefficient and 1 when

_{n}*μ*denotes the scattering coefficient. Defining another operator

_{n}**𝒦**

_{1}, whose effect on the distribution function is given by we can rewrite Eq. (35) in operator notation as Thus, the third term in Eq. (21) can be written as Substituting Eq. (34) and (38) in Eq. (21), we get Comparing with Eq. (17), we realize that the above equation is just the RTE, with the source term given by Thus, to evaluate the gradient of the photon distribution function with respect to

*μ*, we just have to compute each term of the RTE with the source term as −

_{n}*ϕ*

_{n}c_{m}

*w*+

*βϕ*

_{n}**𝒦**

_{1}

*w*, i.e. the distribution function (−

*c*

_{m}

*w*+

*β*

**𝒦**

_{1}

*w*) that exists over the spatial support of the

*n*

^{th}basis function.

### 2.4. Implementation

32. A. K. Jha, M. A. Kupinski, T. Masumura, E. Clarkson, A. A. Maslov, and H. H. Barrett, “Simulating photon-transport in uniform media using the radiative transfer equation: A study using the Neumann-series approach,” J. Opt. Soc. Amer. A **29**, 1741–1757 (2012). [CrossRef]

33. A. K. Jha, M. A. Kupinski, H. H. Barrett, E. Clarkson, and J. H. Hartman, “Three-dimensional Neumann-series approach to model light transport in nonuniform media,” J. Opt. Soc. Am. A **29**, 1885–1899 (2012). [CrossRef]

33. A. K. Jha, M. A. Kupinski, H. H. Barrett, E. Clarkson, and J. H. Hartman, “Three-dimensional Neumann-series approach to model light transport in nonuniform media,” J. Opt. Soc. Am. A **29**, 1885–1899 (2012). [CrossRef]

**29**, 1741–1757 (2012). [CrossRef]

**s**

*is the source term as given by Eq. (40). To implement this equation, we further rewrite in a spherical harmonic and voxel basis as below [32*

_{n}**29**, 1741–1757 (2012). [CrossRef]

**29**, 1885–1899 (2012). [CrossRef]

**A**and

**D**denote the attenuation and scatter operators in voxel and spherical harmonic basis respectively, and

**S**

*and*

_{n}**W**

_{d}′ are the source term and the derivative of the photon distribution function in the spherical harmonic and voxel basis, respectively. We have derived the expressions and implemented the attenuation and scattering operators on the GPU for a nonuniform anisotropic scattering media [33

**29**, 1885–1899 (2012). [CrossRef]

**29**, 1885–1899 (2012). [CrossRef]

### 2.5. Methods for the specific SKE/BKE task

*ϕ*

_{0}(

**r**) and

*ϕ*

_{1}(

**r**), respectively. Also, let the absorption coefficient of the background and signal be denoted by

*μ*

_{a,0}and

*μ*

_{a,1}, respectively. The signal and the background are assumed to have the same scattering coefficient. Let us also denote the difference between

*μ*

_{a,0}and

*μ*

_{a,1}by Δ

*μ*

_{a}. Using Eq. (7), the expression for detectability in this case will be given by where, using Eq. (13),

*F*

_{a1,a1}is given by Using Eq. (20), the gradient of the image data with respect to

*μ*

_{a,1}is given by To evaluate the gradient of the photon distribution with respect to

*μ*

_{a,1}, in accordance with Eq. (39) we execute the RTE with the source term as Therefore, the mean output image

**and the gradient of the mean output image are evaluated using the RTE. To vary the signal depth, we vary the spatial support of the signal, given by**

*ḡ**ϕ*

_{1}(

**r**). The variation of detectability with other parameters, such as the size and contrast of the signal, and the scattering coefficient of the background can also be studied using this framework.

**29**, 1741–1757 (2012). [CrossRef]

37. T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Hybrid radiative-transfer-diffusion model for optical tomography,” Appl. Opt. **44**, 876–886 (2005). [CrossRef] [PubMed]

38. T. Spott and L. O. Svaasand, “Collimated light sources in the diffusion approximation,” Appl. Opt. **39**, 6453–6465 (2000). [CrossRef]

**29**, 1741–1757 (2012). [CrossRef]

39. Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, “Wave transport in random media: the ballistic to diffusive transition,” Phys. Rev. E **60**, 4843–4850 (1999). [CrossRef]

**29**, 1885–1899 (2012). [CrossRef]

40. E. Aydin, C. de Oliveira, and A. Goddard, “A finite element-spherical harmonics radiation transport model for photon migration in turbid media,” J. Quant. Spectr. Rad. Trans. **84**, 247–260 (2004). [CrossRef]

**29**, 1885–1899 (2012). [CrossRef]

41. A. Klose and E. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. **220**, 441–470 (2006). [CrossRef]

16. B. W. Pogue, S. C. Davis, X. Song, B. A. Brooksby, H. Dehghani, and K. D. Paulsen, “Image analysis methods for diffuse optical tomography,” J. Biomed. Opt. **11**, 33001 (2006). [CrossRef] [PubMed]

## 3. Experiments and results

*x*−

*y*coordinates as (0.5 mm, 0.5 mm). The scattering medium is a 3-D phantom of dimensions 2 × 2 × 2 cm

^{3}, characterized by a reduced scattering coefficient

*μ′*

_{s}=

*μ*

_{s}(1 −

*g*). The background of the phantom is homogeneous with absorption and scattering coefficients given by 0.01 cm

^{−1}and 1 cm

^{−1}, respectively. This is a small-geometry media where the light source is collimated, which is a case where the diffusion approximation breaks down [37

37. T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Hybrid radiative-transfer-diffusion model for optical tomography,” Appl. Opt. **44**, 876–886 (2005). [CrossRef] [PubMed]

39. Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, “Wave transport in random media: the ballistic to diffusive transition,” Phys. Rev. E **60**, 4843–4850 (1999). [CrossRef]

*α*of the media is set to 0, partly because it is easier to predict the expected observer output with such media. The signal is spherical in shape with a radius of 1.5 mm and has the same scattering coefficient as the background, but a higher absorption coefficient of 0.011 cm

^{−1}. The DOT setup consists of a pixellated detector in the

*x*−

*y*plane, consisting of 20×20 pixels, and of dimensions 2×2 cm

^{2}, which is in contact with the entrance face of the tissue and measures the reflected intensity. In the Neumann-series method, for all the experiments, the highest degree in the spherical harmonic expansion is 3. For this DOT setup and the specified signal and phantom properties, the computation time to determine the SFoM was close to a minute.

### 3.1. Signal detectability as a function of depth

*x*−

*y*plane. The

*x*−

*y*coordinates of the center of the signal are (0.5 mm, 0.5 mm), and its location is varied along the

*z*dimension. We would expect that since we are using only the reflected intensity to make inferences about the presence of the signal, the detectability would decrease as the signal depth increases. The detectability is computed using the developed software and plotted as a function of the signal location in Fig. 2, and we observe the expected trend.

*x,y*) = (5.5 mm, 5.5 mm), and repeat the above experiment. The plot of detectability vs. signal depth is plotted in Fig. 2. As we would expect, when the depth increases, the signal detectability reduces. Also, we expect that the detectability should be lower for an off-axis signal compared to an on-axis signal, and the plot shows this trend.

### 3.2. Signal detectability as a function of the scattering coefficient of the tissue

### 3.3. Signal detectability as a function of signal size and signal depth

### 3.4. Signal detectability as a function of signal size and signal contrast

**11**, 33001 (2006). [CrossRef] [PubMed]

^{−1}to 0.05 cm

^{−1}. Simultaneously, the radius of the signal is also increased from 0.5 mm to 2.5 mm. The signal is placed at a depth of 2.5 mm from the exit face of the tissue. From the results plotted in Fig. 5, we observe that as the signal contrast increases, the detectability also increases, which is an expected result.

## 4. Discussions and conclusions

41. A. Klose and E. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. **220**, 441–470 (2006). [CrossRef]

43. S. R. Arridge and W. R. B. Lionheart, “Nonuniqueness in diffusion-based optical tomography,” Opt. Lett. **23**, 882–884 (1998). [CrossRef]

44. S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. **20**, 299–309 (1993). [CrossRef] [PubMed]

45. Q. Fang and D. A. Boas, “Monte Carlo simulation of photon migration in 3D turbid media accelerated by graphics processing units,” Opt. Express **17**, 20178–20190 (2009). [CrossRef] [PubMed]

**29**, 1885–1899 (2012). [CrossRef]

*et al.*[46

46. M. A. Kupinski, E. Clarkson, K. Gross, and J. W. Hoppin, “Optimizing imaging hardware for estimation tasks,” in “Proc. SPIE Medical Imaging ,” (2003), pp. 309–313. [CrossRef]

*μ*, where

_{s}H*μ*denotes the scattering coefficient of the phantom and

_{s}*H*denotes the length of the tissue [32

**29**, 1741–1757 (2012). [CrossRef]

**29**, 1885–1899 (2012). [CrossRef]

**19**, 12261–12274 (2011). [CrossRef] [PubMed]

*β*= 1. The computed gradient value can then be used to evaluate the Fisher information with respect to the scattering coefficients using Eq. (13). Following this, the detectability for a scattering inhomogeneity can be computed using Eq. (7).

**19**, 12261–12274 (2011). [CrossRef] [PubMed]

34. V. Toronov, E. D’Amico, D. Hueber, E. Gratton, B. Barbieri, and A. Webb, “Optimization of the signal-to-noise ratio of frequency-domain instrumentation for near-infrared spectro-imaging of the human brain,” Opt. Express **11**, 2717–2729 (2003). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. |

2. | D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. |

3. | A. Gibson and H. Dehghani, “Diffuse optical imaging,” Phil. Tran. A. Math. Phys. Eng. Sci. |

4. | H. Dehghani, S. Srinivasan, B. W. Pogue, and A. Gibson, “Numerical modelling and image reconstruction in diffuse optical tomography,” Phil. Trans. Royal Soc. A |

5. | H. Dehghani, B. W. Pogue, S. P. Poplack, and K. D. Paulsen, “Multiwavelength three-dimensional near-infrared tomography of the breast: initial simulation, phantom, and clinical results,” App. Optics |

6. | S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “In vivo hemoglobin and water concentrations, oxygen saturation, and scattering estimates from near-infrared breast tomography using spectral reconstruction,” Acad. Radiol. |

7. | T. Austin, A. P. Gibson, G. Branco, R. M. Yusof, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, “Three dimensional optical imaging of blood volume and oxygenation in the neonatal brain,” Neuroimage |

8. | B. W. Zeff, B. R. White, H. Dehghani, B. L. Schlaggar, and J. P. Culver, “Retinotopic mapping of adult human visual cortex with high-density diffuse optical tomography,” Proc. Nat. Acad. Sciences |

9. | A. H. Hielscher, A. D. Klose, A. K. Scheel, B. Moa-Anderson, M. Backhaus, U. Netz, and J. Beuthan, “Sagittal laser optical tomography for imaging of rheumatoid finger joints,” Phys. Med. Biol. |

10. | A. H. Hielscher, “Optical tomographic imaging of small animals,” Curr. Opinion in Biotech. |

11. | A. Li, E. L. Miller, M. E. Kilmer, T. J. Brukilacchio, T. Chaves, J. Stott, Q. Zhang, T. Wu, M. Chorlton, R. H. Moore, D. B. Kopans, and D. A. Boas, “Tomographic optical breast imaging guided by three-dimensional mammography,” Appl. Opt. |

12. | X. Intes, J. Yu, A. Yodh, and B. Chance, “Development and evaluation of a multi-wavelength multi-channel time resolved optical instrument for NIR/MRI mammography co-registration,” in “Proceedings of the IEEE 28th Annual Northeast Bioengineering Conference,” (2002), pp. 91–92. |

13. | G. Gulsen, O. Birgul, M. B. Unlu, R. Shafiiha, and O. Nalcioglu, “Combined diffuse optical tomography (DOT) and MRI system for cancer imaging in small animals,” Tech. Cancer Res. Treatment |

14. | N. Biswal, Y. Xu, and Q. Zhu, “Imaging tumor oxyhemoglobin and deoxyhemoglobin concentrations with ultrasound-guided diffuse optical tomography.” Tech. Cancer Res. Treatment |

15. | S. van de Ven, S. Elias, A. Wiethoff, M. van der Voort, A. Leproux, T. Nielsen, B. Brendel, L. Bakker, M. van der Mark, W. Mali, and P. Luijten, “Diffuse optical tomography of the breast: initial validation in benign cysts,” Mol. Imaging Biol. |

16. | B. W. Pogue, S. C. Davis, X. Song, B. A. Brooksby, H. Dehghani, and K. D. Paulsen, “Image analysis methods for diffuse optical tomography,” J. Biomed. Opt. |

17. | V. C. Kavuri, Z. J. Lin, F. Tian, and H. Liu, “Sparsity enhanced spatial resolution and depth localization in diffuse optical tomography,” Biomed. Opt. Express |

18. | H. Niu, Z. J. Lin, F. Tian, S. Dhamne, and H. Liu, “Comprehensive investigation of three-dimensional diffuse optical tomography with depth compensation algorithm,” J. Biomed Opt. |

19. | D. Kang and M. A. Kupinski, “Signal detectability in diffusive media using phased arrays in conjunction with detector arrays,” Opt. Express |

20. | S. P. Morgan, “Detection performance of a diffusive wave phased array,” Appl. Opt. |

21. | Y. Chen, C. Mu, X. Intes, and B. Chance, “Signal-to-noise analysis for detection sensitivity of small absorbing heterogeneity in turbid media with single-source and dual-interfering-source,” Opt. Express |

22. | S. Morgan and K. Yong, “Controlling the phase response of a diffusive wave phased array system,” Opt. Express |

23. | J. P. Culver, A. M. Siegel, J. J. Stott, and D. A. Boas, “Volumetric diffuse optical tomography of brain activity,” Opt. Lett. |

24. | H. Niu, P. Guo, X. Song, and T. Jiang, “Improving depth resolution of diffuse optical tomography with an exponential adjustment method based on maximum singular value of layered sensitivity,” Chin. Opt. Lett. |

25. | H. H. Barrett and K. J. Myers, |

26. | F. Shen and E. Clarkson, “Using Fisher information to approximate ideal-observer performance on detection tasks for lumpy-background images,” J. Opt. Soc. Am. A |

27. | E. Clarkson and F. Shen, “Fisher information and surrogate figures of merit for the task-based assessment of image quality,” J. Opt. Soc. Am. A |

28. | Y. Zhan, A. T. Eggebrecht, J. P. Culver, and H. Dehghani, “Image quality analysis of high-density diffuse optical tomography incorporating a subject-specific head model,” Front Neuroenergetics |

29. | R. Ziegler, B. Brendel, A. Schipper, R. Harbers, M. v. Beek, H. Rinneberg, and T. Nielsen, “Investigation of detection limits for diffuse optical tomography systems: I. Theory and experiment,” Phys. Med. Biol. |

30. | R. Ziegler, B. Brendel, H. Rinneberg, and T. Nielsen, “Investigation of detection limits for diffuse optical tomography systems: II. Analysis of slab and cup geometry for breast imaging,” Phys. Med. Biol. |

31. | S. Young, M. A. Kupinski, and A. K. Jha, “Estimating signal detectability in a model diffuse optical imaging system,” in “ |

32. | A. K. Jha, M. A. Kupinski, T. Masumura, E. Clarkson, A. A. Maslov, and H. H. Barrett, “Simulating photon-transport in uniform media using the radiative transfer equation: A study using the Neumann-series approach,” J. Opt. Soc. Amer. A |

33. | A. K. Jha, M. A. Kupinski, H. H. Barrett, E. Clarkson, and J. H. Hartman, “Three-dimensional Neumann-series approach to model light transport in nonuniform media,” J. Opt. Soc. Am. A |

34. | V. Toronov, E. D’Amico, D. Hueber, E. Gratton, B. Barbieri, and A. Webb, “Optimization of the signal-to-noise ratio of frequency-domain instrumentation for near-infrared spectro-imaging of the human brain,” Opt. Express |

35. | L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. |

36. | M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, “Light transport in biological tissue using three-dimensional frequency-domain simplified spherical harmonics equations,” Phys. Med. Biol. |

37. | T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Hybrid radiative-transfer-diffusion model for optical tomography,” Appl. Opt. |

38. | T. Spott and L. O. Svaasand, “Collimated light sources in the diffusion approximation,” Appl. Opt. |

39. | Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, “Wave transport in random media: the ballistic to diffusive transition,” Phys. Rev. E |

40. | E. Aydin, C. de Oliveira, and A. Goddard, “A finite element-spherical harmonics radiation transport model for photon migration in turbid media,” J. Quant. Spectr. Rad. Trans. |

41. | A. Klose and E. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. |

42. | A. H. Hielscher and R. E. Alcouffe, “Discrete-ordinate transport simulations of light
propagation in highly forward scattering heterogeneous media,” in
“ |

43. | S. R. Arridge and W. R. B. Lionheart, “Nonuniqueness in diffusion-based optical tomography,” Opt. Lett. |

44. | S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. |

45. | Q. Fang and D. A. Boas, “Monte Carlo simulation of photon migration in 3D turbid media accelerated by graphics processing units,” Opt. Express |

46. | M. A. Kupinski, E. Clarkson, K. Gross, and J. W. Hoppin, “Optimizing imaging hardware for estimation tasks,” in “Proc. SPIE Medical Imaging ,” (2003), pp. 309–313. [CrossRef] |

47. | A. K. Jha, “Retrieving Information from Scattered Photons in Medical Imaging,” Ph.D. thesis, College of Optical Sciences, University of Arizona, Tucson, AZ, USA (2013). |

48. | B. W. Miller, “High-Resolution Gamma-Ray Imaging with Columnar Scintillators,” Ph.D. thesis, College of Optical Sciences, University of Arizona, Tucson, AZ, USA (2011). |

**OCIS Codes**

(110.3000) Imaging systems : Image quality assessment

(110.7050) Imaging systems : Turbid media

(170.3880) Medical optics and biotechnology : Medical and biological imaging

(110.3055) Imaging systems : Information theoretical analysis

**ToC Category:**

Image Reconstruction and Inverse Problems

**History**

Original Manuscript: June 12, 2013

Revised Manuscript: August 21, 2013

Manuscript Accepted: August 26, 2013

Published: September 9, 2013

**Citation**

Abhinav K. Jha, Eric Clarkson, and Matthew A. Kupinski, "An ideal-observer framework to investigate signal detectability in diffuse optical imaging," Biomed. Opt. Express **4**, 2107-2123 (2013)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-4-10-2107

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### References

- A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol.50, 1–43 (2005). [CrossRef]
- D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag.18, 57–75 (2001). [CrossRef]
- A. Gibson and H. Dehghani, “Diffuse optical imaging,” Phil. Tran. A. Math. Phys. Eng. Sci.367, 3055–3072 (2009). [CrossRef]
- H. Dehghani, S. Srinivasan, B. W. Pogue, and A. Gibson, “Numerical modelling and image reconstruction in diffuse optical tomography,” Phil. Trans. Royal Soc. A367, 3073–3093 (2009). [CrossRef]
- H. Dehghani, B. W. Pogue, S. P. Poplack, and K. D. Paulsen, “Multiwavelength three-dimensional near-infrared tomography of the breast: initial simulation, phantom, and clinical results,” App. Optics42, 135–146 (2003). [CrossRef]
- S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “In vivo hemoglobin and water concentrations, oxygen saturation, and scattering estimates from near-infrared breast tomography using spectral reconstruction,” Acad. Radiol.13, 195–202 (2006). [CrossRef] [PubMed]
- T. Austin, A. P. Gibson, G. Branco, R. M. Yusof, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, “Three dimensional optical imaging of blood volume and oxygenation in the neonatal brain,” Neuroimage31, 1426–1433 (2006). [CrossRef] [PubMed]
- B. W. Zeff, B. R. White, H. Dehghani, B. L. Schlaggar, and J. P. Culver, “Retinotopic mapping of adult human visual cortex with high-density diffuse optical tomography,” Proc. Nat. Acad. Sciences104, 12169–12174 (2007). [CrossRef]
- A. H. Hielscher, A. D. Klose, A. K. Scheel, B. Moa-Anderson, M. Backhaus, U. Netz, and J. Beuthan, “Sagittal laser optical tomography for imaging of rheumatoid finger joints,” Phys. Med. Biol.49, 1147–1163 (2004). [CrossRef] [PubMed]
- A. H. Hielscher, “Optical tomographic imaging of small animals,” Curr. Opinion in Biotech.16, 79–88 (2005). [CrossRef]
- A. Li, E. L. Miller, M. E. Kilmer, T. J. Brukilacchio, T. Chaves, J. Stott, Q. Zhang, T. Wu, M. Chorlton, R. H. Moore, D. B. Kopans, and D. A. Boas, “Tomographic optical breast imaging guided by three-dimensional mammography,” Appl. Opt.42, 5181–5190 (2003). [CrossRef] [PubMed]
- X. Intes, J. Yu, A. Yodh, and B. Chance, “Development and evaluation of a multi-wavelength multi-channel time resolved optical instrument for NIR/MRI mammography co-registration,” in “Proceedings of the IEEE 28th Annual Northeast Bioengineering Conference,” (2002), pp. 91–92.
- G. Gulsen, O. Birgul, M. B. Unlu, R. Shafiiha, and O. Nalcioglu, “Combined diffuse optical tomography (DOT) and MRI system for cancer imaging in small animals,” Tech. Cancer Res. Treatment5, 351–363 (2006).
- N. Biswal, Y. Xu, and Q. Zhu, “Imaging tumor oxyhemoglobin and deoxyhemoglobin concentrations with ultrasound-guided diffuse optical tomography.” Tech. Cancer Res. Treatment10, 417 (2011).
- S. van de Ven, S. Elias, A. Wiethoff, M. van der Voort, A. Leproux, T. Nielsen, B. Brendel, L. Bakker, M. van der Mark, W. Mali, and P. Luijten, “Diffuse optical tomography of the breast: initial validation in benign cysts,” Mol. Imaging Biol.11, 64–70 (2009). [CrossRef]
- B. W. Pogue, S. C. Davis, X. Song, B. A. Brooksby, H. Dehghani, and K. D. Paulsen, “Image analysis methods for diffuse optical tomography,” J. Biomed. Opt.11, 33001 (2006). [CrossRef] [PubMed]
- V. C. Kavuri, Z. J. Lin, F. Tian, and H. Liu, “Sparsity enhanced spatial resolution and depth localization in diffuse optical tomography,” Biomed. Opt. Express3, 943–957 (2012). [CrossRef] [PubMed]
- H. Niu, Z. J. Lin, F. Tian, S. Dhamne, and H. Liu, “Comprehensive investigation of three-dimensional diffuse optical tomography with depth compensation algorithm,” J. Biomed Opt.15, 046005 (2010). [CrossRef] [PubMed]
- D. Kang and M. A. Kupinski, “Signal detectability in diffusive media using phased arrays in conjunction with detector arrays,” Opt. Express19, 12261–12274 (2011). [CrossRef] [PubMed]
- S. P. Morgan, “Detection performance of a diffusive wave phased array,” Appl. Opt.43, 2071–2078 (2004). [CrossRef] [PubMed]
- Y. Chen, C. Mu, X. Intes, and B. Chance, “Signal-to-noise analysis for detection sensitivity of small absorbing heterogeneity in turbid media with single-source and dual-interfering-source,” Opt. Express9, 212–224 (2001). [CrossRef] [PubMed]
- S. Morgan and K. Yong, “Controlling the phase response of a diffusive wave phased array system,” Opt. Express7, 540–546 (2000). [CrossRef] [PubMed]
- J. P. Culver, A. M. Siegel, J. J. Stott, and D. A. Boas, “Volumetric diffuse optical tomography of brain activity,” Opt. Lett.28, 2061–2063 (2003). [CrossRef] [PubMed]
- H. Niu, P. Guo, X. Song, and T. Jiang, “Improving depth resolution of diffuse optical tomography with an exponential adjustment method based on maximum singular value of layered sensitivity,” Chin. Opt. Lett.6, 886–888 (2008). [CrossRef]
- H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004), 1st ed.
- F. Shen and E. Clarkson, “Using Fisher information to approximate ideal-observer performance on detection tasks for lumpy-background images,” J. Opt. Soc. Am. A23, 2406–2414 (2006). [CrossRef]
- E. Clarkson and F. Shen, “Fisher information and surrogate figures of merit for the task-based assessment of image quality,” J. Opt. Soc. Am. A27, 2313–2326 (2010). [CrossRef]
- Y. Zhan, A. T. Eggebrecht, J. P. Culver, and H. Dehghani, “Image quality analysis of high-density diffuse optical tomography incorporating a subject-specific head model,” Front Neuroenergetics4, 6 (2012). [CrossRef] [PubMed]
- R. Ziegler, B. Brendel, A. Schipper, R. Harbers, M. v. Beek, H. Rinneberg, and T. Nielsen, “Investigation of detection limits for diffuse optical tomography systems: I. Theory and experiment,” Phys. Med. Biol.54, 399–412 (2009). [CrossRef]
- R. Ziegler, B. Brendel, H. Rinneberg, and T. Nielsen, “Investigation of detection limits for diffuse optical tomography systems: II. Analysis of slab and cup geometry for breast imaging,” Phys. Med. Biol.54, 413–431 (2009). [CrossRef]
- S. Young, M. A. Kupinski, and A. K. Jha, “Estimating signal detectability in a model diffuse optical imaging system,” in “Biomedical Optics,” (Optical Society of America, 2010), p. BSuD26.
- A. K. Jha, M. A. Kupinski, T. Masumura, E. Clarkson, A. A. Maslov, and H. H. Barrett, “Simulating photon-transport in uniform media using the radiative transfer equation: A study using the Neumann-series approach,” J. Opt. Soc. Amer. A29, 1741–1757 (2012). [CrossRef]
- A. K. Jha, M. A. Kupinski, H. H. Barrett, E. Clarkson, and J. H. Hartman, “Three-dimensional Neumann-series approach to model light transport in nonuniform media,” J. Opt. Soc. Am. A29, 1885–1899 (2012). [CrossRef]
- V. Toronov, E. D’Amico, D. Hueber, E. Gratton, B. Barbieri, and A. Webb, “Optimization of the signal-to-noise ratio of frequency-domain instrumentation for near-infrared spectro-imaging of the human brain,” Opt. Express11, 2717–2729 (2003). [CrossRef] [PubMed]
- L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J.93, 70–83 (1941). [CrossRef]
- M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, “Light transport in biological tissue using three-dimensional frequency-domain simplified spherical harmonics equations,” Phys. Med. Biol.54, 2493–2509 (2009). [CrossRef] [PubMed]
- T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Hybrid radiative-transfer-diffusion model for optical tomography,” Appl. Opt.44, 876–886 (2005). [CrossRef] [PubMed]
- T. Spott and L. O. Svaasand, “Collimated light sources in the diffusion approximation,” Appl. Opt.39, 6453–6465 (2000). [CrossRef]
- Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, “Wave transport in random media: the ballistic to diffusive transition,” Phys. Rev. E60, 4843–4850 (1999). [CrossRef]
- E. Aydin, C. de Oliveira, and A. Goddard, “A finite element-spherical harmonics radiation transport model for photon migration in turbid media,” J. Quant. Spectr. Rad. Trans.84, 247–260 (2004). [CrossRef]
- A. Klose and E. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys.220, 441–470 (2006). [CrossRef]
- A. H. Hielscher and R. E. Alcouffe, “Discrete-ordinate transport simulations of light propagation in highly forward scattering heterogeneous media,” in “Advances in Optical Imaging and Photon Migration,” (Optical Society of America, 1998), p. ATuC2.
- S. R. Arridge and W. R. B. Lionheart, “Nonuniqueness in diffusion-based optical tomography,” Opt. Lett.23, 882–884 (1998). [CrossRef]
- S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys.20, 299–309 (1993). [CrossRef] [PubMed]
- Q. Fang and D. A. Boas, “Monte Carlo simulation of photon migration in 3D turbid media accelerated by graphics processing units,” Opt. Express17, 20178–20190 (2009). [CrossRef] [PubMed]
- M. A. Kupinski, E. Clarkson, K. Gross, and J. W. Hoppin, “Optimizing imaging hardware for estimation tasks,” in “Proc. SPIE Medical Imaging,” (2003), pp. 309–313. [CrossRef]
- A. K. Jha, “Retrieving Information from Scattered Photons in Medical Imaging,” Ph.D. thesis, College of Optical Sciences, University of Arizona, Tucson, AZ, USA (2013).
- B. W. Miller, “High-Resolution Gamma-Ray Imaging with Columnar Scintillators,” Ph.D. thesis, College of Optical Sciences, University of Arizona, Tucson, AZ, USA (2011).

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