## PDE-constrained multispectral imaging of tissue chromophores with the equation of radiative transfer |

Biomedical Optics Express, Vol. 1, Issue 3, pp. 812-824 (2010)

http://dx.doi.org/10.1364/BOE.1.000812

Acrobat PDF (1045 KB)

### Abstract

We introduce a transport-theory-based PDE-constrained multispectral model for direct imaging of the spatial distributions of chromophores concentrations in biological tissue. The method solves the forward problem (boundary radiance at each wavelength) and the inverse problem (spatial distribution of chromophores concentrations), in an *all-at-once* manner in the framework of a reduced Hessian sequential quadratic programming method. To illustrate the code’s performance, we present numerical and experimental studies involving tumor bearing mice. It is shown that the PDE-constrained multispectral method accelerates the reconstruction process by up to 15 times compared to unconstrained reconstruction algorithms and provides more accurate results as compared to the so-called two-step approach to multi-wavelength imaging.

© 2010 OSA

## 1. Introduction

9. A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. **43**(5), 1285–1302 (1998). [CrossRef] [PubMed]

## 2. Equation of radiative transfer as a light propagation model

10. K. Ren, G. Bal, and A. H. Hielscher, “Frequency domain optical tomography based on the equation of radiative transfer,” SIAM J. Sci. Comput. **28**(4), 1463–1489 (2006). [CrossRef]

11. H. K. Kim and A. H. Hielscher, “A PDE-constrained SQP algorithm for optical tomography based on the frequency-domain equation of radiative transfer,” Inverse Probl. **25**(1), 015010 (2009). [CrossRef]

*u*is the spectral radiation intensity in unit [W/cm

_{λ}^{2}/sr./nm],

*Q*is the interior spectral radiation source in unit [W/cm

_{λ}^{3}/sr./nm], and μ

_{a}(λ) and μ

_{s}(λ) are the wavelength-dependent absorption and scattering coefficient in unit [1/cm], and Φ is the phase function describing scattering from direction Ω

^{+}into direction Ω. We use here the Henyey-Greenstein phase function that is commonly used in tissue optics. The above equation can be solved with appropriate boundary conditions. In this work, we implemented a partially-reflective boundary condition [12] that allows us to consider the refractive index mismatch between the tissue and air.

11. H. K. Kim and A. H. Hielscher, “A PDE-constrained SQP algorithm for optical tomography based on the frequency-domain equation of radiative transfer,” Inverse Probl. **25**(1), 015010 (2009). [CrossRef]

**where the sparse matrix**

*Au = b***contains all coefficients resulting from discretization and the vector**

*A***represents the discretized boundary conditions, respectively. The algebraic equation**

*b***, is solved with a complex version of the GMRES (**

*Au = b**m*) linear solver [13

13. Y. Saad and M. H. Schultz, “GMRES: A generalized minimum residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Comput. **7**(3), 856–869 (1986). [CrossRef]

*m*denotes the iteration number after which GMRES is restarted. Thus, the spectral intensity and phase shift are obtained by solving the forward problem [Eq. (1)] with a distribution of known optical properties at wavelength λ.

*i*-th chromophore concentration C

*to the absorption coefficient μ*

_{i}_{a}(λ) is given with the

*i*-th absorption extinction coefficient ε

*(λ) aswhere*

_{i}*N*is the number of total chromophores that contribute to the absorption at wavelength λ. In the matrix form, Eq. (2) is given as [

_{c}*M*] = [

*E*][

*C*] where the coefficient matrix [

*E*] denotes the molar extinction coefficients of chromophores whose values are available in the literature [14

14. S. Prahl, “Optical properties spectra,” retrieved 16 March 2003, http://omlc.ogi.edu/spectra/index.html, 2001.

*C*] denotes the concentration of chromphores.

## 3. Two-step and multi-spectral PDE-constrained models

### 3.1 Two step Method

_{a}(λ) for each wavelength λ which corresponds to minimizing the objective function defined at wavelength λ given aswhere

*Q*is the measurement operator that projects the angular intensity distribution

*ND*is the total number of measurements. Then these images are converted into the chromophore concentration images. For example, consider four main absorbers (HbO

_{2}, Hb, Water and Lipid) with four wavelengths measurements. In this case, we can decompose four absorption coefficient maps into into the chromophore images by using Eq. (2) as

*N*<

_{c}*N*, the chromophore concentrations may be the least-squared solutions to Eq. (2).

_{λ}*N*is the number of measurement wavelengths and

_{λ}*C*is the

_{i}*i*-th chromophore concentration that can be either [HbO

_{2}] or [Hb] or [Water] or [Lipid] concentrations. Note that the reconstruction variables in Eq. (5) are the chromophore concentrations themselves, not the absorption coefficients. Equation (5) also shows that the chromophore concentrations are updated by checking variations in all spectral data through the minimization process. Since the multispectral method can make use of as many wavelength data as available for the same number of unknowns, it improves the nature of the problem, thus making a highly ill-posed problem less ill-posed. This cannot be achieved by the two-step method, which just increases the number of unknowns as the number of measurement wavelengths is increased.

### 3.2 PDE-constrained multispectral method

**and**

*u***denote the vectors of spectral intensity predictions and measurements, respectively,**

*z*^{obs}*R*denotes appropriate regularization with a regularization parameter

*β*, and

**denotes a vector of unknown wavelength-dependent absorption coefficients. The equation**

*μ*_{λ}**is a discretized version of the forward transport equation. The problem given by Eq. (6) is often referred to as “PDE-constrained” since the optimal solution at minimum of**

*Au = b**f*has to satisfy the partial differential equations (i.e., ERT) represented by

**.**

*Au = b***as a dependent variable of the inverse variable**

*u***, which makes it possible to replace the prediction vector**

*μ*_{λ}**in**

*u**f*of Eq. (6) by its forward solution vector

**. As a result, the problem Eq. (6) is reformulated aswhich is often referred to as “unconstrained” because equality**

*A*^{−1}b**no longer appears in Eq. (7), i.e.,**

*Au = b**f**is now a function of

**only. Thus the accurate forward solution**

*μ*_{λ}**has to be obtained for each wavelength λ for evaluation of the objective function Eq. (7), which makes the unconstrained code computationally very expensive both with respect to time and memory. Nonetheless, this approach has been widely used in optical tomography because of the ease of implementation. The existing DA-based multispectral or two-step schemes [2**

*A*^{−1}b2. G. Boverman, Q. Fang, S. A. Carp, E. L. Miller, D. H. Brooks, J. Selb, R. H. Moore, D. B. Kopans, and D. A. Boas, “Spatio-temporal imaging of the hemoglobin in the compressed breast with diffuse optical tomography,” Phys. Med. Biol. **52**(12), 3619–3641 (2007). [CrossRef] [PubMed]

7. A. Corlu, T. Durduran, R. Choe, M. Schweiger, E. M. C. Hillman, S. R. Arridge, and A. G. Yodh, “Uniqueness and wavelength optimization in continuous-wave multispectral diffuse optical tomography,” Opt. Lett. **28**(23), 2339–2341 (2003). [CrossRef] [PubMed]

**and the inverse variable**

*u***independently. This enables solving the equality-constrained problem [Eq. (6)] directly by updating the forward and inverse variables simultaneously at each of optimization iterations. Typically the PDE-constrained inverse problem [Eq. (6)] can be reformulated into the framework of the following extended objective function called “Lagrangian” as:**

*μ*_{λ}**is called the vector of Lagrange multipliers. The simultaneous solutions of forward and inverse problems can be achieved at points satisfying the so-called first order Karush-Khun-Tucker (KKT) conditions [11**

*η*11. H. K. Kim and A. H. Hielscher, “A PDE-constrained SQP algorithm for optical tomography based on the frequency-domain equation of radiative transfer,” Inverse Probl. **25**(1), 015010 (2009). [CrossRef]

15. G. S. Abdoulaev, K. Ren, and A. H. Hielscher, “Optical tomography as a PDE-constrained optimization problem,” Inverse Probl. **21**(5), 1507–1530 (2005). [CrossRef]

*L*in Eq. (8) vanishes with respect to

**,**

*η***and**

*u***, respectively. One major advantage of this PDE-constrained approach is that the solution of the forward problem does not have to be accurate until optimization reaches convergence. So the inexact solution obtained with a loose criterion can be used into optimization, which leads to a significant saving in the total reconstruction time. Recently, our group has for the first time used this PDE-constrained optimization into imaging of absorption and scattering maps in tissue: the augmented Lagrangian (AL) method [15**

*μ*_{λ}15. G. S. Abdoulaev, K. Ren, and A. H. Hielscher, “Optical tomography as a PDE-constrained optimization problem,” Inverse Probl. **21**(5), 1507–1530 (2005). [CrossRef]

**25**(1), 015010 (2009). [CrossRef]

**of all unknown chromophore concentrations, i.e.,**

*x***= (C**

*x*_{j}). We employ here the rSQP method to solve the PDE-constrained multispectral problem [Eq. (9)] by minimizing the multispectral Lagrangian function

*L*[Eq. (10)] with respect to

**,**

*η***and**

*u***. A multispectral version of the rSQP algorithm is described in the following section.**

*x*### 3.3 Reduced Sequential Quadratic Programming (rSQP)

*p*= (Δ

*x*, Δ

*u*)

_{λ}^{T}at (

*x*

^{k},

*u*

_{λ}^{k}),

*p*= (

*x*,

*u*), given by

_{λ}*x*, we finally can obtain the following forms for the updates of forward and inverse variables as:where the matrix-vector product

*j*-th chromophore concentration as:

## 4. Results

### 4.1 Numerical study

_{2}] and the one only in [Hb]. The corresponding chromophore concentrations are given in Table 1 for the background medium and the objects. The background medium has [HbO

_{2}] of 40[μM] and [Hb] of 20[μM]. The object 1 has [HbO

_{2}] of 80 while the object 2 has [Hb] of 40[μM]. These values are chosen here to mimic absorption coefficients of 0.1~0.2 (cm

^{−1}), which correspond to the typical transport regime.

*z*in Eq. (5), we used a mesh of 11538 triangle elements with a S

_{d}_{10}quadrature. The prediction

*Qu*in Eq. (5) is obtained on a coarser mesh of 2854 triangular elements with a S

_{d}_{6}quadrature. Measurements containing noise are simulated by adding an error term to

*z*in the form

_{d}*σ*is the standard deviation of measurement errors and

*ϖ*is the random variable with normal distribution. Here we chose a noise level of 15dB which represents the typical noise level encountered in optical tomography [17

17. A. D. Klose and A. H. Hielscher, “Quasi-newton methods in optical tomographic image reconstruction,” Inverse Probl. **19**(2), 387 (2003). [CrossRef]

_{2}] and [Hb], we only need two wavelengths to distinguish between these two chromophores. However, different sets of two wavelengths may affect the reconstruction accuracy. To illustrate the influence of the wavelengths set chosen, we consider two different sets of measurement wavelengths which are selected here from commonly available diode lasers: one set with (650nm, 830nm), the other set with (760nm, 830nm). Another reason for this set comes from the fact that, as shown in Fig. 1, the molar extinction coefficients of HbO

_{2}and Hb are different from each other for chosen wavelengths, which is thus believed to be favorable to reconstructing HbO

_{2}and Hb concentrations.

*ρ*

_{c}and a deviations factor

*ρ*

_{d}, as defined in [17

17. A. D. Klose and A. H. Hielscher, “Quasi-newton methods in optical tomographic image reconstruction,” Inverse Probl. **19**(2), 387 (2003). [CrossRef]

_{2}(Hb) concentration divided by the Hb (HbO

_{2}) concentration in Hb (HbO

_{2}) position of the known object. Besides the effects of different wavelength sets, we also investigate the CPU times in both PDE-constrained and conventional multispectral methods, which is another important metric for evaluation of the performance of the proposed algorithm. All reconstructions are performed using two methods: the PDE-constrained one-step multispectral method and the unconstrained two-step method. Note that the unconstrained two-step method is based on the quasi-Newton (BFGS) updating scheme [16,17

17. A. D. Klose and A. H. Hielscher, “Quasi-newton methods in optical tomographic image reconstruction,” Inverse Probl. **19**(2), 387 (2003). [CrossRef]

^{−2}can be used to solve for the linearized forward problem. Since we use a GMRES iterative solver, the usage of a loose tolerance (of 10

^{−2}) takes a much smaller number of iterations to a stopping criterion than the tight tolerance (of 10

^{−10}) used in the unconstrained two-step method. As a consequence, this leads to a significant time savings through the overall optimization process.

*all*forward [Fig. 2(a)] and inverse [Fig. 2(b)] errors

*at once*in each of optimization iterations.

_{2}and Hb obtained with two wavelength sets. Each row corresponds to the reconstructed images of HbO

_{2}and Hb concentrations, respectively, and the columns display the reconstructed images for wavelength sets 1 and 2, for the PDE-constrained and unconstrained two-step methods, respectively. In terms of image quality, the two methods with sets 1 and 2 give different results. For set 1, it can be seen from Table 2 and Fig. 3 that the PDE-constrained method gives more accurate results than the conventional method. The correlation factor of the PDE-constrained method is 0.62 and 0.71 for [HbO

_{2}] and [Hb], respectively. This is almost 20% and 40% better than when the two-step method is used (

*ρ*

_{c}= 0.54 and 0.56, respectively). For set 2, both methods give similar results for both [HbO

_{2}] and [Hb]. When both wavelength sets are compared against the method, the two-step method works better with the 760-830 set rather than the 650-830 set. On the other hand, the PDE-constrained method outperforms the unconstrained two-step method for both wavelength sets. In particular, it gives best results for the 650-830 set.

_{2}or Hb or both of the two (see Fig. 3). More evidences supporting this statement can be found from results obtained with the (760nm, 830nm) set. As compared to the results with the (650nm, 830nm) set, the two methods give lower accuracy both with respect to the correlation coefficient and the deviation factor as given in Table 2. Especially some cross-talk between two hemoglobin concentrations is observed in both of the two methods: the false Hb (HbO

_{2}) perturbation is retrieved in the position of object 2 Eq. (1) where the medium has only inhomogeneity in HbO

_{2}(Hb). In addition to cross-talk, the conventional method reveals larger artifacts and overestimation or underestimation in both [HbO

_{2}] and [Hb]. This nature of the (760nm, 830nm) set may be explained by the concept of a condition number

*κ*(

*E*) of the molar extinction coefficient matrix

*E*of HbO

_{2}and Hb that represents a measure of how ‘well-posed’ a problem is: the larger

*κ*(

*E*) is, the more ‘ill-posed’ is the system. In other words, if the condition number is large, even a small error in the estimated absorption coefficient may cause a large error in the reconstructed hemoglobin concentration, and vice versa. In our cases, the condition numbers for the (650nm, 830nm) and (760nm, 830nm) sets are 1.76 and 3.17 respectively. Therefore it is evident that the (760nm, 830nm) set exhibits the nature of more ill-posed noise-sensitive problem, consequently making it more difficult to distinguish between the two unknowns, as compared to the (650nm, 830nm) set.

### 4.2 Experimental study

^{6}cultured human Ewing sarcoma cells engineered to express luciferase (SK-NEP1-luc) were implanted intra-renally in NCR nude mice and allowed to grow until the tumor size reached about 1g as determined by weekly bioluminescence measurements. The experimental data was acquired with a continuous-wave digital optical tomography system that illuminates the target simultaneously with two wavelengths (λ = 760 nm and 830nm) modulated at 5 kHz and 7 kHz respectively. The system’s 16 sources and 32 detectors are configured in two rings, separated by 1.25cm, around a 3.175cm Delrin cylinder. Each ring contains 8 sources and 16 detectors arranged in a source-detector-detector-source configuration. The animal, anesthetized using isofluorane gas, was placed in the cylinder with the kidney and tumor located between the two rings of sources and detectors [see Fig. 4(a) ]. Then 1% Intralipid fluid was used to fill the remaining space in the cylinder to help reduce edge effects. The system uses silicon photodiodes to detect the transmitted and reflected photons passing through the cylinder. A digital-signal-processor (DSP) chip provides fast demodulation and control of the system giving an imaging frame rate of over 5 frames per second. More detailed information about the measurement system can be found elsewhere [18

18. J. M. Lasker, J. M. Masciotti, M. Schoenecker, C. H. Schmitz, and A. H. Hielscher, “Digital-signal-processor-based dynamic imaging system for optical tomography,” Rev. Sci. Instrum. **78**(8), 083706 (2007). [CrossRef] [PubMed]

19. Y. Pei, H. L. Graber, and R. L. Barbour, “Influence of systematic errors in reference states on image quality and on stability of derived information for DC optical imaging,” Appl. Opt. **40**(31), 5755–5769 (2001). [CrossRef] [PubMed]

*s*and

*d*denote the numbers of sources and detectors.

20. A. Yokoi, K. W. McCrudden, J. Huang, E. S. Kim, S. Z. Soffer, J. S. Frischer, A. Serur, T. New, J. Yuan, M. Mansukhani, K. O’toole, D. J. Yamashiro, and J. J. Kandel, “Human epidermal growth factor receptor signaling contributes to tumor growth via angiogenesis in her2/neu-expressing experimental Wilms’ tumor,” J. Pediatr. Surg. **38**(11), 1569–1573 (2003). [CrossRef] [PubMed]

21. J. Glade Bender, E. M. Cooney, J. J. Kandel, and D. J. Yamashiro, “Vascular remodeling and clinical resistance to antiangiogenic cancer therapy,” Drug Resist. Updat. **7**(4-5), 289–300 (2004). [CrossRef] [PubMed]

_{2}) and (Hb) as well as their sum the total hemoglobin (THb). As with the numerical studies, we compare the performance of the two methods (PDE-constrained multispectral and conventional two-step methods) on the experimental data. We measure the CPU times of the two methods, and since the exact distributions of chromophores are not known for the mouse, we just provide a qualitative assessment of the results based on the expected biological change.

_{2}] and [Hb] are shown in Figs. 5 and 6 . Note that Figs. 5 and 6 show the results for the 2cmx3cmx3cm volume in the cylinder of height 8cm and diameter 3.175cm. We present here a drawing of the 3D volume contour of the reconstructed values above the given threshold, which can provide a better way for visual inspection of the time-trace results than the 2D cross-section maps. In terms of the CPU times, the PDE-constrained multispectral method took approximately 2 hrs to converge, while the conventional two-step method reached convergence in about 24 hours on a Dual Core Intel Xeon 3.33GHz processor: this constitutes acceleration factor of about 10. Furthermore, the two methods gave different trends of reconstructed chromophores over time. As can be seen in Fig. 5, the images generated by the PDE-constrained multispectral method clearly show that [HbO

_{2}], [Hb], and [THb] increase in volume and value over time. This is expected as the tumor volume and vascular density increase over time as well [20

20. A. Yokoi, K. W. McCrudden, J. Huang, E. S. Kim, S. Z. Soffer, J. S. Frischer, A. Serur, T. New, J. Yuan, M. Mansukhani, K. O’toole, D. J. Yamashiro, and J. J. Kandel, “Human epidermal growth factor receptor signaling contributes to tumor growth via angiogenesis in her2/neu-expressing experimental Wilms’ tumor,” J. Pediatr. Surg. **38**(11), 1569–1573 (2003). [CrossRef] [PubMed]

21. J. Glade Bender, E. M. Cooney, J. J. Kandel, and D. J. Yamashiro, “Vascular remodeling and clinical resistance to antiangiogenic cancer therapy,” Drug Resist. Updat. **7**(4-5), 289–300 (2004). [CrossRef] [PubMed]

_{2}] and [Hb] values decrease at 72 hrs and 5 days, respectively. We believe that this is an artifact introduced by the two-step method. Since the two-step method does not take any spectral constraints into account, it has been observed to produce less reliable results before [4

4. A. Corlu, R. Choe, T. Durduran, K. Lee, M. Schweiger, S. R. Arridge, E. M. C. Hillman, and A. G. Yodh, “Diffuse optical tomography with spectral constraints and wavelength optimization,” Appl. Opt. **44**(11), 2082–2093 (2005). [CrossRef] [PubMed]

6. A. Li, Q. Zhang, J. P. Culver, E. L. Miller, and D. A. Boas, “Reconstructing chromosphere concentration images directly by continuous-wave diffuse optical tomography,” Opt. Lett. **29**(3), 256–258 (2004). [CrossRef] [PubMed]

7. A. Corlu, T. Durduran, R. Choe, M. Schweiger, E. M. C. Hillman, S. R. Arridge, and A. G. Yodh, “Uniqueness and wavelength optimization in continuous-wave multispectral diffuse optical tomography,” Opt. Lett. **28**(23), 2339–2341 (2003). [CrossRef] [PubMed]

## 5. Conclusions

*all-at-once*manner, which leads to a significant saving in the total image reconstruction time. We have evaluated the performance of the proposed algorithm through numerical studies and with experimental data obtained from tumor bearing mice.

## Acknowledgments

## References and links

1. | L.C. Enfield, A.P. Gibson, J.C. Hebden, and M. Douek, “Optical tomography of breast cancer—monitoring response to primary medical therapy,” Targ. Oncol. (2009) at DOI 10.1007/s11523-009-0115-z. [CrossRef] |

2. | G. Boverman, Q. Fang, S. A. Carp, E. L. Miller, D. H. Brooks, J. Selb, R. H. Moore, D. B. Kopans, and D. A. Boas, “Spatio-temporal imaging of the hemoglobin in the compressed breast with diffuse optical tomography,” Phys. Med. Biol. |

3. | J. Masciotti, F. Provenzano, J. Papa, A. Klose, J. Hur, X. Gu, D. Yamashiro, J. Kandel, and A. H. Hielscher, “Monitoring tumor growth and treatment in small animals with magnetic resonance optical tomographic imaging,” in Multimodal Biomedical Imaging; Fred S. Azar, Dimitris N. Metaxas, eds., SPIE International Symposium on Biomedical Optics, Proc. SPIE 6081, #608105 (2006). |

4. | A. Corlu, R. Choe, T. Durduran, K. Lee, M. Schweiger, S. R. Arridge, E. M. C. Hillman, and A. G. Yodh, “Diffuse optical tomography with spectral constraints and wavelength optimization,” Appl. Opt. |

5. | A. Y. Bluestone, M. Stewart, J. Lasker, G. S. Abdoulaev, A. H. Hielscher, G. S. Abdoulaev, and A. H. Hielscher, “Three-dimensional optical tomographic brain imaging in small animals, part 1: hypercapnia,” J. Biomed. Opt. |

6. | A. Li, Q. Zhang, J. P. Culver, E. L. Miller, and D. A. Boas, “Reconstructing chromosphere concentration images directly by continuous-wave diffuse optical tomography,” Opt. Lett. |

7. | A. Corlu, T. Durduran, R. Choe, M. Schweiger, E. M. C. Hillman, S. R. Arridge, and A. G. Yodh, “Uniqueness and wavelength optimization in continuous-wave multispectral diffuse optical tomography,” Opt. Lett. |

8. | A. H. Hielscher, A. Y. Bluestone, G. S. Abdoulaev, A. D. Klose, J. Lasker, M. Stewart, U. Netz, and J. Beuthan, “Near-infrared diffuse optical tomography,” Dis. Markers |

9. | A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. |

10. | K. Ren, G. Bal, and A. H. Hielscher, “Frequency domain optical tomography based on the equation of radiative transfer,” SIAM J. Sci. Comput. |

11. | H. K. Kim and A. H. Hielscher, “A PDE-constrained SQP algorithm for optical tomography based on the frequency-domain equation of radiative transfer,” Inverse Probl. |

12. | M. Modest, Riative heat transfer, MacGraw-Hill Inc., New York, 1993. |

13. | Y. Saad and M. H. Schultz, “GMRES: A generalized minimum residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Comput. |

14. | S. Prahl, “Optical properties spectra,” retrieved 16 March 2003, http://omlc.ogi.edu/spectra/index.html, 2001. |

15. | G. S. Abdoulaev, K. Ren, and A. H. Hielscher, “Optical tomography as a PDE-constrained optimization problem,” Inverse Probl. |

16. | J. Nocedal and S. J. Wright, |

17. | A. D. Klose and A. H. Hielscher, “Quasi-newton methods in optical tomographic image reconstruction,” Inverse Probl. |

18. | J. M. Lasker, J. M. Masciotti, M. Schoenecker, C. H. Schmitz, and A. H. Hielscher, “Digital-signal-processor-based dynamic imaging system for optical tomography,” Rev. Sci. Instrum. |

19. | Y. Pei, H. L. Graber, and R. L. Barbour, “Influence of systematic errors in reference states on image quality and on stability of derived information for DC optical imaging,” Appl. Opt. |

20. | A. Yokoi, K. W. McCrudden, J. Huang, E. S. Kim, S. Z. Soffer, J. S. Frischer, A. Serur, T. New, J. Yuan, M. Mansukhani, K. O’toole, D. J. Yamashiro, and J. J. Kandel, “Human epidermal growth factor receptor signaling contributes to tumor growth via angiogenesis in her2/neu-expressing experimental Wilms’ tumor,” J. Pediatr. Surg. |

21. | J. Glade Bender, E. M. Cooney, J. J. Kandel, and D. J. Yamashiro, “Vascular remodeling and clinical resistance to antiangiogenic cancer therapy,” Drug Resist. Updat. |

**OCIS Codes**

(170.5280) Medical optics and biotechnology : Photon migration

(110.4234) Imaging systems : Multispectral and hyperspectral imaging

(110.6955) Imaging systems : Tomographic imaging

**ToC Category:**

Image Reconstruction and Inverse Problems

**History**

Original Manuscript: June 14, 2010

Revised Manuscript: August 25, 2010

Manuscript Accepted: September 7, 2010

Published: September 8, 2010

**Virtual Issues**

Optical Imaging and Spectroscopy (2010) *Biomedical Optics Express*

**Citation**

Hyun Keol Kim, Molly Flexman, Darrell J. Yamashiro, Jessica J. Kandel, and Andreas H. Hielscher, "PDE-constrained multispectral imaging of tissue chromophores with the equation of radiative transfer," Biomed. Opt. Express **1**, 812-824 (2010)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-1-3-812

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### References

- L. C. Enfield, A. P. Gibson, J. C. Hebden, and M. Douek, “Optical tomography of breast cancer—monitoring response to primary medical therapy,” Targ. Oncol. 4(3) 219-233 (2009). [CrossRef]
- G. Boverman, Q. Fang, S. A. Carp, E. L. Miller, D. H. Brooks, J. Selb, R. H. Moore, D. B. Kopans, and D. A. Boas, “Spatio-temporal imaging of the hemoglobin in the compressed breast with diffuse optical tomography,” Phys. Med. Biol. 52(12), 3619–3641 (2007). [CrossRef] [PubMed]
- J. Masciotti, F. Provenzano, J. Papa, A. Klose, J. Hur, X. Gu, D. Yamashiro, J. Kandel, and A. H. Hielscher, “Monitoring tumor growth and treatment in small animals with magnetic resonance optical tomographic imaging,” in Multimodal Biomedical Imaging; Fred S. Azar, Dimitris N. Metaxas, eds., SPIE International Symposium on Biomedical Optics, Proc. SPIE 6081, #608105 (2006).
- A. Corlu, R. Choe, T. Durduran, K. Lee, M. Schweiger, S. R. Arridge, E. M. C. Hillman, and A. G. Yodh, “Diffuse optical tomography with spectral constraints and wavelength optimization,” Appl. Opt. 44(11), 2082–2093 (2005). [CrossRef] [PubMed]
- A. Y. Bluestone, M. Stewart, J. Lasker, G. S. Abdoulaev, A. H. Hielscher, G. S. Abdoulaev, and A. H. Hielscher, “Three-dimensional optical tomographic brain imaging in small animals, part 1: hypercapnia,” J. Biomed. Opt. 9(5), 1046–1062 (2004). [CrossRef] [PubMed]
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