## A deterministic approach to the adapted optode placement for illumination of highly scattering tissue |

Biomedical Optics Express, Vol. 3, Issue 7, pp. 1732-1743 (2012)

http://dx.doi.org/10.1364/BOE.3.001732

Acrobat PDF (14765 KB)

### Abstract

A novel approach is presented for computing optode placements that are adapted to specific geometries and tissue characteristics, e.g., in optical tomography and photodynamic cancer therapy. The method is based on optimal control techniques together with a sparsity-promoting penalty that favors pointwise solutions, yielding both locations and magnitudes of light sources. In contrast to current discrete approaches, the need for specifying an initial set of candidate configurations as well as the exponential increase in complexity with the number of optodes are avoided. This is demonstrated with computational examples from photodynamic therapy.

© 2012 OSA

## 1. Introduction

1. J. P. Culver, V. Ntziachristos, M. J. Holboke, and A. G. Yodh, “Optimization of optode arrangements for diffuse optical tomography: A singular-value analysis,” Opt. Lett. **26**, 701–703 (2001). [CrossRef]

2. H. Xu, H. Dehghani, B. W. Pogue, R. Springett, K. D. Paulsen, and J. F. Dunn, “Near-infrared imaging in the small animal brain: optimization of fiber positions,” J. Biomed. Opt. **8**, 102–110 (2003). [CrossRef] [PubMed]

3. E. E. Graves, J. P. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, “Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomography,” J. Opt. Soc. Am. A **21**, 231–241 (2004). [CrossRef]

4. T. Lasser and V. Ntziachristos, “Optimization of 360° projection fluorescence molecular tomography,” Med. Image Anal. **11**, 389–399 (2007). [CrossRef] [PubMed]

5. D. Dolmans, D. Fukumura, and R. Jain, “Photodynamic therapy for cancer,” Nat. Rev. Cancer **3**, 380–387 (2003). [CrossRef] [PubMed]

6. H. Schouwink and P. Baas, “Foscan-mediated photodynamic therapy and operation for malignant pleural mesothelioma,” Ann. Thorac. Surg. **78**, 388; author reply 388–388; author reply 389 (2004). [CrossRef] [PubMed]

7. P. J. Dwyer, W. M. White, R. L. Fabian, and R. R. Anderson, “Optical integrating balloon device for photodynamic therapy,” Lasers Surg. Med. **26**, 58–66 (2000). [CrossRef] [PubMed]

10. P. Baas, L. Murrer, F. A. Zoetmulder, F. A. Stewart, H. B. Ris, N. van Zandwijk, J. L. Peterse, and E. J. Rutgers, “Photodynamic therapy as adjuvant therapy in surgically treated pleural malignancies,” Br. J. Cancer **76**, 819–826 (1997). [CrossRef] [PubMed]

11. P. van Veen, J. H. Schouwink, W. M. Star, H. J. Sterenborg, J. R. van der Sijp, F. A. Stewart, and P. Baas, “Wedge-shaped applicator for additional light delivery and dosimetry in the diaphragmal sinus during photodynamic therapy for malignant pleural mesothelioma,” Phys. Med. Biol. **46**, 1873–1883 (2001). [CrossRef] [PubMed]

8. J. S. Friedberg, R. Mick, J. Stevenson, J. Metz, T. Zhu, J. Buyske, D. H. Sterman, H. I. Pass, E. Glatstein, and S. M. Hahn, “A phase I study of Foscan-mediated photodynamic therapy and surgery in patients with mesothelioma,” Ann. Thorac. Surg. **75**, 952–959 (2003). [CrossRef] [PubMed]

12. B. Selm, M. Rothmaier, M. Camenzind, T. Khan, and H. Walt, “Novel flexible light diffuser and irradiation properties for photodynamic therapy,” J. Biomed. Opt. **12**, 034024 (2007). [CrossRef] [PubMed]

13. M. Rothmaier, B. Selm, S. Spichtig, D. Haensse, and M. Wolf, “Photonic textiles for pulse oximetry,” Opt. Express **16**, 12973–12986 (2008). [CrossRef] [PubMed]

14. Y. Hu, K. Wang, and T. C. Zhu, “A light blanket for intraoperative photodynamic therapy,” Proc. SPIE **7380**, 73801W (2009). [CrossRef]

15. Y. Hu, K. Wang, and T. C. Zhu, “Pre-clinic study of uniformity of light blanket for intraoperative photodynamic therapy,” Proc. SPIE **7551**, 755112 (2010). [CrossRef]

16. G. Stadler, “Elliptic optimal control problems with *L*^{1}-control cost and applications for the placement of control devices,” Comput. Optim. Appl. **44**, 159–181 (2009). [CrossRef]

## 2. Theory

17. B. W. Henderson, T. M. Busch, L. A. Vaughan, N. P. Frawley, D. Babich, T. A. Sosa, J. D. Zollo, A. S. Dee, M. T. Cooper, D. A. Bellnier, W. R. Greco, and A. R. Oseroff, “Photofrin photodynamic therapy can significantly deplete or preserve oxygenation in human basal cell carcinomas during treatment, depending on fluence rate,” Canc. Treat. **60**, 525–529 (2000).

### 2.1. Mathematical model

18. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. **15**, R41–R93 (1999). [CrossRef]

*φ*∈

*H*

^{1}(Ω), The geometry of the object is given by the domain Ω ⊂ ℝ

*,*

^{d}*d*∈ {2,3} being the number of spatial dimensions, with boundary Γ whose outward normal vector is denoted by

*n⃗*. The medium is characterized by the absorption coefficient

*μ*, the reduced scattering coefficient

_{a}*μ*′

*, and the diffusion coefficient*

_{s}*ρ*models the reflection of a part of the photons at the boundary due to a mismatch in the index of refraction. Finally, the source term

*q*models the light emission of the embedded optodes.

### 2.2. Optode placement optimization

*q*∈ ℝ

_{j}_{+}and

*x*∈ Ω, 1 ≤

_{j}*j*≤

*N*, where

*δ*denotes the Dirac distribution (i.e., ∫

*f dδ*(

*x*) =

*f*(0) for all continuous functions

*f*). A straightforward approach for optimizing the placement of the optodes (as was done, e.g., in [19

19. M. Freiberger, C. Clason, and H. Scharfetter, “Total variation regularization for nonlinear fluorescence tomography with an augmented Lagrangian splitting approach,” Appl. Opt. **49**, 3741–3747 (2010). [CrossRef] [PubMed]

*M*≫

*N*possible optode locations

*x*

_{1},...,

*x*and chose the best

_{M}*N*locations such that a certain performance criterion

*J*(

*q*) is minimized. The corresponding optimal source magnitudes

*q*would then be computed in a second step.

_{j}*z*in an observation region

*ω*⊂ Ω such that

_{o}*φ*|

_{ωo}denotes the restriction of

*φ*to

*ω*. Due to the linearity of the forward problem, we can take

_{o}*z*= 1 Wm

^{−2}without loss of generality. After optimization, the magnitude of the resultant sources can be linearly scaled to achieve the required illumination

*z*. In addition, we restrict the possible light source locations to a control region

*ω*⊂ Ω, which does not overlap with the observation region

_{q}*ω*(i.e.,

_{o}*ω*̄

*∩*

_{q}*ω*̄

*= ∅︀), and enforce non-negativity of the source term*

_{o}*q*(which represents the optodes). This leads to the following optimization problem: It was shown in [21] that this problem has a solution

*q*

^{*}∈

*ℳ*(

*ω*), which can be approximated by a sequence of functions

_{q}*q*∈

_{γ}*L*

^{2}(

*ω*) for

_{q}*γ*→ ∞ satisfying where

*p*is the solution of (2.2) with right hand side

_{γ}*f*:=

*φ*−

_{γ}*z*and

*φ*is the solution of (2.1) with right hand side

_{γ}*q*. Equation (2.4) can be solved using a semismooth Newton method which is superlinearly convergent; see [21]. To globalize the Newton method and closely approximate the solution

_{γ}*q*

^{*}of (2.3), we use a continuation scheme in

*γ*where we iteratively solve the problem for an increasing sequence

*γ*, using the previous solution as initial guess.

_{n}### 2.3. Finite element discretization

*q*converge to measures as

_{γ}*γ*increases. We therefore employ the finite element discretization proposed in [22], where the photon density

*φ*and the adjoint variable

_{γ}*p*are discretized using piecewise linear elements on a given triangulation

_{γ}*T*, while the source term

*q*is discretized using linear combinations of Dirac distributions centered at the interior nodes

_{γ}*x*, 1 ≤

_{i}*i*≤

*N*(

*T*), of

*T*: In practice, the number of nodes

*N*(

*T*) will be determined by the need to resolve the geometry of the domain and the required accuracy of the solution of the forward model (2.1). Although further refinement of the triangulation increases the number of possible optode locations, the sparsity-promoting property of the minimized functional discourages placing additional optodes. In fact, it was shown in [22] that for a given discretization of the forward model, the computed sources (for

*γ*→ ∞) are optimal among all (non-discretized) measures.

*e*becomes i.e., the mass matrix is the identity. Introducing the stiffness matrix

_{j}*A*corresponding to (2.1) and the observation mass matrix

*M*with entries

_{o}*M*= ∫

_{ij}_{ωo}

*e*

_{i}e_{j}*dx*, we obtain the discrete optimality system Eliminating

*q*using the last equation and applying a semismooth Newton method, cf. [21], we have to solve for (

_{γ}*φ*

^{k}^{+1},

*p*

^{k}^{+1}) the block system where

*D*is a diagonal matrix with the entries of the vector

_{k}*d*, on the diagonal. It can be shown that the semismooth Newton method has converged once

^{k}*d*

^{k}^{+1}=

*d*holds. After the final

^{k}*p*has been computed, the corresponding control can be obtained from (2.4). The complete procedure is given in Algorithm 1.

^{k}## 3. Materials and methods

23. A. Logg, K.-A. Mardal, and G. N. Wells, Automated Solution of Differential Equations by the Finite Element Method (Springer, 2012); software available from http://fenicsproject.org. [CrossRef]

*m*

^{*}= 34 (such that

*γ*

^{*}≈ 10

^{10}) and

*k*

^{*}= 20. To model a textile-based diffuser, the material parameters in (2.1) are taken as

*μ*= 10

_{a}^{−4}mm

^{−1},

*μ*′

*= 10*

_{s}^{−1}mm

^{−1}, and

*ρ*= 0.1992. The influence of the parameter

*α*is illustrated by comparing the results for different values of

*α*specified below.

*κ*were created as shown in Fig. 1. The dimensions correspond approximately to a width of 10 mm and height of 120 mm. In all cases, the region

*ω*in which the illumination should be homogenized are the left and right outer lines (indicated in orange in Fig. 1). The region

_{o}*ω*where optodes are allowed to be placed is a single line equidistant from both (indicated by a dashed line in Fig. 1). The meshes for the single-curved models of curvature

_{q}*κ*= 5, 10, 20, 40, and 60 consist of 61038, 61789, 67160, 80664, and 105322 finite elements, respectively. The double-curved models of curvature

*κ*= 5, 10, 15, and 20 are comprised of 62349, 70735, 82119, and 104220 finite elements, respectively.

*ω*is defined as the outer and inner surface of the model, and

_{o}*ω*is an interior manifold equidistant from both (see Fig. 2;

_{q}*ω*is indicated in purple). The generated mesh consists of 81770 elements.

_{q}*α*. The coefficient of variation

*c*of the resultant photon density

_{v}*φ*over the observation region

_{γ}*ω*and the number

_{o}*N*of sources after the optimization procedure serve as quality measures. For the latter, the nodes in the control region

*ω*satisfying

_{q}*q*> 10

_{γ}^{−16}are counted. We compare the results for

*α*∈ {0.1, 0.01, 0.001} for the two-dimensional models and

*α*∈ {0.2, 0.4,...,1.8} for the three-dimensional model.

## 4. Results

*N*with the total number of nodes for each model, the algorithm indeed produces discrete sources that can be used as optode positions. The obtained coefficients of variation

*c*indicate that a homogeneous illumination of the desired region is possible at least for

_{v}*α*< 0.1, demonstrating the feasibility of the proposed approach. The robustness of the algorithm with respect to geometry is illustrated by the fact that the achieved variations do not depend very much on the curvature. It can also be observed how the penalty parameter

*α*determines the tradeoff between the number of active optodes and the homogeneity of the illumination in the region of interest: larger values of

*α*yield fewer optodes but less homogeneous illumination, again independent of curvature.

*α*is shown in Fig. 3(a) and Fig. 3(b) for a representative single-curved (

*κ*= 20) and double-curved model (

*κ*= 15), respectively, where the relative strength of the sources is coded by height. (Note when comparing Tables 1 and 2 with Fig. 3 that neighboring active nodes appear as a single peak and thus can be taken as a single optode.) While for the single-curved model and

*α*= 0.1, the distribution of optodes agrees well with the intuitive choice of equally spaced optodes of approximately equal magnitude, the other values indicate that a better illumination can be achieved with stronger sources towards the tips of the model. It should be pointed out that even in the former case, the number of optodes to be distributed is not obvious. For the double-curved models, the results indicate that optodes should be placed preferentially in regions where the curvature changes.

*φ*(in Wm

_{γ}^{−2}, normalized to unit mean) plotted along part of the observation region (left line in Fig. 1), illustrating how the parameter

*α*and the model geometry influence the homogeneity of the illumination in this region. As expected, photon fluence shows the most pronounced inhomogeneities close to the borders. In the case of the single curved model, a nearly sinusoidal ripple pattern arises in more than 80 % of the target region, while in the double curved model the ripple is superimposed on a step-profile with the steps located approximately at the zero-crossing points of the curvature. With

*α*= 0.1, the peak–peak fluctuations are still around 40 % of the mean value even far away from the borders, which may be considered as unsatisfactory. However, when decreasing alpha to 0.01 or less, the ripple remains within a few percent, which is sufficient, especially when comparing this value to other sources of fluctuations of the irradiation such as local absorption changes by tissue inhomogeneities, bleeding, or inhomogeneities of the distribution of the photosensitizer.

*α*= 1.8, no controls are placed and thus the photon density is zero. This is consistent with the theory, which predicts that there is a threshold value for

*α*above which the optimal control is identically zero; cf. [22

1. J. P. Culver, V. Ntziachristos, M. J. Holboke, and A. G. Yodh, “Optimization of optode arrangements for diffuse optical tomography: A singular-value analysis,” Opt. Lett. **26**, 701–703 (2001). [CrossRef]

^{−2}, normalized to unit mean) for

*α*= 1.2,

*α*= 0.8, and

*α*= 0.4. Due to the nonuniform curvature of the model, a homogeneous illumination is harder to achieve than in the two-dimensional case, especially at the borders of the target region. However, for

*α*< 1.2, the inhomogeneities in the interior are usually within 10 %, and the few hot spots of 30 % would still be acceptable. Although of course the specific placement may be difficult to realize in practice, the qualitative distribution can be useful information in the initial design process.

## 5. Discussion

*κ*= 5. Our approach could therefore also be used in an interactive setting, where the engineer will adapt design parameters, such as the optical coefficients of the diffuser, based on the outcome of an optimization run.

*J*(

*q*). In principle, the approach can be applied to the problem of optimal experiment design for optical tomography, if the objective

*J*(

*q*) is based on a suitable sensitivity term. However, this extension of our method is subject to future work.

## Acknowledgments

## References and links

1. | J. P. Culver, V. Ntziachristos, M. J. Holboke, and A. G. Yodh, “Optimization of optode arrangements for diffuse optical tomography: A singular-value analysis,” Opt. Lett. |

2. | H. Xu, H. Dehghani, B. W. Pogue, R. Springett, K. D. Paulsen, and J. F. Dunn, “Near-infrared imaging in the small animal brain: optimization of fiber positions,” J. Biomed. Opt. |

3. | E. E. Graves, J. P. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, “Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomography,” J. Opt. Soc. Am. A |

4. | T. Lasser and V. Ntziachristos, “Optimization of 360° projection fluorescence molecular tomography,” Med. Image Anal. |

5. | D. Dolmans, D. Fukumura, and R. Jain, “Photodynamic therapy for cancer,” Nat. Rev. Cancer |

6. | H. Schouwink and P. Baas, “Foscan-mediated photodynamic therapy and operation for malignant pleural mesothelioma,” Ann. Thorac. Surg. |

7. | P. J. Dwyer, W. M. White, R. L. Fabian, and R. R. Anderson, “Optical integrating balloon device for photodynamic therapy,” Lasers Surg. Med. |

8. | J. S. Friedberg, R. Mick, J. Stevenson, J. Metz, T. Zhu, J. Buyske, D. H. Sterman, H. I. Pass, E. Glatstein, and S. M. Hahn, “A phase I study of Foscan-mediated photodynamic therapy and surgery in patients with mesothelioma,” Ann. Thorac. Surg. |

9. | T. Krueger, H. J. Altermatt, D. Mettler, B. Scholl, L. Magnusson, and H.-B. Ris, “Experimental photodynamic therapy for malignant pleural mesothelioma with pegylated mTHPC,” Lasers Surg. Med. |

10. | P. Baas, L. Murrer, F. A. Zoetmulder, F. A. Stewart, H. B. Ris, N. van Zandwijk, J. L. Peterse, and E. J. Rutgers, “Photodynamic therapy as adjuvant therapy in surgically treated pleural malignancies,” Br. J. Cancer |

11. | P. van Veen, J. H. Schouwink, W. M. Star, H. J. Sterenborg, J. R. van der Sijp, F. A. Stewart, and P. Baas, “Wedge-shaped applicator for additional light delivery and dosimetry in the diaphragmal sinus during photodynamic therapy for malignant pleural mesothelioma,” Phys. Med. Biol. |

12. | B. Selm, M. Rothmaier, M. Camenzind, T. Khan, and H. Walt, “Novel flexible light diffuser and irradiation properties for photodynamic therapy,” J. Biomed. Opt. |

13. | M. Rothmaier, B. Selm, S. Spichtig, D. Haensse, and M. Wolf, “Photonic textiles for pulse oximetry,” Opt. Express |

14. | Y. Hu, K. Wang, and T. C. Zhu, “A light blanket for intraoperative photodynamic therapy,” Proc. SPIE |

15. | Y. Hu, K. Wang, and T. C. Zhu, “Pre-clinic study of uniformity of light blanket for intraoperative photodynamic therapy,” Proc. SPIE |

16. | G. Stadler, “Elliptic optimal control problems with |

17. | B. W. Henderson, T. M. Busch, L. A. Vaughan, N. P. Frawley, D. Babich, T. A. Sosa, J. D. Zollo, A. S. Dee, M. T. Cooper, D. A. Bellnier, W. R. Greco, and A. R. Oseroff, “Photofrin photodynamic therapy can significantly deplete or preserve oxygenation in human basal cell carcinomas during treatment, depending on fluence rate,” Canc. Treat. |

18. | S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. |

19. | M. Freiberger, C. Clason, and H. Scharfetter, “Total variation regularization for nonlinear fluorescence tomography with an augmented Lagrangian splitting approach,” Appl. Opt. |

20. | C. Clason and K. Kunisch, “A duality-based approach to elliptic control problems in non-reflexive Banach spaces,” ESAIM Control Optim. Calc. Var. |

21. | C. Clason and K. Kunisch, “A measure space approach to optimal source placement,” Comput. Optim. Appl. (online first, Nov. 9, 2011). |

22. | E. Casas, C. Clason, and K. Kunisch, “Approximation of elliptic control problems in measure spaces with sparse solutions,” SIAM J. Control Optim. (to be published). |

23. | A. Logg, K.-A. Mardal, and G. N. Wells, Automated Solution of Differential Equations by the Finite Element Method (Springer, 2012); software available from http://fenicsproject.org. [CrossRef] |

**OCIS Codes**

(060.2380) Fiber optics and optical communications : Fiber optics sources and detectors

(120.4570) Instrumentation, measurement, and metrology : Optical design of instruments

(170.3890) Medical optics and biotechnology : Medical optics instrumentation

(170.5180) Medical optics and biotechnology : Photodynamic therapy

(220.2945) Optical design and fabrication : Illumination design

**ToC Category:**

Optics of Tissue and Turbid Media

**History**

Original Manuscript: April 20, 2012

Revised Manuscript: June 5, 2012

Manuscript Accepted: June 5, 2012

Published: June 26, 2012

**Citation**

Patricia Brunner, Christian Clason, Manuel Freiberger, and Hermann Scharfetter, "A deterministic approach to the adapted optode placement for illumination of highly scattering tissue," Biomed. Opt. Express **3**, 1732-1743 (2012)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-3-7-1732

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### References

- J. P. Culver, V. Ntziachristos, M. J. Holboke, and A. G. Yodh, “Optimization of optode arrangements for diffuse optical tomography: A singular-value analysis,” Opt. Lett.26, 701–703 (2001). [CrossRef]
- H. Xu, H. Dehghani, B. W. Pogue, R. Springett, K. D. Paulsen, and J. F. Dunn, “Near-infrared imaging in the small animal brain: optimization of fiber positions,” J. Biomed. Opt.8, 102–110 (2003). [CrossRef] [PubMed]
- E. E. Graves, J. P. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, “Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomography,” J. Opt. Soc. Am. A21, 231–241 (2004). [CrossRef]
- T. Lasser and V. Ntziachristos, “Optimization of 360° projection fluorescence molecular tomography,” Med. Image Anal.11, 389–399 (2007). [CrossRef] [PubMed]
- D. Dolmans, D. Fukumura, and R. Jain, “Photodynamic therapy for cancer,” Nat. Rev. Cancer3, 380–387 (2003). [CrossRef] [PubMed]
- H. Schouwink and P. Baas, “Foscan-mediated photodynamic therapy and operation for malignant pleural mesothelioma,” Ann. Thorac. Surg.78, 388; author reply 388–388; author reply 389 (2004). [CrossRef] [PubMed]
- P. J. Dwyer, W. M. White, R. L. Fabian, and R. R. Anderson, “Optical integrating balloon device for photodynamic therapy,” Lasers Surg. Med.26, 58–66 (2000). [CrossRef] [PubMed]
- J. S. Friedberg, R. Mick, J. Stevenson, J. Metz, T. Zhu, J. Buyske, D. H. Sterman, H. I. Pass, E. Glatstein, and S. M. Hahn, “A phase I study of Foscan-mediated photodynamic therapy and surgery in patients with mesothelioma,” Ann. Thorac. Surg.75, 952–959 (2003). [CrossRef] [PubMed]
- T. Krueger, H. J. Altermatt, D. Mettler, B. Scholl, L. Magnusson, and H.-B. Ris, “Experimental photodynamic therapy for malignant pleural mesothelioma with pegylated mTHPC,” Lasers Surg. Med.32, 61–68 (2003). [CrossRef] [PubMed]
- P. Baas, L. Murrer, F. A. Zoetmulder, F. A. Stewart, H. B. Ris, N. van Zandwijk, J. L. Peterse, and E. J. Rutgers, “Photodynamic therapy as adjuvant therapy in surgically treated pleural malignancies,” Br. J. Cancer76, 819–826 (1997). [CrossRef] [PubMed]
- P. van Veen, J. H. Schouwink, W. M. Star, H. J. Sterenborg, J. R. van der Sijp, F. A. Stewart, and P. Baas, “Wedge-shaped applicator for additional light delivery and dosimetry in the diaphragmal sinus during photodynamic therapy for malignant pleural mesothelioma,” Phys. Med. Biol.46, 1873–1883 (2001). [CrossRef] [PubMed]
- B. Selm, M. Rothmaier, M. Camenzind, T. Khan, and H. Walt, “Novel flexible light diffuser and irradiation properties for photodynamic therapy,” J. Biomed. Opt.12, 034024 (2007). [CrossRef] [PubMed]
- M. Rothmaier, B. Selm, S. Spichtig, D. Haensse, and M. Wolf, “Photonic textiles for pulse oximetry,” Opt. Express16, 12973–12986 (2008). [CrossRef] [PubMed]
- Y. Hu, K. Wang, and T. C. Zhu, “A light blanket for intraoperative photodynamic therapy,” Proc. SPIE7380, 73801W (2009). [CrossRef]
- Y. Hu, K. Wang, and T. C. Zhu, “Pre-clinic study of uniformity of light blanket for intraoperative photodynamic therapy,” Proc. SPIE7551, 755112 (2010). [CrossRef]
- G. Stadler, “Elliptic optimal control problems with L1-control cost and applications for the placement of control devices,” Comput. Optim. Appl.44, 159–181 (2009). [CrossRef]
- B. W. Henderson, T. M. Busch, L. A. Vaughan, N. P. Frawley, D. Babich, T. A. Sosa, J. D. Zollo, A. S. Dee, M. T. Cooper, D. A. Bellnier, W. R. Greco, and A. R. Oseroff, “Photofrin photodynamic therapy can significantly deplete or preserve oxygenation in human basal cell carcinomas during treatment, depending on fluence rate,” Canc. Treat.60, 525–529 (2000).
- S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl.15, R41–R93 (1999). [CrossRef]
- M. Freiberger, C. Clason, and H. Scharfetter, “Total variation regularization for nonlinear fluorescence tomography with an augmented Lagrangian splitting approach,” Appl. Opt.49, 3741–3747 (2010). [CrossRef] [PubMed]
- C. Clason and K. Kunisch, “A duality-based approach to elliptic control problems in non-reflexive Banach spaces,” ESAIM Control Optim. Calc. Var.17, 243–266 (2011). [CrossRef]
- C. Clason and K. Kunisch, “A measure space approach to optimal source placement,” Comput. Optim. Appl. (online first, Nov. 9, 2011).
- E. Casas, C. Clason, and K. Kunisch, “Approximation of elliptic control problems in measure spaces with sparse solutions,” SIAM J. Control Optim. (to be published).
- A. Logg, K.-A. Mardal, G. N. Wells, and , Automated Solution of Differential Equations by the Finite Element Method (Springer, 2012); software available from http://fenicsproject.org . [CrossRef]

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