Modeling the forward problem based on the adaptive FEMs framework in bioluminescence tomography Yujie Lva , Jie Tiana , Hui Lia,c , Jie Luoa , Wenxiang Congb , Ge Wangb and Durairaj Kumarb a
Medical Image Processing Group, Institute of Automation, Chinese Academy of Sciences, P. O. Box 2728, Beijing 100080, China b Bioluminescence Tomography Laboratory, Department of Radiology, University of Iowa, Iowa City, Iowa 52242, USA c Department of Education Technology, Capital Normal University, Beijing 100037, China ABSTRACT
Bioluminescence tomography (BLT) is a novel technique in vivo which may localize and quantify bioluminescent source to reveal the molecular and cellular information, and therefore it can monitor the growth and regression of tumor non-invasively. In complicated biological tissue, the accuracy improvement of numerical solution to the forward problem of BLT is beneficial to achieve better spatial resolution of source distribution. In this paper, we introduce the adaptive FEMs framework based on the diffusion equation to enhance the solution accuracy of the forward problem, and the bioluminescence imaging experiment has been performed with the heterogenous physical phantom which is also scanned by microCT scanner to generate the volumetric mesh as the initial finite element mesh. Finally, The effectiveness of the adaptive FEMs framework is demonstrated with the comparison between the experimental results and the simulation solution. Keywords: bioluminescence tomography/imaging, diffusion approximation, adaptive finite element methods (FEMs), mesh generation
1. INTRODUCTION Small-animal molecular imaging is a rapidly emerging biomedical research field where is involved in the adaptation of current clinical imaging methods and new imaging technologies based on photonics.1 Bioluminescence tomography (BLT)2 and fluorescence molecular tomography (FMT)1 are the important tool for small animal study in vivo at the molecular and cellular level. The mechanism of BLT and FMT is to identify light source distribution from the light flux detected on the surface of small animal, which has been used for tumorigenesis and metastasis, drug discovery, and gene therapy etc.3 Compared with the traditional structural and functional imaging modalities such as computed tomography (CT), magnetic resonance imaging (MRI) and positron emission tomography (PET), BLT and FMT can obtain functional and spatial information simultaneously with much higher imaging contrast and sensitivity,4 which is beneficial from the optical reporters linked to the tissue-specific target.3 In fluorescence molecular tomography, external exciting light sources result in the autofluorescence on the surface of small animal, so bioluminescence imaging has low background noise and can detect the surface light signals primely emitted by about 500 cells subcutaneously.5 As far as solving the forward problem of BLT is concerned, there are three solved methods, that is, analytical modeling techniques, statistical modeling techniques and numerical modeling techniques.4 However, analytical methods is only suitable for the homogeneous object and the single spherical perturbed media, and statistical methods spend a great deal of time,6 numerical techniques are required for modeling the complicated biological tissue. Bioluminescent photon propagation in biological tissue is governed by the radiative transfer equation Further author information: (Send correspondence to Jie Tian and Ge Wang) Jie Tian: E-mail:
[email protected], Telephone: +86 (10)-82618465 Ge Wang: E-mail:
[email protected], Telephone: +1 (319)-356-2930
(RTE).7 The radiative transfer equation is computationally expensive in practical BLT, the diffusion equation is a relative accurate approximation, and has been widely adopted to describe photon propagation when the biological tissue has characteristic of high scattering optical property. General finite element method has been developed to solve the diffusion equation with a priori optimal fixed mesh for the physical phantom or tissue. However, tremendous time and memory cost compels researcher to look for a novel method for dealing with the contradiction between the numerical resolution and the mesh discretization degree to the given domain. Recently, in near infrared (NIR) fluorescence enhanced imaging field, the adaptive FEMs have been performed to reduce the computational complexity of the forward problem, which proposes a preferable strategy to solve the above contradiction.8 In this work, we propose the adaptive FEMs framework to model photon transportation based on the diffusion equation. Using the hierarchical defect correction based a posteriori error estimate, the evolution of the mesh is adaptively performed. In order to further accelerate the simulation procedure, BPX-preconditioner, which can remarkably reduce the condition number of the solved linear equation, is introduced to solve the finiteelement based linear equation. In addition, a heterogeneous physical phantom is fabricated in bioluminescent imaging experiment and the finite-element volumetric mesh of the phantom is also built through microCT slices via a series of operations. Finally, we also compare the simulation solution with the experimental results, and demonstrate the validity of the adaptive FEMs framework.
2. DIFFUSION APPROXIMATION AND BOUNDARY CONDITION 2.1. Diffusion approximation and boundary condition In bioluminescence tomography, the measured optical signal is continuously emitted by internal bioluminescent source, so the diffusive photon propagation is depicted by the steady-state diffusion equation and Robin boundary condition:9 −∇· D(x)∇Φ(x) + µa (x)Φ(x) = S(x) (x ∈ Ω) (1-1) Φ(x) + 2A(x; n, n0 )D(x) v(x)·∇Φ(x) = 0 (x ∈ ∂Ω) (1-2) where Ω and ∂Ω are the given domain and the domain boundary; Φ(x) denotes the photon flux density[W atts/mm2 ]; S(x) is the source energy density [W atts/mm3 ]; µa (x) is the absorption coefficient[mm−1 ]; D(x)=1/(3(µa (x) + (1 − g)µs (x))) is the optical diffusion coefficient[mm−1 ], µs (x) the scattering coefficient[mm−1 ], g the anisotropy parameter. In the following bioluminescence imaging experiment, the optical parameters µa (x), µs (x) and g are measured by optical sensing techniques.10 When the bioluminescence imaging experiment is carried out in a dark lightproof environment, Eq.(1-2) represents the boundary condition between the medium and the small-animal or phantom, where v is the unit outer normal on ∂Ω. Considering the mismatch between the refractive indices n for Ω and n0 for the external medium, A(x; n, n0 ) can be approximately represented: A(x; n, n0 ) ≈
1 + R(x) 1 − R(x)
(2)
n0 is close to 1.0 when the experimental environment is in air, R(x) can be approximated by9 R(x) ≈ −1.4399n−2 + 0.7099n−1 + 0.6681 + 0.0636n The outgoing flux density Q(x) will be quantitatively measured by a highly sensitive charge-coupled device (CCD) camera, that is: Φ(x) (x ∈ ∂Ω) (3) Q(x) = −D(x) v·∇Φ(x) = 2A(x; n, n0 )
2.2. Finite element discretization Based on finite element theory,11 the weak form formulation of Eqs.(1-1) and (1-2) can be represented as follows: Z D(x) ∇Φ(x) · ∇Ψ(x) + µa (x)Φ(x)Ψ(x) dx Ω Z Z 1 Φ(x)Ψ(x)dx = S(x)Ψ(x)dx ∀Ψ(x)∈H 1 (Ω) (4) + 2A (x) n Ω ∂Ω where H 1 (Ω) is the Sobolev space and Ψ(x) denotes an arbitrary piece-wise continuous test function. Let ϕk (x) SNe (l) Ω , where Ne elements represent denote the shape functions for the finite element space defined over Ω = l=1 Ω(l) (l = 1, 2, . . . , Ne ). Φ(x) will be approximated as: Φ(x) ≈ Φh (x) =
T X
φk (x)ϕk (x)
(5)
k=1
where T denotes the discretized node number of Ω, φk (x) is the nodal value of Φ(x) on the k-th node. By incorporating Eq.(4) with Eq.(5) based on the shape functions ϕk (x), the matrix form to be calculated is as follows:12 ([Kh ] + [Ch ] + [Bh ])Φh = [Mh ]Φh = Sh (6) where the components of the matrices Kh , Ch , Bh are obtained by (h) R (h) (h) kij = Ω D(x) ∇ϕi (x) · ∇ϕj (x) dx c(h) = R µ (x)ϕ(h) (x)ϕ(h) (x)dx a ij RΩ (h) i (h) j (h) 0 b = ϕ ij i (x)ϕj (x)/ 2A(x; n, n ) dx ∂Ω (h) R (h) (h) (h) sij = Ω si ϕi (x)ϕj (x)dx
3. THE ADAPTIVE FEMS FRAMEWORK Taking into account the error e between the approximation and exact solution of Φ, it is well known that the error is bounded by13 kek=kΦ − Φhk≤ Chmin(m−1,p) (7) where k·k denotes the Euclidean norm, h is the largest element diameter, p is the degree of the shape functions ϕk , and m is a measure of the order of the singularity in the problem. In order to minish the error e, h-adaptivity, p-adaptivity and r-adaptivity methods corresponding to reducing the element diameter, increasing the degree of the shape functions and swapping the edge and face of the mesh may be carried out.14 We adopt the h-adaptivity methods which reduce the element diameter h to perform the local mesh refinement, including three phases, i.e. solving the linear equation Mh Φh = Sh , error estimation and local mesh refinement.
3.1. Solving the linear equation Because Mh is a sparse symmetric positive definite (SPD) matrix, there are a large group of algorithms for solving the problem15 , such as Gauss-Seidel, Jacobi iterations, diagonal preconditioning techniques, etc. Regarding the nested sequence characteristic of the mesh during the local mesh refinement, BPX preconditioner16 utilized in the framework is an effective method to accelerate the iterative convergence.
3.2. Error estimation it is more effective and robust that a-posteriori error estimation based on the real-time information of the numerical solution relative to a-priori error assessment based on the real solution and/or the given geometries. A-posteriori error estimation using hierarchical defect correction techniques17 is implemented to select the mesh to be refined where the error of Φh is relatively large.
3.3. Local mesh refinement The anatomical structure of the small animal can be obtained through the CT/MRI scanner or some alternative methods. In biomedical research field, the surface and volumetric mesh of the biological tissue can be effectively generated based on triangular and tetrahedral elements,,18–20 so we choose tetrahedral element as the initial mesh which is helpful to the future work. The process of the local mesh refinement is to divide the father element into eight son tetrahedrons using the bisection of all the edges.21
4. HETEROGENEOUS PHANTOM EXPERIMENT
Figure 1. The sketch map of the multiple-view bioluminescence imaging experiment.
In the bioluminescence imaging experiment, a heterogeneous physical phantom has been designed and fabricated with the dimension of 30mm height and 15mm radius. The phantom, displayed in Fig.2(a), was made up of four different materials, i.e. high-density polyethylene (8624K16), nylon 6/6 (8538K23), delrin (8579K21) and polypropylene (8658K11) to represent muscle (M), lungs (L), heart (H) and bone (B) respectively. The two luminescent sources of about 1.9mm height and 0.56mm diameter were embedded in the left lung region of the phantom with the centers at (-9.0, 1.5, 0.0) and (-9.0, -1.5, 0.0). The power of the two sources were 105.1nano-Watts and 97.4nano-Watts respectively, and the emission wavelength was at the range of ∼ 650nm and ∼ 700nm. The slice of the phantom was obtained by microCT scanner to produce the volumetric finite element mesh, as shown in Fig.2(b).
(a)
(b)
Figure 2. Physical heterogeneous phantom. (a) The phantom including bone (B), heart (H), lungs (L) and muscle (M); (b) A slice was scanned by microCT scanner.
Material µa [mm−1 ] µ0s [mm−1 ]
Muscle(M) 0.007 1.031
Lung(L) 0.023 2.000
Heart(H) 0.011 1.096
Bone(B) 0.001 0.060
Table 1. Optical property parameters of the physical heterogeneous phantom
The optical property parameters of four materials were indispensable to simulate the forward problem of BLT. Cylinders of the materials, which were modeled as a semi-infinite homogenous medium, were made for determining the optical parameters. The absorption coefficient µa and the reduced scattering coefficient µ0s were calculated by a nonlinear least-square fitting,10 as shown in Table 1. A CCD camera was applied to record the flux density emitted from the cylindrical surface of the phantom fixed on a sample holder. The multiple-view detection mode was essential to reduce the influence of the curved surface of the phantom on the measured value on the CCD camera. In the experiment, four views which were separated by 90 degrees along radial directions were shown in Fig.1. Detected data on the CCD camera, shown in Fig.3, was transformed from the recorded pixel gray levels by22 φ = pix × 0.377 pico−W atts/mm2 where φ is the photon density and pix denotes the pixel value.
(a) Front view
(b) Right view
(c) Left view
(d) Back view
Figure 3. The detected photon energy distribution on the phantom surface by CCD camera.
5. RESULTS The above-mentioned adaptive FEMs framework utilize the microCT slices based mesh of the phantom to model the photon propagation. The initial mesh, consisting of 1865 nodes and 9350 tetrahedral elements, was obtained in turn through manual segmentation, surface reconstruction,19 surface smooth,23 surface simplification,24 and volumetric finite-element generation.25 When setting the global precision at 0.01, the evolution of the three adaptive mesh refinement has been performed and displayed in Fig.5, which spends 233 seconds on an Intel processor (Pentium 4 2.8GHz) PC with 1GB of RAM. The eventual mesh comprises 69381 nodes and 405361 tetrahedral elements. The refined elements surround the two sources in the left lung region, which have relative high error under the error estimation by means of hierarchical defect correction techniques. There is a good agreement between experimental data and simulation results with the average relative error being about 20%, as shown in Fig.6, which are normalized to the corresponding maximal flux density. The relative error (RE) is defined by: |Φmeas − Φsim | RE = Φsim
Figure 4. The initial finite element mesh used in the adaptive FEMs framework: muscle (white), bone (wine), lungs (gray), heart (blue), luminescent sources (red).
(a) Initial Mesh
(b) Second Mesh
(c) Third Mesh
(d) Fourth Mesh
Figure 5. The evolution from the initial mesh (a) to the final mesh (d) via (b) and (c) by the local mesh refinement.
where Φmeas is the boundary measured flux density and Φsim is the simulation one. In the simulaton environment, the flux density of the pixels on the CCD detector has been obtained by the interpolation between the points of the phantom surface.
6. DISCUSSION AND CONCLUSION We represent the adaptive FEMs framework for simulating the forward problem of the bioluminescence tomography. The bioluminescence imaging experiment with the heterogenous physical phantom has been also carried out. The results show that the proposed adaptive FEMs framework is more accurate and effective to describe the photon density distribution on the surface of the phantom than traditional FEM. In addition, BPX preconditioner, a posteriori error estimation and local mesh refinement can improve the simulation speed and reduce the computational memory consumption. We may incorporate certain a prior knowledge into the adaptive FEMs framework which may reduce the ill-posedness of the inverse problem to localize source distribution in BLT. The corresponding research is going on.
(a) −4.9mm
(b) 0.2mm
(c) 5.1mm
Figure 6. Comparison between the simulated and experimental flux density along the detection circle located in (a) −4.9mm, (b) 0.2mm, (c) 5.1mm from the middle cross-section of the phantom
7. ACKNOWLEDGEMENTS This paper is supported by the Project for National Science Fund for Distinguished Young Scholars of China under Grant No. 60225008, the Joint Research Fund for Overseas Chinese Young Scholars under Grant No.30528027, the National Natural Science Foundation of China under Grant No. 30370418, 90209008, 60302016, 60532050, 30500131, Beijing Natural Science Fund under Grant No. 4051002, 4042024.
REFERENCES 1. V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weisslder, “Looking and listening to light: the evolution of whole body photonic imaging,” Nature Biotechnology 23, pp. 313–320, March 2005. 2. G. Wang, E. A. Hoffman, G. McLennan, L. V. Wang, M. Suter, and J. F. Meinel, “Development of the first bioluminescence ct scanner,” Radiology 566, p. 229, 2003. 3. T. F. Massoud and S. S. Gambhir, “Molecular imaging in living subjects: seeing fundamental biological processes in a new light,” Genes and development 17, pp. 545–580, 2003. 4. A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, pp. R1–R43, 2005. 5. B. W. Rice, M. D. Cable, and M. B. Nelson, “In vivo imaging of light-emitting probles,” Journal of Biomedical Optics 6, pp. 432–440, October 2001. 6. H. Li, J. Tian, F. Zhu, W. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation enviroment(mose) to investigate bioluminescent phenomena in the living mouse with the monte carlo method,” Academic Radiology 11, pp. 1029–1038, September 2004. 7. A. D. Klose, V. Ntziachristos, and A. H. Hiescher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys 202, pp. 323–345, 2005. 8. A. Joshi, W. Bangerth, A. B. Thompson, and E. M. Sevick-Muraca, “Adaptive finite element methods for fluorescence enhanced frequency domain optical tomography: Forward imaging problem,” in Proceedings of the 2004 IEEE International Symposium on Biomedical imaging, 2, pp. 1103–1106, IEEE, April 2004. 9. M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22, pp. 1779–1792, 1995. 10. M. Guven, B. Yazici, X. Intes, and B. Chance, “Diffuse optical tomography with a priori annatomical information,” Phys. Med. Biol. 50, pp. 2837–2858, 2005. 11. S. S. Rao, The finite element method in engineering, Butterworth-Heinemann, Boston, 1999. 12. W. Cong, D. Kumar, Y. Liu, A. Cong, and G. Wang, “A practical method to determine the light source distribution in bioluminescent imaging,” in Developments in X-Ray Tomography IV, U. Bonse, ed., 5535, pp. 679–686, SPIE, 2004.
13. O. C. Zienkiewicz and A. Craig, Adaptive refinement, error estimates, multigrid solution, and hierarchic finite element method concepts, ch. 2, pp. 25–59. John Wiley & Sons Ltd., 1986. 14. R. Mahmood, Multilevel Mesh adaptivity for elliptic boundary value problems in two and three space dimensions. Ph.D., The University of Leeds, January 2002. 15. R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. V. der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd Edition, SIAM, Philadelphia, PA, 1994. 16. J. H. Bramble, J. E. Pasciak, and J. Xu, “Parallel multilevel preconditioners,” Mathematics of Computation 55, pp. 1–22, July 1990. 17. O. C. Zienkiewicz, J. P. de S. R. Gago, and D. W. Kelly, “The hierarchical concept in finite element analysis,” Computers and Structures 16, 1983. 18. W. E. Lorensen and H. E. Cline, “Marching cubes: a high resolution 3d surface construction algorithm,” Proceedings of SIGGRAPH 21(4), pp. 163–169, 1987. 19. Z. Wu and J. M. Sullivan, Jr., “Multiple material marching cubes algorithm,” International Journal for Numerical Methods in Engineering 58, pp. 189–207, July 2003. 20. A. Cong, Y. Liu, D.Kumar, W. Cong, and G. Wang, “Geometrical modeling using multiregional marching tetrahedra for bioluminescence tomography,” in Medical Imaging 2005: Visualization, Image-Guided Procedures, and Display, K. R. C. Robert L. Galloway, Jr., ed., 5744, pp. 756–763, 2005. 21. R. E. Bank, A. H. Sherman, and H. Weiser, “Refinement algorithms and data structures for regular local mesh refinement,” Scientific Computing , pp. 3–17, 1983. 22. W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Optics Express 13, pp. 6756–6771, August 2005. 23. W. Schroeder, K. Martin, and B. Lorensen, The Visualization Toolkit, An Object-Oriented Approach to 3D Graphics, Prentice Hall, 2nd ed., 1997. 24. M. Garland and P. S. Heckbert, “Surface simplification using quadric error metrics,” in SIGGRAPH 97 Conference Proceedings, pp. 209–216, ACM SIGGRAPH, August 1997. 25. H. Jin and R. I. Tanner, “Generation of unstructured tetrahedral meshes by advancing front technique,” Int. J. Numer. Methods. Eng. 36, pp. 217–246, 1993.
