Optimization and Engineering

, Volume 19, Issue 3, pp 765–812 | Cite as

PDE-constrained optimization in medical image analysis

  • Andreas MangEmail author
  • Amir Gholami
  • Christos Davatzikos
  • George Biros
Research Article


PDE-constrained optimization problems find many applications in medical image analysis, for example, neuroimaging, cardiovascular imaging, and oncologic imaging. We review the related literature and give examples of the formulation, discretization, and numerical solution of PDE-constrained optimization problems for medical imaging. We discuss three examples. The first is image registration, the second is data assimilation for brain tumor patients, and the third is data assimilation in cardiovascular imaging. The image registration problem is a classical task in medical image analysis and seeks to find pointwise correspondences between two or more images. Data assimilation problems use a PDE-constrained formulation to link a biophysical model to patient-specific data obtained from medical images. The associated optimality systems turn out to be sets of nonlinear, multicomponent PDEs that are challenging to solve in an efficient way. The ultimate goal of our work is the design of inversion methods that integrate complementary data, and rigorously follow mathematical and physical principles, in an attempt to support clinical decision making. This requires reliable, high-fidelity algorithms with a short time-to-solution. This task is complicated by model and data uncertainties, and by the fact that PDE-constrained optimization problems are ill-posed in nature, and in general yield high-dimensional, severely ill-conditioned systems after discretization. These features make regularization, effective preconditioners, and iterative solvers that, in many cases, have to be implemented on distributed-memory architectures to be practical, a prerequisite. We showcase state-of-the-art techniques in scientific computing to tackle these challenges.


Optimal control Medical imaging PDE-constrained optimization Memory-distributed algorithms 

Mathematics Subject Classification

49K20 65Y05 65M32 65K10 76D55 68U10 35M10 



The authors thank the anonymous reviewers for their careful reading and their insightful comments.


This material is based upon work supported by AFOSR grants FA9550-17-1-0190; by NSF Grant CCF-1337393; by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Award Numbers DE-SC0010518 and DE-SC0009286; by NIH Grant 10042242; by DARPA Grant W911NF-115-2-0121; and by the Technische Universität München—Institute for Advanced Study, funded by the German Excellence Initiative (and the European Union Seventh Framework Programme under grant agreement 291763). Any opinions, findings, and conclusions or recommendations expressed herein are those of the authors and do not necessarily reflect the views of the AFOSR, DOE, NIH, DARPA, and NSF. Computing time on the Texas Advanced Computing Center’s (TACC) systems was provided by an allocation from TACC and the NSF. Computing time on the High-Performance Computing Center’s (HLRS) Hazel Hen system (Stuttgart, Germany) was provided by an allocation of the federal project application ACID-44104. This work was completed in part with resources provided by the University of Houston Center for Advanced Computing and Data Science (CACDS)


  1. Akcelik V, Biros G, Ghattas O (2002) Parallel multiscale Gauss–Newton–Krylov methods for inverse wave propagation. In: Proceeding of the ACM/IEEE conference on supercomputing, pp 1–15Google Scholar
  2. Akcelik V, Biros G, Ghattas O, Hill J, Keyes D, van Bloemen Wanders B (2006) Parallel algorithms for PDE constrained optimization. In: Parallel processing for scientific computing, vol 20, chap. 16. SIAM, Philadelphia, pp 291–322Google Scholar
  3. Alexanderian A, Petra N, Stadler G, Ghattas O (2016) A fast and scalable method for A-optimal design of experiments for infinite-dimensional Bayesian nonlinear inverse problems. SIAM J Sci Comput 38(1):A243–A272MathSciNetzbMATHGoogle Scholar
  4. Amit Y (1994) A nonlinear variational problem for image matching. SIAM J Sci Comput 15(1):207–224MathSciNetzbMATHGoogle Scholar
  5. Andreev R, Scherzer O, Zulehner W (2015) Simultaneous optical flow and source estimation: space-time discretization and preconditioning. Appl Numer Math 96:72–81MathSciNetzbMATHGoogle Scholar
  6. Angenent S, Haker S, Tannenbaum A (2003) Minimizing flows for the Monge-Kantrovich problem. SIAM J Math Anal 35(1):61–97MathSciNetzbMATHGoogle Scholar
  7. Arridge SR (1999) Optical tomography in medical imaging. Inverse Probl 15:R41–R93MathSciNetzbMATHGoogle Scholar
  8. Arridge SR, Schotland JC (2009) Optical tomography: forward and inverse problems. Inverse Probl 25(12):123,010MathSciNetGoogle Scholar
  9. Arsigny V, Commowick O, Pennec X, Ayache N (2006) A Log-Euclidean framework for statistics on diffeomorphisms. Proceedings of the medical image computing and computer-assisted intervention, vol LNCS 4190:924–931Google Scholar
  10. Ashburner J (2007) A fast diffeomorphic image registration algorithm. NeuroImage 38(1):95–113Google Scholar
  11. Ashburner J, Friston KJ (2011) Diffeomorphic registration using geodesic shooting and Gauss-Newton optimisation. NeuroImage 55(3):954–967Google Scholar
  12. Atuegwu NC, Colvin DC, Loveless ME, Xu L, Gore JC, Yankeelov TE (2012) Incorporation of diffusion-weighted magnetic resonance imaging data into a simple mathematical model of tumor growth. Phys Med Biol 57(1):225Google Scholar
  13. Avants B, Schoenemann PT, Gee JC (2006) Lagrangian frame diffeomorphic image registration: morphometric comparison of human and chimpanzee cortex. Med Image Anal 10:397–412Google Scholar
  14. Avants BB, Epstein CL, Brossman M, Gee JC (2008) Symmetric diffeomorphic image registration with cross-correlation: evaluating automated labeling of elderly and neurodegenerative brain. Med Image Anal 12(1):26–41Google Scholar
  15. Avants BB, Tustison NJ, Song G, Cook PA, Klein A, Gee JC (2011) A reproducible evaluation of ANTs similarity metric performance in brain image registration. NeuroImage 54:2033–2044Google Scholar
  16. Axel L (2002) Biomechanical dynamics of the heart with MRI. Annu Rev Biomed Eng 4:321–347Google Scholar
  17. Balay S, Abhyankar S, Adams MF, Brown J, Brune P, Buschelman K, Eijkhout V, Gropp WD, Kaushik D, Knepley MG, McInnes LC, Rupp K, Smith BF, Zhang H (2016) PETSc users manual. Tech. Rep. ANL-95/11-Revision 3.7. Argonne National LaboratoryGoogle Scholar
  18. Barbu V, Marinoschi G (2016) An optimal control approach to the optical flow problem. Syst Control Lett 87:1–9MathSciNetzbMATHGoogle Scholar
  19. Barthel W, John C, Tröltsch F (2010) Optimal boundary control of a system of reaction diffusion equations. Z Angew Math Mech 90:966–982MathSciNetzbMATHGoogle Scholar
  20. Beg MF, Miller MI, Trouvé A, Younes L (2005) Computing large deformation metric mappings via geodesic flows of diffeomorphisms. Int J Comput Vis 61(2):139–157Google Scholar
  21. Bellomo N, Li NK, Maini PK (2008) On the foundations of cancer modelling: selected topics, speculations, and perspectives. Math Models Methods Appl Sci 18(4):593–646MathSciNetzbMATHGoogle Scholar
  22. Benamou JD, Brenier Y (2000) A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer Math 84:375–393MathSciNetzbMATHGoogle Scholar
  23. Benzi M, Golub GH, Liesen J (2005) Numerical solution of saddle point problems. Acta Numer 14:1–137MathSciNetzbMATHGoogle Scholar
  24. Benzi M, Haber E, Taralli L (2009) Multilevel algorithms for large-scale interior point methods. SIAM J Sci Comput 31(6):4152–4175MathSciNetzbMATHGoogle Scholar
  25. Benzi M, Haber E, Taralli L (2011) A preconditioning technique for a class of PDE-constrained optimization problems. Adv Comput Math 35(2–4):149–173MathSciNetzbMATHGoogle Scholar
  26. Biegler LT, Ghattas O, Heinkenschloss M, van Bloemen Waanders B (2003) Large-scale PDE-constrained optimization. Springer, BerlinzbMATHGoogle Scholar
  27. Biegler LT, Ghattas O, Heinkenschloss M, Keyes D, van Bloemen Waanders B (2007) Real-time PDE-constrained optimization. SIAM, PhiladelphiazbMATHGoogle Scholar
  28. Biros G, Ghattas O (1999) Parallel Newton–Krylov methods for PDE-constrained optimization. In: Proceedings of the ACM/IEEE conference on supercomputing, pp 28–40Google Scholar
  29. Biros G, Ghattas O (2005a) Parallel Lagrange–Newton–Krylov–Schur methods for PDE-constrained optimization-part I: the Krylov-Schur solver. SIAM J Sci Comput 27(2):687–713MathSciNetzbMATHGoogle Scholar
  30. Biros G, Ghattas O (2005b) Parallel Lagrange–Newton–Krylov–Schur methods for PDE-constrained optimization-part II: the Lagrange-Newton solver and its application to optimal control of steady viscous flows. SIAM J Sci Comput 27(2):714–739MathSciNetzbMATHGoogle Scholar
  31. Bistoquet A, Parks WJ, Skrinjar O (2006) Myocardial deformation recovery using a 3D biventricular incompressible model. In: Proceedings of the biomedical image registration (Lecture Notes in computer science), vol 4057. Springer, Berlin, pp 110–119Google Scholar
  32. Bluemke DA, Krupinski EA, Ovitt T, Gear K, Unger E, Axel L, Boxt LM, Casolo G, Ferrari VA, Funaki B, Globits S, Higgins CB, Julsrud P, Lipton M, Mawson J, Nygren A, Pennell DJ, Stillman A, White RD, Wichter T, Marcus F (2003) MR imaging of arrhythmogenic right ventricular cardiomyopathy: morphologic findings and interobserver reliability. Cardiology 99(3):153–162Google Scholar
  33. Borzì A, Ito K, Kunisch K (2002) Optimal control formulation for determining optical flow. SIAM J Sci Comput 24(3):818–847MathSciNetzbMATHGoogle Scholar
  34. Borzì A, Schulz V (2012) Computational optimization of systems governed by partial differential equations. SIAM, PhiladelphiazbMATHGoogle Scholar
  35. Brezzi F, Fortin M (eds) (1991) Mixed and hybrid finite element methods. Springer, BerlinzbMATHGoogle Scholar
  36. Burger M, Modersitzki J, Ruthotto L (2013) A hyperelastic regularization energy for image registration. SIAM J Sci Comput 35(1):B132–B148MathSciNetzbMATHGoogle Scholar
  37. Castillo E, Lima JAC, Bluemke DA (2003) Regional myocardial function: advances in MR imaging and analysis. Radiographics 23:S127–S140Google Scholar
  38. Chen K (2011) Optimal control based image sequence interpolation. Ph.D. thesis, University of BremenGoogle Scholar
  39. Chen K, Lorenz DA (2011) Image sequence interpolation using optimal control. J Math Imaging Vis 41:222–238MathSciNetzbMATHGoogle Scholar
  40. Chen X, Summers RM, Yoa J (2012) Kidney tumor growth prediction by coupling reaction-diffusion and biomechanical model. IEEE Trans Biomed Eng 60(1):169–173Google Scholar
  41. Christensen GE, Rabbitt RD, Miller MI (1994) 3D brain mapping using a deformable neuroanatomy. Phys Med Biol 39(3):609–618Google Scholar
  42. Christensen GE, Rabbitt RD, Miller MI (1996) Deformable templates using large deformation kinematics. IEEE Trans Image Process 5(10):1435–1447Google Scholar
  43. Christensen GE, Geng X, Kuhl JG, Bruss J, Grabowski TJ, Pirwani IA, Vannier MW, Allen JS, Damasio H (2006) Introduction to the non-rigid image registration evaluation project. Proceedings of the biomedical image registration, vol LNCS 4057:128–135Google Scholar
  44. Chung K, Deisseroth K (2013) CLARITY for mapping the nverous system. Nat Methods 10:508–513Google Scholar
  45. Clatz O, Sermesant M, Bondiau PY, Delingette H, Warfield SK, Malandain G, Ayache N (2005) Realistic simulation of the 3D growth of brain tumors in MR images coupling diffusion with biomechanical deformation. IEEE Trans Med Imaging 24(10):1334–1346Google Scholar
  46. Cocosco C, Kollokian V, Kwan RKS, Evans AC (1997) Brainweb: online interface to a 3D MRI simulated brain database. NeuroImage 5:425Google Scholar
  47. Colin T, Iollo A, Lagaert JB, Saut O (2014) An inverse problem for the recovery of the vascularization of a tumor. J Inverse Ill Posed Probl 22(6):759–786MathSciNetzbMATHGoogle Scholar
  48. Collis J, Connor AJ, Paczkowski M, Kannan P, Pitt-Francis J, Byrne HM, Hubbard ME (2017) Bayesian calibration, validation and uncertainty quantification for predictive modelling of tumour growth: a tutorial. Bull Math Biol 79(4):939–974MathSciNetzbMATHGoogle Scholar
  49. Crippa G (2007) The flow associated to weakly differentiable vector fields. Ph.D. thesis, University of ZürichGoogle Scholar
  50. Croft W, Elliott CM, Ladds G, Stinner B, Venkataraman C, Weston C (2015) Parameter identification problems in the modelling of cell motility. J Math Biol 71(2):399–436MathSciNetzbMATHGoogle Scholar
  51. Czechowski K, Battaglino C, McClanahan C, Iyer K, Yeung PK, Vuduc R (2012) On the communication complexity of 3D FFTs and its implications for exascale. In: Proceedings of the ACM/IEEE Conference on supercomputing, pp 205–214Google Scholar
  52. Delingette H, Billet F, Wong KCL, Sermesant M, Rhode K, Ginks M, Rinaldi C, Razavi R, Ayache N (2012) Personalization of cardiac motion and contractility from images using variational data assimilation. IEEE Trans Biomed Eng 59(1):20–24Google Scholar
  53. Dembo RS, Steihaug T (1983) Truncated-Newton algorithms for large-scale unconstrained optimization. Math Program 26(2):190–212MathSciNetzbMATHGoogle Scholar
  54. Dontchev AL, Hager WW, Veliov VM (2000) Second-order Runge-kutta approximations in control constrained optimal control. SIAM J Numer Anal 38(1):202–226MathSciNetzbMATHGoogle Scholar
  55. Dupuis P, Gernander U, Miller MI (1998) Variational problems on flows of diffeomorphisms for image matching. Q Appl Math 56(3):587–600MathSciNetzbMATHGoogle Scholar
  56. Eisentat SC, Walker HF (1996) Choosing the forcing terms in an inexact Newton method. SIAM J Sci Comput 17(1):16–32MathSciNetzbMATHGoogle Scholar
  57. Eklund A, Dufort P, Forsberg D, LaConte SM (2013) Medical image processing on the GPU-past, present and future. Med Image Anal 17(8):1073–1094Google Scholar
  58. Ellingwood ND, Yin Y, Smith M, Lin CL (2016) Efficient methods for implementation of multi-level nonrigid mass-preserving image registration on GPUs and multi-threaded CPUs. Comput Methods Program Biomed 127:290–300Google Scholar
  59. Engl H, Hanke M, Neubauer A (1996) Regularization of inverse problems. Kluwer Academic Publishers, DordrechtzbMATHGoogle Scholar
  60. Falcone M, Ferretti R (1998) Convergence analysis for a class of high-order semi-Lagrangian advection schemes. SIAM J Numer Anal 35(3):909–940MathSciNetzbMATHGoogle Scholar
  61. Figueiredo IN, Leal C (2013) Physiologic parameter estimation using inverse problems. SIAM J Appl Math 73(3):1164–1182MathSciNetzbMATHGoogle Scholar
  62. Figueiredo IN, Figueiredo PN, Almeida N (2011) Image-driven parameter estimation in absorption-diffusion models of chromoscopy. SIAM J Imaging Sci 4(3):884–904MathSciNetzbMATHGoogle Scholar
  63. Fischer B, Modersitzki J (2008) Ill-posed medicine–an introduction to image registration. Inverse Probl 24(3):1–16MathSciNetzbMATHGoogle Scholar
  64. Fluck O, Vetter C, Wein W, Kamen A, Preim B, Westermann R (2011) A survey of medical image registration on graphics hardware. Comput Methods Programs Biomed 104(3):e45–e57Google Scholar
  65. Fohring J, Haber E, Ruthotto L (2014) Geophysical imaging for fluid flow in porous media. SIAM J Sci Comput 36(5):S218–S236MathSciNetzbMATHGoogle Scholar
  66. Garcke H, Lam KF, Sitka E, Styles V (2016) A Cahn-Hilliard-Darcy model for tumour growth with chemotaxis and active transport. Math Methods Appl Sci 26(6):1095–1148MathSciNetzbMATHGoogle Scholar
  67. Geweke J, Tanizaki H (1999) On Markov chain Monte Carlo methods for nonlinear and non-Gaussian state-space models. Commun Stat Simul Comput 28(4):867–894zbMATHGoogle Scholar
  68. Geweke J, Tanizaki H (2003) Note on the sampling distribution for the Metropolis-Hastings algorithm. Commun Stat Theory Methods 32(4):2003Google Scholar
  69. Gholami A, Hill J, Malhotra D, Biros G (2016a) AccFFT: a library for distributed-memory FFT on CPU and GPU architectures. In review (arXiv:1506.07933)
  70. Gholami A, Mang A, Biros G (2016b) An inverse problem formulation for parameter estimation of a reaction-diffusion model of low grade gliomas. J Math Biol 72(1):409–433. MathSciNetzbMATHGoogle Scholar
  71. Gholami A, Mang A, Scheufele K, Davatzikos C, Mehl M, Biros G (2017) A framework for scalable biophysics-based image analysis. Proc ACM/IEEE Conf Supercomput 19:1–13. Google Scholar
  72. Goenezen S, Dord JF, Sink Z, Barbone PE, Jiang J, Hall TJ, Oberai AA (2012) Linear and nonlinear elastic modulus imaging: an application to breast cancer diagnosis. IEEE Trans Med Imaging 31(8):1628–1637Google Scholar
  73. Gooya A, Pohl KM, Bilello M, Cirillo L, Biros G, Melhem ER, Davatzikos C (2013) GLISTR: glioma image segmentation and registration. IEEE Trans Med Imaging 31(10):1941–1954Google Scholar
  74. Grama A, Gupta A, Karypis G, Kumar V (2003) An Introduction to parallel computing: design and analysis of algorithms, 2nd edn. Addison Wesley, BostonzbMATHGoogle Scholar
  75. Griewank A (1992) Achieving logarithmic growth of temporal and spatial complexity in teverse automatic differentiation. Optim Methods Softw 1:35–54Google Scholar
  76. Gu X, Pand H, Liang Y, Castillo R, Yang D, Choi D, Castillo E, Majumdar A, Guerrero T, Jiang SB (2010) Implementation and evaluation of various demons deformable image registration algorithms on a GPU. Phys Med Biol 55(1):207–219Google Scholar
  77. Gu S, Chakraborty G, Champley K, Alessio AM, Claridge J, Rockne R, Muzi M, Krohn KA, Spence AM, Alvord EC, Anderson ARA, Kinahan PE, Swanson KR (2012) Applying a patient-specific bio-mathematical model of glioma growth to develop virtual [18F]-FMISO-PET images. Math Med Biol 29(1):31–48zbMATHGoogle Scholar
  78. Gunzburger MD (2003) Perspectives in flow control and optimization. SIAM, PhiladelphiazbMATHGoogle Scholar
  79. Gurtin ME (1981) An introduction to continuum mechanics, Mathematics in Science and Engineering, vol 158. Academic Press, CambridgeGoogle Scholar
  80. Ha L, Krüger J, Joshi S, Silva CT (2011) Multiscale unbiased diffeomorphic atlas construction on multi-GPUs. In: CPU computing gems Emerald edition, chap. 48. Elsevier Inc, New York City, pp 771–791Google Scholar
  81. Ha LK, Krüger J, Fletcher PT, Joshi S, Silva CT (2009) Fast parallel unbiased diffeomorphic atlas construction on multi-graphics processing units. In: Proceedings of the eurographics conference on parallel grphics and visualization, pp 41–48 (2009)Google Scholar
  82. Ha L, Krüger J, Joshi S, Silva TC (2010) Multi-scale unbiased diffeomorphic atlas construction on multi-GPUs. GPU Comput Gems Emerald Ed 1:771–791Google Scholar
  83. Haber E, Ascher UM (2001) Preconditioned all-at-once methods for large, sparse parameter estimation problems. Inverse Probl 17(6):1847–1864MathSciNetzbMATHGoogle Scholar
  84. Haber E, Horesh R (2015) A multilevel method for the solution of time dependent optimal transport. Numer Math Theory Methods Appl 8(1):97–111MathSciNetzbMATHGoogle Scholar
  85. Haber E, Modersitzki J (2006) A multilevel method for image registration. SIAM J Sci Comput 27(5):1594–1607MathSciNetzbMATHGoogle Scholar
  86. Haber E, Ascher UM, Oldenburg D (2000) On optimization techniques for solving nonlinear inverse problems. Inverse Probl 16:1263–1280MathSciNetzbMATHGoogle Scholar
  87. Hager WW (2000) Runge-Kutta methods in optimal control and the transformed adjoint system. Numer Math 87:247–282MathSciNetzbMATHGoogle Scholar
  88. Hansen C (1992) Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev 34(4):561–580MathSciNetzbMATHGoogle Scholar
  89. Harpold HLP, Alvord EC, Swanson KR (2007) The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol 66(1):1–9Google Scholar
  90. Hart GL, Zach C, Niethammer M (2009) An optimal control approach for deformable registration. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 9–16Google Scholar
  91. Hawkins-Daarud A, Prudhomme S, van der Zee KG, Oden JT (2013a) Bayesian calibration, validation, and uncertainty quantification of diffuse interface models of tumour growth. J Math Biol 67(6–7):1457–1485MathSciNetzbMATHGoogle Scholar
  92. Hawkins-Daarud A, Rockne RC, Anderson ARA, Swanson KR (2013b) Modeling tumor-associated edema in gliomas during anti-angiogenic therapy and its impact on imageable tumor. Front Oncol 3(66):1–12Google Scholar
  93. Heinkenschloss M (2005) A time-domain decomposition iterative method for the solution of distributed linear quadratic optimal control problems. J Comput Appl Math 1(1):169–198MathSciNetzbMATHGoogle Scholar
  94. Hernandez M, Bossa MN, Olmos S (2009) Registration of anatomical images using paths of diffeomorphisms parameterized with stationary vector field flows. Int J Comput Vis 85(3):291–306Google Scholar
  95. Herzog R, Kunisch K (2010) Algorithms for PDE-constrained optimization. GAMM Mitt 33(2):163–176Google Scholar
  96. Herzog R, Pearson JW, Stoll M (2018) Fast iterative solvers for an optimal transport problem. arXiv: 1801.04172
  97. Hinze M, Pinnau R, Ulbrich M, Ulbrich S (2009) Optimization with PDE constraints. Springer, BerlinzbMATHGoogle Scholar
  98. Hogea C, Davatzikos C, Biros G (2007) Modeling glioma growth and mass effect in 3D MR images of the brain. In: Proceedings of the medical image computing and computer-assisted intervention, pp 642–650Google Scholar
  99. Hogea C, Davatzikos C, Biros G (2008a) Brain-tumor interaction biophysical models for medical image registration. SIAM J Imaging Sci 30(6):3050–3072MathSciNetzbMATHGoogle Scholar
  100. Hogea C, Davatzikos C, Biros G (2008b) An image-driven parameter estimation problem for a reaction-diffusion glioma growth model with mass effects. J Math Biol 56(6):793–825MathSciNetzbMATHGoogle Scholar
  101. Hormuth II DA, Weis JA, Barnes SL, Miga MI, Rericha EC, Quaranta V, Yankeelov TE (2015) Predicting in vivo glioma growth wit the reaction diffusion equation constrained by quantitative magnetic resonance imaging data. Phys Biol 12(4):046006Google Scholar
  102. Horn BKP, Shunck BG (1981) Determining optical flow. Artif Intell 17(1–3):185–203Google Scholar
  103. Hu Z, Metaxas D, Axel L (2003) In vivo strain and stress estimation of the heart left and right ventricles from MRI images. Med Image Anal 7(4):435–444Google Scholar
  104. Jackson PR, Juliano J, Hawkins-Daarud A, Rockne RC, Swanson KR (2015) Patient-specific mathematical neuro-oncology: Using a simple proliferation and invasion tumor model to inform clinical practice. Bull Math Biol 77(5):846–856MathSciNetzbMATHGoogle Scholar
  105. Joshi A, Bangerth W, Sevick-Muraca EM (2004) Adaptive finite element based tomography for fluorescence optical imaging in tissue. Opt Express 12(22):5402–5417Google Scholar
  106. Joshi S, Davis B, Jornier M, Gerig G (2005) Unbiased diffeomorphic atlas construction for computational anatomy. NeuroImage 23(1):S151–S160Google Scholar
  107. Kaipio J, Somersalo E (2005) Statistical and computational inverse problems. Springer, BerlinzbMATHGoogle Scholar
  108. Kalmoun EM, Garrido L, Caselles V (2011) Line search multilevel optimization as computational methods for dense optical flow. SIAM J Imaging Sci 4(2):695–722MathSciNetzbMATHGoogle Scholar
  109. Klein S, Staring M, Murphy K, Viergever MA, Pluim JPW (2010) ELASTIX: a tollbox for intensity-based medical image registration. IEEE Trans Med Imaging 29(1):196–205Google Scholar
  110. Knopoff DA, Fernández DR, Torres GA, Turner CV (2013) Adjoint method for a tumor growth PDE-constrained optimization problem. Comput Math Appl 66(6):1104–1119MathSciNetGoogle Scholar
  111. Knopoff D, Fernández DR, Torres GA, Turner CV (2017) A mathematical method for parameter estimation in a tumor growth model. Comput Appl Math 36(1):733–748MathSciNetzbMATHGoogle Scholar
  112. Kø N, Tanderup K, Lindegaard JC, Grau C, Søorensen TS (2008) GPU accelerated viscous-fluid deformable registration for radiotherapy. Stud Health Technol Inform 132:327–332Google Scholar
  113. Konukoglu E, Clatz O, Bondiau PY, Delingette H, Ayache N (2010) Extrapolating glioma invasion margin in brain magnetic resonance images: suggesting new irradiation margins. Med Image Anal 14(2):111–125Google Scholar
  114. Konukoglu E, Clatz O, Menze BH, Stieltjes B, Weber MA, Mandonnet E, Delingette H, Ayache N (2010) Image guided personalization of reaction-diffusion type tumor growth models using modified anisotropic eikonal equations. IEEE Trans Med Imaging 29(1):77–95Google Scholar
  115. Kutten KS, Charon N, Miller MI, Ratnanather JT, Deisseroth K, Ye L, Vogelstein JT (2017) A diffeomorphic approach to multimodal registration with mutual information: Applications to CLARITY mouse brain images. Proceedings of the medical image computing and computer-assisted intervention, vol LNCS 10433:275–282Google Scholar
  116. Lê M, Delingette H, Kalpathy-Cramer J, Gerstner ER, Batchelor T, Unkelbach J, Ayache N (2015) Bayesian personalization of brain tumor growth model. In: Proceedings of the medical image computing and computer-assisted intervention, pp 424–432Google Scholar
  117. Lê M, Delingette H, Kalpathy-Cramer J, Gerstner ER, Batchelor T, Unkelbach J, Ayache N (2017) Personalized radiotherapy planning based on a computational tumor growth model. IEEE Trans Med Imaging 36(3):815–825Google Scholar
  118. Lee E, Gunzburger M (2010) An optimal control formulation of an image registration problem. J Math Imaging Vis 36(1):69–80MathSciNetGoogle Scholar
  119. Lee E, Gunzburger M (2011) Anaysis of finite element discretization of an optimal control formulation of the image registration problem. SIAM J Numer Anal 9(4):1321–1349zbMATHGoogle Scholar
  120. Leugering G, Benner P, Engell S, Griewank A, Harbrecht H, Hinze M, Rannacher R, Ulbrich S (eds) (2014) Trends in PDE constrained optimization. Springer, BerlinzbMATHGoogle Scholar
  121. Lima E, Oden J, Almeida R (2014) A hybrid ten-species phase-field model of tumor growth. Math Models Methods Appl Sci 24(13):2569–2599MathSciNetzbMATHGoogle Scholar
  122. Lima EABF, Oden JT, Hormuth DA, Yankeelov TE, Almeida RC (2016) Selection, calibration, and validation of models of tumor growth. Math Models Methods Appl Sci 26(12):2341–2368MathSciNetzbMATHGoogle Scholar
  123. Lima E, Oden JT, Wohlmuth B, Shahmoradi A, Hormuth DA, Yankeelov TE (2017) Selection and validation of predictive models of radiation effects on tumor growth based on noninvasive imaging data. Comput Methods Appl Mech Eng 327:277–305MathSciNetGoogle Scholar
  124. Lions JL (1971) Optimal control of systems governed by partial differential equations. Springer, BerlinzbMATHGoogle Scholar
  125. Liu Y, Sadowki SM, Weisbrod AB, Kebebew E, Summers RM, Yao J (2014) Patient specific tumor growth prediction using multimodal images. Med Image Anal 18(3):555–566Google Scholar
  126. Luo Y, Liu P, Shi L, Luo Y, Yi L, Li A, Qin J, Heng PA, Wang D (2015) Accelerating neuroimage registration through parallel computation of similarity metric. PLoS ONE 10(9): e0136,718 (2015)Google Scholar
  127. Mang A (2014) Methoden zur numerischen Simulation der Progression von Gliomen: Modellentwicklung. Springer Fachmedien Wiesbaden, Wiesbaden, Numerik und Parameteridentifikation. zbMATHGoogle Scholar
  128. Mang A, Biros G (2015) An inexact Newton-Krylov algorithm for constrained diffeomorphic image registration. SIAM J Imaging Sci 8(2):1030–1069. MathSciNetzbMATHGoogle Scholar
  129. Mang A, Biros G (2016) Constrained \(H^1\)-regularization schemes for diffeomorphic image registration. SIAM J Imaging Sci 9(3):1154–1194. MathSciNetzbMATHGoogle Scholar
  130. Mang A, Biros G (2017) A semi-Lagrangian two-level preconditioned Newton-Krylov solver for constrained diffeomorphic image registration. SIAM J Sci Comput 39(5):B860–B885. MathSciNetzbMATHGoogle Scholar
  131. Mang A, Ruthotto L (2017) A Lagrangian Gauss–Newton–Krylov solver for mass- and intensity-preserving diffeomorphic image registration. SIAM J Sci Comput 39(5):B860–B885. MathSciNetzbMATHGoogle Scholar
  132. Mang A, Schuetz TA, Becker S, Toma A, Buzug TM (2012a) Cyclic numerical time integration in variational non-rigid image registration based on quadratic regularisation. In: Proceedings of the vision, modeling and visualization workshop, pp 143–150.
  133. Mang A, Toma A, Schuetz TA, Becker S, Eckey T, Mohr C, Petersen D, Buzug TM (2012b) Biophysical modeling of brain tumor progression: from unconditionally stable explicit time integration to an inverse problem with parabolic PDE constraints for model calibration. Med Phys 39(7):4444–4459. Google Scholar
  134. Mang A, Gholami A, Biros G (2016) Distributed-memory large-deformation diffeomorphic 3D image registration. In: Proceedings of the ACM/IEEE conference on supercomputing, p 72.
  135. Mang A, Tharakan S, Gholami A, Nimthani N, Subramanian S, Levitt J, Azmat M, Scheufele K, Mehl M, Davatzikos C, Barth B, Biros G (2017) SIBIA-GlS: scalable biophysics-based image analysis for glioma segmentation. In: Proceedings of the BraTS 2017 workshop, pp 197–204Google Scholar
  136. Martin J, Wilcox LC, Burstedde C, Ghattas O (2012) A stochastic Newton MCMC method for large-scale statistical inverse problems with application to seismic inversion. SIAM J Sci Comput 34(3):A1460–A1487MathSciNetzbMATHGoogle Scholar
  137. Menze BH, Jakab A, Bauer S, Kalpathy-Cramer J, Farahani K, Kirby J, Burren Y, Porz N, Slotboom J, Wiest R, Lanczi L, Gerstner E, Weber MA, Arbel T, Avants BB, Ayache N, Buendia P, Collins DL, Cordier N, Corso JJ, Criminisi A, Das T, Delingette H, Demiralp Ç, Durst CR, Dojat M, Doyle S, Festa J, Forbes F, Geremia E, Glocker B, Golland P, Guo X, Hamamci A, Iftekharuddin KM, Jena R, John NM, Konukoglu E, Lashkari D, Mariz JA, Meier R, Pereira S, Precup D, Price SJ, Raviv TR, Reza SMS, Ryan M, Sarikaya D, Schwartz L, Shin HC, Shotton J, Silva CA, Sousa N, Subbanna NK, Szekely G, Taylor TJ, Thomas OM, Tustison NJ, Unal G, Vasseur F, Wintermark M, Ye DH, Zhao L, Zhao B, Zikic D, Prastawa M, Reyes M, Leemput KV (2015) The multimodal brain tumor image segmentation benchmark (BRATS). IEEE Trans Med Imaging 34(10):1993–2024Google Scholar
  138. Menze BH, Van Leemput K, Honkela A, Konukoglu E, Weber MA, Ayache N, Golland P (2011) A generative approach for image-based modeling of tumor growth. In: Information processing in medical imaging (IPMI 2011). Lecture notes in computer science, vol 6801. pp 735–747Google Scholar
  139. Mi H, Petitjean C, Dubray B, Vera P, Ruan S (2014) Prediction of lung tumor evolution during radiotherapy in individual patients with PET. IEEE Trans Med Imaging 33(4):995–1003Google Scholar
  140. Miller MI (2004) Computational anatomy: shape, growth and atrophy comparison via diffeomorphisms. NeuroImage 23(1):S19–S33Google Scholar
  141. Miller MI, Younes L (2001) Group actions, homeomorphism, and matching: a general framework. Int J Comput Vis 41(1/2):61–81zbMATHGoogle Scholar
  142. Miller MI, Trouvé A, Younes L (2006) Geodesic shooting for computational anatomy. J Math Imaging Vis 24:209–228MathSciNetGoogle Scholar
  143. Modat M, Ridgway GR, Taylor ZA, Lehmann M, Barnes J, Hawkes DJ, Fox NC, Ourselin S (2010) Fast free-form deformation using graphics processing units. Comput Methods Programs Biomed 98(3):278–284Google Scholar
  144. Modersitzki J (2004) Numerical methods for image registration. Oxford University Press, New YorkzbMATHGoogle Scholar
  145. Modersitzki J (2009) FAIR: flexible algorithms for image registration. SIAM, PhiladelphiazbMATHGoogle Scholar
  146. Mohamed A, Davatzikos C (2005) Finite element modeling of brain tumor mass-effect from 3D medical images. In: Proceedings of the medical image computing and computer-assisted intervention, pp 400–408Google Scholar
  147. Mori S, Oishi K, Jiang H, Jiang L, Li X, Akhter K, Hua K, Faria AV, Mahmood A, Woods R, Toga AW, Pike GB, Neto PR, Evans A, Zhang J, Huang H, Miller MI, van Zijl P, Mazziotta J (2008) Stereotaxic white matter atlas based on diffusion tensor imaging in an ICBM template. NeuroImage 40(2):570–582Google Scholar
  148. Mosayebi P, Cobzas D, Murtha A, Jagersand M (2012) Tumor invasion margin on the Riemannian space of brain fibers. Med Image Anal 16(2):361–373Google Scholar
  149. Munson T, Sarich J, Wild S, Benson S, McInnes LC (2017) TAO 3.7 users manual. Argonne National Laboratory, Mathematics and Computer Science Division, IllinoisGoogle Scholar
  150. Murray JD (1989) Mathematical biology. Springer, New YorkzbMATHGoogle Scholar
  151. Muyan-Ozcelik P, Owens JD, Xia J, Samant SS (2008) Fast deformable registration on the GPU: A CUDA implementation of demons. In: IEEE international conference on computational sciences and its applications, pp 223–233Google Scholar
  152. Nocedal J, Wright SJ (2006) Numerical optimization. Springer, New YorkzbMATHGoogle Scholar
  153. Oberai AA, Gokhale NH, Feijóo RG (2003) Solution of inverse problems in elasticity imaging using the adjoint method. Inverse Probl 19(2):297MathSciNetzbMATHGoogle Scholar
  154. Oden JT, Prudencio EE, Hawkins-Daarud A (2013) Selection and assessment of phenomenological models of tumor growth. Math Models Methods Appl Sci 23(7):1309–1338MathSciNetzbMATHGoogle Scholar
  155. Ophir J, Alam SK, Garra B, Kallel F, Konofagou E, Krouskop T, Varghese T (1999) Elastography: ultrasonic estimation and imaging of the elastic properties of tissues. J Eng Med 213(3):203–233Google Scholar
  156. Ou Y, Sotiras A, Paragios N, Davatzikos C (2011) DRAMMS: deformable registration via attribute matching and mutual-saliency weighting. Med Image Anal 15(4):622–639Google Scholar
  157. Papademetris X, Sinusas A, Dione D, Duncan J (2001) Estimation of 3D left ventricular deformation from echocardiography. Med Image Anal 5(1):17–28Google Scholar
  158. Papademetris X, Sinusas AJ, Dione DP, Constable RT, Duncan JS (2002) Estimation of 3D left ventricular deformation from medical images using biomechanical models. IEEE Trans Med Imaging 21(7):786–800Google Scholar
  159. Pearson JW, Stoll M (2013) Fast iterative solution of reaction-diffusion control problems arising from chemical processes. SIAM J Sci Comput 35(5):B987–B1009MathSciNetzbMATHGoogle Scholar
  160. Perperidis D, Mohiaddin R, Rueckert D (2005a) Construction of a 4D statistical atlas of the cardiac anatomy and its use in classification. In: Proceedings of the medical image computing and computer-assisted intervention (Lecture notes in computer science), vol 3750. Springer, Berlin, pp 402–410Google Scholar
  161. Perperidis D, Mohiaddin RH, Rueckert D (2005b) Spatio-temporal free-form registration of cardiac MR image sequences. Med Image Anal 9(5):441–456Google Scholar
  162. Petra N, Martin J, Stadler G, Ghattas O (2014) A computational framework for infinite-dimensional Bayesian inverse problems Part II: stochastic Newton MCMC with application to ice sheet flow inverse problems. SIAM J Sci Comput 36(4):A1525–A1555MathSciNetzbMATHGoogle Scholar
  163. Pock T, Urschler M, Zach C, Beichel R, Bischof H (2007) A duality based algorithm for TV-L\(^1\)-optical-flow image registration. Proceedings of the medical image computing and computer-assisted intervention, vol LNCS 4792:511–518Google Scholar
  164. Powathil G, Kohandel M, Sivaloganathan S, Oza A, Milosevic M (2007) Mathematical modeling of brain tumors: effects of radiotherapy and chemotherapy. Phys Med Biol 52(11):3291Google Scholar
  165. Quiroga AAI, Fernández D, Torres GA, Turner CV (2015) Adjoint method for a tumor invasion PDE-constrained optimization problem in 2D using adaptive finite element method. Appl Math Comput 270:358–368MathSciNetGoogle Scholar
  166. Rahman MM, Feng Y, Yankeelov TE, Oden JT (2017) A fully coupled space-time multiscale modeling framework for predicting tumor growth. Comput Methods Appl Mech Eng 320:261–286MathSciNetGoogle Scholar
  167. Rekik I, Allassonnière S, Clatz O, Geremia E, Stretton E, Delingette H, Ayache N (2013) Tumor growth parameters estimation and source localization from a unique time point: Application to low-grade gliomas. Comput Vis Image Underst 117(3):238–249Google Scholar
  168. Ren K, Bal G, Hielscher AH (2006) Frequency domain optical tomography based on the equation of radiative transfer. SIAM J Sci Comput 28(4):1463–1489MathSciNetzbMATHGoogle Scholar
  169. Rockne R, Rockhill JK, Mrugala M, Spence AM, Kalet I, Hendrickson K, Lai A, Cloughesy T, Alvord EC, Swanson KR (2010) Predicting the efficacy of radiotherapy in individual glioblastoma patients in vivo: a mathematical modeling approach. Phys Med Biol 55(12):3271Google Scholar
  170. Roose T, Chapman SJ, Maini PK (2007) Mathematical models of avascular tumor growth. SIAM Rev 49(2):179–208MathSciNetzbMATHGoogle Scholar
  171. Rühaak J, König L, Tramnitzke F, Köstler H, Modersitzki J (2017) A matrix-free approach to efficient affine-linear image registration on CPU and GPU. J Real Time Image Proc 13(1):205–225Google Scholar
  172. Ruhnau P, Schnörr C (2007) Optical Stokes flow estimation: an imaging-based control approach. Exp Fluids 42:61–78Google Scholar
  173. Saratoon T, Tarvainen, T, Cox BT, Arridge SR (2013) A gradient-based method for quantitative photoacoustic tomography using the radiative transfer equation. Inverse Probl 29(7): 075,006Google Scholar
  174. Scheufele K, Mang A, Gholami A, Davatzikos C, Biros G, Mehl M (2018) Coupling brain-tumor biophysical models and diffeomorphic image registration. arXiv: 1710.06420
  175. Schuetz TA, Becker S, Mang A, Toma A, Buzug TM (2013) Modelling of glioblastoma growth by linking a molecular interaction network with an agent based model. Math Comput Model Dyn Syst 19(5):417–433. MathSciNetzbMATHGoogle Scholar
  176. Sermesant M, Delingette H, Ayache N (2006a) An electromechanical model of the heart for image analysis and simulation. IEEE Trans Med Imaging 25(5):612–625Google Scholar
  177. Sermesant M, Moireau P, Camara O, Sainte-Marie J, Andriantsimiavona R, Cimrman R, Hill DL, Chapelle D, Razavi R (2006b) Cardiac function estimation from MRI using a heart model and data assimilation: advances and difficulties. Med Image Anal 10(4):642–656zbMATHGoogle Scholar
  178. Shackleford J, Kandasamy N, Sharp G (2013) High performance deformable image registration algorithms for manycore processors. Morgan Kaufmann, WalthamGoogle Scholar
  179. Shah DJ, Judd RM, Kim RJ (2005) Technology insight: MRI of the myocardium. Nat Clin Pract Cardiovasc Med 2(11):597–605Google Scholar
  180. Shams R, Sadeghi P, Kennedy R, Hartley R (2010a) Parallel computation of mutual information on the GPU with application to real-time registration of 3D medical images. Comput Methods Programs Biomed 99:133–146Google Scholar
  181. Shams R, Sadeghi P, Kennedy RA, Hartley RI (2010b) A survey of medical image registration on multicore and the GPU. Signal Process Mag IEEE 27(2):50–60Google Scholar
  182. Shen DG, Sundar H, Xue Z, Fan Y, Litt H (2005) Consistent estimation of cardiac motions by 4D image registration. In: Proceedings of the Medical image computing and computer-assisted intervention (Lecture notes in computer science), vol 3750. Springer, Berlin, pp 902–910Google Scholar
  183. Shenk O, Manguoglu M, Sameh A, Christen M, Sathe M (2009) Parallel scalable PDE-constrained optimization: antenna identification in hyperthermia cancer treatment planning. Comput Sci Res Dev 23(3–4):177–183Google Scholar
  184. Simoncini V (2012) Reduced order solution of structured linear systems arising in certain PDE-constrained optimization problems. Comput Optim Appl 53(2):591–617MathSciNetzbMATHGoogle Scholar
  185. Sommer S (2008) Accelerating multi-scale flows for LDDKBM diffeomorphic registration. In: Proceedings of the IEEE international conference on computer visions workshops, pp 499–505Google Scholar
  186. Sotiras A, Davatzikos C, Paragios N (2013) Deformable medical image registration: a survey. IEEE Trans Med Imaging 32(7):1153–1190Google Scholar
  187. Sullivan TJ (2015) Introduction to uncertainty quantification. Springer, BerlinzbMATHGoogle Scholar
  188. Sundar H, Davatzikos C, Biros G (2009) Biomechanically constrained 4D estimation of mycardial motion. In: Proceedings of the medical image computing and computer-assisted intervention, vol LNCS 5762, pp 257–265Google Scholar
  189. Swanson KR, Alvord EC, Murray JD (2000) A quantitative model for differential motility of gliomas in grey and white matter. Cell Prolif 33(5):317–330Google Scholar
  190. Swanson KR, Alvord EC, Murray JD (2002) Virtual brain tumours (gliomas) enhance the reality of medical imaging and highlight inadequacies of current therapy. Br J Cancer 86(1):14–18Google Scholar
  191. Swanson KR, Rostomily RC, Alvord EC (2008) A mathematical modelling tool for predicting survival of individual patients following resection of glioblastoma: a proof of principle. Br J Cancer 98(1):113–119Google Scholar
  192. Tarantola A (2005) Inverse problem theory and methods for model parameter estimation. SIAM, PhiladelphiazbMATHGoogle Scholar
  193. Toma A, Mang A, Schuetz TA, Becker S, Buzug TM (2012) A novel method for simulating the extracellular matrix in models of tumour growth. Comput Math Methods Med 2012:960,256-1–960,256-11. MathSciNetzbMATHGoogle Scholar
  194. Tomer R, Ye L, Hsueh B, Deisseroth K (2014) Advanced CLARITY for rapid and high-resolution imaging of intact tissues. Nat Protoc 9(7):1682–1697Google Scholar
  195. Trouvé A (1998) Diffeomorphism groups and pattern matching in image analysis. Int J Comput Vis 28(3):213–221Google Scholar
  196. Tuyisenge V, Sarry L, Corpetti T, Innorta-Coupez E, Ouchchane L, Cassagnes L (2016) Estimation of myocardial strain and contraction phase from cine MRI using variational data assimilation. IEEE Trans Med Imag 35(2):442–455Google Scholar
  197. ur Rehman T, Haber E, Pryor G, Melonakos J, Tannenbaum A (2009) 3D nonrigid registration via optimal mass transport on the GPU. Med Image Anal 13(6):931–940Google Scholar
  198. Valero-Lara P (2014) Multi-GPU acceleration of DARTEL (early detection of Alzheimer). In: Proceedings of the IEEE international conference on cluster computing, pp 346–354Google Scholar
  199. Vercauteren T, Pennec X, Perchant A, Ayache N (2008) Symmetric log-domain diffeomorphic registration: a demons-based approach. Proceedings of the medical image computing and computer-assisted intervention, vol LNCS 5241:754–761Google Scholar
  200. Vercauteren T, Pennec X, Perchant A, Ayache N (2009) Diffeomorphic demons: efficient non-parametric image registration. NeuroImage 45(1):S61–S72Google Scholar
  201. Vialard FX, Risser L, Rueckert D, Cotter CJ (2012) Diffeomorphic 3D image registration via geodesic shooting using an efficient adjoint calculation. Int J Comput Vis 97:229–241MathSciNetzbMATHGoogle Scholar
  202. Wang Z, Deisboeck TS (2008) Computational modeling of brain tumors: discrete, continuum or hybrid. Sci Model Simul SMNS 15(1–3):381zbMATHGoogle Scholar
  203. Weis JA, Miga MI, Yankeelov TE (2017) Three-dimensional image-based mechanical modeling for predicting the response of breast cancer to neoadjuvant therapy. Comput Methods Appl Mech Eng 314:494–512MathSciNetGoogle Scholar
  204. Wilcox LC, Stadler G, Bui-Thanh T, Ghattas O (2015) Discretely exact derivatives for hyperbolic PDE-constrained optimization problems discretized by the discontinuous Galerkin method. J Sci Comput 63(1):138–162MathSciNetzbMATHGoogle Scholar
  205. Wlazło J, Fessler R, Pinnau R, Siedow N, Tse O (2016) Elastic image registration with exact mass preservation. arXiv: 1609.04043
  206. Wong KCL, Summers RM, Kebebew E, Yao J (2015) Pancreatic tumor growth prediction with multiplicative growth and image-derived motion. Proceedings of the information processing in medical imaging, vol LNCS 9123:501–513Google Scholar
  207. Wong KCL, Summers RM, Kebebew E, Yoa J (2017) Pancreatic tumor growth prediction with elastic-growth decomposition, image-derived motion, and FDM-FEM coupling. IEEE Trans Med Imaging 36(1):111–123Google Scholar
  208. Younes L (2007) Jacobi fields in groups of diffeomorphisms and applications. Q Appl Math 650(1):113–134MathSciNetzbMATHGoogle Scholar
  209. Younes L (2010) Shapes and diffeomorphisms. Springer, BerlinzbMATHGoogle Scholar
  210. Zacharaki EI, Hogea CS, Biros G, Davatzikos C (2008a) A comparative study of biomechanical simulators in deformable registration of brain tumor images. IEEE Trans Biomed Eng 55(3):1233–1236Google Scholar
  211. Zacharaki EI, Hogea CS, Shen D, Biros G, Davatzikos C (2008b) Parallel optimization of tumor model parameters for fast registration of brain tumor images. In: Proceedings of the SPIE medical imaging, pp 69,140K1–69,140K10Google Scholar
  212. Zacharaki EI, Hogea CS, Shen D, Biros G, Davatzikos C (2009) Non-diffeomorphic registration of brain tumor images by simulating tissue loss and tumor growth. NeuroImage 46(3):762–774Google Scholar
  213. Zhang M, Fletcher PT (2015) Bayesian principal geodesic analysis for estimating intrinsic diffeomorphic image variability. Med Image Anal 25(1):37–44Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Andreas Mang
    • 1
    Email author
  • Amir Gholami
    • 2
  • Christos Davatzikos
    • 3
  • George Biros
    • 4
  1. 1.Department of MathematicsUniversity of HoustonHoustonUSA
  2. 2.Department of Electrical Engineering and Computer SciencesUniversity of CaliforniaBerkeleyUSA
  3. 3.Center for Biomedical Image Computing and Analytics, Department of RadiologyUniversity of PennsylvaniaPhiladelphiaUSA
  4. 4.Institute for Computational Engineering and SciencesUniversity of Texas at AustinAustinUSA

Personalised recommendations