PDE-constrained optimization in medical image analysis

  • Andreas Mang
  • Amir Gholami
  • Christos Davatzikos
  • George Biros
Research Article
  • 12 Downloads

Abstract

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.

Keywords

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

Mathematics Subject Classification

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

Notes

Acknowledgements

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

Funding

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)

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Copyright information

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

Authors and Affiliations

  • Andreas Mang
    • 1
  • 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

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