Invited Session Thu.2.H 0111

Thursday, 13:15 - 14:45 h, Room: H 0111

Cluster 19: PDE-constrained optimization & multi-level/multi-grid methods [...]

PDE optimization in medicine II


Chair: Anton Schiela



Thursday, 13:15 - 13:40 h, Room: H 0111, Talk 1

Richard Barnard
Optimal radiotherapy treatment planning using minimum entropy models

Coauthors: Martin Frank, Michael Herty


We study the problem of finding an optimal radiotherapy treatment plan. A time-dependent Boltzmann particle transport model is used to model the interaction between radiative particles with tissue. This model allows for the modeling of inhomogeneities in the body and allows for anisotropic sources modeling distributed radiation and external beam sources. We study two optimization problems: minimizing the deviation from a spatially-dependent prescribed dose through a quadratic tracking functional; and minimizing the survival of tumor cells through the use of the linear-quadratic model of radiobiological cell response. For each problem, we derive the optimality systems. In order to solve the state and adjoint equations, we use the minimum entropy approximation; the advantages of this method are discussed. Numerical results for real patient data are presented.



Thursday, 13:45 - 14:10 h, Room: H 0111, Talk 2

Chamakuri Nagaiah
Numerical solutions for boundary control of bidomain equations in cardiac electrophysiology

Coauthors: Karl Kunisch, Gernot Plank


The bidomain equations are widely
accepted as one of the most complete descriptions of the cardiac
bioelectric activity at the tissue and organ level.
The model consist of a system of elliptic partial differential equations
coupled with a non-linear parabolic equation of reaction-diffusion type,
where the reaction term, modeling ionic transport is described by a set of
ordinary differential equations. The optimal control
approach is based on minimizing a properly chosen cost functional
J(v,Ie) depending on the extracellular current Ie as
input, which must be determined in such a way
that wave-fronts of transmembrane voltage v are smoothed in an optimal manner.
The boundary control formulation is presented.
The numerical realization of the optimality system
is described in detail and numerical experiments, which demonstrate the
capability of influencing and terminating reentry phenomena, are presented.
We employ the parallelization techniques to
enhance the solution process of the optimality system
and a numerical feasibility study of the Lagrange-Newton-Kryloy method
in a parallel environment will be shown.



Thursday, 14:15 - 14:40 h, Room: H 0111, Talk 3

Malik Kirchner
Large deformation diffeomorphic metric mapping using conforming adaptive finite elements

Coauthors: Andreas G√ľnther, Hans Lamecker, Martin Weiser


Automatic registration of anatomical objects is an important task in medical imaging.
One crucial prerequisite is finding a pointwise mapping between different shapes.

Currents are linear functionals providing a unified description of those shapes of any positive
integer dimension m ≤ d embedded in Rd.
The Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework [Joshi and Miller,
IEEE Transactions on Image Processing, 2000] solves the
correspondence problem between them by evolving a displacement field
along a velocity field.

In this talk we propose three aspects making this ODE/PDE optimization problem numerically practical.
We compute the temporal propagation of m-currents using a spatially discretized velocity field on
conforming adaptive finite elements. A hierarchical approach from coarse to fine lattices improves
performance and robustness of our method. The adaptive refinement process is driven by some residual
estimator based on the Riesz representative of shape differences.


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