Project Details
Projekt
Numerical Methods for Diagnosis and Therapy Design of Cerebral Palsy by Bilevel Optimal Control of Constrained Biomechanical Multi-Body Systems
Subject Area
Mathematics
Term
from 2016 to 2021
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 314150787
This project emerged from a long-standing collaboration with the MotionLab of the Heidelberg University Hospital and intends to develop a mathematical model for the human gait and numerical methods for the solution of inverse and robustified optimal control problems to support a detailed diagnosis and a subsequent systematic planning of a therapy for patients with cerebral palsy (CP). CP is a movement disorder, caused by abnormal development of the brain in an early infancy, that effects muscle coordination, leads to deformed bones and can be characterized by a crouched gait. As the basis of this model a constraint biomechanical multi-body system is developed. As the solution of the variational problem with state constraints already the differential equations exhibit non-smooth and discontinuous MPEC type switching dynamics. The gait of the patient is modeled as a solution of an optimal control problem characterizing the patient attempts to optimize criteria like efficiency or stability. Two different bilevel optimal control problems, or infinite MPECs, and their numerical solution are the core of the project. An inverse optimal control problem calibrates the OCP model to measured marker data of the patient's gait, thus individualizing the model. Together with a sensitivity analysis this provides the physician with much more detailed information for a diagnosis of causes of the disorder. In a second stage, this individualized patient model is then used to systematically plan - and optimize - a therapy by surgical intervention or physiotherapy. Here, the OCP must be robustified as a worst case optimization to account for uncertainties in the parameters and inexact realizations of the controls. Numerous mathematical challenges need to be answered to arrive at an efficient solution for this non-smooth and complementarity based problems: effective ways to deal with a lack of constraint qualification, structure exploitation in the discretized multi-level problems and generation of higher-order derivatives, new strategies for globalizing convergence, techniques to avoid weak stationary points and sensitivity analysis of non-smooth optimal control solutions. A part not to be underestimated is an adequate translation of the mathematical results back into the world of the physician, e.g., by adequate visualization tools, supporting diagnosis as well as systematic therapy planning.
DFG Programme
Priority Programmes
Subproject of
SPP 1962:
Non-smooth and Complementarity-Based Distributed Parameter Systems: Simulation and Hierarchical Optimization
Co-Investigator
Dr. Johannes Schlöder
