Numerical Methods for Diagnosis and Therapy Design of Cerebral Palsy by Bilevel Optimal Control of Constrained Biomechanical Multi-Body Systems

Description

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 technique for these 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.

Publications

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Preprints

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Research Area

Modeling, problem analysis, algorithm design and convergence analysis

The focus of this area is on the development and analysis of genuinely non-smooth models in the sciences in order to properly capture real-world effects and to avoid comprising smoothing approaches. In simulation and optimization this requires to advance set-valued analysis and the design of robust algorithms for non-smooth problems.

Realization of algorithms, adaptive discretization and model reduction

As the target applications of this SPP involve non-smooth structures and partial differential operators, the discretization of the associated problems and robust error estimation are important issues to be address, and proper model-reduction techniques need to be developed.

Incorporation of parameter dependencies and robustness

In many applications the robustness of solutions with respect to a given parameter range (uncertainty set) is highly relevant. Correspondingly, in this research area of the SPP, bi- or multilevel optimization approaches will be studied in order to robustify problem solutions against uncertain parameters.

Members

  • member's portrait

    Prof. Hans Georg Bock

    Universität Heidelberg
    Principal Investigator
  • member's portrait

    Prof. Ekaterina Kostina

    Universität Heidelberg
    Principal Investigator
  • member's portrait

    Dr. Johannes Schlöder

    Universität Heidelberg
    Co-PI
  • member's portrait

    Marta Sauter

    Universität Heidelberg
    Research Assistant
  • member's portrait

    Matthias Schlöder

    Universität Heidelberg
    Research Assistant

Project Related News

  • 10. 10. 2016 : Welcome to our new project member

    Matthias Schlöder joins project 3.

  • 26. 04. 2017 : Welcome to our new project member

    Marta Sauter joins Project 3