Optimization and Thermoregulation

In the regional hyperthermia cancer therapy, radio frequency radiation is used to heat the tumor in order to make it more susceptible to an accompanying radio- or chemotherapy. Heat transport by blood flow has a major impact onto the attainable tumor temperature. In this project, a multiscale model of the heat transport in the human vascular system is developed and implemented in order to obtain more accurate simulations, and in order to find optimal antenna parameters. For the solution of the arising stationary, periodic, or transient optimization problems we consider function space oriented interior point methods. These are inexact Newton pathfollowing methods in function space. We study control reduced primal interior point methods for optimal control problems with PDEs and pointwise control or state constraints, and Lavretiev regularization. For interior point methods applied to optimal control problems with state constraints the importance of rational barrier functionals has been discovered. In the case of finite dimensional control a generically optimal rational order could be determined. A path-following method in function space has been developed that exploits problem structure by a pointwise damping step and problem suited adaptivity. The combination of Lavrentiev regularization and control reduction turned standard software (COMSOL) for the solution of non-linear PDEs into an efficient solver for optimal control problems.

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Short Description

For many years heat has been studied as a therapeutic option in medicine, e.g. in the various realizations of the cancer therapy hyperthermia. Part of the members have been successfully involved in therapy planning for regional hyperthermia in close cooperation with radiologists and surgeons at the Charité Berlin. In the course of this cooperation a number of important mathematical problems have come up that could not be treated within that medicine dominated project, but, when successfully solved, would have the potential to open a new reliable perspective for the general application of heat within medical treatment.

These problems include (1) a more realistic modelling of the temperature distribution within the body and (2) an optimization of antenna parameters in the setting of refined temperature distribution models and a more adequate objective function. The solution of these problems requires both a firm analytical derivation and an efficient algorithmic realization. In this close context, (3) interior point methods in function space are also developed further for the benefit of a class of optimal control problems in PDEs.

Multiscale models of thermoregulation

The transport of heat by blood flow contributes significantly to the final temperature distribution in the human body. Up to now, the geometric structure of the vascular system has been neglected by imposing an isotropic, homogeneous Helmholtz term that accounts for the heat transport.

Since the directional transport of heat by the vascular system is considered significant, it should be taken into account. For computational tractability, a multiscale model of the vascular system has to be developed, that treats large vessels directly as 3D objects, medium size vessels as 1D simplifications, and small vessels by a homogenized Helmholtz term as before. The development of multiscale perfusion models has been pursued mainly on the scale of medium size vessels with a diameter between 10mm and 2mm. These have a non-negligible thermal influence and their geometry can be acquired individually from MR and CT scans. However, they are too small to be represented as 3D objects in a coarse mesh at reasonable computational cost. A representation of medium scale vessels as sequences of 1D edges in tetrahedral meshes and their coupling to the surrounding tissue based on conservation of mass flow has been analyzed.

vessels medium size vessels
Large size vessels. Embedding of medium size vessels.

Another important effect of medium scale vessels is their influence on the tissue perfusion due to the flow balances in the vascular system. In particular, the high reduction of peripheral resistance in heated muscle may lead to decreased blood supply to a tumor fed by the same vessel. This steal-effect is observed in clinical practice. A hierarchical model of flow balances coupled to the bio-heat-transfer equation is developed.

The steal effect

The steal effect.

The perfusion of the tumor is highly individual, and a-priori models are unreliable. Blood flow to and from the tumor can be measured by MRI using contrast agents. We model the tumor vasculature as a porous medium, giving the resulting blood flow to be used in therapy planning.

Interior point methods for elliptic and parabolic PDE

Interior point methods have been proven to be a very efficient class of methods for inequality constrained finite dimensional optimization. We consider adaptive interior point methods for the solution of optimization problems in infinite dimensional function spaces. To obtain efficient optimization algorithms it is important to analyse and exploit the analytical structure of the problem. Once a proveably convergent method in function space is found, an inexact Newton pathfollowing algorithm can be constructed on the base of an efficient discretization scheme. The framework of inexact Newton methods yields accuracy requirements on the discretization error used to control the adaptive mesh refinement.

Structure of the algorithm

Path-following in function space.

Our work envolves various topics in control theory. We developed a primal interior point method for parabolic problems with linear equation, observation at the final time in the objective functional, and pointwise state constraints and proved convergence for a logarithmic barrier function. This idea gains particular importance in the context of transient and periodic heating strategies.

One highlight of our theoretical approach for problems with state constraints is a new technique to analyze them directly, without continuity assumptions for the state. This idea has the potential to extend the derivation of optimality conditions to a large class of problems.

We also investigate different regularization techniques for steady-state heating. In particular, the application of interior point methods and their coupling with a Lavrentiev type regularization have been studied. By this regularization technique, the original state constraints, e.g. $\theta_{min}\leq \theta \leq \theta_{max}$, are replaced by $\theta_{min}\leq \theta +\varepsilon u \leq \theta_{max}$, with a small regularization parameter $\varepsilon$, where $u$ is the control function. Based on the analysis of the barrier homotopy path an adaptive path-following algorithm in function space was constructed and analysed. In particular, it has been shown that this algorithm produces a sequence of iterates that converges to the solution of the original state constrained problem. An implementation for our hyperthermia treatment planning problem shows that this regularization improves the performance of our algorithms significantly.

Moreover, we consider control reduced primal interior point methods where in the case of bound constrained control, the control can be eliminated from the optimality system. Then the reduced system only contains smooth variables. On the one hand this leads to optimal error estimates, on the other hand the analytical properties of the underlying differential equation can be exploited more efficiently, and we obtain a superlinearly convergent algorithm. Besides, pathfollowing can be performed on relatively coarse discretizations, which makes the resulting algorithm very efficient.

A further topic of interest is the minimization of an $L^{\infty}$-functional constrained by an elliptic PDE. After transformation into a problem with a linear objective functional and additional state constraints, this problem can be handled without additional regularization.

L2-objective maximum-norm objective

Comparison of $L^{2}$ (left) and maximum-norm objective (right) on a stationary problem with discontinuous coefficients.
The circular target area is covered more evenly by the solution of the $L^{\infty}$ objective.

State constrained problems and their regularization

State constraints are an important and difficult aspect of optimal control with PDEs. Analytically, they are well understood in the case of continuous states, while steps towards an understanding in the remaining cases have only been taken recently. Their high non-linearity necessitates the use of regularization and path-following methods for their solution. We focused on two regularization approaches and their interplay. For problems with continuous states, interior point techniques can be applied. Here the state constraints are replaced by a logarithmic or rational barrier term, and the original problem is tackled by solving a sequence of non-linear problems. For cases, where discontinuous states cannot be excluded in advance, these techniques have been complemented by Lavrentiev type regularization. Here the state constrained problem is transformed into a problem with mixed control-state constraints, which is more regular and easier to solve.

Interior point methods for stationary problems with state constraints

Most state constrained stationary problems can be treated directly with interior point techniques. We gained a thorough analytic understanding of the effect of barrier regularizations and their corresponding homotopies. In particular, the homotopy path enjoys simple and strong convergence and continuity properties. Discretization schemes for central path solutions were developed and analyzed. It turned out that the use of appropriate rational barrier functions, as opposed to the classical logarithmic barrier function, leads to a strictly feasible path. This in turn, is the basis for a convergence theory of a Newton path-following method, a result which is up to now unique in the literature. For a finite dimensional control space, we could determine an optimal rational order, depending on the spatial dimension of the problem, which leads to linear convergence in generic situations. The arrier approach was extended to a problem with maximum-norm objective. This setting is of special interest in the formulation of optimization criteria in local hyperthermia. Further, we extended our barrier techniques to gradient bounds on the state.

Interior point methods for time-dependent problems with state constraints

Based on the expertise obtained for elliptic control problems, particular emphasis was laid on the construction and analysis of solution algorithms for parabolic problems. These will gain particular importance in the context of transient and periodic heating strategies. In most relevant cases additional Lavrentiev regularization is necessary. A conceptual primal interior point method was developed for time-dependent problems with pointwise state constraints. After regularization, convergence was shown for a logarithmic barrier function. Here, adaptive time discretization techniques were decisive to deal with irregular Lagrange multipliers. The analysis is similar to the one for the elliptic case, but in most time dependent problems continuity of the states is not guaranteed and thus a classical theoretical foundation is not available. This obstacle is completely overcome by the Lavrentiev type regularization, which permits to derive KKT-conditions for any dimension of the domain.

Function space oriented optimization of treatment parameters

Up to now, superposition of temperature fields corresponding to the antenna parameters had been exploited for fast and easy optimization of the antenna parameters. This scheme depends inherently on the linearity of the BHTE and fails if more accurate nonlinear perfusion models are used. Moreover, precomputation of the temperature fields must be done without knowledge about the superposed temperature field, such that adaptive mesh refinement oriented towards the final solution is impossible. In addition, the temperature constraints, which have been treated by an outer penalty approach, should be treated as inequality constraints.


Heat distribution after optimization.

Algorithmic development

Based on the analysis of the barrier homotopy path an adaptive path-following algorithm was constructed and analyzed. The algorithm exploits the special problem structure by a pointwise damping step, which increases the efficiency and robustness of the algorithm significantly, compared to the standard variant.

Choice of step-sizes of interior point algorithms with and without pointwise damping

An implementation has been applied successfully to hyperthermia treatment planning problems. In particular it has been shown to retain its good numerical performance also for application problems with complex geometries and irregular data. Currently, we are extending this solver for clinical use. A powerful approach to exploit the function space structure of our problem are inexact Newton path-following methods in function space. This approach integrates non-linear algorithms and adaptive refinement by a-posteriori error control of the Newton steps. The steps are performed inexactly in function space. A crucial issue is the choice of an appropriate scaled norm, which governs the convergence behavior of Newton's method. We were able to construct an a-posteriori error estimator fits exactly into this framework. The resulting algorithms shows excellent numerical behavior, since most of the iterations can be performed on coarse computational grids. This happens, although the radius of convergence of the Newton corrector shrinks to zero in the limit of the homotopy. A promising variant of goal-oriented error estimation and mesh adaptivity for optimization problems based on the actual cost reduction has been developed. First results show a significant improvement over existing approaches. One possibility to address optimal control problems is to apply standard solvers to discretized problems. The NLP code IpOpt has been used to solve hyperthermia treatment problems on fixed discretizations. The performance has not been satisfying compared to the function space oriented code developed in A1.

Left: Optimal control for a model problem. Right: Adaptively chosen grid.

The research on interior point methods was complemented by a new numerical approach for solving optimality systems in state-constrained PDE control. Here, a Lavrentiev type regularization and control reduction are appropriate tools to convert the optimality system into a Newton-differentiable coupled system of nonlinear PDEs that can be solved by available codes for PDEs. This method was successfully applied to parabolic problems up to space dimension 2 and nonlinear elliptic problems of dimension 3 related to regional hyperthermia.

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    Perfusionsmodellierung in menschlichen Tumoren
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    1D-Reduktion thermal signifikanter Aderstränge in der Hyperthermie-Modellierung
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  • Poster for Urania presentation
  • Posters for 30. Evangelischer Kirchentag ( 1, 2, 3)
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    Organizational Details

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