# Seminars Academic Year 2013-2014

- 22/08/2014

Voronov Dmitriy Andreevich (Novosibirsk State University, Russia)

Numerical solutions of the inverse problem of pharmacokinetics - 21/02/2014

E. Jan W. ter Maten (TU Eindhoven and Bergische Univ. Wuppertal) and Marc Timme (University of Goettingen Max Planck Institute for Dynamics and Self-Organization)

IUAP workshop "Mathematical modelling of electrical networks and devices" - 7/10/2013

Cristiana Sebu (Oxford Brookes University)

Sparse 3D reconstructions in Electrical Impedance Tomography using real data

## Numerical solutions of the inverse problem of pharmacokinetics

Pharmacokinetics deals with kinetics of absorption, distribution, metabolism and excretion of drugs and their corresponding pharmacologic, therapeutic or toxic responses in man and animals.

What happens to the drug in the body can be visualized by considering the body as being made up of a large number of compartments, each of which has a volume where the drug is well mixed. Drug is then transferred between these compartments, either transported by the blood from one to another, or by passing an interior membrane in some body organ. We can visualize this whole process as a dynamic system described by a system of ordinary differential equations.

Parameter identifiability analysis for dynamic system ODE models addresses the question of which unknown parameters can be quantifies from given input-output data. The linear compartment models that we focus on in this report are never identifiable, except in the trivial case of a model with only one compartment. This forces us to look for identifiable reparametrization of our model. In this report we consider scaling reparametrization that is obtained by replacing an unobserved variable by a scaling version of itself, and updating coefficients accordingly. In real life we determine a series of time points at which blood samples are taken and plasma concentrations are measured. Here inverse problem arises: it is required to find rate constants (entries of matrix) knowing concentration of a drug at the given moments of a time in one compartment. The quality of those data depends on our choice of time points. An inappropriate choice may make up miss the peak concentration or we may not have sampled long enough to obtain a good estimate of the rate constants. It is demonstrated that the resolving ability of the inverse problem can be improved by varying of the location of measurement data points. The Frechet derivative matrix was constructed. Different types of three-compartment models with central elimination and two-compartment model with extravascular drug administration with absorption are covered in this report. Also an algorithm for solving inverse problem in case of n-compartment is covered in this report.

Inverse problem is solved by different algorithms: Landweber iterations method, Newton-Kantorovich method and Singular Value Decomposition. The question of choosing initial approximations is covered in this report. It is shown that physical properties of initial approximations strongly affect on obtained solutions. The results of numerical experiments are presented.

## IUAP workshop "Mathematical modelling of electrical networks and devices"

Talks Prof. Dr. E. Jan W. ter Maten

1) Advanced techniques in time-domain circuit simulation

2) Sampling, Failure Probabilities and Uncertainty Quantification

Talks Prof. Dr. Marc Timme

1) Synchronization and Collective Dynamics in Oscillator Networks

2) Modern Power Grids: (In-)Stability, Braess' Paradox and Power Outage

## Sparse 3D reconstructions in Electrical Impedance Tomography using real data

We present a 3D reconstruction algorithm with sparsity constraints for Electrical Impedance Tomography (EIT). EIT is the inverse problem problem of determining the distribution of conductivity in the interior of an object from simultaneous measurements of currents and voltages on its boundary. The feasibility of the sparsity reconstruction approach is tested with real data obtained from a new planar EIT device developed at the Institut of Physics, Johannes Gutenberg University, Mainz, Germany. The complete electrode model is adapted for the given device to handle incomplete measurements and the inhomogeneities of the conductivity are a priori assumed to be sparse with respect to a certain basis. This prior information is incorporated into a Tikhonov-type functional by including a sparsity-promoting l1-regularization term. The functional is minimized with an iterative soft shrinkage-type algorithm.