# Mathematical Biosciences

Identification Of Parameters In Systems Biology

Systems Biology is an actively emerging interdisciplinary area between biology and applied mathematics, based on the idea of treating biological systems as a whole entity which is more than the sum of its interrelated components. These systems are networks with emerging properties generated by a complex interaction of a large number of cells and organisms. One of the major goals of systems biology is to reveal, understand, and predict such properties through the development of mathematical models based on experimental data. In many cases, predictive models of systems biology are described by large systems of nonlinear differential equations. Quantitative identification of such systems requires the solution of inverse problems on the identification of parameters of the system. In a recent project, we consider the inverse problem for the identification of parameters for systems of nonlinear ODEs arising in systems biology. A new numerical method that combines Pontryagin optimization, Bellman's quazilinearization with sensitivity analysis, and Tikhonov's regularization is implemented. The method is applied to various biological models such as the Lotka-Volterra system, bistable switch model in genetic regulatory networks, gene regulation, and repressilator models from synthetic biology. The numerical results and application to real data demonstrate that the method is very well adapted to canonical models of systems biology with moderate size parameter sets and has quadratic convergence. Software package qlopt is developed to implement the method and posted in GitHub under the GNU General Public License v3.0. The recent paper is listed below.

To address adaptation and the scalability of the method to inverse problems with the significantly larger size of parameter sets we developed a modification of the method by embedding a method of staggered corrector for sensitivity analysis and by enhancing multi-objective optimization which enables the application of the method to large-scale models with practically non-identifiable parameters based on multiple data sets, possibly with partial and noisy measurements. Application of the modified method to benchmark model of a three-step pathway modeled by 8 nonlinear ODEs with 36 unknown parameters demonstrate geometric convergence with a minimum of five data sets and with minimum measurements per data set. MATLAB package AMIGO2 is used to demonstrate the advantage of qlopt over most popular methods/software such as lsqnonlin, fmincon and nl2sol. Here is a current paper:

Identification of Parameters for Large-scale Kinetic Models, Journal of Computational Physics, Volume 429, 15 March 2021, 110026

## Cancer Detection Through EIT And Optimal Control Theory

EIT is a non-invasive imaging technique recently gaining popularity in various medical applications including breast screening and cancer detection. Our recent project is on the inverse EIT problem on recovering electrical conductivity tensor and potential in the body based on the measurement of the boundary voltages on the electrodes for a given electrode current. The inverse EIT problem presents an effective mathematical model of breast cancer detection based on the experimental fact that the electrical conductivity of malignant tumors of the breast may significantly differ from conductivity of the surrounding normal tissue. We analyze the inverse EIT problem in a PDE constrained optimal control framework in Besov space, where the electrical conductivity tensor and boundary voltages are control parameters, and the cost functional is the norm declinations of the boundary electrode current from the given current pattern and boundary electrode voltages from the measurements. The state vector is a solution of the second order elliptic PDE in divergence form with bounded measurable coefficients under mixed Neumann/Robin type boundary condition. To address the highly ill-posed nature of the inverse EIT problem, we develop a "variational formulation with additional data" which is well adapted to clinical situation. We prove existence of the optimal control problem and Frechet differentiability in the Besov space setting. Based on the Frechet gradient and optimality condition effective numerical method based on the projective gradient method in Besov spaces is developed. Below is the recent paper.

Discretization and Convergence of the EIT Optimal Control Problem in 2D and 3D Domains, to appear in Inverse and Ill-posed Problems, 2021.