Research and Recent Publications

Mathematical Biosciences

1. 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 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 which 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 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 the quadratic convergence. Software package qlopt is developed to implement the method and posted in GitHub under the GNU General Public License v3.0. Here is a recent paper:

2. Breast Cancer Detection through Electrical Impedance Tomography (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 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. Effective numerical analysis in a carefully constructed model example which adequately represents the diagnosis of breast cancer in reality, demonstrate accurate reconstruction of the electrical conductivity function of the body in the frame of the model based on "variational formulation with additional data". Here is the recent preprint:

  • U.G. Abdulla, V. Bukshtynov, S. Seif Breast Cancer Detection through Electrical Impedance Tomography and Optimal Control Theory: Theoretical and Computational Analysis, Math Arxiv:1809.05936, September 16, 2018