Abstract
The paper considers a time-adaptive finite element method for determination of drug efficacy in a parameter identification problem (PIP) for a system of ordinary differential equations (ODE) that describes dynamics of the primary human immunodeficiency virus (HIV) infection with drug therapy. Tikhonov’s regularization method, optimization approach and finite element method to solve this problem are presented. A posteriori error estimates in the Tikhonov’s functional and reconstructed parameter are derived. Based on these estimates a time adaptive algorithm is formulated and numerically tested for different scenarios of noisy observations of virus population function. Numerical results show a significant improvement of reconstruction of drug efficacy parameter using the local time-adaptive mesh refinement method compared to the gradient method applied on a uniform time mesh.
Highlights
Mathematical modeling of the immune processes is an essential part of the research in immunology [15, 16, 22]
We note that the main goal of our work is to present mathematical framework of a posteriori error estimation for solution of parameter identification problem (PIP)’s and to show usefulness of the time-adaptive error control for determination of parameters in PIP which we demonstrate on the example of the model of human immunodeficiency virus (HIV) infection
The exact knowledge of the personal drug efficacy can aid in the determination of the most suitable drug as well as the most optimal dose to an individual, in the long run resulting in a personalized treatment with maximum efficacy and minimum adverse drug reactions
Summary
Mathematical modeling of the immune processes is an essential part of the research in immunology [15, 16, 22]. Despite the emergence of a great amount of new high-performance methods for experimental analysis of the immunity, the results of mathematical modeling are relevant, in particular, in clinical practice in order to work out optimal, individually customized strategies for treatment of the pathological process (bacterial/viral infections or tumor growth). The computational algorithms for parameter identification that we work on, are based on an adaptive time-mesh refinement [13] for coefficient inverse problems (CIPs). The main idea of our work is adoptation of results for space mesh refinement developed in [13] for solution of hyperbolic CIPs, to the parameter identification problem (PIP) for reconstruction of parameters on the time mesh. The adaptive finite element method has shown that it significantly improves reconstruction of parameters when solving the coefficient inverse problems for hyperbolic PDE [5, 7, 8, 12, 13]
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
