Efficient Methods for Parameter Estimation of Ordinary and Partial Differential Equation Models of Viral Hepatitis Kinetics

Abstract

Parameter estimation in mathematical models that are based on differential equations is known to be of fundamental importance. For sophisticated models such as age-structured models that simulate biological agents, parameter estimation that addresses all cases of data points available presents a formidable challenge and efficiency considerations need to be employed in order for the method to become practical. In the case of age-structured models of viral hepatitis dynamics under antiviral treatment that deal with partial differential equations, a fully numerical parameter estimation method was developed that does not require an analytical approximation of the solution to the multiscale model equations, avoiding the necessity to derive the long-term approximation for each model. However, the method is considerably slow because of precision problems in estimating derivatives with respect to the parameters near their boundary values, making it almost impractical for general use. In order to overcome this limitation, two steps have been taken that significantly reduce the running time by orders of magnitude and thereby lead to a practical method. First, constrained optimization is used, letting the user add constraints relating to the boundary values of each parameter before the method is executed. Second, optimization is performed by derivative-free methods, eliminating the need to evaluate expensive numerical derivative approximations. The newly efficient methods that were developed as a result of the above approach are described for hepatitis C virus kinetic models during antiviral therapy. Illustrations are provided using a user-friendly simulator that incorporates the efficient methods for both the ordinary and partial differential equation models.

Intracellular “In Silico Microscopes” – Fully 3D Spatio-Temporal Virus Replication Model Simulations

“…Also kinetic mathematical models [16][17][18][19][20][21][22][23][24] using ordinary differential equation (ODE) descriptions were developed for HCV infection to supply new findings, such as to figure out optimal doses of DAAs and other antiviral agents.…”

Understanding the antiviral effects of RNAi-based therapy in HBeAg-positive chronic hepatitis B infection

“…In this section, we include the dynamics of uninfected and infected cell populations and investigate changes in predictions for increased killing rate We incorporate uninfected hepatocytes T which get infected by free virus at rate , as modeled previously 26 , 39 , 63 . Note that we ignore the age of the infection and assume that once a cell becomes infected, it is producing virus (for a PDE model extension in a hepatitis C virus infection, see 64 , 65 ). Both uninfected and infected hepatocytes proliferate according to a logistic term with maximal growth rate and and carrying capacity .…”

Intracellular “In Silico Microscopes”—Comprehensive 3D Spatio-Temporal Virus Replication Model Simulations

“…However, there is a lack of basic quantitative biophysical understanding and detailed experiments. To support clinical and virological research, kinetic mathematical models for HCV infection have been developed [ 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 ] using ordinary differential equation (ODE) models. These models enable the simulation of treatment effects and aid in optimizing the dosing of antiviral agents.…”