Biological Models’ Parameter Estimation Based on Discrete Measurements and Adjoint Sensitivity Analysis

Mathematical models of biological processes are usually continuous time (CT) and take the form of non-linear ordinary differential equations. On the other hand the estimation of model parameters is done based on discrete time (DT), relatively rare, measurements. Hence, overall problem of parameter estimation has hybrid, continuous-discrete form: it uses CT model and minimise DT performance index depending on DT prediction errors. In our previous works we have published Generalized Back Propagation Through Time (GBPPT) method—a method allowing us to use the adjoint sensitivity analysis for obtained hybrid system, and giving as a result a computationally effective recipe for calculating gradient of the performance index in parameter space. GBPTT specifies rules for construction of the adjoint system, in particular it specifies how to manage elements interfacing between CT and DT parts of the system: ideal sampler (IS) and ideal pulser (IP). Such rules for isolated IS and IP elements has been proposed without strict formal rationale. In this article we deliver a proof of correctness of such rules. Additionally, as an illustration, we present an example of application of GBPTT to parameter estimation of chemical enzymatic reaction which is one of basic biochemical reaction.