Indirect Measurement of Hepatic Drug Clearance by Fitting Dynamical Models

We present an indirect signal processing-based measurement method for biological quantities in humans that cannot be directly measured. We develop the method by focusing on estimating hepatic enzyme and drug transporter activity through breath-biopsy samples clinically obtained via the erythromycin breath test (EBT): a small dose of radio-labeled drug is injected and the subsequent content of radio-labeled CO$_2$ is measured repeatedly in exhaled breath; the resulting time series is analyzed. To model EBT we developed a 14-variable non-linear reduced order dynamical model that describes the behavior of the drug and its metabolites in the human body well enough to capture all biological phenomena of interest. Based on this system of coupled non-linear ordinary differential equations (ODEs) we treat the measurement problem as inverse problem: we estimate the ODE parameters of individual patients from the measured EBT time series. These estimates then provide a measurement of the liver activity of interest. The parameters are hard to estimate as the ODEs are stiff and the problem needs to be regularized to ensure stable convergence. We develop a formal operator framework to capture and treat the specific non-linearities present, and perform perturbation analysis to establish properties of the estimation procedure and its solution. Development of the method required 150,000 CPU hours at a supercomputing center, and a single production run takes CPU 24 hours. We introduce and analyze the method in the context of future precision dosing of drugs for vulnerable patients (e.g., oncology, nephrology, or pediatrics) to eventually ensure efficacy and avoid toxicity.