  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.
