Analyzing Multi-response Data Using Forcing Functions: illustrated in pharmacokinetic physiological flow modeling

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Objectives: To analyze multi-response data, a multivariate output model can be fit to all the response components simultaneously (SIM), or each response component can be fit separately to a univariate output model, conditioning on the non-modeled components, the so-called forcing function approach (FFA). Focusing on a special case of multi-response model corresponding to a pharmacokinetic physiological flow model (PFM), the aims of this study are to provide an algorithm for applying FFA, examine its performance, and make recommendations regarding its use.

Methods: The basic PFM has 4 homogenous compartments. All are sampled: arterial blood (A), non-eliminating tissue (N), eliminating tissue (E), and venous blood (V), which is also the drug dosing site. Parameters are blood flow rates to E and N, volumes of distribution of A, E, N, V, elimination rate constant from E, and observation error variances. Observations from a generic individual under various study designs and parameter values are simulated. Using data-analytic models both the same as, and different than the data simulation model, SIM fits the PFM to all data simultaneously; FFA first fits each type of response (one per tissue) separately, approximating the tissue’s input by linearly interpolating the observed concentrations from the donor tissue(s), estimates the identifiable parameter combinations for the response type, and then solves the simultaneous equations linking these across tissues, to obtain the primary model parameters of interest. This simulation and analysis steps are repeated to generate reliable performance statistics. Performance measures include parameter estimation error, prediction error, and the ability to identify the correct analytic model.

Results and Conclusions: When data-analytic model is correct, FFA’s parameter estimation errors are generally about 2 times greater than those with SIM, and FFA’s prediction errors are about 10 times greater than those of SIM. When data-analytic model is misspecified, FFA’s prediction errors are about 3 times greater than those of SIM. However, SIM fails to identify the correct analytical model twice as often as FFA. The study suggests FFA’s final parameter estimates cannot be trusted when the multi-response system being modeled involves feedback, despite its greater convenience for model building, and its clear advantages for model identification. A test is proposed to indicate when FFA’s final estimates may be trustworthy.