Analyzing multi-response data using forcing functions

Two analytic strategies can be taken to the analysis of 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 in some way 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 f low model (PFM), the aims of this study are to (i) provide an algorithm for applying FFA to multi-response data from a PFM; (ii) examine the performance of FFA vs. SIM under optimal conditions for both, and in the presence of model misspecification; (iii) make recommendations regarding the use of FFA for multi-response data analysis.

The basic PFM we use (variants of the basic model are used for simulation) has four homogenous compartments among which drug distributes. 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 f low rates to E and N, volumes of distribution of A, E, N, and 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 (DAM) both the same as, and different than the data simulation model (DSM), 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. Performances are compared with respect to parameter estimation error (when DAM and DSM are identical), and interpolated prediction error (when DAM and DSM are/are-not identical). The ability of SIM and FFA to identify the correct analytic model is also examined by comparing their failure rates in rejecting the wrong DAM.

The parameter estimation errors with FFA are generally about two times greater than those with SIM when the DAM is identical to the DSM. The prediction errors of FFA are about ten times greater than those of SIM when the DAM is identical to the DSM, and are about three times greater when the two are different. However, SIM fails to identify the correct model twice as often as FFA.

Despite its greater convenience for model building, and its clear advantages for model identification, FFA's final parameter estimates cannot be trusted when the multi-response system being modeled involves feedback. The size of the ratio of the two FFA residuals (obtained from the response-specific fits and from predictions made with the final FFA parameters) can, however, be used to indicate when FFA's final estimates may be trustworthy.