Sensitivity functions in the estimation of parameters of cellular exchange

Abstract

The analysis of multiple-indicator dilution curves to estimate the rates of transport of ions and substrates across the sarcolemma of myocardial cells requires the formulation of models for the blood-interstitial fluid-cell exchanges. The fitting of models to the sets of experimental data is dependent on acquiring a large enough data set, in one physiological state, that there is at least as much information in the data as there are unknown model parameters to be determined. Inasmuch as data are necessarily noisy, redundancy of data and overdetermination of the unknowns are highly desirable. Sensitivity functions are useful in demonstrating which portions of the data relate to which unknown parameters. They are also useful in adjusting model parameters to fit the model to the data, and therefore in parameter evaluation.

Figure 1

Solutions and sensitivity functions for a model of blood-tissue exchange in the heart. Upper panel) Impulse responses for a system consisting of arteries and veins and a set of capillary-tissue units in parallel. The large-vessel transport functions are described by a unimodal density function with mean transit time 7.9 s, relative dispersion 0.34, and skewness 0.75 (3a). The curve for the intravascular reference h_R(t) is the large-vessel transport function convoluted with the weighted sum of 2, fifth-order Poisson operators representing the intracapillary transport and dispersion. The mean capillary transit time was 3.0 s. The curve h_D(t) for the permeating tracer has parameters suitable for D-glucose in the heart. The flow F_s was 1 ml · g⁻¹ · min⁻¹; the parameters governing the exchange and the intracellular sequestration K_Seq are listed. Lower panel) Sensitivity functions for each of the governing transport parameters. Note that all are different from the others, and that they are ordered in time in accordance with the sequence of solute transport from blood into the cells. The functions S(t) are for 1 % changes in parameter values and multiplied by scalars for display purposes: the scalars are 1 for PS_c, 160 for V′_I, 5 for PS_Cell, 500 for $V_{\text{Cell}}^{\prime }$, and 1500 for K_Seq.

Figure 2

Calculation of adjustment in parameter vector for fitting a model to data. There are two unknowns, PS_C and V′₁. The intravascular reference data curve is h_R(t). The model curve ĥ_D(t) differs from the data curve h_D(t): the differences in each of the two time windows are given by the average differences D̄₁ and D̄₂, the area divided by the window width. The average values of the sensitivity functions in the two time windows are given by the S̄’s. These give the coefficients defining the influences of the two parameters in each of the two regions along h_D(t).