Efficient Generation and Selection of Virtual Populations in Quantitative Systems Pharmacology Models

Abstract

Quantitative systems pharmacology models mechanistically describe a biological system and the effect of drug treatment on system behavior. Because these models rarely are identifiable from the available data, the uncertainty in physiological parameters may be sampled to create alternative parameterizations of the model, sometimes termed "virtual patients." In order to reproduce the statistics of a clinical population, virtual patients are often weighted to form a virtual population that reflects the baseline characteristics of the clinical cohort. Here we introduce a novel technique to efficiently generate virtual patients and, from this ensemble, demonstrate how to select a virtual population that matches the observed data without the need for weighting. This approach improves confidence in model predictions by mitigating the risk that spurious virtual patients become overrepresented in virtual populations.

Figure 1

Overview of algorithm for efficient generation and prevalence‐based selection of virtual patients. To generate virtual patients from a model, the prior information (green boxes) is used to define physiologically reasonable ranges for model outputs and parameter values. An initial parameter guess is optimized until model outputs are physiologically plausible. This is repeated multiple times to form a plausible population. A virtual population is constructed by selecting from this population with probability proportional to the prevalence in the real population relative to the prevalence in the plausible population. This selection is optimized to produce the best virtual population given the patients in the plausible population.

Figure 2

Cost function transformation for convergence to plausible virtual patients. Outputs of the model contribute to the cost function to be minimized by considering the sum of squared errors (SSE) from an associated experimental observation. For each observation we define a physiologically plausible range (arrows in a,b) and shift the SSE associated with that observation so that it is zero if the model output is in this range (a,b). Combining these transformations in each dimension leads to a broader cost function that is minimized by many points, rather than one (black rectangle in c).

Figure 3

Comparison of the initial plausible population (N = 300,000) with NHANES multivariate distribution ((a‐c) black dotted lines estimated PDF, Supplementary Figure 1a‐c. (d‐f) 2D projection of the 95% confidence surface of the estimated probability density function).

Figure 4

Histogram of plausible population selection probability. The probability of inclusion into the virtual population is calculated by optimized relative prevalence. The red histogram (main figure, and figure inset) is a virtual population that matches NHANES data, and is selected from the plausible population (blue histogram) based on displayed probability.

Figure 5

Comparison of a virtual population with NHANES multivariate distribution (dotted black lines). The virtual population (red dots and red histogram) matches the mean, variance, and covariance of the multivariate experimental distribution ((a‐c) black dotted lines estimated probability density function, Supplementary Figure 1a‐c. (d‐f) 2D projection of the 95% confidence surface of the estimated PDF).

Figure 6

Degeneracy of the virtual population. Violin plots of the plausible and virtual populations (a,b, respectively) parameter values (normalized to each parameter's upper and lower bounds) and correlation matrix of the plausible and virtual population (c,d, respectively).