Computation of the geometric mean and variance of the AUC using polynomial chaos
Yevgen Ryeznik (1, 2), Andrew C. Hooker (2)
(1) Department of Mathematics, Uppsala University, (2) Department of Pharmaceutical Biosciences, Uppsala University
Objectives: In many applications, a drugs pharmacokinetics (PK) can be expressed in terms of a system of ordinary differential equations (ODEs) with random parameters. One of the important PK parameters is the Area Under the Curve (AUC) which is obtained as a definite integral of a concentration function over time interval. Usually, concentration is measured at several time points, and the AUC is approximated by some approximation formula (e.g. trapezoidal rule). This result may be inaccurate, for example, due to a small number of time points taken into the approximation. Another approach is to include AUC into the system of ODEs as one of the unknown functions. One advantage of such an approach is that the (partial) AUC can be evaluated from the integration start time to any later time point. In the case of linear models, an analytical solution can be obtained for the AUC at infinity, otherwise, an appropriate numerical technique must be applied. In this work, we compare two methods of calculating the mean and the variance of the population model-based AUC. One approach is the most common Markov Chain Monte Carlo (MCMC), and another is a Polynomial Chaos (PC) method [1] which is based on a polynomial expansion of a function with respect to an appropriate system of orthogonal polynomials. There are two issues connected with MCMC: accuracy and computational time. If the sample size from parameter space is small, then there is a loss of accuracy. Otherwise, if the sample size is large then the result becomes more accurate, but the process of computation is time-consuming. The PC approach does not require any sampling techniques and does not contain any stochastic component. This allows computations to achieve similar or better accuracy in relatively less time compared to MCMC.
Methods: We consider a one-compartment model with linear oral absorption, which is represented as a system of two ODEs with three parameters (Ka, CL, V). AUC(t) is also included in the system as an unknown function: dAUC(t)/dt = C(t), where C(t) is a concentration function. Population parameters are assumed to be known, individual parameters are assumed to be log-normally distributed. In this model, no correlations between random effects are assumed. In odder to calculate partial AUCs with the MCMC approach we sample from the individual parameter distribution, compute the AUC by solving the system of ODEs from the start of integration to each sample time point, then repeat the process many times. Finally, the mean and the variance are calculated. With the PC approach, all the unknown functions in the system are replaced with their PC expansions. After substituting these expansions into the ODEs, the initial system of ODEs with random parameters is transformed to a system of ODEs with fixed parameters which has to be solved only once. In this new system, unknown functions are the coefficients of the polynomial expansions, and the mean and the variance of the AUC are expressed in terms of these coefficients. To explore how these two approaches work the Python package chaospy [2] has been used as well as the R package reticulate for making the methods work in the R environment. Currently, the chaospy package only allows log-normally distributed individual parameters and non-correlated random effects. But, from a mathematical perspective, there are no limitations to extend the PC approach to more general cases.
Results: Numerical results show that the PC approach is more robust and is much faster than classical MCMC. PC has the same accuracy when the highest degree in the polynomial expansion is larger than or equals to two, while the MCMC method may have large fluctuations in accuracy depending on sample size. Computational time for PC with a degree of expansion equal to three is twenty five times faster than MCMC with the same accuracy.
Conclusions: The PC approach can improve efficiency of AUC estimation. It requires less time to achieve the same accuracy as MCMC methods.
References:
[1] Dongbin Xiu “Numerical Methods for Stochastic Computations.A spectral Method Approach”, 2010, Princeton University Press.
[2] Jonathan Feinberg, Hans Petter Langtangen “Chaospy: An open source tool for designing methods of uncertainty quantification”, 2015, Journal of Computational Sciences, Vol. 11, p. 46-57.