Analytical Solution of Multicompartment Solute Kinetics for Hemodialysis

Objective. To provide an exact solution for variable-volume multicompartment kinetic models with linear volume change, and to apply this solution to a 4-compartment diffusion-adjusted regional blood flow model for both urea and creatinine kinetics in hemodialysis. Methods. A matrix-based approach applicable to linear models encompassing any number of compartments is presented. The procedure requires the inversion of a square matrix and the computation of its eigenvalues λ, assuming they are all distinct. This novel approach bypasses the evaluation of the definite integral to solve the inhomogeneous ordinary differential equation. Results. For urea two out of four eigenvalues describing the changes of concentrations in time are about 105 times larger than the other eigenvalues indicating that the 4-compartment model essentially reduces to the 2-compartment regional blood flow model. In case of creatinine, however, the distribution of eigenvalues is more balanced (a factor of 102 between the largest and the smallest eigenvalue) indicating that all four compartments contribute to creatinine kinetics in hemodialysis. Interpretation. Apart from providing an exact analytic solution for practical applications such as the identification of relevant model and treatment parameters, the matrix-based approach reveals characteristic details on model symmetry and complexity for different solutes. 1. Introduction Compartment models are popular in pharmacokinetics and, as a special application, in hemodialysis, where such models serve to quantify treatment dose [1–3]. Most of that kinetic analysis has been done for urea usually described by 2-compartment models. However, unlike most pharmacokinetic models, the volume of compartments cannot be assumed as constant because of ultrafiltration of excess volume within and accumulation of volume between hemodialysis treatments. The effects on solute concentrations and solute balance caused by volume changes are not negligible. Still, the problem can be expressed as 2-dimensional, inhomogeneous ordinary differential equations (ODE) [4], and the closed form solution to this problem is known. Furthermore, for the variable-volume 2-compartment model urea concentrations have been presented as explicit functions of time and model parameters so that the concentrations in the two compartments at any time can be computed in a single step [5, 6]. That approach was based on the variation of the constants method. While urea, a solute of little toxicity, is a useful marker of uremia, solutes with limited transfer between compartments such