# On mass balance and the use-reuse of the rate constant matrix in pharmacokinetics

Introduction

Assurance of mass balance is a critical part of pharmacometric model formulation (i.e. model creation and coding).  In many cases, for simple mamillary compartmental models this can be assessed informally.  However, in more complicated models (e.g. physiologically based models) the proof of mass balance can be more complicated.  A method to evaluate mass balance is introduced using a variation on the rate constant matrix (RCM).  The RCM is a 2-dimensional matrix of rate constants that completely specifies the movement or reactions associated within a pharmacokinetic model.  Each position within the matrix describes the movement of mass from one location to another or reaction from 1 species to another.  For example, the rate constant in position 1,2 specifies the movement (or reaction) from compartment 2 to 1.  At steady-state the values of the rate constants are not themselves of importance and the matrix can be simplified to a matrix of indicators: 1s (movements or reactions), -1s (losses) and 0s (no change).

In this work, a reuse of the RCM is described that provides an explicit solution to evaluation of mass balance.

Methods

The steady-state indicator RCM (RCMi) is introduced for an open 3-compartment mamillary model with iv-bolus input.  For this model the dimension of the RCMi is 3×3.  The RCMi comprises two components, the input-output and the distribution matrices. The input-output RCMi consists of +1 for input compartments and -1 for loss compartments.  The distribution RCMi consists of off-diagonal 1 or 0 values that signify movement of mass from one compartment to another (or reaction of one species to another) and -1 diagonal elements signifying movement from that compartment.  Mass balance was defined as (1) when the Σ(rows) = Σ(columns)=0 (conservation of mass) and (2) all columns or rows have at least one location with a value > 0 (e.g. do not consist of zeros).  This signifies the existence of interconnectedness between compartments and that input and output are specified correctly. This method was applied to a nonlinear PBPK model for bosenten .  This model consisted of 14 compartments with 32 disposition processes.  This model was coded in R (version 4.0.4) and the RCM created.  The RCM was converted into a steady-state RCMi and partitioned into the input-output and distribution matrices.  The system had 1 input (intravenous bolus dose of bosenten into compartment 1) and 5 output compartments (the central compartment and 5 hepatocellular compartments).

Results

The utility of the RCMi for evaluating mass balance was explored based on various three-compartment model structures which included, mamillary and catenary structures as well as arbitrary model structures with multiple inputs, multiple outputs and a series of models that do not display mass balance. The two elements of the criteria for mass balance were important.  The first element (Σ(rows) = Σ(columns)=0) identified coding errors, e.g. a missing connection.  The second element identified mass balance errors: such as an island state (a compartment with no input but potentially with output) which was shown by a row of 0s, or a terminus state (a compartment with input but no output) shown by a column of 0s [see  for a description of these states].

Coding the bosenten example identified transcription errors in coding the differential equations as well as proving that the system (when correctly coded) follows mass balance.

Conclusions

Mass balance is of critical importance in the formulation of pharmacokinetic models.  Despite this there are no simple formal methods that have been developed to aid the pharmacometrician.  The reuse of the rate constant matrix appears to provide an acceptable and practical approach to determining mass balance.

References

1. Sato et al. Drug Metab Dispos 2018;46:740-748.
2. Siripuram VJ. Understanding azathioprine metabolism using a systems pharmacology approach. PhD Thesis, University of Otago. 2019 