Introduction: Quantitative systems pharmacology models are used in PKPD and pharmacometrics to describe the effects of drugs as modulators of system behaviour. In most cases the systems that are described in these models can be constructed of modular units that can be assembled en-masse to create the overall system. An example of a modular component might be an enzyme catalysed reaction. Assessing the module for mass and energy balance to ensure its properties are consistent with the system and other modules is a critical step in the modular framework. The aim of this work is to introduce concepts of mass and energy balance and bond graph theory.
Methods: Bond graph theory arose from electrical engineering. It consists of junctions (nodes) representing conservation of mass and conservation of energy and edges representing the energy transmission, i.e. the flow of current and causality. There are also storage elements (capacitors) which are the storage (integral) of the molar concentration of the substrate. In this work we will apply bond graph theory to describe a simple Michaelis-Menten enzyme process. Here the capacitance are the states of the system and the edges are the flux, which represent the energy storage of the system given by the integral of flux (i.e. concentration nmol/L, q) and the gain or loss of mass over time (i.e. nmol/L/s), respectively. We balance the system on internal energy (a component of Gibbs free energy) and mass. The internal energy (chemical potential) represents the driving force for the flux. We use CellML to construct and simulate the system.
Results: The Michaelis-Menten system is described in a kinetic framework with substrate (q1), enzyme (q3), substrate-enzyme intermediate (q4), and product (q2) system. In this example we limit the system to be closed (not in mass exchange with the outside) and isolated (not in energy exchange with the outside). Micro-rate constants are described for the reaction processes. A bond graph is derived that shows flow and causality (mass and energy potential). Calculation of the energy and mass balance yield the well established equilibrium constant, Km and maximum velocity, Vmax. The system is shown to be balanced on mass and energy.
Discussion: In this work we show a simple application of bond graph theory to a pharmacokinetic system. We see that we can assess both mass and energy balance to ensure the system is complete and therefore modular. This work can be extended to other more complicated systems, e.g. open pharmacokinetic systems in which mass balance cannot be enforced, or ion channels in which special attention to energy balance is required.