Background
Mathematical modelling of clinical pharmacology processes is fundamental to understanding the time course of the determinants of patient response and inform key decisions in the discovery, development, and clinical use of drugs. In drug development, modelling and simulation (M&S) applications span a wide spectrum from target identification and validation through analysis of preclinical data, determining the best first-in-human dose, and analysis of phase I, II, and III data to optimise doses for safety and efficacy [1]. M&S also plays an important role in clinical practice, serving purposes such as dose individualisation based on covariates or measured biomarkers, optimising doses for special populations or off-label use, and optimising clinical trial design [2]. A wide variety of models and approaches are available, ranging from empirical and conceptual models that include no specific representation of biologic mechanisms to fully mechanistic models that describe biochemical, (patho)physiological, and pharmacological mechanisms in detail [3] [4].
Quantitative systems pharmacology (QSP) is a discipline within M&S that creates mathematical models that focus on the quantitative interplay between biological and pharmacological mechanisms (see [5-7] for detailed descriptions). QSP has roots in engineering, systems biology, and pharmacology. A key feature of QSP models is the integration of information and knowledge from various disciplines with experimental data into a single quantitative model that describes drug response often at multiple spatial scales (e.g. spanning from cellular to whole body effects), and temporal scales (e.g. from milliseconds to decades). Due to their mechanistic structure, QSP models are capable of extrapolating knowledge to predict outcomes in scenarios that have not been tested experimentally [4]. The ability to use QSP models to extrapolate findings has many important applications for the development and clinical use of drugs. For example, extrapolating efficacy from preclinical in-vitro or in-vivo studies to predict efficacy in humans [8] and extrapolating from adults to paediatric patients [9]. Integration of data collected from preclinical and clinical studies [10], and from various drugs that act on the same or similar targets [11], into a single mathematical framework can expedite drug development by cumulating knowledge gained throughout development of both successful and failed drug candidates [12].
Despite the benefits offered by the QSP approach, there are several challenges in their application. The level of detail in which pharmacological and biological mechanisms are represented in QSP models, usually results in models with a large number of states (in pharmacokinetics states represent a concentration in a particular compartment) and parameters. We show a (relatively simple) hypothetical QSP model schematic in figure 1. This schematic is illustrated with 16 circles (called nodes), that may represent different pharmacological species or physiological measures. Each node is connected to another node using arrows or dashed lines with a bar at the end (collectively called edges). We use an arrow to illustrate movement based on mass (or molar) balance and a dashed line with a bar to illustrate mass (or molar) action through positive or negative feedback. The size (dimension) of the model is a function of the number of nodes (circles) and edges (lines). Feedback mechanisms relate to homeostasis through damping (e.g. blood pressure control [13]) or amplification (e.g. formation of a blood clot [14]). These mechanisms make it difficult to understand which component(s) of a system are important (or unimportant) with respect to observing a phenomenon of interest or interpreting a response of the system given a change (e.g. administration of a dose). Both damping and amplification behaviours result in nonlinearity in QSP models which makes them mathematically difficult to work with and solve computationally [15]. Additionally, the large number of parameters and the often-limited measurable responses makes traditional modelling techniques infeasible due to structural identifiability [16]. Thus, even if an infinite amount of observational data are available (e.g. infinite INR values), it may not be possible to reliably estimate all parameters in a QSP model nor determine which parameters can be estimated [17].
An alternative approach is to use simpler models, such as compartmental models, that are built based on available data. These models are readily amenable for use in simulation and estimation analyses. However, since they are built based on data they do not necessarily provide mechanistic insights into how the data arose and may not be appropriate for extrapolation (see for example [18,19]). Therefore, a hybrid approach to building simpler but mechanistically accurate models would be of significant benefit. Model-order reduction techniques provide a set of methods to harness the mechanistic characteristics of large QSP models but render them into simpler models that are amenable for use in simulation, estimation and design. Simpler models using these methods have been used to extrapolate beyond the data used to build them in predicting changes in bone mineral density [18,19], be used to predict response to warfarin therapy [20], individualise treatment for children with acute lymphoblastic leukaemia [21], predict fibrinogen kinetics in snake envenomed patients [22], and design of clinical trials [23].
The aim of this article is to provide a framework to illustrate the utility of various methods that can be used for simplifying QSP models. These methods are globally referred to here as model-order reduction approaches. We will focus on the concepts and potential applications of various model-order reduction methods in the context of QSP with relevant references provided for readers who are interested to delve into more technical details. We denote the original QSP model as full-order and a simpler model (produced by model-order reduction) as a reduced-order model. The order of a model represents its size, i.e. number of compartments and parameters, which is also termed the degrees of freedom, i.e. the number of independent parameters in the model. A reduced-order model will, therefore, by definition, have fewer parameters than the (original) full-order model. We provide an explanation of terms in Appendix 1.