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.