1.2.1 Time-Scale Separation
The existence of multiple time-scales is an inherent property of
biological systems. Multi-time scale systems contain reactions that
occur over several orders of magnitude. A clear example is the effect of
denosumab on bone mineral density which is mediated by the high affinity
binding to receptor activator of NF-κB ligand (RANKL). Denosumab-RANKL
binding kinetics occur over a time scale of seconds-minutes. However,
the effect of this binding on bone mineral density evolves over months
to years [39]. These separations in time-scales can be exploited to
reduce the model by setting the slowly reacting components to be
constant relative to the rapidly reacting components or by setting the
rapidly reacting species to equilibrate instantaneously. The choice
depends on the response variable of interest (in this case binding of
RANKL or bone mineral density).
The assumption that rapid reactions are at equilibrium is known as Rapid
Equilibrium Approximation (REA) or Quasi-Steady State Approximation
(QSSA). The most widely used application of QSSA is the derivation of
Michealis-Menten equation of the enzyme kinetics [40,41]. It has
also been used to simplify kinetic models for drugs exhibiting
target-mediated drug disposition (TMDD) [42].
For large scale QSP models, finding the appropriate partitioning of
reactions into fast and slow classes can be challenging. Holland et al.
[43] applied the approach to reduce a 25-dimensional model of
cardiac β1-adrenergic signalling to a 6-dimensional reduced model with a
reasonable predictive performance. Biswal et al. applied a variant of
the approach to simplify a 27-state stiff model of calcium homeostasis
and bone remodelling and was able to reduce it to “very slow”,
“slow”, and “fast” models that describe different time scales of the
full-order model.