Modeling and simulation has become an indispensable part of drug development. Historically, the type of modeling employed changes as a drug moves through the stages of drug development—from discovery to pre-clinical testing to clinical trials.
At the early stages of drug discovery, we may use systems biology models to identify drug targets and possible compounds, establish their activity, and select clinical candidates. As we move forward from discovery to pre-clinical and clinical development, we tend to switch to traditional pharmacokinetic/pharmacodynamic (PK/PD) models. And these emerge as the forefront type models in those areas. Quantitative systems pharmacology (QSP) models have the advantage of combining modeling frameworks to enable one framework to inform the entire drug development process.
Unfortunately, QSP models are generally too complicated to use in the clinic. In this blog, I’ll discuss how model reduction approaches can be used to “zoom in and out” of your QSP models relative to its current application. This approach has the advantage of supporting a single model wherein all experiments contribute to the learning and validation of the reduced model. Conversely, information from reduced models can be incorporated into the original QSP model.
Defining model reduction
What is model reduction, and how can we use it to get greater utility from QSP models? In brief, this is how this approach works. We start with the physiological system that we wish to model. We then convert it to a mathematical representation, which can be a system of differential equations. Next, we apply model reduction methods to reduce the system of differential equations to the point where it is simple enough to be linked to a dataset. Finally, we analyze the reduced system.
The goal of model reduction is to reproduce the behavior of the original model (within reasonable error) while reducing the number of species, reactions, or complexes.
Model reduction methods
Modelers can use seven distinct classes of methods to make models easier to manage. The first four are considered model reduction methods, whereas the last three are considered model simplification.
- Time-scale exploitation refers to any method that exploits the often large differences in reaction rates that can occur within a biochemical system to create a reduced model that is accurate at a given time-scale of interest.
- Optimization/sensitivity analysis refers to methods that evaluate the relative importance of model components and eliminate uninfluential ones. A typical procedure might involve “switching off” or fixing reactions or species deemed uninfluential.
- Lumping is a method that constructs a reduced system with new state-variables corresponding to subsets of the original species. Most commonly, it refers to proper lumping, where the subsets are simply summed together and each of the original state-variables corresponds to, at most, one of the lumped state-variables.
- Balanced truncation looks at systems with defined inputs (dosing, for example) and outputs (typically representing endpoints of interest), and how all of the states in the system co-vary with respect to these inputs and the outputs. The reduced dynamical description then focuses on maintaining the input-output behavior of the original system. Traditionally, the method only applies to linear systems, but it can be generalized to nonlinear models via empirical approaches.
- Conservation analysis can reduce the number of modeled differential equations by recasting the model as a system of differential-algebraic equations by exploiting the conserved moieties in the system.
- Nondimensionalization can expose ratios of parameters key to the dynamics and potentially reduce the number of parameters explicitly represented in the model.
- Model decomposition methods partition the network into key, often functionally related, groups and which can be used to guide model reduction approaches.
Why should we use model reduction?
Model reduction provides multiple benefits. It is a rigorous means of extracting a practical, usable model from our best biological understanding. Starting with complex systems, these methods then allow us to extract smaller, simpler models that are more in line with the traditional pharmacometric scale of modeling and the typically available scope of clinical data.
Moreover, model reduction can improve numerical difficulties and parameter identifiability properties. It can expose the most influential components of the model on the behaviors of interest. This approach also yields systems with fewer model parameters and species, which means fewer experiments will be needed to validate and fit those models.
Finally, model reduction allows us to create modeling frameworks that last through the duration of the drug development pipeline and across compounds. These frameworks help us learn more from failure thus increasing R&D productivity.
Model reduction is not a magic bullet
Like any tool, model reduction requires care and asking clear questions to be best used. Its methods are highly technical and require a good grasp of the mathematics to be used effectively. In addition, the relationship between reduction accuracy and model parameterization is complex and has knock-on implications for identifiability. Model reduction always incurs some loss of information. As such, its use is often a compromise between accuracy and practicality. A compromise that cannot always be reached.
To learn more about how model reduction can help you get more out of your models, please watch a webinar I gave on this topic.