The pharmacometrician's dilemma: the tension between mechanistic and empirical approaches in mathematical modelling and simulation – a continuation of the age‐old dispute between rationalism and empiricism?

There is an inherent tension between the mechanistic approach to developing mathematical models and the empirical approach to capturing observed data in a generalized mathematical framework. This tension often co‐exists within the context of developing mathematical models and the question of whether a given approach is ‘fit‐for‐purpose’ is raised. In this editorial, we will outline the contentions and highlight the challenges faced by pharmacometricians from the perspective of the mechanism driven ‘rationalist’ modeller and the data driven ‘empiricist’ modeller. For simplicity, we will refer to the mechanistic approach as one that reflects the rationalist and the empirical approach as one that represents the empiricist.

Rationalism can be defined as ‘the theory that the exercise of reason, rather than experience, authority or spiritual revelation, provides the primary basis for knowledge’ 1.

Empiricism can be defined as ‘the view that experience, especially of the senses, is the only source of knowledge. From this flows: a. Employment of empirical methods, as in science. b. An empirical conclusion. c. The practice of medicine that disregards scientific theory and relies solely on practical experience’ 1.

The inherent tension between rationalism and empiricism is articulated in the following quotation from the Stanford Encyclopedia of Philosophy 2:

‘The dispute between rationalism and empiricism concerns the extent to which we are dependent upon sense experience in our effort to gain knowledge. Rationalists claim that there are significant ways in which our concepts and knowledge are gained independently of sense experience. Empiricists claim that sense experience is the ultimate source of all our concepts and knowledge.

Rationalists generally develop their view in two ways. First, they argue that there are cases where the content of our concepts or knowledge outstrips the information that sense experience can provide. Second, they construct accounts of how reason in some form or other provides that additional information about the world. Empiricists present complementary lines of thought. First, they develop accounts of how experience provides the information that rationalists cite, insofar as we have it in the first place. (Empiricists will at times opt for skepticism as an alternative to rationalism: if experience cannot provide the concepts or knowledge the rationalists cite, then these concepts cannot exist). Second, empiricists attack the rationalists' accounts of how reason is a source of concepts or knowledge’.

In addition, conflict can arise as new experimental data may not necessarily align with prior constructs of mechanism and hence there is a discordance between the empiricist who models the data and the rationalist who applies the construct. This is compounded by an additional tension as to whether the data that has been generated by the experiment can necessarily be used to test the prior construct (i.e. can the construct be falsified?). A lack of empirical falsifiability forms the basis of this conflict in that the new experiment was not designed, and is therefore not optimal, to falsify the construct. Indeed, a theory or hypothesis may not be falsifiable at a given time or with available methods although may be formally testable at some future point. Such has been the case for many theories in physics that have only been tested formally or experimentally many years after being proposed.

Pharmacometricians exist in a special intermediate space – that of the rational empiricist. To extend upon the above quotation 2, perhaps the rational component relates to the mechanistic model based on a synthesis of knowledge from first principles and prior information that is then used to predict an outcome that has yet to be observed. The empirical component flows from the capture of already observed or ‘sensed’ measures that are described using mathematical constructs. The pharmacometrician exists at the nexus of mechanistic, ‘rational’ information and the generation of outputs that could be considered predictions of what the empiricist would observe if the experiment were performed. More specifically, the outputs from the mechanistic model represent a testable set of predictions that can move the findings more squarely into the empirical category. If the predictions are testable then the underlying model is falsifiable. Pharmacometricians, therefore, face an inherent tension in serving as the mathematical formalizer of empirical findings and then perhaps ascribing a mechanistic underpinning to those observations.

The tension is evident in the compromises necessarily made when attempting to identify signals in data that are suggestive of particular underlying causes whilst also generating a rational, mechanistic framework that can produce similar outputs (predictions) for those observations, albeit with many more degrees of freedom. It is often this increase in the degrees of freedom that leads to calls for how to define the generalizability or utility of the model parameters under conditions so overdetermined that one could have any number of parameter values that result in the same output. How does that characteristic reflect a directly interpretable physiological meaning? Claiming a ‘good fit to the data’ in an overdetermined system does a disservice to predictive mechanistic models. The rational mechanistic framework, however, can be used to propose hypotheses and subsequent designs for assessing falsifiability of internal mechanistic elements.

Perhaps all of these concerns feed into what may be considered ‘The pharmacometrician's dilemma’. As the integrators of information across rational and empirical domains, pharmacometricians face many potential areas of conflict if any of the constituencies involved in a particular area become too inflexible in interpreting a particular approach. The pharmacometrician's dilemma is illustrated in Figure 1.

Figure 1.

Increasingly rationalist approaches are shown at top from left to right, informing the mechanistic domain of a problem. Increasingly empirical approaches are shown at bottom from right to left. Along these continuous scales, there is potential for the pharmacometrician to interact with the experimentalist, who is in either the rationalist domain and/or the empirical domain. The pharmacometrician can serve as a bridge to facilitate the incorporation of mechanistic information into empirical approaches as well as the incorporation of empirically defined properties for the rationalist or mechanistically oriented experimentalist to develop further

If one is attempting to capture fundamental mechanisms and make predictions outside the experimental conditions that were used to develop the model, the rationalist approach becomes more important. If the purpose is to re‐articulate findings or observations in a context that is not far from the conditions where observations were generated, then the pharmacometrician serves more on the empiricist's side of the rational empiricist equation. In both of these situations, the pharmacometrician may make predictions of phenomena that have not yet been observed and pass these along to the experimentalist, who will conduct experiments to refine further mechanistic understanding of the system. Alternatively, the experimentalist may pass along findings that are then synthesized by the pharmacometrician to make predicted ‘empirical’ outcomes from the model, which can be further tested by the empiricist. This may be viewed as a cyclical and adaptive process.

To provide a more concrete example, consider the coagulation times of blood. One may develop a simple model based on empirical observations of blood clotting times. Such a model would have narrowly bounded operating characteristics limited to those situations that mimicked the context under which the data were generated (e.g. assessing correlation). This simple model would be severely constrained with respect to making new predictions or being generalizable across a range of conditions. Consider now a more detailed representation of the coagulation network, similar to that published by Wajima et al 3. This mechanistically more detailed representation of the system provides a framework where a broader range of ‘what‐if’ questions can be asked and predictions made, in particular, predictions of observations that have not yet been made. These predictions represent the synthesis of current mechanistic knowledge on this topic (i.e. akin to the rationalist) and, despite not being constrained by well‐estimated parameters from a model‐fitting perspective, such predictions can lead to new insights and experiments. Additionally, predictions can be tested experimentally and the model updated accordingly. A specific scenario could involve a new anticoagulant medication with a novel mechanism of action. This novel mechanism of action could be interrogated with the more detailed model to predict and explore a) what the effect of the drug is likely to be, b) how best to monitor the effects of the new drug and/or c) how may we best adjust the treatment regimen to provide an optimal response. It is unlikely that the aforementioned questions could be adequately addressed by the simpler empirical model.

Another example relates to the theoretical basis for allometric exponents. While a more specific estimate of an allometric exponent may be obtainable from the data, in doing so the underlying scientific principles for the ¾ exponent relating size to drug clearance, for example, would be ignored 4. A major problem, however, is that the ¾ power law relating size to clearance cannot reasonably be expected to be determined from a single pharmacological experiment and hence is not generally amenable to empirical falsifiability. It is instead based upon a derivation that arises from the theoretically optimum size structure of a mammalian organism for maximizing the delivery of nutrients and removal of wastes with a fractal branching tree structure. This theorem has been later tested experimentally, across many non‐human species, and found to be a reasonable description of metabolic rate (i.e. the evaluation failed to falsify the theorem). Utilization of the ¾ power law allows for the probing of other mechanistic processes that may be ongoing besides a strictly size related change. These processes could include maturation of enzyme or transporter systems that would remain hidden in an estimate of an exponent not constrained by the underlying fractal size relationship in biological organisms. Ultimately this situation mirrors the concept of deterministic identifiability, in that the parameters of the system under question are not able to be estimated with high precision due to the experimental conditions. This does not mean that allometry is non‐falsifiable.

The pharmacometrician must operate in these seemingly contradictory contexts: that of the most precise description and that of the rationalist invoking underlying principles that may not be empirically falsifiable with the current experimental design. Thus, the challenge to the field of pharmacometrics and to individual pharmacometricians is to balance rationalism and empiricism appropriately for the task at hand and also to evaluate model performance adequately for the ultimate purpose of the model.

Bies, R. , Cook, S. , and Duffull, S. (2016) The pharmacometrician's dilemma: the tension between mechanistic and empirical approaches in mathematical modelling and simulation – a continuation of the age‐old dispute between rationalism and empiricism?. Br J Clin Pharmacol, 82: 580–582. doi: 10.1111/bcp.13040.