Pharmacometrics: so much mathematics and why planes achieve their destinations with almost perfect results …

The mention of population analysis or pharmacokinetic and pharmacodynamic (PKPD) modelling often sends clinicians and clinical pharmacologists running, not wanting to know about mathematics and understanding complexity. Individualized dosing, optimal design of studies and mechanistic models of physiological processes are all ‘too hard’ often leading to another refrain – ‘and how do we apply this to our clinical practice?’ However, we are prepared to use complex technology and equipment in the practice of medicine that requires high level mathematical complexity, but we fear the use of similar methods to investigate, explore and understand drug therapy, drug effects and disease progression. Should we not be using the most modern methods to quantify and understand clinical pharmacology? Mathematics underlies much of how we function, specifically that we expect pilots and airlines to use the most advanced computers, simulations and analytic techniques to make flying safe. We often rely on global position system (GPS) devices to assist us in arriving at our destinations; require that our computer and smartphone operating systems provide fundamental conveniences and coordinate our schedules, phone calls and information and utilize Google™ to search for information on any number of topics. So why do we struggle with them in understanding drug effects?

Modern medicine now demands the best evidence for treatments, but usually confines this to empirical evidence such as randomized controlled trials, where evidence is based on statistical inference from the analysis of large clinical trials. This is often limited to testing a single hypothesis, is rarely based on disease or drug mechanisms and is often difficult to apply to individuals in a population. Lewis Sheiner used the terms learning and confirmatory trials, the latter referring to empirical or hypothesis testing studies [1]. The former are exploratory studies that aim to understand mechanisms and are not limited to single hypotheses, and will ultimately generate hypotheses to develop confirmatory studies. Population analysis provides a means to undertake exploratory studies, which measure drug effects including quantifying the typical effect of a drug in a population and the variability around this mean effect within the population [2]. The latter is probably one of the most useful elements of population analysis, allowing evidence about drug treatments to be individualized, such as individualizing dosing regimens [3]. This was in fact the motivating reason for Lewis Sheiner who developed the initial algorithms for computer guided dosing of digitalis therapy [4] and these algorithms were further refined [5] and eventually became the basis of the software package nonmem.

It is now over three decades since population analysis was introduced in the late 1970s and there have been major advances in computing speed and capacity, and approaches to improving model selection. There has been an exponential increase in publications of population analysis (Figure 1, [2]) and improvements in the efficiency and capacity to analyze quantitatively data. Sale & Sherer describe modern approaches to model development and selection and how global search strategies can be used in PKPD modelling. They describe how these are superior to traditional methods where a major problem is algorithms finding local solutions (minima), rather than global minima [6]. They propose an elegant genetic algorithm global search strategy based on natural processes of evolution and survival of the fittest – considering models as organisms. With increased computer speed and capacity such global search strategies can be used. However, they still show that human insight and interpretation must continue to play an important role in modelling and that we cannot simply rely on blind automatic analysis. This is analogous to planes still requiring pilots in unusual situations or drivers using human judgement when the GPS gives directions that are non-sensical.

Holford discusses the use of population analysis to understand and quantify disease progression and the effect of drug therapy (or other interventions) on disease progression [7]. This adds another level to the PKPD model with a model describing disease status and progression based pathophysiology and clinical effects. This appears to be important in understanding the disease modifying effects of drugs in slow progressive illnesses such as Parkinson's disease or obstructive airways disease. The World Health Organization has recommended the development of mathematical models to help understand malarial drug resistance and improve drug treatment regimens [8]. Patel et al. review ways in which mechanistic models of the malarial parasite life cycle and PKPD models of anti-malaria drugs such as artemisinin can improve the treatment of malaria, individualizing drug dosing particularly in susceptible populations such as children, pregnant women and patients with human immunodeficiency virus [8].

With the increasing use of population analysis techniques and the requirement now for these by drug regulatory bodies, we are often confronted with multiple population analyses. Duffull & Wright discuss what benefits we gain from repeated analyses [2]. Taking enoxaparin as an example they demonstrate that a drug is well characterized after a few population analyses in terms of the base model and covariate inclusion. However, further modelling appears to be important for special populations such as the elderly, pregnant or particular disease states. Further modelling also confirms findings and refines the determinations of significant effects on PK and PD characteristics over time.

PK models can range from almost completely empirical models to highly complex physiologically based pharmacokinetic (PBPK) models. In other words, PK modelling can be based mainly on clinical data using a more empirical approach and estimating parameters using regression techniques, a ‘top down’ approach, or based on a more detailed understanding of the human body, a ‘bottom up’ approach [9]. Tsamandouras et al. review the evolution of PBPK models, the difficulties and approaches to resolving issues related to the complexities of these models [9]. They describe the ‘middle out’ approach which uses both rationally driven modelling and parameter estimation (empiricism) as an approach that includes both physiologically developed models and information from observed clinical data. Their overview also covers some key limitations with the PBPK approach and provides an important insight into the evolution of this approach over the last 45 years. Kimko & Pinheiro then describe the value of PBPK models in early drug development and the increasing availability of software which has been used by industry [10].

Population PKPD analyses have been used in many different specialist applications and require different approaches because of different time courses, disease processes and drug properties. Bender et al. provide an in depth review of PKPD analyses in clinical oncology and some of the difficulties with modelling tumour growth and tumour biomarkers [11], the lack of placebo data and the limited dosing data. De Buck et al. provide a specific example of using population PKPD analysis to investigate the effect of a new phosphoinositide 3-kinase antagonist on tumour growth in solid malignancies [12]. Studies in oncology differ from PKPD modelling in anaesthesia with much more rapid time courses and the ability to have rich blood sampling and continuous monitoring of pharmacodynamic parameters [13]. Gambus & Troconiz discuss the many applications of PKPD modelling to anaesthetics and the value of derived parameters to clinicians in comparing different agents [13].

A new application for population analysis has been in clinical toxicology where there are uncertainties in the dose ingested and the time of ingestion, and there is usually sparse sample collection, particularly in the absorption phase. The population approach has been used in a number of studies of drugs in overdose to define the PKPD and use this to develop clinical guidelines for treatment [14]. A more difficult situation of paraquat poisoning is investigated by Wunnapuk et al. using a model based approach to predict paraquat exposure, clinical outcomes and effect of antidotal treatment [15].

An important and more recent addition to population analysis has been optimal design which is being increasingly used in early clinical drug development in phase I and II studies [16]. The use of optimal design improves the accuracy and precision of parameter estimates [10,17] and therefore makes early phase trials more efficient. In addition, optimal design should be used in clinical studies. This requires the additional consideration of the practicalities of sample collection in busy and chaotic environments. A recent study of the pharmacokinetics of droperidol in agitated patients in the emergency department demonstrated that you can use sampling windows to address practical issues of blood collection while maintaining good efficiency. (Duffull, personal communication). In this issue Nyberg et al. compare the available software for optimal design and encouragingly demonstrate that they all produce similar results, despite the complexities in estimating the Fisher information matrix [16].

Constructing these models with an eye to identifying important patient specific characteristics that may modulate either exposure, response or outcome is often more complex than would be assumed on face value. Hutmacher & Kowalski review techniques for approaching this important aspect of pharmacometrics and one that has the potential to have a significant impact on understanding patient exposure and response in their review of model methods for covariate identification [18].

Our end game needs to be planes safely arriving at their destination and customer satisfaction. In this case we need to offer the best pharmacometric tools to improve the individualization of drug therapy. Barrett discusses model-based decision support at the bedside in children and not only the importance of using population analysis but the engagement of clinicians and care-givers to make this work [19]. The aim of the issue is to explore the possibilities of population analysis in clinical pharmacology and to understand the benefits of mathematical modelling in understanding complex physiological systems as well as being able to apply the results of these approaches to patient care.