Changing mindsets to embrace “fuzzy thinking” in pharmacokinetics and

pharmacodynamics can be difficult, but the returns may be worth the trouble

**Mark Greener***Greener is a freelance writer based in Cambridge, UK*

It is a pharmaceutical industry truism that the cost of bringing a new chemical entity (NCE) to market continues to rise, while the likelihood of success steadily declines. So every pharmaceutical and biotechnology company aims to reduce costs and maximize productivity. Determining if a NCE shows appropriate pharmacokinetics (PK) and pharmacodynamics (PD) is one approach to improving the chances of success. Unfortunately, the industry’s track record at optimizing PK and PD isn’t especially impressive. At least 95% of new drugs and biopharmaceuticals show a poor PK profile and other properties that could transform “a possible blockbuster into just another has-been” [G. Orive *et al., Trends Pharmacol. Sci*., vol. 7, pp. 382-387 (2004)]. Furthermore, candidate drugs are becoming larger, more flexible and increasingly lipophilic, partly to meet the demands of high-throughput screens, but these properties tend to compromise PK.

Crocheted model of the Lorenz manifold. (Source: H. M. Osinga and B. Krauskopf, University of Bristol, UK) |

Against this background, increasingly sophisticated methods–including fractal geometry, chaos theory and fuzzy logic–offer the prospect of honing critical PK and PD models, and the needed math is within the grasp of most drug development scientists. The main stumbling block may be whether researchers can sufficiently change their mind-set to make the most of the new tools.

A decade ago, many companies didn’t ascribe studies of a drug’s PK and PD a particularly high priority, especially in early development. Today that’s changed, driven by economic necessity and regulatory scrutiny. “PK/PD modeling, which can be costly in terms of time, can be perceived by management as an unnecessary effort, unless it is recognized as an equally important part of development as animal studies or *in vitro* studies,” says Panos Macheras, PhD, of the School of Pharmacy at the University of Athens, Greece. He says this perception is changing, especially as regulatory agencies look favorably on NCEs with well-characterized profiles.

**Modeling outcomes**

As a result, many pharma and biotech companies now invest heavily in computer modeling to predict outcomes and help decide whether to continue an NCE’s development. “As in silico models improve and their use becomes more widespread and accepted by researchers and regulatory bodies, drug development will come to rely more heavily on the information presented by such models,” says Joseph Turner, PhD, research affiliate of the School of Medicine at University of Queensland, Brisbane, Australia. “Their relevance to many areas of drug development, including PK and PD modeling, combined with their speed, versatility, and cost-effectiveness will see them making vital contributions as they are integrated into the drug-development process.”

This investment in computer models and other approaches to optimise “druggability” may already be paying dividends. William Jusko, PhD, professor of pharmacokinetics and pharmacodynamics, School of Pharmacy and Pharmaceutical Sciences at the University of Buffalo, N.Y. says that the growing recognition of physicochemical, absorption, and metabolic properties that are not conducive to druggability reduced the number of failures over recent years. “I have heard presentations that indicate poor PK properties led to discontinuation of development of perhaps 40% of compounds 10 years ago, while the current rate is about 10%,” says Jusko.

Nevertheless, predicting outcomes is difficult because of the numerous factors that affect PK and PD, including age, gender, weight, concurrent disease, concomitant medication, and genetic polymorphisms in metabolic enzymes and drug transporters [B. A. Sproule *et al., Trends Pharmacol. Sci*., vol. 23, pp. 412-417 (2002)]. Furthermore, little direct evidence clearly correlates a drug’s blood levels with efficacy or tolerability [P. Meredith, *Clinical Therapeutics*, vol. 25, pp. 2875-2890 (2003)]. In many cases, oral bioavailability assessed in rodents, dogs and primates correlate relatively poorly with that in humans [I. R. Wilding, D. V. Prior, *Crit. Rev. Ther. Drug Carr. Systems*, vol. 20, pp. 405-431 (2003)].

Indeed, Macheras says PK, and PD in particular, rest on relatively weak theoretical foundations. “There have been no serious attempts [to produce] a meta-pharmaceutical frame,” he says. “Rather, the urgent and continuous need for application imposes overwhelming empiricism. PK is mostly considered as occurring in ad hoc homogeneous compartments and PD still draws from classic *in vitro* biochemistry, such as the law of mass action for receptor binding.”

Against this background, researchers are assessing several approaches to model PK and PD more accurately. Together these approaches may mark the start of a paradigm shift in PK and PD from deterministic models toward nonlinear approaches, a shift that has already occurred in the physical sciences.

Whole-Body Pharmacokinetic ModelingPharmacologists divide pharmacokinetic models into the broad categories of empirical, compartmental, and mechanistic. Empirical models offer a mathematical description of the drug concentration in a sample of biological tissue or fluid over time. Compartmental models are also empirical. However, they incorporate some physiological hypotheses about pharmacokinetic behavior, so they use compartments for absorption, distribution, and elimination. In a two-compartment model, for example, all the tissues represent the peripheral compartment and the plasma, the central compartment. Drugs enter and leave the peripheral compartment via the central compartment. This means that the model has a fast and slow phase. “These empirical models, being entirely data-driven, do not lend themselves well to extrapolation beyond the data range—such as later or earlier time points—interspecies scaling, and they do not provide an insight into specific tissues’ concentration-time profiles, which are not always reflected by blood concentrations,” says Ivelina Gueorguieva, PhD, of the Centre for Applied Pharmacokinetic Research at the University of Manchester, UK. Over the last 30 years, pharmacologists developed a new approach, whole-body physiologically based pharmacokinetic (WBPBPK) models, in an attempt to overcome these problems. “WBPBPK models are mechanistic and do not have the drawbacks of the other two types,” says Gueorguieva. “WBPBPK models are simplified representations of a complex system, i.e., the mammalian body, and they use physiological and biochemical information to help describe and predict drug disposition.” WBPBPK models can integrate data from several sources. For example, they can use principles of mass conservation, physiological parameters (including tissue volumes and blood flows), as well as drug specific data, such as clearance, tissue-partition coefficients and protein binding. Each organ can be represented in three ways. Firstly, the analysis can represent the organ as a well-stirred model, which assumes instantaneous distribution of the drug in the organ. Alternatively, the analysis can assume that the drug flows in particular direction along a tube. The final type of analysis, the dispersion model, combines the other two. Additional organ models can take nonlinear tissue kinetics into account. Furthermore, a drug’s physicochemical and structural properties make important contributions to the PK profile. “An exciting new research area is examining the interrelationships between physiochemical properties and pharmacokinetic behavior,” Gueorguieva says. This research raises the prospect of predicting PK and PD from quantitative structure property relationships (QSPR). “Integrated QSPR-WBPBPK modeling should facilitate the identification of chemicals of a family that possess desired properties of bioaccumulation and blood concentration profile in both test animals and humans,” says Gueorguieva. “Such information will aid better design and selection of compounds.” |

**Butterfly Effect?**

In 1961, meteorologist Edward Lorenz wanted to improve long-range weather forecasting. Using one of the first commercially available computers, he modeled a change in the weather using 12 equations. Lorenz assumed the weather was deterministic—the prediction’s accuracy depended on the precision with which he could define the starting state. But when checking his results he noticed something strange. Two predictions differed markedly, yet all he had done was round up the results, entering 0.506 rather than 0.506127. This tiny difference, of one part in 5,000, led to the now-classic idea that a butterfly flapping its wings in Beijing could change the weather in New York a month later. In other words, complex systems such as the weather, epidemics, human physiology, for example, comprise multiple components that interact in nonlinear ways. Any uncertainty in the initial state is magnified dramatically, so it is impossible to predict behavior more than a few steps ahead.

Such insights can aid attempts to model biological processes and improve pharmaceutical production. Bernd Krauskopf, PhD, and Hinke Osinga, PhD, developed a hands-on model of the complicated behavior of the Lorenz equations. Krauskopf and Osinga both teach in the Department of Engineering Mathematics at the University of Bristol, UK. Consider, for example, how a leaf floating in a turbulent river passes either to the left or right around a rock downstream. The leaves that end up clinging to the rock followed a unique path. Translated to the Lorenz equations, such special paths define a complicated surface, called the Lorenz manifold, that helps explain the system’s chaotic nature.

Essentially, says Krauskopf, the Lorenz manifold (See figure on page 55) forms a boundary surface between points that end up behaving very differently after some time, even if they started very close together but on different sides of the surface. Krauskopf says that computer modeling of the Lorenz manifold reveals numerous applications. “For example, small changes in the concentration of calcium on either side of a cell membrane can lead to radically different actions.” Other applications include ensuring the optimal mixing of chemicals, important in pharmaceutical production, and the design of flight paths for space vehicles.

At about the same time that Lorenz was trying to predict the weather, Benoit Mandelbrot, PhD, emeritus professor of mathematics at Yale University, was studying variations in cotton prices. He noted that the pattern of peaks and troughs looked the same irrespective of the time frame. If he removed the time label he couldn’t tell if the charts referred to a week or a year. These so-called self-similar systems follow power laws, and implicit in this is that large changes are more common in these distributions than in classical Gaussian bell curves [J. Giles, *Nature*, vol. 432, pp. 266-267 (2004)].

Based on this insight, Mandelbrot used some simple equations to develop fractals. Magnifying one part of a fractal reveals the same pattern, and this applies even across several orders of magnitude. Fractal properties, which are linked to the dynamics of chaotic systems, are common in nature, from coastlines to the branches of the vasculature and lungs (Giles, 2004). Fractal geometry may offer a more accurate model of the pattern of electric activity in cardiac condition tissue, for example.

**Fractals and Homogeneity**

Furthermore, many traditional PK models assume homogeneity (See “Whole-Body Pharmacokinetic Modeling,” page 58). Conventional dissolution testing, for example, uses Fick’s laws of diffusion and assumes that a well-stirred medium mimics the gastrointestinal lumen. But gastrointestinal dissolution, release, transit, and uptake are heterogeneous and occur in different phases under variable conditions. There may be, for example, inadequate mixing around the site of absorption, which invalidates Fick’s laws. In such cases, fractal-like kinetics may be more appropriate and offer, for example, a better comparison of absorption from various formulations than conventional models [A. Dokoumetzidis et al., *Trends Pharmacol. Sci*., vol. 25, pp. 140-146 (2004)].

As a final example, drugs are distributed through the body via the arterial and venular trees, which are fractal networks. Modeling this fractal architecture helps build a more physiological model for drug distribution. Similarly, fractal kinetics help explain the anomalous pharmacokinetics of the antiarryhtmic drug amiodarone. Nonlinear dynamics aids mathematical modeling of the heart’s electrical activity, which may help explain the poor efficacy of some anti-arrhythmic drug. Nonlinear approaches also aid in the analysis of electroencephalogram data from studies of anticonvulsants

(Dokoumetzidis, 2004).

The pharmaceutical applications of chaos theory and fractal geometry are part of a move away from deterministic approaches to PK and PD, a trend further exemplified by researchers’ increasing interest in Bayesian statistics (See page 35 in the March 2004 issue of *Genomics & Proteomics*) and fuzzy pharmacology. “Fuzzy sets theory and stochastic processes, including traditional and Bayesian statistics, are all aimed at acknowledging uncertainty and intra- and inter-subject variability of PK/PD,” says Ivelina Gueorguieva, PhD, of the Centre for Applied Pharmacokinetic Research, University of Manchester, UK. “Although there are differences in the methodological approaches, their purpose is to move away from deterministic PK/PD prediction. Classical compartments are still useful, although there is a need for a more mechanistic approach to PK/PD analysis acknowledging uncertainty and variability.”

Conventional analyses take a binary view: a drug response is either good or bad. Fuzzy logic assumes that everything is a matter of degree. For example, assume the maximum drug response is 1. A patient with a 0.6 response has elements of good (0.6) and bad (0.4) responses. Using a fuzzy approach allows the transition between terms to be gradual. The binary options (good or bad) become the extreme ends of a continuum. Fuzzy logic differs from probability, which estimates, for example, the likelihood that a result is true (Sproule, 2002).

Fuzzy systems are more than academic mathematical abstractions. Fuzzy logic is an engineering mainstay in systems as diverse as washing machines, vacuum cleaners and subway trains. Fuzzy systems also control mechanical drug delivery devices, such as those administering neuromuscular blocking agents during surgery or insulin to critically ill diabetic patients. Fuzzy control titrates the dose based on individual response more accurately than conventional systems.

Moreover, initial studies of fuzzy logic in PK suggest that the approach may predict drug concentrations using fewer data points from larger populations than conventional analyses. Fuzzy simulations also help deal with uncertain model parameters. The latter are represented as fuzzy sets rather than means. Fuzzy logic’s ability to deal with uncertainty may allow companies to consider PK analyses very early in drug development. Initial studies applying fuzzy logic to PD suggest the approach can help optimize dose-response relationships, antibiotic combinations, and vaccination strategies. More work is needed, but the prospects are intriguing (Sproule, 2002).

**Impact of Chaos, Fractals**

Nevertheless, how forcibly chaos theory, fractal geometry, fuzzy logic, and other nonlinear approaches will impact on PK and PD remains to be seen. “Most PK/PD modeling will likely continue on the premise that there are myriad but consistent ‘rules of biology’ that govern biochemical, physiological, and pharmacologic processes and can be incorporated into deterministic and mechanistic PK/PD models,” says the University of Buffalo’s Jusko. “There may be occasions when complex [nonlinear] approaches will help to understand some aspect of the PK or PD of certain drugs, particularly when the investigator has extensive data and an unclear understanding of basic factors governing the system.”

Jusko says there are numerous examples of nonlinear PK arising from, for example, saturable metabolism, nonlinear binding and target-mediated disposition. “Thus, clinical studies must always be undertaken to ascertain whether the PK is linear or not and to learn how to deal with any nonlinearities.”

At the very least, it seems that nonlinear approaches will offer a valuable addition to the PK toolbox. At most, nonlinear approaches could catalyze a paradigm shift away from deterministic analysis. The impact depends, in part, on how willingly drug-development scientists embrace the new approach.

Bristol’s Krauskopf says that the level of mathematics for many applications of chaos theory is within the grasp of most practical scientists. Furthermore, several software tools are available. “It’s not that hard to manipulate the equations. The more difficult step is to change one’s way of thinking to include nonlinear effects. In most cases, this means taking a new view of the problem. One needs to be willing to embrace nonlinearity and invest in some mathematical background. But the returns can be truly amazing.”

Filed Under: Drug Discovery