Developing Pharmacokinetic and Pharmacodynamic Models in Simulink

Developing Pharmacokinetic and Pharmacodynamic Models in Simulink

By Sam Roberts


It takes from 10 to 15 years to bring a new drug to market—at a cost rapidly approaching $1 billion. A key contributor to these staggering costs is late-stage failure: Of every 250 compounds that enter preclinical testing, only five proceed into clinical trials, and only one will be approved by the U.S. Food and Drug Administration (FDA). Many potential new drugs fail because researchers lack reliable information about their behavior.


With only a partial understanding of the link between dose and response, and knowing little about the drug’s concentration in the body over time, researchers often cannot guarantee a drug’s safety until too late in the development process.
The FDA’s 2004 Critical Path Report proposed, among other solutions, the increased use of model-based approaches to drug development—including pharmacokinetic and pharmacodynamic (PK/PD) modeling.
Pharmacokinetics focuses on how the body processes a drug, resulting in a drug concentration. Pharmacodynamics is concerned with how the drug acts on the body, resulting in a measurable drug effect. Through PK/PD modeling and simulation, pharmaceutical scientists acquire an earlier understanding of the link between drug and response, and can better characterize a drug’s absorption, distribution, and elimination properties.
PK/PD Modeling Approaches

PK/PD modelers typically work with a variety of software packages. They might use Microsoft Excel as a base file format and for simple data processing, a statistical package for more advanced exploratory data analysis and modeling, and specialized programs (often derived from academic projects) to solve differential equations, estimate parameters, and model nonlinear mixed effects. These programs often use a text-based input file and a specialized command language that is hard to learn and to communicate to the nonspecialist.
PK/PD modelers are increasingly using MATLAB and Simulink to overcome these difficulties. These products provide a unified environment for performing statistical analysis, visualizing data, and building models that represent subsystems as graphical blocks rather than as complex differential equations—making them easy for biologists to understand and use.
Implementing a PK/PD Model

Compare the processes of implementing a PK/PD model of a biological system in a traditional package and in Simulink. The model captures the behavior of a drug that exhibits saturable absorption kinetics: as the dose increases, the rate of drug elimination from the gastrointestinal tract (GIT) levels off to a saturated maximum.
The model in Figure 1 is captured in Microsoft PowerPoint, a familiar environment for many biologists. It describes the body in terms of three compartments: the GIT, the central compartment (containing the blood plasma and other highly perfused organs, such as the liver or kidney), and the peripheral compartment (containing less perfused organs, such as muscle and tissue). A set of three differential equations describes the rate of elimination from the GIT and the rates of change of drug concentration in the central and peripheral compartments.
Figure 1. PK/PD model of drug absorption and elimination, captured in Microsoft PowerPoint.
Images © LifeArt. Click on image to see enlarged view.


When implementing this model using traditional software packages, the researcher must create the differential equations that describe the rate of elimination from the GIT using an application-specific language and a text file (Figure 2). The text file must be fed into a specialized package that executes the simulation commands.
Figure 2. Differential equations describing the rate of elimination from the GIT. Click on image to see enlarged view.

The Simulink version of the model (Figure 3) represents the system in graphical blocks rather than text-based differential equations. Three interconnected blocks correspond to the three compartments of the system. These blocks mask subsystems: by selecting, for example, the peripheral compartment, the researcher can access a graphical representation of the internal details of that compartment. Because the Simulink model is as executable as a list of commands, there is no need to edit equations or program instructions.
Figure 3. Drug absorption and elimination modeled in Simulink. Images © LifeArt. Click on image to see enlarged view.

During simulation, a scope block automatically produces a time-course of the concentration of drug in the model compartments over a specified period. The time-course reveals a leveling off of concentrations in the central compartment. To obtain the same information using a traditional PK/PD modeling package would require a knowledge of the plotting commands in the software.
Example: Building a PK/PD Model in Simulink

We will model the transdermal input of nicotine from a nicotine patch applied to the skin for 16 hours. The nicotine patch has two layers. The first releases nicotine quickly and in high doses. The second gives a slower, controlled release of a lower dose. The single-compartment model is governed by the differential equation

dC/dt = Dfast + Dslow – Cl*C
where C is the plasma concentration, Cl is the clearance rate from the plasma, and Dfast and Dslow are the fast and slow doses of nicotine, respectively.

Modeling the Nicotine Doses

In Simulink we can complete the entire process of building and simulating the model without using a specialized command language. We open a new Simulink model and drag and drop a Signal Builder block from the Simulink Library Browser onto it. A graphical user interface (GUI) lets us set the properties of the signal. We interactively add lines to the Signal Builder GUI to create a multilevel signal that captures the fast and slow doses of nicotine.
Returning to the top-level model, we add an integrator block, an add/subtract block, and a multiplication block, linking all three to form a circuit that represents the differential equation (Figure 4). Finally, we add a scope block to display the simulation results—in this case, the plasma concentration of nicotine over time (Figure 5).
Figure 4. Building a Simulink model of a nicotine patch. Click on image to see enlarged view.
Figure 5. Scope block showing plasma concentrations of nicotine over time.


Estimating Model Parameters

The general form of the model is correct, but we need to estimate the correct values of the model parameters, such as the clearance rate, Cl. To do this we use an experimental data set derived from measurements of nicotine levels in a patch subject over time. To estimate parameters in a traditional package would require knowledge of optimization techniques­­—to most biologists a step change in the math that they need to understand. In Simulink we import this data into the model from an Excel file or another source and then use Simulink Parameter Estimation, a GUI that leads us through a choice of algorithms to find parameter values that optimize the fit of the model to the measured data (Figure 6).
Figure 6. Optimizing the fit of the model (yellow line) to the measured data (pink line).


Looking Ahead

Technologies are evolving to measure genes, proteins, and metabolites in the body on a systems level. PK/PD researchers will want to incorporate that knowledge into their models.
The need to model systems with an increased number of variables requires sophisticated new analysis and visualization techniques The MathWorks will support these developments with Simulink and with SimBiology, a new product that focuses on modeling and simulation of biochemical pathways at a systems level.
References

DiMasi, Joseph A, Ronald W. Hansen, and Henry G. Grabowski, "The price of innovation: new estimates of drug development costs," Journal of Health Economics 22 (2003): 151–185.
Pharmaceutical Research and Manufacturers of America, Pharmaceutical Industry Profile 2006, PhRMA, Washington, DC. March 2006.
Food and Drug Administration, Challenge and Opportunity on the Critical Path to New Medical Products, March 2004.

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