Pharmacokinetic-pharmacodynamic equations

Dose-response models describe a cause-effect relationship. There are a wide range of mathematical models that have been used for this purpose. The complexity of a dose-response model can range from a simple one-parameter equation to complex multicompartment pharmacokinetic/pharmacodynamic models. Many dose-response models, including most cancer risk assessment models, are population models that predict the frequency of a disease in a population. Such dose-response models typically employ one or more frequency distributions as part of the equation. Dose-response may also operate at an individual level and predict the severity of a health outcome as a function of dose. Particularly complex dose-response models may model both severity of outcome and population variability, and perhaps even recognize the influence of multiple causal factors. 

Biomarker models that integrate pharmacokinetics, pharmacodynamics, and biomarkers are complex because they are based on sets of differential equations, parts of the models are nonlinear, and there are multiple levels of random effects. Therefore, advanced methods from numerical analysis and applied mathematics are needed to estimate these complex models. When the model is estimated, one seeks a model that is appropriate for its intended use (see Chapter 8). 

With the complexity of modern pharmacokinetic-pharmacodynamic models, analytical derivation of sensitivity indexes is rarely possible because rarely can these models be expressed as an equation. More often these models are written as a matrix of derivatives and the solution to finding the sensitivity index for these models requires a software package that can do symbolic differentiation of the Jacobian matrix. Hence, the current methodology for sensitivity analysis of complex models is empirical and done by systematically varying the model parameters one at a time and observing how the model outputs change. While easy to do, this approach cannot handle the case where there are interactions between model parameters. For example, two... 

Once the health-effect endpoint and data points describing the exposure concentration-duration relationship have been selected, the values are plotted and fit to a mathematical equation from which the AEGL values are developed. There may be issues regarding the placement of the exponential function in the equation describing the concentration-duration relationship (e.g., C x t = k vs C X t = k2 vs X E = k3>. It is clear that the exposure concentration-duration relationship for a given chemical is directly related to its pharmacokinetic and pharmacodynamic properties. Hence, the use and proper placement of an exponent or exponents to describe these properties quantitatively is highly complex and not completely understood for all materials of concern. 

W. Krzyzanski and W. J. Jusko, Note Caution in the use of empirical equations for pharmacodynamic indirect response models. J Pharmacokinet Biopharm 26 735-741 (1998). 

This section contains the differential equations that define the pharmacokinetics and the pharmacodynamics of the drug. Here, the effect is stimulatory on icdeg, which will result in a transient reduction of the biomarker. 

The choice of model should be based on biological, physiological, and pharmacokinetic plausibility. For example, compartmental models may be used because of their basis in theory and plausibility. It is easy to conceptualize that a drug that distributes more slowly into poorly perfused tissues than rapidly perfused tissues will show multi-phasic concentration-time profiles. Alternatively, the Emax equation, one of the most commonly used equations in pharmacodynamic modeling, can be developed based on mass balance principles and receptor binding kinetics. 

For some dmgs, we can link the parameters and equations of pharmacokinetics to those of pharmacodynamics, resulting in a PKPD model which can predict pharmacological effect over time. This concept is discussed in more detail later in this chapter. (Equation 17.7 is a typical PKPD equation.) Figure 17.3 depicts the relationship of effect versus time for the dmg albuterol (salbuta-mol) and contrasts this with a superimposed plot of plasma dmg concentration versus time. 

Poulin P. 2015b. Albumin and uptake of drugs in cells additional validation exercises of a recently published equation that quantifies the albumin-facUitated uptake mechanism(s) in physiologically based pharmacokinetic and pharmacodynamic modeling research. J Pharm Sci 104. doi 10.1002/jps.24676. 

The goal of pharmacokinetics is the quantitative description of drug entry, distribution, and elimination in the body. These processes are typically integrated into a mathematical model which uses a system of equations such as Eqs. (l)-(20) to calculate the systemic concentration due to a certain dose administered to the patient. If the pharmacodynamic characteristics of the drug can be clearly defined, the desired concentration at the target or in the plasma can be specified, and the pharmacokinetic model can be used to calculate the dose to attain the effective concentration. 

Relatively few PBPK models have been developed to describe the pharmacokinetics and pharmacodynamics of chemical warfare nerve agents. Maxwell et aV developed a PBPK-PD model for GD in the rat, describing the inhibition of AChE and carboxylesterase (CaE) in blood and tissues with mass balance equations based on parameters for blood flow, tissue volumes, GD metabolism and tissue/plasma partition coefficients. The resulting model gives accurate predictions of AChE activity in the blood and seven different tissues following intramuscular dosing with 90 pg GD kg bodyweight (BW), and was able to reproduce dose-response AChE inhibition from 10 to 100% in the brain. 

---