📒 Felmlee 2013

Contents

Mechanism-based pharmacodynamic modeling1

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Introduction

• diseases and both types of drug responses may emerge from perturbations of singular complex interconnected networks

Modeling Requirements

• Useful pharmacodynamic models are based on plausible mathematical and pharmacological exposure–response relationships
• relevant biological fluid (e.g., plasma, Cp) or the biophase (Ce)

Practical Modeling Approaches

• A good graphical analysis (along with a priori knowledge of drug mechanisms) may be used to narrow down the number of structural models
• good initial parameter estimates can reduce the likelihood of falling into local minima
• fitting a model to concentration–time profiles in relevant biological fluids
• Although simultaneous PK/PD modeling is desirable, this can still be a formidable challenge for complex models

Simple Direct Effect Models

• Emax model: The full Hill equation, or sigmoid Emax model, incorporates a curve-fitting parameter, γ, which describes the steepness of the concentration–effect relationship

$$E=E_{0} \pm \frac{E_{m a x} \times C_{\mathrm{p}}^{\gamma}}{\mathrm{EC}_{50}^{\gamma}+C_{\mathrm{p}}^{\gamma}}$$

• linear slope of the effect: $$m=\frac{E_{\max } \times \gamma}{4}$$

Biophase Distribution

• in vivo pharmacological effects will lag behind plasma drug concentrations: hysteresis, or a temporal disconnect in effect versus concentration plots

• drug effect through a hypothetical effect compartment

• The biophase model is only suitable for describing delayed responses due to drug distribution

Indirect Response Models

• The four basic models include inhibition of production (Model I) or dissipation (Model II) of response or stimulation of production (Model III) or dissipation of response (Model IV)

Model 1

$$\frac{\mathrm{d} R}{\mathrm{d} t}=k_{\mathrm{in}}\left(1-\frac{I_{\mathrm{max}} \times C_{\mathrm{p}}}{\mathrm{IC}_{50}+C_{\mathrm{p}}}\right)-k_{\mathrm{out}} \times R$$

Model 2

$$\frac{\mathrm{d} R}{\mathrm{d} t}=k_{\mathrm{in}}-k_{\mathrm{out}}\left(1-\frac{I_{\mathrm{max}} \times C_{\mathrm{p}}}{\mathrm{IC}_{50}+C_{\mathrm{p}}}\right) R$$

Model 3

$$\frac{\mathrm{d} R}{\mathrm{d} t}=k_{\mathrm{in}}\left(1+\frac{S_{m a x} \times C_{\mathrm{p}}}{\mathrm{SC}_{50}+C_{\mathrm{p}}}\right)-k_{\mathrm{out}} \times R$$

Model 4

$$\frac{\mathrm{d} R}{\mathrm{d} t}=k_{\mathrm{in}}-k_{\mathrm{out}}\left(1+\frac{S_{\mathrm{max}} \times C_{\mathrm{p}}}{\mathrm{SC}_{50}+C_{\mathrm{p}}}\right) R$$

• The basic indirect response models can be extended to incorporate a precursor compartment (P)

Signal Transduction Models

• a lag between drug concentration and observed effects owing to time-dependent signal transduction: delayed differential equations (DDEs)?
• rate of initial transit compartment (M1) $$\frac{\mathrm{d} M_{1}}{\mathrm{d} t}=\frac{1}{\tau}\left(\frac{E_{m a x} \times C_{\mathrm{p}}}{\mathrm{EC}_{50}+C_{\mathrm{p}}}-M_{1}\right)$$ For the ith compartment: $$\frac{\mathrm{d} M_{i}}{\mathrm{d} t}=\frac{1}{\tau}\left(M_{i-1}-M_{i}\right)$$
• e.g. Chemotherapy-induced myelosuppression

Irreversible Effect Models

$$\frac{\mathrm{d} R}{\mathrm{d} t}=-k \times C \times R$$

• This approach is only applicable for non-proliferating cell populations, but may be extended to incorporate cell growth $$\frac{\mathrm{d} R}{\mathrm{d} t}=k_{s} \times R-k \times C \times R$$
• The irreversible effect model can also be adapted to include the turnover or production and loss of a biomarker $$\frac{\mathrm{d} R}{\mathrm{d} t}=k_{\mathrm{in}}-k_{\mathrm{out}} R-k \times C \times R$$

More Complex Models

The time-course of paraoxon inactivation of in vitro whole blood cholinesterase (WBChE)

$$\frac{\mathrm{d} E_{\mathrm{A}}}{\mathrm{d} t}=-\left(\frac{k C_{\mathrm{PO}}}{\mathrm{EC}_{50, \mathrm{PO}}+C_{\mathrm{PO}}}\right) E_{\mathrm{A}}+k_{r} E_{\mathrm{I}}$$ $$\frac{\mathrm{d} E_{I}}{\mathrm{d} t}=\left(\frac{k C_{\mathrm{PO}}}{\mathrm{EC}_{50, \mathrm{PO}}+C_{\mathrm{PO}}}\right) E_{A}-\left(k_{r}+k_{\mathrm{age}}\right) E_{1}$$

• The reactivation of this in vitro system by PRX was modeled as an indirect response $$\frac{\mathrm{d} E_{\mathrm{A}}}{\mathrm{d} t}=-\left(\frac{k C_{\mathrm{PO}}}{\mathrm{EC}_{50, \mathrm{PO}}+C_{\mathrm{PO}}}\right) E_{\mathrm{A}}+k_{\mathrm{r}}\left(1+\frac{E_{\mathrm{max}} C_{\mathrm{PRX}}^{h}}{\mathrm{EC}_{50, \mathrm{PRX}}^{h}+C_{\mathrm{PRX}}^{h}}\right) E_{\mathrm{I}}$$ \begin{aligned} \frac{\mathrm{d} E_{\mathrm{I}}}{\mathrm{d} t}=\left(\frac{k C_{\mathrm{PO}}}{\mathrm{EC}_{50, \mathrm{PO}}+C_{\mathrm{PO}}}\right) & E_{\mathrm{A}}-k_{\mathrm{r}}\left(1+\frac{E_{\mathrm{max}} C_{\mathrm{PRX}}^{h}}{\mathrm{EC}_{50, \mathrm{PRX}}^{h}+C_{\mathrm{PRX}}^{h}}\right) E_{\mathrm{I}} \cr &-\left(k_{\mathrm{age}} E_{\mathrm{I}}\right) \end{aligned}
• the toxicodynamic biomarker, expiratory time (TE), was linked to apparent active enzyme (EA) according to the following nonlinear transfer function $$T_{\mathrm{E}}=T_{\mathrm{E}}^{0}+\frac{E_{\mathrm{max}, T_{\mathrm{E}}}\left(\frac{E_{0}}{E_{\mathrm{A}}}-1\right)^{n}}{E_{50}^{n}+\left(\frac{E_{0}}{E_{\mathrm{A}}}-1\right)^{n}}$$

Prospects

• The future of mechanism-based pharmacodynamic modeling for both therapeutic and adverse drug responses is promising for model-based drug development and therapeutics
• A diverse array of models is available with a minimal number of identifiable parameters to mimic mechanisms and the time-course of therapeutic and adverse drug effects
• Network-based systems pharmacology models have shown utility for understanding drug-induced adverse events (e.g. FBA, FVA)

1. Felmlee MA, Morris ME, Mager DE. Mechanism-based pharmacodynamic modeling. Methods Mol Biol. 2012;929:583–600. PMC3684160 ↩︎