Pharmacodynamics Equations

This section details the equations that define and support the optional PBPKPlus™ model that Simulations Plus developed for modeling multiple PD effects for any drug record in a database. See:

For details about the internal standards that were used to define the equations in this section, see Equation Standards.
The PD effect can also be referred to as the pharmacological response, or just the response, and is abbreviated as E or R, respectively. These terms and abbreviations are used interchangeably in this chapter.

Direct Response PD Model Equations

Unless explicitly noted otherwise, “E” represents both a stimulated and an inhibited PD effect. If the PD effect is solely inhibition, then “I” is used.

Equation 3-1: Linear model, observed PD effect versus concentration

where:

Variable	Definition
	The PD effect.
	The total or unbound concentration of the drug in plasma.
	The slope of the plot of E versus Cp.

Equation 3-2: Linear model, observed PD effect versus concentration with baseline

Equation 3-3: Log Linear model

Equation 3-4: E_max model, PD effect is stimulated

Equation 3-5: E_max model, PD effect is inhibited

where:

Variable	Definition
	The PD effect.
	The baseline of the PD effect.
	The maximum PD response.
	The total or unbound concentration of the drug in plasma.
	The concentration of the drug at 50% of the maximum PD response.

Equation 3-6: Hill equation

where:

Variable	Definition
	The Hill parameter.
	The PD effect.
	The baseline of the PD effect.
	The maximum PD response.
	The total or unbound concentration of the drug in plasma.
	The concentration of the drug at 50% of the maximum PD response.

Indirect Response PD Model Equations

Equation 3-7: Drug transfer rate for the Effect Compartment model

where:

Variable	Definition
	The amount of drug in the plasma compartment.
	The amount of drug in the effect compartment.
	The drug transfer rate constant from the plasma compartment to the effect compartment.
	The drug transfer rate constant from the effect compartment to the plasma compartment.

Equation 3-8: Drug mass in the Effect Compartment model

where:

Variable	Definition
	The rate of distribution of drug from the plasma compartment to the effect compartment, referred to as the inter-compartmental distribution rate.
	The total or unbound concentration of drug in the plasma compartment.
	The total or unbound concentration of drug in the effect compartment.

Equation 3-9: Rate of change of PD effect variable in the absence of drug

Under steady state conditions, k_in = k_outR₀, where R₀ is the baseline PD effect.

Equation 3-10: Standard inhibitory equation

Equation 3-11: Change in PD effect variable over time, Class I indirect response model

Equation 3-12: Change in PD effect variable over time, Class II indirect response model

Equation 3-13: Standard stimulatory equation

Equation 3-14: Change in PD effect variable over time, Class III indirect response model

Equation 3-15: Change in PD effect variable over time, Class IV indirect response model

Equation 3-16: Rate of change in quantity of target cells (Cell killing model, phase non-specific)

where:

Variable	Definition
	The cell growth rate constant, which is the difference between the natural mitotic rate and physiologic degradation.
	The rate of the irreversible reaction.

Equation 3-17: Initial state of the system, BKG model

where:

Variable	Definition
	The initial number of bacteria.
	The percent of pre-existing antibiotic-resistant bacteria.
	The percent of bacterial cells that are in the resting state.
	Resting state of sub-population 1.
	Resting state of sub-population 2.

Equation 3-18: Transfer rate from S population to R population, BKG model

where:

Variable	Definition
	The maximum number of bacteria in the stationary phase.
	The growth rate constant for antibiotic-susceptible bacteria.
	The rate constant for bacterial natural death.
	The total number of bacteria across all sub-populations.

Equation 3-19: Effect of the drug on each bacterial sub-population, BKG model

where k_drug,mut is the growth rate constant for pre-existing antibiotic-resistant bacteria.

Equation 3-20: Effect of an antibiotic drug with Power model or Sigmoidal model

where:

Variable	Definition
	The drug concentration.
	The Hill factor in the drug-effect relationship.
	The maximum kill rate constant.
	The concentration of the drug that produces 50% of E_max.
	The shift in the concentration of the drug that is required for the drug to have the same effect on the mutant bacteria as it does on the antibiotic-susceptible bacteria,

Equation 3-21: Calculations for a precursor-dependent indirect response model

where:

Variable	Definition
	The amount of precursor.
	The zero order rate constant for precursor production.
	The first order rate constant for response production.
	The first order rate constant for loss of precursor.
	The first order rate constant for loss of response.
	The inhibition (Class VII) or stimulation (Class VIII) of precursor production.
	The inhibition (Class V) or stimulation (Class VI) of response production.

Equation 3-22: Incorporating a circadian rhythm in the baseline response, precursor-dependent indirect response model

where:

Variable	Definition
	The mean production of response rate.
	The amplitude of rate.
	The time of occurrence of the peak production rate.