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 |
We don't have a way to export this macro. | The PD effect. |
We don't have a way to export this macro. | The total or unbound concentration of the drug in plasma. |
We don't have a way to export this macro. | 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: Emax model, PD effect is stimulated
Equation 3-5: Emax model, PD effect is inhibited
where:
Variable | Definition |
We don't have a way to export this macro. | The PD effect. |
We don't have a way to export this macro. | The baseline of the PD effect. |
We don't have a way to export this macro. | The maximum PD response. |
We don't have a way to export this macro. | The total or unbound concentration of the drug in plasma. |
We don't have a way to export this macro. | The concentration of the drug at 50% of the maximum PD response. |
Equation 3-6: Hill equation
where:
Variable | Definition |
We don't have a way to export this macro. | The Hill parameter. |
We don't have a way to export this macro. | The PD effect. |
We don't have a way to export this macro. | The baseline of the PD effect. |
We don't have a way to export this macro. | The maximum PD response. |
We don't have a way to export this macro. | The total or unbound concentration of the drug in plasma. |
We don't have a way to export this macro. | 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 |
We don't have a way to export this macro. | The amount of drug in the plasma compartment. |
We don't have a way to export this macro. | The amount of drug in the effect compartment. |
We don't have a way to export this macro. | The drug transfer rate constant from the plasma compartment to the effect compartment. |
We don't have a way to export this macro. | 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 |
We don't have a way to export this macro. | The rate of distribution of drug from the plasma compartment to the effect compartment, referred to as the inter-compartmental distribution rate. |
We don't have a way to export this macro. | The total or unbound concentration of drug in the plasma compartment. |
We don't have a way to export this macro. | 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, kin = koutR0, 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 |
We don't have a way to export this macro. | The cell growth rate constant, which is the difference between the natural mitotic rate and physiologic degradation. |
We don't have a way to export this macro. | The rate of the irreversible reaction. |
Equation 3-17: Initial state of the system, BKG model
where:
Variable | Definition |
We don't have a way to export this macro. | The initial number of bacteria. |
We don't have a way to export this macro. | The percent of pre-existing antibiotic-resistant bacteria. |
We don't have a way to export this macro. | The percent of bacterial cells that are in the resting state. |
We don't have a way to export this macro. | Resting state of sub-population 1. |
We don't have a way to export this macro. | Resting state of sub-population 2. |
Equation 3-18: Transfer rate from S population to R population, BKG model
where:
Variable | Definition |
We don't have a way to export this macro. | The maximum number of bacteria in the stationary phase. |
We don't have a way to export this macro. | The growth rate constant for antibiotic-susceptible bacteria. |
We don't have a way to export this macro. | The rate constant for bacterial natural death. |
We don't have a way to export this macro. | The total number of bacteria across all sub-populations. |
Equation 3-19: Effect of the drug on each bacterial sub-population, BKG model
where kdrug,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 |
We don't have a way to export this macro. | The drug concentration. |
We don't have a way to export this macro. | The Hill factor in the drug-effect relationship. |
We don't have a way to export this macro. | The maximum kill rate constant. |
We don't have a way to export this macro. | The concentration of the drug that produces 50% of Emax. |
We don't have a way to export this macro. | 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 |
We don't have a way to export this macro. | The amount of precursor. |
We don't have a way to export this macro. | The zero order rate constant for precursor production. |
We don't have a way to export this macro. | The first order rate constant for response production. |
We don't have a way to export this macro. | The first order rate constant for loss of precursor. |
We don't have a way to export this macro. | The first order rate constant for loss of response. |
We don't have a way to export this macro. | The inhibition (Class VII) or stimulation (Class VIII) of precursor production. |
We don't have a way to export this macro. | 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 |
We don't have a way to export this macro. | The mean production of response rate. |
We don't have a way to export this macro. | The amplitude of rate. |
We don't have a way to export this macro. | The time of occurrence of the peak production rate. |