Pharmacokinetics Equations

This section details the equations that define and support the PBPKPlus™ module that Simulations Plus developed for the simulation of physiologically-based pharmacokinetics (PBPK) of drug candidates. See:

For details about the internal standards that were used to define the equations in this section, see Equation Standards.

PBPK Tissue Model Equations

Equation 2-1: Perfusion-limited tissue model, non-eliminating tissue

Equation 2-2: Perfusion-limited tissue model, eliminating tissue with intrinsic clearance

Equation 2-3: Unbound intrinsic clearance rate calculated from the sum of all tissue enzyme kinetics

where C_tissue,u is the concentration of unbound drug in the current tissue.

Equation 2-4: Unbound intrinsic clearance converted to systemic clearance

where:

Variable	Definition
	The plasma and blood clearance rates, respectively.
	The blood/plasma drug concentration ratio.
	The unbound intrinsic clearance rate for the current tissue.
	The local tissue blood flow rate.
	The fraction of drug unbound in the plasma.

Equation 2-5: Permeability-limited tissue model, rate calculations

Equation 2-6: Permeability-limited tissue model (no lysosomes), partitioning calculations

where:

Variable	Definition
	The saturable clearance rate of drug calculated using Michaelis-Menten kinetic.
	The saturable, carrier-mediated transport rate of drug into the tissue using Michaelis-Menten kinetics.
	The saturable, carrier-mediated transport rate of drug out of the tissue using Michaelis-Menten kinetics.
	The number of enzymes specified for the tissue of interest.
	The number of influx transporters between the extracellular space and the intracellular space in the tissue compartment.
	The number of efflux transporters between the extracellular space and the intracellular space in the tissue compartment.
	The maximum velocity of the i^th transporter/enzyme.
	The Michaelis-Menten constant of the i^th transporter/ enzyme.
	The total concentration of drug in the extracellular space in the tissue compartment and the concentration of unbound drug in the extracellular space in the tissue compartment, respectively.
	The exchange of drug between the extracellular space and the intracellular space via a combination of passive processes and transporter-mediated processes.
	The (permeability*surface area) product for the tissue.
	The total concentration of drug in the intracellular space in the tissue compartment and the concentration of unbound drug in the intracellular space in the tissue compartment, respectively.
	The rate of clearance of the unbound drug in the intracellular space in the tissue compartment.
Note: All other variables have been defined previously for Figure 2-2.

Equation 2-7: Permeability-limited tissue model (w/ lysosomes) partitioning calculations

where:

Variable	Definition
	The concentration of neutral, unbound drug in the intracellular space in the tissue compartment.
	The concentration of neutral, unbound drug in the extracellular space in the tissue compartment.
	The concentration of neutral, unbound drug in the lysosomal compartment.
	The total volume of the lysosomal compartment.
	The volume fraction of lysosomes in tissue.
	The total concentration of drug in the lysosomal compartments.
Note: All other variables have been defined previously for Figure 2-2. and Equation 2-6.

PBPK Model Tissues Type Equations

Equation 2-8: Biliary clearance as a fraction of total liver clearance

where:

Variable	Definition
	The biliary clearance.
	The total liver clearance.
	The biliary clearance fraction, which is the fraction of total liver clearance that is the result of the extraction of the drug to bile.

Equation 2-9: Biliary clearance as the linear transport of drug from hepatocytes to bile

where PStc_apical is the permeability-surface area product for the apical membrane.

Equation 2-10: Biliary clearance as the non-linear transport of drug from hepatocytes to bile (perfusion-limited tissue model)

Equation 2-11: Biliary clearance as the non-linear transport of drug from hepatocytes to bile (permeability-limited tissue)

where:

Variable	Definition
	The biliary clearance.
	The total number of apical transporters that are involved in biliary clearance.
	The maximum velocity of the i^th transporter/enzyme.
	The Michaelis-Menten constant of the i^th transporter/enzyme.
	The concentration of unbound drug in the tissue.
	The concentration of unbound drug in the intracellular space in the tissue compartment.

Equation 2-12: Total amount of drug cleared by excretion (perfusion-limited tissue model)

Equation 2-13: Total amount of drug cleared by excretion (permeability-limited tissue model)

where:

Variable	Definition
	The total mass of the drug that is cleared by biliary excretion.
	The expression of the i^th transporter on the liver apical (canalicular) membrane.
	The maximum velocity of the i^th transporter/enzyme.
	The Michaelis-Menten constant of the i^th transporter/enzyme.
	The concentration of unbound drug in the tissue.
	The concentration of unbound drug in the intracellular space in the tissue compartment.
	The (permeability*surface area) product for the liver apical membrane.
	The total mass of the drug that is cleared by the liver.
	The biliary clearance fraction, which is the fraction of total liver clearance that is the result of the extraction of the drug to bile.

Equation 2-14: General calculation for kidney clearance

where:

Variable	Definition
	Kidney filtration clearance.
	Clearance due to active and passive secretion.
	Clearance due to active and passive reabsorption.
	Clearance due to metabolism.

Equation 2-15: Perfusion-limited GastroPlus® PBPK lung tissue model calculations

where:

Variable	Definition
	The total concentration of drug in the lung tissue, at time t.
	The total concentration of drug in lung tissue ELF, at time t.
	The saturable clearance rate of drug calculated using Michaelis-Menten kinetics. See Equation 2-3.

Equation 2-16: Hybrid perfusion-limited GastroPlus® PBPK lung tissue model calculations

where:

Variable	Definition
	The total concentration of drug in the lung tissue, at time t.
	The total concentration of drug in ELF, at time t.
	The process of drug exchange between tissue and ELF via a combination of passive processes and transporter-mediated processes.
	Represent saturable carrier-mediated transport of drug between the lung tissue and ELF calculated using Michaelis-Menten kinetics.
	The neutral, unbound concentration of drug in the tissue.
	The neutral unbound concentration of drug in ELF.
Note: All other variables have been defined previously for Equation 2-15 and Figure 2-6.

Equation 2-17: Permeability-limited GastroPlus® PBPK lung tissue model calculations

where:

Variable	Definition
	The saturable clearance rate of drug calculated using Michaelis-Menten kinetics.
	The total concentration of drug in the lysosomal compartments, at time t.
	The concentration of neutral, unbound drug in the intracellular space in the tissue compartment.
	The concentration of neutral, unbound drug in the lysosomal compartment.
	The total concentration of drug in ELF, at time t.
	The exchange of drug between tissue and ELF via a combination of passive processes and transporter-mediated processes.
	The concentration of unbound drug in ELF.
	The basolateral (permeability*surface area) product for passive diffusion between extracellular water and the intracellular space in the tissue compartment.
	The apical (permeability*surface area) product for passive diffusion between the intracellular space in the tissue compartment and ELF.
	The saturable, carrier-mediated transport of drug (into tissue and out of tissue, respectively) using Michaelis-Menten kinetics.
	The saturable, carrier-mediated transport of drug between the lung tissue and ELF calculated using Michaelis-Menten kinetics.

Equation 2-18: Default value for the fraction of drug unbound in ELF calculation

where:

Variable	Definition
	The fraction of neutral, unbound drug at the pH of plasma.
	The fraction of neutral, unbound drug at the pH of ELF.

Tissue Distribution Calculations

Equation 2-19: Fraction of drug unbound in tissue calculation

where:

Variable

Definition

The fraction of unbound drug in plasma.

The ratio of albumin concentration in tissue to the albumin concentration in plasma.

Note: When using Poulin’s method of tissue:plasma partition coefficients, RAt:p = 0.15 for adipose and RAt:p = 0.50 for all other tissues.

Equation 2-20: Calculation of tissue:plasma partition coefficients based on the Poulin homogeneous method

where:

Variable	Definition
	The fraction of unbound drug in tissue.
	The fraction of unbound drug in plasma.
	The solvent/water partition coefficient or distribution coefficient for the drug, where: For adipose and yellow marrow, K = the distribution coefficient at pH = 7.4 in vegetable oil and water (logD_vo/w). For all other normal tissue, K = the partition coefficient in octanol and water (logP_oct/w).
	The volume fraction of neutral lipids in tissue.
	The volume fraction of neutral phospholipids in tissue.
	The volume fraction of total water in tissue.
	The volume fraction of neutral lipids in plasma.
	The volume fraction of neutral phospholipids in plasma.
	The volume fraction of total water in plasma.

Equation 2-21: Default GastroPlus® conversion of logP_oct/w to logD_vo/w

Equation 2-22: Kp calculation for perfusion-limited tissues

Equation 2-23: Kp calculation for permeability-limited tissues

where:

Variable	Definition
	The volume of the extracellular space as a fraction of the total tissue volume.
	The fraction of unbound drug in tissue.
	The fraction of unbound drug in plasma.

Equation 2-24: Berezhkovskiy method of Kp calculation (modified Poulin calculation)

Equation 2-25: Kpu calculation with acidic phospholipid binding for compounds with a base pKa > 7

where:

Variable	Definition
	The volume fraction of extracellular water in individual tissues.
	The volume fraction of intracellular water in individual tissues.
	The solvent/water partition coefficient for the drug, where: For adipose and yellow marrow, K = the partition coefficient in vegetable oil and water (logPvo/w). For all other normal tissues, K = the partition coefficient in octanol and water (logPoct/w).
	The volume fraction of neutral lipids in individual tissues.
	The volume fraction of neutral phospholipids in individual tissues.
	The fraction of neutral drug in plasma (pH = 7.4).
	The association constant of a basic compound with acidic phospholipids. See Equation 2-27.
	The concentration of acidic phospholipids in individual tissues.
	The fraction of neutral drug in the intracellular space (water) of individual tissues (pH = 7).

Equation 2-26: Calculation of Kpu_{RBC/plasma water}

where:

Variable	Definition
	The hematocrit.
	The blood/plasma drug concentration ratio.
	The fraction of unbound drug in plasma.

Equation 2-27: Calculation of Ka from Kpu_{RBC/plasma water}

where:

Variable	Definition
	The fraction of neutral drug in the intracellular space of red blood cells (pH = 7.22).
	The fraction of neutral drug in plasma (pH = 7.4).
	The volume fraction of intracellular water in red blood cells.
	The volume fraction of neutral lipids in red blood cells.
	The volume fraction of neutral phospholipids in red blood cells.
	The concentration of acidic phospholipids in red blood cells.
	The solvent/water partition coefficient for the drug, where: For adipose and yellow marrow, K = the partition coefficient in vegetable oil and water (logP_vo/w). For all other normal tissues, K = the partition coefficient in octanol and water (logP_oct/w).

Equation 2-28: Kpu calculation with tissue protein binding for compounds with all base pKas <7

where:

Variable	Definition
	The fraction of neutral drug in the intracellular space (water) of individual tissues (pH = 7).
	The fraction of neutral drug in plasma (pH = 7.4).
	The volume fraction of intracellular water in individual tissues.
	The volume fraction of extracellular water in individual tissues.
	The volume fraction of neutral lipids in individual tissues.
	The volume fraction of neutral phospholipids in individual tissues.
	The fraction of unbound drug in plasma.
	The volume fraction of neutral lipids in plasma.
	The volume fraction of neutral phospholipids in plasma.
	For ionizable drug compounds: The ratio of albumin concentration in tissue to the albumin concentration in plasma. For non-ionizable drug compounds: The ratio of lipoprotein concentration in tissue to the lipoprotein concentration in plasma.
	The solvent/water partition coefficient for the drug, where: For adipose and yellow marrow, K = the partition coefficient in vegetable oil and water (logPvo/w). For all other normal tissues, K = the partition coefficient in octanol and water (logPoct/w).

Equation 2-29: Final Kp calculation for all compounds

Equation 2-30: Lukacova combined equation for Kp calculation

where:

Variable	Definition
	The volume fraction of extracellular water in individual tissues.
	The fraction of neutral drug in the intracellular space (water) of individual tissues (pH = 7).
	The fraction of neutral drug in plasma (pH = 7.4).
	The solvent/water partition coefficient for the drug, where: For adipose and yellow marrow, K = the partition coefficient in vegetable oil and water (logPvo/w). For all other normal tissues, K = the partition coefficient in octanol and water (logPoct/w).
	The volume fraction of intracellular water in individual tissues.
	The volume fraction of neutral lipids in individual tissues.
	The volume fraction of neutral phospholipids in individual tissues.
	The fraction of drug without positive charge in plasma, where the zwitterions with total charge = 0 are not included in this fraction because they contain positive charge that would be associating with the acidic phospholipids.
	The volume fraction of neutral lipids in plasma.
	The volume fraction of neutral phospholipids in plasma.
	For ionizable drug compounds: The ratio of albumin concentration in tissue to the albumin concentration in plasma. For non-ionizable drug compounds: The ratio of lipoprotein concentration in tissue to the lipoprotein concentration in plasma.
	The fraction of drug with positive charge in plasma, including cations or zwitterions with at least one positive charge on the molecule, regardless of the total charge of the molecule.
	The association constant of a basic compound with acidic phospholipids.
	The concentration of acidic phospholipids in individual tissues.

Equation 2-31: NPI calculation

where:

Variable	Definition
	The normalized error of prediction using the original two Rodgers and Rowland equations.
	The normalized error of prediction using the Lukacova combined equation.

Equation 2-32: Normalized error of prediction calculation

where:

Variable	Definition
	The predicted volume of distribution.
	The experimental volume of distribution.

Equation 2-33: Lukacova method with lysosomes

where:

Variable	Definition
	The fraction of neutral drug in lysosomes.
	The fraction of positive drug in lysosomes.
	The fraction of positive drug in intracellular water.
	The volume fraction of lysosomes in tissue.
Note: All other variables are as previously defined. See Equation 2-30.

Equation 2-34: Fraction unbound in plasma for Kp calculations

where:

Variable	Definition
	The adjusted fraction of unbound drug in plasma that is used in Kp calculations.
	The experimental f_upvalue.
	The octanol/water distribution coefficient at pH=7.4, which is the pH of plasma.
	The volume fraction of total neutral lipid and total neutral phospholipid in plasma.
	The volume fraction of total water in plasma.

Equation 2-35: f_ut calculation for perfusion-limited tissue

Equation 2-36: f_u(ext) calculation for permeability-limited tissue

where:

Variable	Definition
	The fraction of neutral drug in plasma (pH = 7.4).
	The fraction of unbound drug in plasma.
	The tissue/plasma partition coefficient.
	The fraction of neutral drug in intracellular water (pH = 7) for a perfusion-limited tissue.
	The fraction of neutral drug in extracellular water (pH = 7.4) for a permeability-limited tissue.

Equation 2-37: f_u(int) calculation for permeability-limited tissue

where:

Variable	Definition
	The fraction of unbound drug in the intracellular space (water) of the tissue.
	The fraction of cationic drug in plasma (pH = 7.4).
	The fraction of cationic drug in intracellular space (water) of the tissue (pH = 7.0).
Note: For the remainder of the variable definitions, see Equation 2-30.

Equation 2-38: f_ut in lysosomal compartments, permeability-limited tissue

where:

Variable	Definition
	The fraction of cationic drug in plasma (pH = 7.4).
	The fraction of cationic drug in the lysosomal compartments.
	The concentration of acidic phospholipids in individual tissues.

Equation 2-39: Total concentration of drug entering a perfusion-limited tissue

where:

Variable	Definition
	The concentration of non-ionized, unbound drug in extracellular water.
	The concentration of non-ionized, unbound drug in intracellular water.
	The concentration of ionized, unbound drug in extracellular water.
	The concentration of ionized, unbound drug in intracellular water.
	The concentration of neutral or anionic drug bound to tissue albumin in the extracellular space.
	The concentration of cationic drug bound to acidic phospholipids in the intracellular space.
	The concentration of neutral or anionic drug bound to neutral lipids in the intracellular space.
	The concentration of neutral or anionic drug bound to neutral phospholipids in the intracellular space.

Equation 2-40: Non-ionized, unbound concentrations of drug in extracellular and intracellular water

Equation 2-41: Equilibria between non-ionized and ionized unbound concentrations of drug in extracellular and intracellular spaces

Equation 2-42: Fraction of unbound drug in the extracellular space

Equation 2-43: Cationic fraction of drug bound to acidic phospholipids

Equation 2-44: Neutral fraction of drug bound to neutral lipids

where, for simplicity in upcoming calculations, this value is abbreviated as f_nl.

Equation 2-45: Neutral fraction of drug bound to neutral phospholipids

where, for simplicity in upcoming calculations, this value is abbreviated as f_npl.

Equation 2-46: Total fraction of unbound drug in the tissue

Equation 2-47: Ratio of the drug concentrations in the extracellular region and intracellular region to the total drug concentration

Equation 2-48: Total drug concentration in tissue

Equation 2-49: Total concentration of drug in the intracellular and extracellular spaces

Equation 2-50: Unbound, un-ionized concentration of drug in the intracellular and extracellular spaces

Equation 2-51: Total, unbound intracellular concentration of drug (neutral and ionized forms)

Equation 2-52: Total fraction of unbound drug in the intracellular space

Kp Calculation Variable Assumptions

Equation 2-53: Steady state volume distribution calculation

where:

Variable	Definition
	The total volume of plasma in the body.
	The total volume of erythrocytes in the body.
	The ratio of the total concentration of the drug in erythrocytes to the total concentration of the drug in plasma, calculated from the blood/ plasma drug concentration ratio and the hematocrit.
	The volume of a given tissue compartment.
	The tissue/plasma partition coefficient.
	The extraction ratio for a given tissue.

Equation 2-54: McNamara’s equation for infant physiologies

where:

Variable	Definition
	The fraction of drug unbound in pediatric plasma.
	The fraction of drug unbound in adult plasma.
	The protein concentration in pediatric plasma.
	The protein concentration in adult plasma.

Equation 2-55: Infant R_b:p scaling

where:

Variable	Definition
	The drug blood/plasma concentration ratio as indicated for infants (ped) or adults (adult).
	The hematocrit, expressed as a fraction, in pediatric blood.
	The hematocrit, expressed as a fraction, in adult blood.

Equation 2-56: Homeostasis of amniotic fluid

where:

Variable	Definition
	The fetal urinary rate.
	The secretion rate from fetal tissue.
	The fetal swallowing rate.
	The active portion of the intra-membranous pathway.

Equation 2-57: Fetal pathway for perfusion-limited tissue (amniotic fluid)

where:

Variable	Definition
	The total volume of the amniotic fluid.
	The total concentration of drug in the amniotic fluid.
	The concentration of unbound drug in the amniotic fluid.
	The total concentration of drug in the fetal tissue.
	The intra-membranous fluid transfer rate constant (defined in the amniotic fluid compartment) representing osmosis.
	The trans-membranous fluid transfer rate constant between the uterus and the amniotic fluid.
	The fetal tissue/plasma partition coefficient.
	The fraction of unbound drug in the fetal plasma.
	The total concentration of drug on the fetal side of the placenta.
	The tissue/plasma partition coefficient on the fetal side of the placenta.
	The fetal filtration clearance rate.
	The secretion rate from fetal tissue.
	The fetal swallowing rate.
	The total concentration of drug in the uterine tissue.
	The uterine tissue/plasma partition coefficient.
	The general clearance rate for the fetal tissue.

Unless otherwise noted, all variables are defined for the amniotic fluid or fetal tissue.

Equation 2-58: Fetal pathway for perfusion-limited tissue (fetal tissue)

where:

Variable	Definition
	The total volume of the fetal tissue.
	The fetal blood flow.
	The total concentration of drug in arterial (in) blood in fetal tissue.
	The fetal blood/plasma drug concentration ratio.
	The fetal tissue/plasma partition coefficient.
	The intra-membranous fluid transfer rate constant (defined in the amniotic fluid compartment) representing osmosis and vesicular routes, respectively.
	The concentration of unbound drug in the amniotic fluid.
	The total concentration of drug in the fetal tissue.
	The fraction of unbound drug in the fetal plasma.
	The fetal filtration clearance rate.
	The secretion rate from fetal tissue.
	The fetal swallowing rate.
	The general clearance rate for the fetal tissue.

For the equations that describe the modeling of the fetal and maternal placenta tissue, the variables were defined according to the following rules:

Variable	Definition
	Represents concentration, amount (mass) and volume, respectively.
	The total concentration of drug in arterial (in) blood and venous (out) blood in tissue.
	Represents fetal, maternal, and total (across both fetal and maternal), respectively.
	The extracellular fluid to plasma partition coefficient.
	The (permeability*surface area) product for the tissue, which can be either on the apical side (api) or the basolateral side (baso).
	The tissue/plasma partition coefficient between maternal tissue and maternal plasma.
	The tissue/plasma partition coefficient between fetal tissue and maternal plasma.
	The tissue/plasma partition coefficient between fetal tissue and fetal plasma.
	The tissue/plasma partition coefficient between total placenta tissue (maternal and fetal) and maternal plasma.
	The total amount (mass) of drug in the maternal placenta and the fetal placenta, respectively.
	The maternal blood/plasma drug concentration ratio and the fetal blood/plasma concentration ratio, respectively.
	The tissue blood flow from the maternal side and the fetal side, respectively.
	The total maternal placenta tissue volume and the total fetal placenta tissue volume, respectively.
	Represents total, unbound, extracellular space, intracellular space, and the plasma compartment in the tissue, respectively.

Equation 2-59: Mass balance in perfusion-limited placenta tissue

Equation 2-60: Total concentration in the placenta (maternal + fetal)

Equation 2-61: Partition coefficient for total placenta tissue and maternal plasma

Equation 2-62: General equation that describes the change in the amount of drug in the total perfusion-limited placenta tissue (maternal placenta + fetal tissue)

where:

Variable	Definition
	The concentration of unbound drug in the amniotic fluid.
	The total concentration of drug in the fetal side of the placenta.
	The tissue/plasma partition coefficient on the fetal side of the placenta.
	The fraction of unbound drug in the fetal tissue.

Equation 2-63: Change in the amount of drug in the fetal placenta

Equation 2-64: Initial equation for the change in the amount of drug in maternal placenta

Equation 2-65: Fetal tissue placenta concentration

Equation 2-66: Final equation for the change in the amount of drug in the maternal placenta

Equation 2-67: Permeability-limited fetal placenta calculations

Equation 2-68: Permeability-limited maternal placenta calculations

where:

Variable	Definition
	The saturable clearance rate of drug calculated using Michaelis-Menten kinetics.
	The saturable, carrier-mediated transport rate of drug into the tissue using Michaelis-Menten kinetics.
	The saturable, carrier-mediated transport rate of drug out of the tissue using Michaelis-Menten kinetics.