ACAT Model Equations

This section details the equations that define and support the Advanced Compartmental Absorption and Transit (ACAT) model developed by Simulations Plus, which extends the capabilities of the pre-cursor Compartmental Absorption and Transit (CAT) model to include in vivo drug behavior of first-pass metabolism and colonic absorption. See:

Dynamic Fluid Volume Model Equations

Fed State Physiologies Equations

Solubility and Dissolution Models Equations

Precipitation Models Equations

Absorption Models Equations

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

Dynamic Fluid Volume Model Equations

Equation 1-1: Change in fluid volume versus time

where:

Variable	Definition
	The volume of the fluid in the j^thintestinal compartment.
	The volume of the fluid in the previous intestinal compartment (i.e. j-1).
	Time.
	The water secretion rate(s) in the j^thintestinal compartment.
	The water absorption rate(s) in the j^thintestinal compartment.
	The transit rate of the j^th intestinal compartment.
	The transit rate of the previous intestinal compartment (i.e. j-1).

Equation 1-2: Change in drug concentration due to fluid volume versus time

where:

Variable	Definition
	Time.
	The total concentration of the drug in the j^th intestinal compartment at time t.
	The volume of fluid in the j^th intestinal compartment.
	The mass of drug in the j^th intestinal compartment.

Equation 1-3: Secretion rate for a new Dynamic Fluid Volume model

where:

Variable	Definition
	The volume of the fluid in the j^th intestinal compartment.
	The volume of the fluid in the previous intestinal compartment (i.e. j-1).
	Time.
	The new secretion rate in the j^th intestinal compartment.
	The gut compartment.
	The water absorption rate(s) in the j^th intestinal compartment.
	The transit rate of the j^th intestinal compartment.
	The new fluid volume in the j^th intestinal compartment.
	The transit rate of the previous intestinal compartment (i.e. j-1).
	The new fluid volume in the previous intestinal compartment (i.e. j-1).

Fed State Physiologies Equations

Equation 1-4: Second-order polynomial for the prediction of bile salt concentration in the intestinal lumen based on % fat in a meal

Table 1-1: Polynomial coefficients for Equation 1-4

	Duodenum	Jejunum1	Jenjunum2	Ileum1	Ileum2	Ileum3
A2	2.131E-04	-3.716E-04	-1.237E-03	-1.217E-03	-2.631E-03	-7.747E-05
A1	2.957E-01	2.746E-01	2.863E-01	2.177E-01	2.677E-01	1.934E-02
A0	5.379E+00	4.117E+00	2.985E+00	1.844E+00	3.265E-01	2.196E-01

Solubility and Dissolution Models Equations

Equation 1-5: Difference between aqueous and in vivo solubility caused by the presence of bile salts

where:

Variable	Definition
	The aqueous solubility at a given pH.
	The aqueous solubilization capacity calculated as the ratio of moles of drug to moles of water at a concentration that is equal to aqueous solubility.
	The drug molecular weight.
	The bile salts concentration.
	The solubility in the presence of bile salts at concentration [bile] and at the same pH as C_s,aq.
	The bile salt solubilization ratio, where two options are available for estimation of this value. in vitro SR can be obtained from in vitro solubilities measured in biorelevant media with known concentration of bile salts using Equation 1-5. Theoretical predication based on logP ⁵ . See Equation 1-6.

Equation 1-6: Theoretical prediction of SR based on logP

Equation 1-7: Apparent Solubility of Drug in Presence of Surfactant

Note that if , then .

Equation 1-8: Cyclodextrin binding constant

where:

Variable	Definition
	The concentration of cyclodextrin-drug complex.
	The concentration of free, dissolved drug.
	The concentration of free cyclodextrin.

Equation 1-9: Default model equation for dissolution rate

Equation 1-10: Wang-Flanagan model equation for dissolution rate

Equation 1-11: Z-Factor model equation for dissolution rate

where:

Variable	Definition
Note: All symbols in all three model equations represent the same properties or quantities.
	The amount of drug that is dissolved.
	The effective diffusion coefficient.
	The spherical particle radius at the current time.
	The shape factor that accounts for non-spherical shape of the particles and is specified as the ratio of length and diameter of the particle. Note: For spherical particles, S =1.
	The solubility at the pH of the corresponding j^thintestinal compartment.
	The total concentration of drug in the lumen of the j^th intestinal compartment at time t.
	The amount of drug that is undissolved at time = t.
	The amount of drug that is undissolved at time =0.
	The density of the drug.
	The thickness of the diffusion layer.

Equation 1-12: Term represented by the z factor

Equation 1-13: Effective diffusion coefficient calculation

where:

Variable	Definition
	The aqueous diffusion coefficient of drug monomers (free dissolved drug).
	The fraction of free, dissolved drug calculated as the ratio of aqueous solubility and solubility in the presence of bile salts.
	The diffusion coefficient of the bile salt-phospholipid-drug aggregate.

Equation 1-14: Probability density function for the Legacy Log-Normal distribution

where:

Variable

Definition

The mean radius (μ) in log units as calculated from the user-specified value for the radius in microns.

The mean standard deviation in log units as calculated from the user-specified value of standard deviation (σ) in microns.

Equation 1-15: Lower radius limit, Legacy Log-Normal distribution

Equation 1-16: Upper radius limit, Legacy Log-Normal distribution

Equation 1-17: Probability density function for the Normal distribution

where:

Variable	Definition
	The mean radius in microns.
	The mean standard deviation in microns.

Equation 1-18: Lower radius limit, Normal distribution

Equation 1-19: Upper radius limit, Normal distribution

Equation 1-20: Adjusting the particle radius to the remaining undissolved amount

where:

Variable	Definition
	The particle radius of bin k at time t.
	The particle radius of bin k at time 0.
	The total undissolved amount in bin k (across all compartments) at time t.
	The total undissolved amount in bin k (across all compartments) at time 0.

Equation 1-21: Kelvin equation

where:

Variable	Definition
	The solubility of drug particle of radius r at temperature T.
	The solubility with no curvature at temperature T.
	The interfacial surface tension.
	The molar volume of drug calculated as the ratio of molecular weight and density.
	The gas constant.
	The drug particle radius.

Equation 1-22: Solubility with no curvature at temperature T

Equation 1-23: Calculation of K in Equation 1-22

where:

Variable	Definition
	The reference solubility measured at temperature T_ref.
	The melting point of the drug.
	The currently specified temperature.
	The molecular weight of the drug.

Equation 1-24: Nanofactor Effect calculation

where:

Variable	Definition
	The drug particle radius.
	The NanoFactor, which is a user-defined, compound-specific parameter, with the same value being applicable across formulations with differing sizes of drug particles.

Equation 1-25: Fraction of Unbound Drug in Solution

where C_u(t) is the unbound drug concentration at all times and C(t) the total drug concentration at all times. Equilibration is assumed to be instantaneous and so f_u is anticipated to be constant. The affinity to surfactants k_aff, is experimentally determined to apparent solubility vs surfactant concentration (C_surf) in each dissolution medium (See Effect of surfactants on drug solubility).

Equation 1-26: Dissolution Rate Calculation

Where A(t) (m²) is the surface area of the particle at time t, D_u and D_b the diffusion coefficient of the unbound drug and micelle-bound drug respectively (m².s^-1), h_u and h_b the UWL thickness of the unbound drug and micelle-bound drug respectively (m), C_S,u is the unbound drug solubility at the surface of the drug crystal and C_u, the unbound drug concentration in the bulk (kg.m^-3). This equation sums up the flux of drug dissolving from the crystal surface as free and micelle bound drug.

Equation 1-27: Dissolution Rate Expanded Equation

Equation 1-28: Characteristic Length Scale for Diffusion

Equation 1-29: Stokes-Einstein Equation for Diffusion Coefficient

Where k = 1.3806504 10^-23 (J.K^-1) is the Boltzmann constant, T is the absolute temperature in kelvin, η (Pa.s) is the dynamic viscosity of the solvent and r_h (m) is the hydrodynamic radius of the diffusing solute. The hydrodynamic radius of the solute can be estimated, assuming that the molecular shape is a sphere and that the hydration of the solute is negligible, by the following equation:

Equation 1-30: Calculation of Diffusion Coefficient in Water

Where M_W is the molecular weight of the drug (g.mol^-1), N_A = 6.02214179 10²³ (mol^-1) is the Avogadro number and ρ_S is the drug true density in (kg.m^-3). The effect of temperature for dissolution is accounted for by a change in the water viscosity.

Equation 1-31: Solubility of Drug Particles as a Function of Particle Radius

Where C_r stands for the solubility of a drug particle of radius r (m). C_∞is the solubility of the same drug with no curvature (or an infinite radius of curvature) at temperature T. Units for C_∞ and C_rare the same relevant concentration units. R_g = 8.314472 J.K^-1.mol^-1 is the ideal gas constant, T (K) is the absolute temperature, (J.m^-2) is the interfacial tension between the solid and the liquid and V_molar (m³.mol^-1) is the molar volume of the drug.

Equation 1-32: Calculation of Number of Particles per bin

Where ρ_S (kg.m^-3) is the material true density assuming a spherical particle size. At all times, the UWL thickness for the dissolving particles and the unbound drug is given by h_u(t)= r(t) if r(t)<30 µm or h_u(t)=30 µm if r(t) ≥ 30 µm. In each particle bin the mass of particles is reduced by integrating equation 9 over time.

Equation 1-33: Surface Area of Dissolving Particles

Equation 1-34: Rate of Change of Solid Mass During Dissolution and Precipitation

Equation 1-35: First-Order Drug Degradation Rate Equation

Precipitation Models Equations

Equation 1-36: First Order Precipitation model equation to predict precipitation rate in the jth intestinal compartment

where:

Variable	Definition
	The volume of the lumen in the j^thintestinal compartment.
	The precipitation rate constant in the j^th intestinal compartment.
	The precipitation time in the j^th intestinal compartment. Note: You can define compartment-specific precipitation times through a precipitation time versus pH profile.
	The concentration of dissolved drug in the j^th intestinal compartment.
	The drug solubility in the j^th intestinal compartment.

Equation 1-37: Sugano’s CLNT model - Differential equation for nucleation

where:

Variable	Definition
	A fitted lump constant of various factors as shown in Equation 1-38.
	The diffusion coefficient of a free monomer drug.
	Avogadro’s number.
	The concentration of a free monomer drug in aqueous solution and not in bile micelles.
	The solubility of the precipitant in water without bile micelles.
	The interfacial surface tension.
	The Boltzman constant.
	The temperature.
	The molecular volume.

Equation 1-38: Sugano’s CLNT model - Particle radius calculation and fitted lump sum of various factors for the calculation of β

where:

Variable	Definition
	The amount of particle growth (weight or mole) that is generated during a time period, with i representing a particle group that is generated at a different time point.
	The particle radius at time t.
	The density of the drug.
	The effective diffusion coefficient in a bile micelle media.
	The diffusion layer thickness.
	The sum of the concentrations of free monomer and the bile micelle-bound molecule.
	The solubility in the bile micelle media.

Equation 1-39: Simulation Plus’s CLNT model

where:

Variable	Definition
	Assuming spherical geometry, the critical radius of the growing cluster. See Equation 1-40.
	The crystallization parameter, which is a correction factor to account for surface integration processes.
Note: All other variables have been defined previously. See Equation 1-37 and Equation 1-38.

Equation 1-40: Critical radius calculation for use in Equation 1-39

Equation 1-41: Formation rate of new nuclei with the optimizable ECF

Equation 1-42: Fraczkiewicz-Model V2 equation

Absorption Models Equations

Equation 1-43: Effective permeability calculation

where:

Variable	Definition
	The volumetric flow rate.
	The total concentration of drug in the input port.
	The total concentration of drug in the output port.
	The radius of the perfused section.*
	The length of the perfused section.*
Note: *For the experiments described here, values are constant: r = 1.75 cm. L = 10 cm.

Equation 1-44: Absorption rate coefficient calculation

Equation 1-45: Absorption rate calculation for passive transcellular absorption

where:

Variable	Definition
	Indicates a particular intestinal compartment.
	The rate of absorption in the j^th intestinal compartment.
	The volume of the lumen in the j^thintestinal compartment.
	The absorption rate coefficient for the j^th intestinal compartment.
	The total concentration of drug in the lumen for the j^thintestinal compartment.
	The concentration of unbound drug in the enterocyte sub-compartment of the j^th intestinal compartment.

GastroPlus absorption models assume that only dissolved drug is subject to absorption.

Equation 1-46: ASF (small intestine), logD model

where:

Variable	Definition
	The surface area to volume ratio, where A =1.2/r and where r = the compartment radius.
	Fitting constants for the purposes of weighting the ASFs to account for aspects that are not considered in this derivation such as active transport and physiological changes in the small intestine, where:
	Value = 6.25. A fitting constant that avoids the singular condition.
	P_eff at pH 6.5.
	The intrinsic (neutral) P_eff of the compound.

Equation 1-47: ASF (colon), logD model

where:

Variable

Definition

The surface area to volume ratio, where A =4/r and where r = the compartment radius.

Note: This calculation is also applicable for the caecum.

Fitting constants that regulate the degree to which logD can influence ASF. If C3 and C4 were both equal to 1.0, then when logD is positive, ASF would be greater than 1.0. Conversely, when logD is negative, then ASF would be less than 1.0, which accounts for the decreased colonic absorption of polar compounds as described in Ungell’s studies ⁷ .

Equation 1-48: ASF (small intestine), Opt logD model

where:

Variable	Description
	Value = 6.25. A fitting constant that avoids the singular condition.
	A fitting constant that determines the rate of change, or steepness, of the ASF gradient.
	A fitting constant that influences the baseline value of the ASFs.
	The surface area to volume ratio, where = 1.2/r for the small intestine. 4/r for the caecum and the colon. and r = the compartment radius.

The ASF calculation for the caecum and the colon using the Opt logD model is identical to that for the logD model. See Equation 1-35.

Equation 1-49: ASF calculation (small intestine), Opt logD SA/V 6.1 model

Equation 1-50: ASF calculation (caecum, colon), Opt logD SA/V 6.1 model

Equation 1-51: ASF calculation (all compartments), Theoretical model

Equation 1-52: ASF calculation (all compartments), Theoretical SA/V model

Equation 1-53: Paracellular absorption rate prediction

where:

Variable	Definition
	Indicates a particular intestinal compartment.
	The paracellular absorption rate coefficient for the j^th intestinal compartment.
	The volume of the lumen in j^th intestinal compartment.
	The total drug concentration in the lumen for the j^th intestinal compartment.
	The concentration of unbound drug in the portal vein.

Equation 1-54: Paracellular absorption rate coefficient

where:

Variable	Definition
	A physiological parameter calculated as 2/radius(j).
	Estimated according to the published equation ⁸ from the drug properties of size and charge, and the physiological parameters of pore radius and porosity in each intestinal compartment as shown in Equation 1-55.

Equation 1-55: Paracellular permeability calculation

where:

Variable	Definition
	Dimensionless electrochemical energy function of the j^th intestinal compartment.
	The unit charge of an ion.
	The charge of the drug molecule at the pH of the j^th intestinal compartment.
	The electrical potential gradient across the aqueous pore.
	The Boltzman constant.
	The temperature.
	The porosity, or the volume fraction of aqueous pores, in the j^thintestinal compartment.
	The pore length.
	The aqueous diffusion coefficient for the drug.
	A Renkin function that characterizes the diffusion of solute in a micropore. See Renkin functions in paracellular absorption.

Equation 1-56: Total rate of drug absorption

where:

Variable

Definition

The paracellular absorption scale factor of the j^th intestinal compartment.

The paracellular permeability of the j^th intestinal compartment.

The drug concentration in the lumen.

The unbound concentration of drug in the portal vein.

The lumen fluid volume of the j^th intestinal compartment.

The unbound concentration of drug in the enterocyte subcompartment.

The transcellular absorption scale factor of the j^th intestinal compartment.

The transcellular permeability of the j^th intestinal compartment.

All absorption models described in this chapter account for passive transcellular absorption and paracellular absorption; however, if your model also accounts for active transcellular absorption, then you must include up to two additional terms— one for influx and one for efflux—in Equation 1-44.

Equation 1-57: Renkin function (Adson)

Equation 1-58: Renkin function (Zhimin)

where:

Variable	Definition
	The pore radius.
	The effective molecular radius when the molecule is described by a single value of molecular radius.
	The mean projected radius of solute molecule.
	The hydrodynamically equivalent sphere radius.

Equation 1-59: r_eff calculation

where MW is the molecular weight of the drug compound.

Equation 1-60: Influx and efflux calculations for saturable carrier-mediated transport

where:

Variable	Definition
	The index for the i^th influx/efflux transporter.
	The index for the j^th lumen compartment.
	The total concentration of the drug in the lumen of the j^th intestinal compartment.
	The concentration of unbound drug in the enterocyte sub-compartment of the j^th intestinal compartment.
	The fraction of unbound drug in the enterocytes.
	The maximum transport rate for the i^th influx/efflux transporter.
	The Michaelis-Menten constant (the concentration at 1/2 V_max) for the i^thtransporter.
	The distribution factor for the i^th transporter to adjust V_max in the j^th intestinal compartment, which represents the relative amount of the transporter compared to its V_max measurement environment. The default value is 1.0.
	The scale factor for overall influx transport. The default value is 1.0.
	The scale factor for overall efflux transport. The default value is 1.0.
	The scale factor for overall K_m for influx transporters. The default value is 1.0.
	The scale factor for overall K_m for efflux transporters. The default value is 1.0.

Saturable Metabolism in the GI Tract and Liver Equations

Equation 1-61: Saturable GI tract metabolism rate

where:

Variable	Definition
	The index for the i^th gut metabolism enzyme.
	The index for the j^th enterocyte compartment.
	The concentration of unbound drug in the enterocyte sub-compartment of the j^th intestinal compartment.
	The fraction of unbound drug in the enterocytes.
	The maximum velocity of the i^th enzyme.
	The Michaelis-Menten constant of the i^th enzyme.
	The scale factor for the maximum velocity for the i^th enzyme in the enterocyte sub-compartment of the j^th intestinal compartment.
	The scale factor for overall enzyme velocity. The default value is 1.0.
	The scale factor for overall Km for gut enzymes. The default value is 1.0.

Equation 1-62: GastroPlus® Hepatic Extraction model, differential equation

where:

Variable	Definition
	The hepatic flow rate.
	The whole blood to plasma drug concentration ratio.
	The total drug concentration in the portal vein.
	The total drug concentration in the liver.
	The rate of metabolism, which is calculated separately for each liver enzyme at each time point, and then the total rate of metabolism is calculated as the sum of all the rates of metabolism for all the individual enzymes according to Equation 1-63.

Equation 1-63: Sum total rate of metabolism for all liver enzymes

where:

Variable	Definition
	The index for the i^th enzyme.
	The plasma concentration of the drug in the liver.
	The fraction of unbound of the drug in plasma.
	The maximum metabolism rate for the i^th enzyme.
	The Michael-Menten constant, which is the concentration at 1/2 V_max, for the i^thenzyme.
	The scale factor for overall liver V_max. This factor is the same for all enzymes and is set to a default value of 1.0.
	The scale factor for overall liver K_m. This factor is the same for all enzymes and is set to a default value of 1.0.

ACATPlus™ Equations

Equation 1-64: The diffusion process from sublayer i to sublayer i+1

where:

Variable	Definition
	The compound diffusivity in mucus.
	The thickness of the mucus sublayer.
	The surface area of each GI segment.
	The unbound concentration in mucus sublayer i.

Equation 1-65: Mucus regeneration rate incorporated in the calculation of mucus turn over time

Mucus regeneration will cause compound transit from sublayer i to sublayer i-1. The corresponding transit rate has been defined as MucusTurnOverRate. MucusTurnOverRate from the first sublayer represents the shedding off rate, which will transport the compound from mucus to lumen.

Equation 1-66: Mucus Turnover Rate in each Sublayer

where:

Variable	Definition
	The total concentration in mucus sublayer i.
	The mucus volume in each sublayer.

Equation 1-67: The mass balance in each sublayer

Equation 1-68: Mass Transport Model in Mucosa, Sub-Mucosa, and Muscularis Propria

Where:

Variable	Definition
	The blood flow in mucosa, sub-mucosa, and muscularis propria sub-compartments.
	The unbound concentration in the mucosa sub-compartment.
	The unbound concentration in the sub-mucosa sub-compartment.
	The unbound concentration in the muscularis sub-compartment.
	The transport between mucosa and sub-mucosa.
	The transport between sub-mucosa and muscularis propria.
and are governed by the permeability between their sub-compartments.

Equation 1-69: Transcellular and Paracellular Diffusion in Mucosa and Sub-Mucosa

Where P₁₂ and P₂₃ are the permeability coefficients. SA₁₂ and SA₂₃ are the surface area.