Calculation of fraction of dose absorbed: comparison between analytical solution based on one compartment steady state concentration approximation and dynamic seven compartment model

Oral absorption of a drug is modeled by the differential equations for dissolution, permeation and gastrointestinal transit processes. The purpose of the present study was to compare simple approximate analytical solutions with full numerical solutions for the calculation of the fraction of a dose absorbed (Fa). The GI compartment model for numerical integration consisted of 1 stomach, 7 intestine and 1 colon compartments, whereas for analytical solutions a simple one well-stirred compartment was used. Full numerical solutions were obtained by numerically integrating the dissolution, permeation and gastrointestinal transit differential equations. In the numerical integration calculation, the concentration change in the GI tract, particle size reduction, transit of drugs, etc., was dynamically simulated. Precipitation in the GI tract and regional differences of solubility and permeability were not considered. In total, 7056 numerical integrations were performed, sweeping practical drug parameter ranges of solubility (0.001 to 1 mg/mL), diffusion coefficient (0.1 – 10 x 10 -6 cm 2 /sec), dose (1 to 1000 mg), particle diameter (1 to 300 μm) and effective permeability (0.03 – 10 x 10 -4 cm/sec). The analytical solutions investigated were (I) a sequential first order approximation (Fa =1–Pn/(Pn – Dn)exp(–Dn) + Dn/(Pn – Dn)exp(–Pn), Dn: dissolution number, Do: dose number and Pn: permeation number. Dn, Do and Pn are the dimensionless parameters which represent the dissolution time/GI transit time ratio, the solubility/dose ratio, and the permeation time/GI transit time ratio, respectively), (II) a limiting step approximation (the minimum value of Fa = 1–exp(–Pn), Fa = Pn/Do and Fa = 1–exp(–Dn)) and (III) a steady state approximation for the dissolved drug concentration (Fa =1–exp(–1/(1/Dn + Do/Pn)), if Do < 1, Do = 1). Fa values by (I) and (II) were higher than those by numerical integration for low solubility compounds (r 2 = 0.80 and 0.98, root mean square error (RMSE) = 0.28 and 0.079, respectively). By applying the steady state approximation, the correlation was improved (r 2 = 0.99, RMSE = 0.047). The steady state approximation for the dissolved drug concentration was appropriate for Fa calculation.