## Abstract

This study aimed to construct a new local pharmacokinetic model of gastrointestinal absorption, the translocation model (TLM), using an anatomically relevant, minimally segmented structure to explain linear and nonlinear intestinal absorption, metabolism, and transport. The TLM was based on the concept of a single absorption site that flexibly moves, expands, and shrinks along with the length of the gastrointestinal tract after the intake of an oral dose. The structure of the small intestine is continuous, and various time- and location-dependent issues are freely incorporated in the analysis. Since the model has only one absorption site, understanding and modification of factors affecting absorption are simple. The absorption site is composed of four compartments: solid drug in the lumen, solution drug in the lumen, concentration in the enterocytes, and concentration in the lamina propria. The lamina propria includes the blood capillaries. Blood flow in the absorption site of the lamina propria appropriately accounts for the absorption. In the TLM, the permeability of the apical membrane and that of the basolateral membrane are distinct. By considering plicate, villi, and microvilli expansions of the surface area, the apparent permeability measured in Caco-2 experiments was converted to the effective permeability in vivo. The intestinal availability, bioavailability, and dose product of intestinal availability and absorption rate relationship of the model drugs were well explained using the TLM. The TLM would be a useful tool for the consideration of local pharmacokinetics in the gastrointestinal tract in various situations.

## Introduction

In drug discovery, the prediction and understanding of the absorption profile of oral drug candidates are required because they are key information to determine the exposure of the drug to therapeutic targets in the body. Accordingly, the development of a logical physiologically based mathematical model that can predict the absorption process of drugs in the gastrointestinal (GI) tract is important. Several models for the prediction of local pharmacokinetics in the GI tract have been reported so far, including the advanced compartmental absorption and transit (ACAT) model (Yu and Amidon, 1999; Agoram et al., 2001), segregated-flow model (Cong et al., 2000), *Q*_{Gut} model (Yang et al., 2007), advanced dissolution, absorption, and metabolism model (Jamei et al., 2009), and GI transit time model (Bergstrand et al., 2009, 2012; Hénin et al., 2012). Among these, the *Q*_{Gut} model is the simplest one in model description that predicts intestinal availability (*F*_{G}) by using *Q*, which is a virtual hybrid parameter of the membrane permeability clearance and blood flow rate in the intestinal villi (Gertz et al., 2010). Although the fundamental principal of the *Q*_{Gut} model is similar to that of the segregated-flow model (Pang and Chow, 2012), the *Q*_{Gut} model is not capable of analyzing nonlinear pharmacokinetics because the model does not consider drug concentration in the enterocytes, where nonlinear processes, such as metabolism and transport, actually occur (Hisaka et al., 2010).

The ACAT and advanced dissolution, absorption, and metabolism models assume a compartmentalized structure of the intestine, and first-order kinetics is applied to the movement of drugs within it. These models can consider location-dependent longitudinal changes in the physiologic conditions. It has been reported that the ACAT model successfully predicts the oral pharmacokinetics of various drugs (Heikkinen et al., 2012). However, even by incorporating the ACAT model into a sophisticated whole body physiologically based pharmacokinetic model, the oral pharmacokinetics of only 23% of drugs are accurately predicted (Poulin et al., 2011).

Furthermore, it should be noted that diagonal (moves from mucosa to submucosa) changes are taken into consideration less than longitudinal (moves along with the absorption site) changes in these models. The concentration of drugs in the enterocytes is sometimes vague since the necessary distinction of permeability between the apical and basolateral membranes is lacking or only implicitly defined. Furthermore, heterogeneous flow that is caused by the differences in villi abundance in different regions should be adequately considered (Kalampokis et al., 1999).

In most of the past physiologic absorption models, the blood flow rate to the whole small intestine is used for the analysis of absorption. In addition, these models often assume that the basolateral transfer is mostly unidirectional; thus, the effect of the blood stream on the absorption rate is ignored. The consequence of changes in the blood flow on the absorption rate becomes important when the basolateral transfer of drugs from the enterocytes to the blood stream is assumed to be reversible. The importance of blood flow is discussed more appropriately in several articles (Winne, 1978; Schulz and Winne, 1987; Chen and Pang, 1997; Pang and Chow, 2012). A physiologic model was reported by Cong et al. (2001), in which branching of the blood flow to the mucosa and submucosa was considered. Pang and Chow mentioned the importance of intestinal blood flow branching and proposed the adequate mucosal to intestinal blood flow ratio (*f*_{Q}) (Pang and Chow, 2012). When the effect of the blood stream on the absorption of drugs is considered based on the physiologic structure of the intestine, branching in both the diagonal and longitudinal directions should be considered. It is well known that blood flow to the duodenum and jejunum is greater than to the ileum, which is in accord with the physiologic function of the intestine. To date, few absorption models appropriately take into consideration both the diagonal and longitudinal blood flow branching, except the segregated-flow model, which can factor for the longitudinal blood flow in animals by dividing the lumen and enterocytes into three compartments (Tam et al., 2003).

In this study, a new model, the translocation model (TLM), is developed to solve these issues and describe drug absorption in the GI tract in a more physiologic manner (Figs. 1 and 2). Although complicated factors needed to be considered in the new model, we hoped to keep the whole structure as simple as possible. This is because the model should be comprehensive and completely open to users. Considering all of these issues, we aimed to construct a new model that could accurately predict various absorption events with only one absorption site.

## Materials and Methods

#### Theory of Location and Distribution Functions.

In TLM, the location and length of the absorption site in the small intestine were calculated using *λ*(*t*), the location function, and *σ*^{2}(*t*), the variance function, at time *t*. Those functions were obtained by using a fitting analysis of the observed location and its variance for a nonabsorbable index drug in the intestinal tract. We assume that a drug is being dissolved completely here. The dissolution process will be considered later. When a total drug concentration (density per length) at a distance *z* from an exit of the stomach is expressed as *C*_{lumen,total}(*z*), *n*th order moment of distribution of a drug in the intestinal tract is defined by eq. 1, where *L*_{si} is the length of the small intestine from the duodenum to the ileum

The mean and variance of location of the index drug in the intestine are given by *λ* and *σ*^{2}, respectively(2)(3)Equations 2 and 3 correspond to the mean residence time and variance of residence time in the moment analysis of blood drug concentration time profiles. In the TLM, it was assumed that *C*_{lumen}(*z*) at the observation time is constant within the absorption site, and it is otherwise zero for simplification. Based on this hypothesis, the length of the absorption site *L*_{abs} is given by eq. 4, which was derived from eqs. 2 and 3. In the following, subscript abs denotes the absorption site:(4)In theory, the equations for *λ* and *σ* can be arbitrarily selected as capable of explaining the observed movements of a drug in the GI tract. In this study, eqs. 5 and 6 were obtained using a fitting analysis of the reported data (Kimura and Higaki, 2002), which showed the time course of the percentage dose of radioactivity in the jejunum, ileum, and colon after administration of nonabsorbable ^{99m}Tc–diethylenetriamine pentaacetic acid in humans (Fig. 3A)

The subscript bolus means that the functions correspond to bolus inputs of a drug into the system, and *t* is the observation time. As a result of the fitting analysis, *m*_{1}, *m*_{2}, *s*_{1}, *s*_{2}, *s*_{3}, and *s*_{4} were obtained as 415 (1.35%), 1.50 (4.71%), 92.4 (4.68%), 0.605 (4.35%), 45.9 (36.8%), and 3.72 (5.03%), respectively (values in the parentheses are CV values calculated from the fitting analysis). Equations 5 and 6 were used to describe the time-dependent change of the drug absorption site and are shown in Fig. 3.

When a drug enters the GI tract with an input function *I*(*t*) but not as a bolus, such as continuous or intermittent administration, the location function should be given with a convolution of *I*(*t*) and eq. 5(7)To simplify the convolution analysis, another moment analysis is applied to drugs in the intestine to calculate the mean residence time *τ*

where *X*_{lumen,total}(*t*) is the total amount of drug in the lumen at time *t*. Calculation of differentials of *M*_{time,n} is described in the Supplemental Material and applied for calculation of *M*_{time,1} in Scheme 1. When the velocity of a drug in the absorption site is constant, even in the absence or presence of continuous drug input/output, a location of the average of the drug distribution moves in accordance with *τ*. Therefore, the convolution of eq. 7 can be approximated as follows:

Equation 10 means that if we calculate *τ* from the time courses of a drug input/output appropriately, the drug movement is described apparently with *λ*_{bolus}(*τ*). Equation 10 generates errors when velocity of the drug movement is variable in the absorption site. Theoretically, it is possible to cover these errors by calculating higher moments. However, the simplest model was adopted in this study because it showed sufficient performance (Fig. 10). Similarly, *σ* was also approximated using *τ* for simplification(11)In the TLM, parameters, such as permeability and clearance, are determined depending on the location of the absorption site (Table 1 and Fig. 4). Strictly speaking, these parameters are possibly variable even within the absorption site. However, for simplification, it was assumed that representative values at the average location could be applied monotonously for the absorption site in the TLM. In the present model, the stomach and large intestine were modeled separately as well stirred compartments. It was assumed that a drug in the stomach traverses to the entrance of the duodenum with a first-order rate constant of *k*_{GE}. The value for *k*_{GE} for a solution was assumed as 5.46 hours^{−1} from the fitting analysis using eqs. 5 and 6. Input into the large intestine is determined by movements of a drug in the small intestine. In the present study, it was assumed that absorption does not occur from the stomach and large intestine, considering the characteristics of the drugs used for the analysis.

#### Radius and Adjusted Location of the Intestinal Tracts.

The function for the radius (*r*) of the lumina of the small intestine, consisting of the duodenum, jejunum, and ileum, was described by a linear function in accordance with a report by Willmann et al. (Fig. 4A; Willmann et al., 2003, 2004).(12)where *z*_{c} (cm) was the center of the absorption site. It needs to be noted that the intestine should be adequately filled with contents to agree to its true radius to the value calculated by eq. 12. In the process of absorption, this is not always realized. For this reason, the internal volume of the intestine will be calculated based on a different assumption later. The values of *r*_{ini} and *r*_{grad} (1.56 and 0.00265, respectively, with *r* = −1.00, and Akaike’s information criterion = −48.7) were obtained from a regression analysis to the radii that were used in GastroPlus (Table 2).

#### Surface Area and Volume at the Absorption Site.

The surface areas of the absorption sites of the basolateral and apical membranes of the small intestine (*S*_{abs,bas} and *S*_{abs,api}, respectively) were calculated using eqs. 13 and 14

where *r*_{in} and *r*_{out} are the radii at the proximal and distal end of the absorption site; PE is plicate expansion; VE_{abs} is villi expansion; and ME is microvilli expansion (Fig. 5) at the absorption site. Equation 13 was derived from the formula of surface area of the frustum. VE_{abs} was calculated using eq. 15 because of a gradual decrease in the size of the villi, depending on the location in the small intestine(15)where VE_{ini} is villi expansion at the entrance of the duodenum. We assumed that PE and ME are constant in the small intestine. For PE, VE_{ini}, and ME, 3, 10, and 20 were used for the small intestine, respectively (DeSesso and Jacobson, 2001).

The surface area of the whole length of the small intestine (*S*_{si}) was calculated using eq. 16 as a solution of integration of the cross-sectional area of the intestinal lumen from 0 to *L*_{si}(16)*S*_{si} will be used for calculation of the blood flow (eq. 20). The volume of the absorption site for the intestinal lumen, *V*_{lumen,abs}, was not calculated from the sizes of the absorption site described by eqs. 4 and 12 because these equations assume that the lumen is adequately filled with contents. Instead, it was calculated using eq. 17 as the total of the water volume taken with the drug and physiologic intestinal water volume in the absorption site(17)where *V*_{water,ini} is the volume of water that is taken with an orally administered drug (200 ml), and *V*_{cont} represents the volume of contents in the intestine under normal fasted conditions (105 ml) (Schiller et al., 2005). *HLw* is the half-life for the disappearance of water taken with the drug. In this study, we assumed that *HLw* is 1 hour. Several fold changes of *HLw* did not affect the absorption profile of drugs in preliminary simulations (data not shown).

The volume of the enterocytes at the absorption site, *V*_{ent,abs}, was calculated as the product of the thickness of the enterocytes (*T*_{ent} = 0.0018 cm) (Sugano, 2009) and *S*_{abs,bas}(18)where volume of the lamina propria, *V*_{propria,abs}, at the absorption site was calculated based on the assumption of almost complete coverage of the intestinal lumen by villi as the product of the villi height (*H*_{villi} = 0.07 cm) (Sugano, 2009) and the surface area of the intestinal lumen subtracted by *V*_{ent, abs}

#### Plasma Flow at the Absorption Site.

The plasma flow rate to enterocytes at the absorption site, *Q*_{plasma,abs}, was calculated using eq. 20 by multiplying the blood flow rate to all enterocytes, *Q*_{blood,enterocyte}, and the ratio of the surface area of the absorption site (*S*_{abs,api}) to the whole small intestine (*S*_{si}) and then converting to plasma flow considering hematocrit (Ht) (0.45)(20)*Q*_{blood, enterocyte} was calculated to be 18,000 ml/h by assuming that the blood flow into the superior mesenteric artery was 37,200 ml/h, which accounts for 10% of the cardiac output. Eighty percent of the blood flow of the superior mesenteric artery flows into the mucosa, and then 60% of the mucosal blood flows into the epithelium cells of the villi (Jamei et al., 2009).

#### Dissolution of a Solid Formulation.

Solubility in fasted state simulated intestinal fluid (FaSSIF) (Sol_{FaSSIF}) was calculated using eq. 21, as reported by Poulin et al. (2011)(21)where Sol_{Water}, MW_{Water}, MW_{Drug}, log *P*, and *C*_{BS} represent water solubility, molecular weight of water and the drug, *n*-octanol:buffer partition coefficient of the drug, and the bile salt concentration of sodium taurocholate (4 nM), respectively. The dissolution constants for the small intestine were defined using eqs. 22 and 23(22)(23)where Sol, *γ*, *ρ*, *r*_{j}, T, and *C*_{i} represent solubility, diffusion coefficient, density (1.2 mg/ml), particle radius (0.0025 cm), diffusion layer thickness (0.003 cm), and the concentration in the stomach or lumen, respectively (Agoram et al., 2001).

#### Expression Level of Metabolizing Enzymes and Transporters.

It has been reported that cytochrome P450 (CYP) 3A is most abundantly expressed in the duodenum and less expressed in the distal end of the ileum (Paine et al., 1997). Therefore, a linear gradient was assumed for the expression of CYP3A, decreasing from the proximal end of the duodenum to the distal end of the ileum. The expression of CYP3A at the absorption site, *A*_{CYP3A,abs}, was calculated as a product of *L*_{abs} and the density of CYP3A expression (eq. 24; Table 1) for the small intestine (*z* < *L*_{si}; *L*_{si} = 306 cm)

where *A*_{CYP3A,total} is the total amount of CYP3A expressed in the small intestine (70,500 pmol). The factor 2 in eq. 24 indicated that the density is double when *z*_{c} is zero. Conversely, *A*_{CYP3A,abs} decreases to zero as the location moves to the large intestine since no expression of CYP3A was reported in the colon (Bruyere et al., 2010). Metabolic clearance of the substrates of CYP3A was calculated from *A*_{CYP3A,abs} and Michaelis constants for each substrate (Scheme 1).

It has been reported that the expression level of P-glycoprotein (P-gp) increases relatively from the proximal to distal end of small intestine (Stephens et al., 2001; Englund et al., 2006; Bruyere et al., 2010); however, the absolute expression level is rarely reported. Therefore, we assumed that the relative expression level of P-gp in interest (*A*_{rel,Pgp,abs}) increases linearly in the lower part of the intestine. For the lower site below, *A*_{rel,Pgp,abs} was calculated using eq. 25(25)where *A*_{init,Pgp} and *A*_{grad,Pgp} represent the amount of P-gp in the entrance of duodenum, and the slope for the increase in the P-gp expression amount, respectively. In this study, 0 and 1 were used for *A*_{init,Pgp} and *A*_{grad,Pgp}, respectively. To calculate the actual expression level of P-gp, a scaling factor was estimated from in vivo observations, as will be described later.

#### Membrane Permeability.

The components of the permeability constants of the enterocytes, i.e., passive intake of apical, passive efflux of apical, active efflux of apical by P-gp, passive output from basal, and passive reverse intake from basal, were designated as *P*_{1}, *P*_{2}, *P*_{P-gp}, *P*_{3}, and *P*_{4}, respectively (Fig. 5). Based on the structural symmetry and similarity of membranes, values for the passive permeability (*P*_{1}–*P*_{4}) were assumed to be the same as the passive permeability of a single membrane, which is denoted as *P*_{s} (when the pH of the environments is the same for both sides of the membrane). *P*_{s} for the Caco-2 system was calculated from the apparent permeability (*P*_{app}) determined in Caco-2 assays (Gertz et al., 2010) using eq. 26(26)where ME_{Caco-2} is the apical/basolateral relative surface area ratio for Caco-2 cells. Equation 26 was derived from the equation reported by Tachibana et al. (2010). For the equation, *PS*_{1} = *P*_{1,Caco-2}*S*·ME_{Caco-2}, *PS*_{2} = *P*_{2,Caco-2}*S*·ME_{Caco-2}, *PS*_{3} = *P*_{3,Caco-2}*S*, and *PS*_{4} = *P*_{4,Caco-2}*S* were assumed. The terms *K*_{m} and *V*_{max} were removed because we used the equation to calculate passive diffusion.

In the present study, an ME_{Caco-2} value of 4 was adopted as the average of the values determined for propranolol and naproxen as nonsubstrates of P-gp (Ohura et al., 2011). The values of *P*_{app,Caco-2} for the substrates of P-gp (quinidine and verapamil) were calculated as the average values of *P*_{app,Caco-2} at the three highest concentrations (approximately 10–100 *μ*M) because the activity of P-gp was saturated at the three highest concentrations in the cited experiments (Tachibana et al., 2010).

*P*_{s} for the in vivo system was determined by eq. 27(27)where psf_{passive} is the scaling factor, which is obtained from approximation of the simulated and reported *F*_{A} for nine drugs (acyclovir, atenolol, cimetidine, ciprofloxacin, enalaprilat, gabapentin, methotrexate, ranitidine, and sulpiride). These drugs were selected based on the following criteria: 1) *F*_{A} < 0.9; 2) Caco-2 permeability is available in the literature (Thomas et al., 2008); 3) *F*_{H} > 0.7; and 4) *CL*_{H}/*CL*_{tot} < 0.3. First, each value of psf for the individual drug was estimated by fitting the analysis to agree to the reported *F*_{A} and predicted *F*_{A} as possible using TLM (Supplemental Table 1). Then, psf_{passive} was determined as the average of the psf for the nine drugs. Calculation of *P*_{1} will be described later.

For the drugs that are substrates of P-gp, the active permeability by P-gp in vitro (eq. 28) and in vivo (eq. 29) were described as follows:(28)(29)where psf_{Pgp}, *C*_{cell}, and *C*_{Enterocyte,u} represent the scaling factor for P-gp transport and the concentrations in the Caco-2 cell and enterocyte, respectively. Michaelis constants for the transport by P-gp (*K*_{m,Pgp} and *V*_{max,Pgp,Caco-2}) were calculated using a fitting analysis with data from Caco-2 cell assays (Tachibana et al., 2010), as previously described, with a slight modification using eq. 30(30)

where *C*_{a} represents the drug concentration in the apical chamber. The *K*_{m,Pgp} values were 0.113 and 0.283 *μ*g/ml for quinidine and verapamil, respectively. For *P*_{s,Caco-2} values, it was confirmed that the values obtained by fitting the analysis with eq. 30 were practically the same as those obtained from eq. 26.

For apical uptake permeability, *P*_{1} was calculated using eq. 31, considering differences of pH between the Caco-2 assay and intestinal lumen. Caco-2 assays are generally carried out at a pH of 7.4, but it has been reported that the surface of the intestinal lumen is somewhat more acidic (Table 1; Bolger et al., 2009).(31)where pH_{Caco-2}, pKa_{acid}, and pKa_{base} represent the pH for the Caco-2 assay (i.e., 7.4), acidic pKa, and basic pKa, respectively. A value for pH_{lumen} represents the pH of the intestinal lumen of the small intestine that is described by the following equation:

The values for pH_{ini} and pH_{grad} were 5.85 and 0.00455, respectively, with *r* = 0.979 and Akaike’s information criterion = −13.7. The values for a product of psf_{Pgp} and *V*_{max,Pgp} in eq. 26 were obtained simultaneously by a fitting analysis using the TLM. The values were estimated to explain the dose response of *F*_{A}*F*_{G} appropriately for verapamil and quinidine, which is a substrate of P-gp but a weak substrate of CYP3A (Supplemental Fig. 2). Based on the above theory, fundamental simultaneous ordinary differential equations for the TLM were constructed, as shown in Scheme 1.

#### Data Collection for Drug-Related Parameters.

The drug-related parameters for 20 drugs (alfentanil, alprazolam, buspirone, cisapride, cyclosporin, felodipine, lovastatin, midazolam, nifedipine, nisoldipine, repaglinide, rifabutin, saquinavir, sildenafil, simvastatin, trazodone, triazolam, zolpidem, quinidine, and verapamil) used in this model are summarized in Table 3. *K*_{m} values for CYP3A were taken from the literature (Lavrijsen et al., 1988; Rotzinger et al., 1998; Ekins et al., 1999; Kajosaari et al., 2005; Ku et al., 2008; Polasek et al., 2010; Gertz et al., 2011; Heikkinen et al., 2012;). When no information was available for the *K*_{m} values of CYP3A, the *K*_{m} values of CYP3A4 were used. Metabolic intrinsic clearance, membrane permeability, and the blood-free fraction were taken from the literature (Gertz et al., 2011). The observed *F*_{G} was estimated using the drug-drug interaction method reported by Hisaka et al. (2014) and taken from other sources (Varma et al., 2010). The other observed or reported values required for analysis in this study are shown in Tables 3 and 4. Unbounded fractions in the intestinal lumen (*f*_{lumen}), enterocyte (*f*_{ent}), and lamina propria (*f*_{propria}) were assumed to be 1.

#### Calculation of Pharmacokinetics Parameters (*F*_{G} and Bioavailability [*F*]).

*F*_{A}*F*_{G} values were calculated by TLM as a ratio of the cumulative drug amount that flew into the portal vein to the administered dose (eq. 33). *F*_{G} values were calculated by TLM as a ratio of *F*_{A}*F*_{G} to *F*_{A.} The latter was calculated by subtracting the ratio of the cumulative drug amount remaining in the colon and rectum to the administered dose from 1(33)(34)By using in vitro hepatic intrinsic clearance (*CL*_{int,HLM}) estimated using human liver microsomes (Gertz et al., 2011), the blood-free fraction (*f*_{B}), hepatic blood flow rate (*Q*_{H}), and hepatic availability were estimated with in vitro information using the following equation (*F*_{H,in vitro}):(35)By multiplying the predicted *F*_{A}*F*_{G} with *F*_{H,in vitro}, *F* was predicted. The observed *F* values were collected from two reports (Varma et al., 2010; Appendix of Parker et al., 2005).

#### Sensitivity Analysis.

A sensitivity analysis of TLM was performed to estimate the effect of the surface area of the apical membrane of enterocytes by using verapamil as a model drug. Absorptions of 40 mg of verapamil was simulated by TLM, with ME values of 1, 5, and 20. Other parameters for TLM were not changed in this sensitivity analysis. Then, changes in the intestinal lumen concentration, enterocyte concentration, lamina propria concentration in the absorption site, and *F*_{A}*F*_{G} were investigated by comparing the time course of these concentrations.

#### Evaluation of Precision and Accuracy of the Prediction.

Precision and accuracy of the prediction for *F*_{G} and *F* were evaluated by using the within 2-fold error, within ±0.3 error, average fold error (AFE) (eq. 36), and root mean square prediction error (RMSE) (eq. 37)(36)(37)where *N* represents the number of drugs that were used for the calculation.

#### Prediction of Nonlinear Pharmacokinetics.

A prediction of nonlinear pharmacokinetics was performed by simulating the *F*_{A}*F*_{G}-dose relationship for midazolam. Average *F*_{A}*F*_{G} values in the range of 0.01–10,000 mg of the administered dose were plotted followed by confirmation of the similarity of the simulated and observed values, which were calculated by using reported clinical data based on the assumption that the major site to induce the nonlinear systemic exposure after oral administration is the intestine (Bornemann et al., 1985; Misaka et al., 2010). For the calculation, body weight, *R*_{B}, and hepatic blood flow rate were assumed to be 70 kg, 1, and 25.5 ml/min per kg, respectively. Total clearance and the fraction excreted in urine were cited from the following source (Appendix of Parker et al., 2005).

#### Systems and Application.

Calculation of the ordinary differential equations constructed in this study was performed using Napp (http://square.umin.ac.jp/todaiyak/download.htm) with the Runge-Kutta–Fehlberg method (Hisaka and Sugiyama, 1998).

## Results

#### Construction of the TLM.

The TLM was constructed based on the concept that a drug is transferred from the lumen to the bloodstream across the enterocytes and lamina propria at an absorption site, which is relocated, expanded, and decreased in size along the length of the GI tract in a time-dependent manner (Fig. 1). The locatable absorption site consisted of four compartments: solid formulation in the lumen, dissolved drug in the intestinal lumen, concentration in the enterocytes, and concentration in the lamina propria, including the capillaries (Fig. 2). The absorption site continuously traversed from the duodenum to the ileum. In accordance with the physiologic structure, the stomach and colon were defined as a separate compartment.

#### Definition of Location and Distribution Functions.

The location and distribution functions that determine the movement of a drug absorption site in the TLM were obtained using a fitting analysis of the results of the gamma scintigraphy reported by Kimura and Higaki (2002). The movement of a nonabsorbable index drug in the gastrointestinal tract was analyzed by using the location and distribution functions, and then agreement was confirmed between the fitted and reported values (Fig. 3A). By applying a simplified convolution with the mean residence time, these functions were extended to cover any drug input situation in the intestine, such as reduction of the gastric emptying rate, multiple inputs, and so forth (Fig. 10).

#### Definition of Location-Dependent Parameters.

All location-dependent parameters defined for the absorption site of the TLM were described by *z*, the location in the intestine. The equations that defined the location-dependent parameters are represented in Table 1. Changes of the parameters defined by *z* are shown in Fig. 4. The calculated total surface area and the total volume of the lumen, enterocyte, and lamina propria were 751,000 cm^{2} and 311, 67.6, and 378 ml, respectively.

#### Approximation of psf and *V*_{max,Pgp}.

To determine the correspondence of passive permeability between in vitro and in vivo systems, a fitting analysis was performed for the *F*_{A} values of nine drugs (Supplemental Table 1). The value of psf_{passive} was determined as 2.23 from the analysis. This result suggests that the passive membrane permeability of enterocytes in vivo is apparently 2.23-fold higher than that in vitro. By using this psf_{passive}, *F*_{A} was estimated within ±0.3 of the observed value for all compounds (Supplemental Fig. 1).

There were few reports on the in vivo *V*_{max} value for P-gp. Therefore, the product of psf_{Pgp} and *V*_{max,Pgp} was obtained by a simultaneous fitting analysis to the human *F*_{A}*F*_{G} values for quinidine (0.1, 1, 10, and 100 mg) and verapamil (0.1, 3, 16, and 80 mg) (Supplemental Fig. 2). The value was estimated as 1.66 g/h (CV = 11.1%), assuming that *V*_{max} is the same for quinidine and verapamil (Table 3).

#### Sensitivity Analysis for ME.

In the TLM, the model description of the surface area of the enterocytes is different for the apical and basolateral membranes. The ratio of the surface area of the apical to basolateral membrane was defined as ME, and ME was assumed to be 20 in this study. This means the surface area of the apical membrane is 20-fold larger than that of the basolateral membrane. The effect of ME on drug exposure in the lumen, enterocytes, lamina propria, and *F*_{A}*F*_{G} were confirmed by sensitivity analysis using 40 mg of verapamil (Fig. 6). Increasing ME reduced the concentration in the intestinal lumen, whereas *F*_{A}*F*_{G} was increased by an increase in ME.

#### Simulation of Nonlinear Pharmacokinetics and Plasma Time Profile of Midazolam.

*F*_{A}*F*_{G}-dose relationships were simulated using TLM for midazolam (Fig. 7A). The predicted dose-dependent increase of *F*_{A}*F*_{G} agreed well with the observed *F*_{A}*F*_{G}. The plasma time profile was predicted for oral midazolam after administration with a therapeutic dose or microdose by using pharmacokinetic parameters after intravenous administration that were obtained from the same experiment with oral administration (Fig. 7B). As a result, although plasma concentration after the microdose was slightly underestimated, overall, the time course of drug concentration after the therapeutic dose and microdose was well predicted.

#### Prediction of Intestinal Availability and Bioavailability.

To validate the further utility of the TLM after confirming the validity of the model’s construction, *F*_{G} was calculated for 18 drugs and *F* was calculated for 15 drugs so that the information about their affinity for CYP3A and P-gp are available (Table 3). The reported *F*_{G} and *F* values and the predicted *F*_{G} and *F* values are shown in Table 4 and Fig. 8, respectively. The number of drugs within 2-fold errors for *F*_{G} and *F* were 50% and 53%, respectively, of drugs analyzed in this study. The ratio within ±0.3 errors for *F*_{G} and *F* were 72% and 93%, respectively. The AFE and RMSE for *F*_{G} prediction were 1.10 and 1.48, while the AFE and RMSE for *F* prediction were 0.347 and 0.230, respectively (Table 5).

#### Simulation of Dose-Dependent Absorption.

The time course of the drug amount in the stomach, intestinal lumen, enterocyte, and lamina propria, cumulative drug amount in the portal vein and feces, and cumulative amount of the metabolized drug were simulated with input conditions at 40 (clinically effective dose) and 0.1 mg (microdose) of oral verapamil using the TLM (Fig. 9). At 40 mg, most of the unchanged drug was rapidly absorbed and then 32.7% of the dose traversed into the portal vein, while 57.9% was metabolized and 9.4% reached the colon. On the other hand, the extent of the drug absorbed into the portal vein was lower at 0.1 mg. Only 16.6% of the dose traversed into the portal vein, while 32.2% was metabolized and 51.2% reached the colon.

#### Application to Multiple Administration.

The time course of the drug location (Fig. 10A) and drug concentration in the intestinal lumen, enterocyte, and lamina propria (Fig. 10B) were simulated when the location and distribution functions were applied to multiple inputs of a low permeability drug. The simulation revealed that the drug absorption site could move both back and forth as needed. A tendency for the drug concentration to decrease from the intestinal lumen to the lamina propria was observed appropriately.

## Discussion

The TLM was developed to construct a physiologically precise and relatively simple absorption model, and its structure and parameters are easily modifiable if needed. The model structure was simplified by introducing a single absorption site, which is moved in accord with the location and variance functions. Previous models represent location-dependent changes of physiologic parameters with the fragmented intestinal compartments (Yu and Amidon, 1999; Agoram et al., 2001; Jamei et al., 2009). Conversely, the TLM enabled them by describing each parameter with a location-dependent function under various input/administration conditions.

The location and distribution functions set in this study reproduced the delay in gastric emptying under the fed state and the corresponding changes in the distribution of a nonabsorbable index drug reported by Kimura and Higaki (2002). Moreover, the profile of distribution in Fig. 3B was similar to the results reported by Willmann et al. (2003). From these results, it was indicated that the location and variance functions used in this study could represent drug movement in the intestinal lumen.

Most of the reported models use effective permeability (*i*_{eff}) as the parameter for the passive permeability of the enterocytes (Yang et al., 2007; Badhan et al., 2009; Gertz et al., 2011). *P*_{eff} is the parameter that is calculated for human jejunum permeability and can be estimated by *P*_{app} in Caco-2 cells by using the result of the regression analysis between *P*_{app} and *P*_{eff} (Sun et al., 2002). In the TLM, in vivo passive permeability was integrated using the in vitro *P*_{app} of Caco-2 cells and multiplying the constants of surface area expansion (PE for plicate, VE for villi, and ME for microvilli). This direct integration of the passive permeability compared with the empirical approach of the extrapolation from *P*_{app} to *P*_{eff} enables the user to take the intestinal physiologic factor and effect of the changes in the intestinal pH into consideration. To confirm the utility of direct integration from single membrane permeability, *P*_{eff} values from the TLM were simulated by modifying the TLM that mimicked the reported experimental conditions of Loc-I-Gut (Petri et al., 2003). The location and size of the absorption site (10 cm) was placed in the jejunum by TLM in exactly the same way. Finally, the reported and simulated *P*_{eff} values were compared (Supplemental Fig. 3). Our results indicate that the passive permeability could be calculated adequately from the single membrane permeability.

To confirm the effect of ME on drug absorption, a sensitivity analysis was performed for verapamil after the oral administration of a therapeutic dose (40 mg). Results showed that the increase of ME reduced verapamil exposure to the lumen and increased the maximum concentration in the enterocytes in the early time point, which might be important for first-pass metabolism due to the higher expression of CYP3A in the proximal part of the small intestine. *F*_{A}*F*_{G} also increased, depending on the extent of the ME increase (Fig. 6). These results indicate that discrimination of the apical and basolateral membranes by using an adequate ME would be advantageous to avoid potential underestimation of the systemic exposure.

The obtained parameters were used for the prediction of the dose-*F*_{A}*F*_{G} relationship of midazolam. As shown in Fig. 7A, the predicted and observed *F*_{A}*F*_{G} for midazolam were in close agreement, suggesting the appropriateness of parameters obtained from the fitting analyses of *F*_{A} values for nine drugs. In addition to the nonlinearity in *F*_{A}*F*_{G}, the plasma time profile of midazolam was also predicted after oral administration of the therapeutic dose and microdose (Fig. 7B). These results suggest that the time course of plasma concentration as well as nonlinear absorption kinetics by TLM could be well predicted.

In this research, the *F*_{G} for 18 drugs and *F* for 15 drugs were simulated, as they are substrates of CYP3A and P-gp (Fig. 8). As a result, the *F*_{G} and *F* of these drugs were reasonably predicted using the TLM from in vitro parameters. As shown in Table 5, AFE and RMSE in all predictions were acceptable, by which the potential of TLM for the prediction of clinical pharmacokinetics after oral administration of various drug candidates from in vitro data from Caco-2 cell assays and microsomal metabolic kinetics studies during the drug discovery stage was demonstrated.

By using the TLM, the simulation of the local pharmacokinetics of the intestine was performed at the clinical effective dose (40 mg) and microdose (0.1 mg) of verapamil. Lower availability was predicted with a microdose than with 40 mg, which corresponds to the clinical observation (Maeda et al., 2011). The simulated maximum concentration in the enterocytes was approximately 6.7 *μ*M, which was smaller than the *K*_{m} value of CYP3A4 (49.0 *μ*M) and higher than the *K*_{m} value for P-gp (0.622 *μ*M). This would indicate a larger contribution of P-gp to the nonlinear drug absorption after 40 mg of oral verapamil.

The recent commercially available applications for the prediction of drug absorption are generally expensive and not easy to modify because of the complex and veiled structure. The TLM was developed by using Napp, a program for the analysis of pharmacokinetics, which is available for free. This program would assist researchers to perform detailed analyses and predictions of drug absorption. Moreover, an advantage of the TLM is that the model structure is completely transparent to users.

In conclusion, we successfully developed the TLM to describe the physiology of drug absorption as precisely as possible, i.e., the movement of a drug in the intestine, permeation through the apical and basolateral membranes, and the contributing blood flow. The structure of TLM is relatively simple compared with the previously available models. It would be useful for the prediction of drug absorption during new drug development in the future. Moreover, various events during the absorption process would be analyzed more accurately with TLM by adjusting its structure and parameters, if necessary.

## Authorship Contributions

*Participated in research design:* Hisaka, Ando, Suzuki

*Conducted experiments:* Ando, Hisaka.

*Contributed new reagents or analytic tools:* Ando, Hisaka.

*Performed data analysis*: Ando, Hisaka, Suzuki.

*Wrote or contributed to the writing of the manuscript:* Ando, Hisaka, Suzuki.

## Footnotes

- Received July 15, 2014.
- Accepted January 23, 2015.
↵1 Current affiliation: Discovery Research Laboratories, Kyorin Pharmaceutical Co., Ltd., Tochigi, Japan.

↵2 Current affiliation: Geriatric Pharmacology and Therapeutics, Graduate School of Pharmaceutical Sciences, Chiba University, Chiba, Japan.

This research was supported in part by a Grant-in-Aid for Scientific Research on Innovative Areas HD Physiology Project from the Japanese Ministry of Education, Culture, Sports, Science, and Technology [Grant 22136015].

↵This article has supplemental material available at dmd.aspetjournals.org.

## Abbreviations

*A*- protein amount
- ACAT
- advanced compartmental absorption and transit
- AFE
- average fold error
*CL*_{int,H}- hepatic intrinsic clearance
- CYP3A
- cytochrome P450 3A
*F*- bioavailability
*f*- free fraction
*F*_{A}- fraction absorbed
*F*_{A}*F*_{G}- product of
*F*_{A}and*F*_{G} *F*_{G}- intestinal availability
*F*_{H}- hepatic availability
- FaSSIF
- fasted state simulated intestinal fluid
- GI
- gastrointestinal
*H*_{villi}- height of villi
*HLw*- half-life for disappearance of water
- Ht
- hematocrit
*k*_{GE}- gastric emptying rate constant
*K*_{m}- Michaelis constant
*L*- length
- ME
- microvilli expansion
*P*- permeability
*P*_{app}- apparent permeability
- PE
- plicate expansion
- P-gp
- P-glycoprotein
*Q*- blood flow rate
- RMSE
- root mean square prediction error
- Sol
- solubility
*T*_{ent}- thickness of enterocyte
- TLM
- translocation model
*V*- volume
*V*_{max}- maximum rate
- VE
- villi expansion
*z*- location

- Copyright © 2015 by The American Society for Pharmacology and Experimental Therapeutics