### Numerical solution of the Diffusion-Convection (DC) model equations

Ni et. al. [12] have described a model of intestinal transit which combines convection, dispersion and absorption (DC model). The main assumption is that there is an equal volume flow into and out of each intestinal region so that the cross-sectional area (radius = r) and the convective flow (F) remains constant as the solute spreads along the intestine by convection and dispersion. The differential equation describing this DC model is:

\pi {r}^{2}\frac{\partial c}{\partial t}=\pi {r}^{2}D\frac{{\partial}^{2}c}{\partial {x}^{2}}-F\frac{\partial c}{\partial x}-2\mathit{\text{\pi rPc}}

(1)

The left hand side is the time dependent change in the concentration c(x,t) (where x is the distance from the pyloric sphincter). The first term on the right is the dispersive mixing, the second is the convective flow and the third is the absorption term where r is the intestinal radius (cm), D is the dispersion coefficient (cm^{2}/sec), F is the volume flow (cm^{3}/sec) and P is the permeability (cm/sec).

Ni et. al. [12] derived an exact analytical solution to Equation (1) that assumes as a boundary condition an exponential concentration at x = 0. This condition is not physiological because it implies that, in addition to the convective flux out of the stomach, there is also a non-physiological dispersive flux both out of and into the stomach (and out of and into the large intestine). For this reason the analytical solution will not be used here and, instead, a finite difference numerical approximation to Equation (1) will be used in which there is only a convective flux from the stomach to the small intestine and from the small intestine to the large intestine. (Also, the numerical solution is computationally much faster than the analytical solution). The small intestine is divided into N equal sections with the following difference equations:

\begin{array}{l}i=1:\phantom{\rule{0.96em}{0ex}}\mathrm{\Delta}\mathit{V}\phantom{\rule{0.12em}{0ex}}\frac{\mathit{dc}\left[1\right]}{\mathit{dt}}={I}_{G}\left(t\right)-(F+\mathrm{\Delta}\mathit{P}+\mathit{De})\phantom{\rule{0.12em}{0ex}}c\left[1\right]+\mathit{De}\phantom{\rule{0.12em}{0ex}}c\left[2\right]\\ 0<i<N:\phantom{\rule{0.6em}{0ex}}\mathrm{\Delta}\mathit{V}\phantom{\rule{0.12em}{0ex}}\frac{\mathit{dc}\left[i\right]}{\mathit{dt}}=\left(F+\mathit{De}\right)\phantom{\rule{0.12em}{0ex}}c[i-1]-(F+\mathrm{\Delta}\mathit{P}+2\phantom{\rule{0.12em}{0ex}}\mathit{De})\phantom{\rule{0.12em}{0ex}}c\left[i\right]+\mathit{De}\phantom{\rule{0.12em}{0ex}}c[i+1]\\ i=N:\phantom{\rule{1.56em}{0ex}}\mathrm{\Delta}\mathit{V}\phantom{\rule{0.12em}{0ex}}\frac{\mathit{dc}\left[N\right]}{\mathit{dt}}\phantom{\rule{0.36em}{0ex}}=\left(F+\mathit{De}\right)\phantom{\rule{0.12em}{0ex}}c[N-1]-(F+\mathit{De}+\mathrm{\Delta}\mathit{P})\phantom{\rule{0.12em}{0ex}}c\left[N\right]\phantom{\rule{0.24em}{0ex}}\end{array}

(2)

where c[i] is the concentration in the ith compartment at time t, I_{G}(t) is the rate of gastric emptying into the intestine, r = intestinal radius, L = intestinal length, S = surface area = 2πrL, V = volume = πr^{2}L, ∆P = PS/N, ∆V = V/N and De = πr^{2}DN/L_{.} The rate E_{DC}(t) that the unabsorbed solute exits the small intestine and passes into the large intestine is:

{E}_{\mathit{DC}}\left(t\right)=F\phantom{\rule{0.12em}{0ex}}c\left[N\right]

(3)

The cumulative amount A_{DC}(t_{i}) that has entered the large intestine at time t_{i} = i ∆t is:

{A}_{\mathit{DC}}\left({t}_{i}\right)={\displaystyle {\sum}_{j=1}^{i}\phantom{\rule{0.12em}{0ex}}{E}_{\mathit{DC}}}\left({t}_{j}\right)\phantom{\rule{0.24em}{0ex}}\mathrm{\Delta}\mathit{t}

(4)

The absorption rate R_{DC}(t) at time t_{i} is:

{R}_{\mathit{DC}}\left({t}_{i}\right)=\mathrm{\Delta}\mathit{P}\phantom{\rule{0.12em}{0ex}}{\displaystyle {\sum}_{i=1}^{N}c\left[i\right]}

(5)

Gastric emptying in humans of non-caloric fluids is approximately exponential with a half time of about 15 minutes [13, 14] and it will be assumed that I_{G}(t) is exponential:

{I}_{G}\left(t\right)=F{C}_{0}\text{exp}\left(-t/{T}_{G}\right)

(6)

where T_{G} is the time constant for gastric emptying, C_{0} is the gastric concentration at t = 0 and FC_{0} = Dose/T_{G}. In addition, the parameters D, F and P will be described in terms of 3 other time constants:

{T}_{P}=r/\left(2P\right)\phantom{\rule{3em}{0ex}}{T}_{F}=V/F\phantom{\rule{3em}{0ex}}{T}_{D}={L}^{2}/\left(2D\right)

(7)

Equation (2) is solved numerically using N = 50 and the Rosenbrock method as implemented in Maple (Maplesoft™). Some of the figures shown here are Maple plots.

### Derivation and description of the “Averaged Model (AM)”

The DC equation (Equation (1)) has the interesting property that, if the drug is completely absorbed in the small intestine and the amount entering the large intestine can be neglected, it has the same kinetics as a well stirred compartment. This can be seen by integrating Equation (1) over x from 0 (pyloric sphincter) to x = L (the ileocecal junction):

\begin{array}{l}\phantom{\rule{4em}{0ex}}\pi {r}^{2}L\frac{\mathit{dC}}{\mathit{dt}}=\phantom{\rule{0.5em}{0ex}}{I}_{0}\left(t\right)\phantom{\rule{0.5em}{0ex}}-{I}_{L}\left(t\right)\phantom{\rule{0.5em}{0ex}}-2\mathit{\text{\pi rLPC}}\\ C\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}\left(1/L\right){\displaystyle \underset{0}{\overset{L}{\int}}c\left(x,t\right)\mathit{dx}}\phantom{\rule{.5em}{0ex}}{I}_{0}\left(t\right)\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}-\pi {r}^{2}D\frac{\mathit{dc}\left(0,t\right)}{\mathit{dx}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\mathit{Fc}(0,t)\phantom{\rule{.5em}{0ex}}\\ {I}_{L}\left(t\right)\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}-\pi {r}^{2}D\frac{\mathit{dc}\left(L,t\right)}{\mathit{dx}}\phantom{\rule{0.5em}{0ex}}+\mathit{Fc}(L,t)\phantom{\rule{0.12em}{0ex}}\end{array}

(8)

where C is the average intestinal concentration and I_{0}(t) and I_{L}(t) are the inflow and outflow rates. If the outflow term I_{L}(t) is negligible, then this equation reduces to:

\phantom{\rule{0.12em}{0ex}}V\frac{\mathit{dC}}{\mathit{dt}}={I}_{0}\left(t\right)-\mathit{PS}\phantom{\rule{0.12em}{0ex}}C

(9)

This is identical to the case of a well mixed compartment of volume V with arbitrary input I_{0}(t). Assuming that I_{0}(t) = I_{G}(t) (Equation (6)) and solving the differential Equation (9) one obtains the “averaged model” (AM) equation for the case of 100% absorption:

C\left(t\right)=\left(\mathit{\text{Dose}}/V\right)\phantom{\rule{0.24em}{0ex}}{T}_{P}\phantom{\rule{0.12em}{0ex}}\left[\text{exp}\left(-t/{T}_{G}\right)\phantom{\rule{0.36em}{0ex}}-\text{exp}\left(-t/{T}_{P}\right)\right]/\left({T}_{G}-{T}_{P}\right)

(10)

where T_{P} and T_{G} are the permeability and gastric emptying time constants (Equation (6) and (7)). The rate of absorption (R(t)) from the small intestine is:

R\left(t\right)=P\phantom{\rule{0.12em}{0ex}}S\phantom{\rule{0.12em}{0ex}}C\left(t\right)=\mathit{Dose}\phantom{\rule{0.24em}{0ex}}\left[\text{exp}\left(-t/{T}_{G}\right)\phantom{\rule{0.36em}{0ex}}-\text{exp}\left(-t/{T}_{P}\right)\right]/\left({T}_{G}-{T}_{P}\right)

(11)

This AM R(t) is identical to the absorption rate for the DC model for the case where all the solute is absorbed (I_{L}(t) = 0, Equation (8)). It should be emphasized that although Equation (9) is similar to the well-mixed equation it is not physically equivalent because C is the average concentration and it is not assumed that the intestine is well mixed. For example, it would be erroneous to assume that the rate of solute flow into the large intestine was equal to F*C.

As discussed above, Equation (11) is a rigorously accurate description of the intestinal absorption for the DC model only for the case where all of the solute is absorbed in the small intestine. This result can be generalized to the arbitrary permeability case where only a fraction F_{A} of the total Dose is absorbed in the small intestine:

\begin{array}{l}{R}_{M}\left(t\right)=M\phantom{\rule{0.24em}{0ex}}\left[\text{exp}\right(-t/{T}_{G})\phantom{\rule{0.36em}{0ex}}-\text{exp}(-t/{T}_{P}\left)\right]/({T}_{G}-{T}_{P})\\ M={F}_{A}\phantom{\rule{0.12em}{0ex}}\mathit{\text{Dose}}\end{array}

(12)

where M is the total amount absorbed. In addition, the relationship between T_{P} and P must be modified for this general case. The C in Equation (8) is based on the assumption of 100% absorption. If, for example, only 50% were absorbed the actual concentration would be twice this value of C and the value of P would be reduced by half. Thus, the general relationship between T_{P} and the averaged model intestinal permeability (P_{M}) is:

{P}_{M}={F}_{A}\phantom{\rule{0.12em}{0ex}}P={F}_{A}\phantom{\rule{0.12em}{0ex}}r/\left(2{T}_{P}\right)

(13)

The amount absorbed (A_{M}(t)) as a function of time is:

\begin{array}{ll}{A}_{M}\left(t\right)& ={\displaystyle \underset{0}{\overset{t}{\int}}{R}_{M}\left(\tau \right)\mathit{dt}=M\{1+[\phantom{\rule{0.12em}{0ex}}{T}_{P}}\text{exp}\left(-t/{T}_{P}\right)\\ -{T}_{G}\text{exp}\left(-t/{T}_{G}\right)\phantom{\rule{0.12em}{0ex}}]/\left({T}_{G}-{T}_{P}\right)\phantom{\rule{0.12em}{0ex}}\}\end{array}

(14)

These AM model expressions for the intestinal absorption rate R_{M}(t) and P_{M} are only approximations to the exact DC model for this general case where not all the solute is absorbed. The range of validity of this approximation will be evaluated by comparing it to the DC model for a range of experimental parameters (see Results, Comparison of DC and AM models).

R_{M} and M represent the rate and total amount absorbed across the small intestinal epithelial luminal membrane. Assuming a linear system, the rate of solute entering the systemic circulation (R_{SM}) is:

\begin{array}{l}{R}_{\mathit{SM}}\left(t\right)={M}_{S}\phantom{\rule{0.24em}{0ex}}\left[\text{exp}\right(-t/{T}_{G})\phantom{\rule{0.36em}{0ex}}-\text{exp}(-t/{T}_{P}\left)\right]/({T}_{G}-{T}_{P})\\ {M}_{S}={F}_{A}\left(1-{E}_{H}\right)(1-{E}_{I})\phantom{\rule{0.12em}{0ex}}\mathit{Dose}\end{array}

(15)

where E_{H} and E_{I} are the hepatic and intestinal extraction ratios [15]. The hepatic extraction (E_{H}) can be estimated from the liver blood flow (Q_{H}) [15] and the whole blood liver clearance (Cl_{H}):

{E}_{H}=C{l}_{H}/{Q}_{H}

(16)

The liver clearance (Cl_{H}) was estimated by correcting the whole blood clearance following the IV infusion for the fractional renal clearance using data obtained in the same subjects that were used for the permeability estimates.

Equation (15) is a simple 3 parameter function whose parameters (M_{S}, T_{G} and T_{P}) can be determined experimentally by deconvolution (see below for details) of the blood concentration time course following IV and oral doses. The fraction absorbed (F_{A}) can be determined from M_{S} and estimates of E_{H} and E_{I} (Equation (15)). Finally, the AM model intestinal permeability (P_{M}) can be determined from F_{A} and T_{P} (Equation (13)).

Equation (15) is symmetrical in T_{G} and T_{P} so that there is an ambiguity in distinguishing the gastric emptying time constant T_{G} from the permeability time constant T_{P}. Most of the applications described here will be based on data obtained using oral solutions (not tablets) given to fasting subjects and the time constant that is closest to 10 to 15 minutes will be assumed to be T_{G} .

The theoretical accuracy of the AM model absorption rate RM (Equation 12) was evaluated by comparing it with the exact DC model R_{DC} (Equation 5). A set of the 7 DC parameters (Dose, T_{G}, T_{P,} T_{F}, T_{D}, r, L) were selected and the DC model intestinal absorption rate and fraction absorbed was determined. Then, the AM model parameters (M, T_{G}, T_{P}, Equation (12)) that provided the best fit to the DC absorption rate were determined by minimizing the following error function using the optimization routine in Maple (Maplesoft™):

\mathit{Error}=\left(1/N\right)\phantom{\rule{0.12em}{0ex}}{\sum}_{i=1}^{N}{\left({R}_{\mathit{DC}},\left[i\right],-,{R}_{M},\left({t}_{i}\right)\right)}^{2}

(17)

where t_{i} = i ∆t and comparing the AM model parameters (F_{A}, T_{P} and T_{G}) with the actual input DC parameters.

### Experimental determination of the averaged model (AM) parameters by deconvolution

The determination of the 3 AM model parameters (M_{S}, T_{G} and T_{P}) is based on standard procedures that have been described previously [6]. First, the 2 or 3 exponential systemic bolus response function r(t) is determined from the experimental blood concentration time course following the known IV infusion. The blood concentration C_{oral}(t) following the oral dose is equal to the convolution of r(t) and the AM model systemic absorption rate R_{SM}(t) (Equation (15)):

{C}_{\mathit{oral}}\left(t\right)={\displaystyle \underset{0}{\overset{t}{\int}}r\left(t-\tau \right)\phantom{\rule{0.12em}{0ex}}{R}_{\mathit{SM}}\left(\tau \right)\phantom{\rule{0.12em}{0ex}}\mathit{d\tau}}

(18)

The 3 AM model parameters (M_{S}, T_{G} and T_{P}) are then estimated by finding the parameter set that minimizes the error function:

\mathit{\text{Err}}={\displaystyle {\sum}_{k}\frac{\left|\phantom{\rule{0.12em}{0ex}}{C}_{\mathit{\text{oral}}}\left({t}_{k}\right)-{C}_{k}\right|}{{C}_{k}+\mathit{\text{noise}}}}

(19)

where C_{k} is the experimental blood concentration at time t_{k} following the oral dose. The “noise” determines the relative weighting of each data point and can be arbitrarily adjusted but is usually set to 10% of the average blood value. The optimized set of parameters is determined by a non-linear Powell minimization routine [16]. Most of the drugs were administered as oral solutions in fasting subjects and T_{G} was forced to be in the range of 10 to 20 minutes (the normal range for non-caloric fluids [13, 14]) and only the two parameters T_{P} and M_{S} were freely adjusted. For a few solutes that were administered as capsules or tablets, all 3 parameters were adjusted.

These procedures have been implemented in PKQuest Java, a freely distributed software program that has been used previously for pharmacokinetic analysis of more than 30 different solutes in a series of publications [7]. The implementation is designed to be user friendly and simple to use. The user only needs to enter 1) the dose and duration of the constant IV infusion; 2) the experimental blood concentration for the IV dose (which can be copied and pasted from a standard Excel file); and 3) the experimental blood concentration following the oral dose. The program then finds the optimum set of AM parameters. It also outputs 4 plots that are useful for evaluating the results: 1) A comparison of the experimental blood concentration for the IV dose versus the blood concentration predicted by bolus response function (there is usually nearly perfect agreement). 2) The AM model absorption rate as a function of time; 3) AM total absorption as a function of time; and 4) a comparison of the experimental blood concentration following the oral dose versus the AM model prediction (Equation (18)). This last plot is especially useful because it provides the best measure of the quality of the AM model. See Figures 1, 2, 3 and 4 for examples of these plots. PKQuest Java and a detailed tutorial can be freely downloaded from http://www.pkquest.com. Also available for download are the complete data sets for the 90 solutes discussed in this paper. This allows the user to reproduce all of the results.

### Experimental intestinal absorption data

In order to be a candidate for determination of intestinal permeability it was required that the solute met the following 4 conditions: 1) intravenous and oral dose pharmacokinetics in the same subject; 2) the oral dose was in the form of a solution (not tablet) to fasting subjects; 3) the drug’s pharmacokinetics are linear, at least in the concentration range that is investigated; 4) the drug is soluble at the concentrations used in the absorption study. These conditions severely limit the number of experimental results that can be used. Condition #1 is satisfied in only a small fraction of permeability studies. Condition #2 also severely restricts the number of possible candidates because tablets or capsules are used in most oral drug studies. A thorough search of the published literature returned 90 drugs that met these conditions. A few drugs that were administered as tablets have been included if the drug had a high water solubility so that the tablet would be rapidly dissolved and a low permeability (long T_{P}) that could not be confused with the T_{G}. The results and analyses are summarized in the Excel file that is included in the Additional file 1: “Table 2”. Additional file 1: Table 2 lists the solute, a link to the reference publication, the AM model parameters, a subjective measure of the quality of the AM fit to the data and the calculated permeability. The table includes the ionization behavior of the solute (weak acid, base, neutral or always ionized) in the pH range of 4 to 8 and the pKa if it is a weak base or acid. Also listed is an estimate of the experimental log(octanol/water) partition coefficient at pH 7.4 (log D). For most solutes there are multiple reported values of log D that can vary by as much a log unit. For those solutes which are available on the LOGKOW site maintained by James Sangster, the value listed is an approximate average of the listed values. When necessary, the log Pow values were converted from pH1 to a different pH2 using the following relations (this assumes that only the neutral solute has a finite octanol partition) [17]:

\begin{array}{ll}\mathit{\text{Mono}}\phantom{\rule{0.12em}{0ex}}\mathit{\text{protic}}\phantom{\rule{0.12em}{0ex}}\mathit{\text{base}}:\phantom{\rule{0.36em}{0ex}}\text{log}\mathit{Po}{w}_{2}& =\text{log}\mathit{Po}{w}_{1}+\text{log}\left(1+{10}^{\left(\mathit{\text{pKa}}-\mathit{pH}1\right)}\right)\\ -\text{log}(1+{10}^{\left(\mathit{\text{pka}}-\mathit{pH}2\right)})\\ \mathit{\text{Mono}}\phantom{\rule{0.12em}{0ex}}\mathit{\text{protic}}\phantom{\rule{0.12em}{0ex}}\mathit{\text{acid}}:\phantom{\rule{0.36em}{0ex}}\text{log}\mathit{Po}{w}_{2}& =\text{log}\mathit{Po}{w}_{1}+\text{log}\left(1+{10}^{\left(\mathit{pH}1-\mathit{\text{pKa}}\right)}\right)\\ -\text{log}(1+{10}^{\left(\mathit{pH}2-\mathit{pKa}\right)})\end{array}

(20)

The experimental perfused human jejunum permeability [18] and the Caco-2 permeability are also listed in Additional file 1: Table 2 if they were available. The form of the oral dose (solution, tablet, capsule) is listed and solutes which may have solubility limitations are marked in the table. If there is suggestive evidence that the intestinal absorption is protein mediated (either influx or efflux), this is also indicated. The experimental data points were read from the published figures using UN-SCAN-IT (Silk Scientific Corporation).