# Derivation of Closed-Form Analytical Solutions for Drug Absorption Models

**Derivation of Closed-Form Analytical Solutions for Drug Absorption Models**

The details leading to Equations 8, 9 and 12, corresponding to Case 1, 2 and 3 are shown below. For each case the total amount of drug absorbed is found by adding the drug absorbed during each of the three phases of absorption defined in Figure 5.

Case 1: Phase i (= 0 to , time to maximum concentration)

The concentration in solution in vitro () is calculated from the slope and intercept of phase i of the concentration-time profile, Figure 5(a).

/ (16)where is defined by Equation 2 and is the maximum drug concentration recorded during in vitro dissolution.

Case 1 assumes *in vivo mimics in vitro concentration (i.e*., = ). Substituting Equation 16 into 7 and integrating from 0 to , the time profile of the amount of drug absorbed during phase i () is:

The total amount of drug absorbed during phase i () is:

/ (18)Case 1: Phase ii (t = to , time from peak to plateau region)

During phase ii, the concentration of drug in solution in vitro()is described by the slope and intercept of phase ii of the concentration-time profile (Figure 5(a)).

/ (19)Where , the time in phase ii.

Again, substituting Equation 19 in 7where , integratingfrom 0 to and replacing with (), the time profile for absorption of drug during phase ii is:

/ (20)The total amount of drug absorbedduring phase ii is:

/ (21)Case 1: Phase iii ( = to , the plateau region)

When there is excess solid drug remaining, the concentration of dissolved drug in vivo ()is assumed to be equal to that in vitro () during all of phase iii.

/ (22)Substituting Equation 22 into 7where = and integrating from to , the time profile for drug absorption in Case 1, phase iii becomes:

/ (23)The total drug absorbed during phase iii is:

/ (24)*Case 1: Total Amount of Drug Absorption (* = to )

The total amount of drug absorbed using the assumptions in Case 1is found by adding the drug absorbed during each phase found in Equations 18, 21 and24.

/ (25)Substituting for , (Equation 3) and rearranging yields Equation 8 in the text.

Case 2: Phase i ( = 0 to , time to peak)

The concentration in vivoin the absence of absorption() is assumed to equal the in vitro drug concentration () in phase i, which is defined by Equation 26.

/ (26)The concentration of drug remaining in solution in vivo() is equal to the difference between the concentration in solution in vivoin the absence of absorption() and the amount absorbed () divided by the volume of the intestinal lumen () as shown in Equation 27.

/ (27)Combining equations 26 and 27, the concentration of drug remaining in solution in the intestinal lumen becomes:

/ (28)Substituting equation 28 into Equation 7 and integrating from 0 to yields the time profile for the amount of drug absorbedduring phase i ().

/ (29)Recalling that in Case 2, the time to reach the peak concentration in vivo() is equal to that in vitro(), the total amount of drug absorbed for Case 2 during phase i is:

/ (30)At , the maximum concentration in solution, =, is lower than the concentration in the absence of absorption() as shown in Figure 5(b).At , Equation 27 can be rearranged to yield an expression for the peak drug concentration in vivo () (Equation 31).

/ (31)Substituting the expression for (Equation 30) in Equation 31 yields an expression for , the maximum concentration reached in solution in vivo which is the drug concentration at the start of phase ii (Equation 10 in text).

Case 2: Phase ii ( = to , peak to plateau)

The concentration of drug in solution in vivo in the absence of absorption ()can be described by the slope and intercept of phase ii of the dissolution-time profile (Figure 5(b)).

/ (32)where , the time in phase ii.

Taking absorption into account, the in vivo drug concentration () can be obtained by substituting (Equation 32) into Equation 27.

/ (33)Substituting Equation 33 into Equation 7, integrating from 0 to and replacing with () yields the time profile for the amount of drug absorbed during phase ii ().

/ (34)The total drug absorbed during phase ii of Case 2 is:

/ (35)The end of phase ii is defined as the time the drug concentration falls to , the concentration at the plateau. Thus, the time at which the plateau concentration is reached () can be found by setting equal to at in Equation 33. Substituting the expression for (Equation 35) into Equation 33 and rearranging yields Equation 11 in text.

Case 2: Phase iii ( = to , the plateau region)

Making the same assumption for phase iii as for phase iii of Case 1, Equation 22 can be substituted into 7where = and integrated from to . The time profile for drug absorption in Case 2, phase iii becomes:

/ (36)The total amount of drug absorbedduring phase iii of Case 2 is:

/ (37)*Case 2: Total Amount of Drug Absorption (* = to )

There are two equations (9a and 9b) for calculating the total amount of drug absorbed for Case 2. When the supersaturation ratio () is greater than the product of and , drug absorption occurs during all three phases so that the sum of the amounts absorbed in each phase (Equations 30, 35 and 37) equals the total amount absorbed for Case 2 (Equations 9(a)). However, when drops below the product of and , then , eliminating phase ii. Therefore, the total amount of drug absorbed is calculated from phase i and iii only (Equations 30 and 37) (Equation 9(b)). Before rearrangement and simplification of Equation 9(b) by substituting (Equation 10), the total amount of drug absorbed is:

/ (38)Case 3: Phase i ( = 0 to , time to peak)

In Case 3, phase i follows the same derivation and behavior as in Case 2, phase i, except that phase i of Case 3 ends at. The time profile for the amount of drug absorbed is described by Equation 29. The total drug absorption during phase i is similar to Equation 30, except that replaces to yield:

/ (39)By definition, at , in vivo drug concentration in solution , is equal to the peak drug concentration from in vitro dissolution, . Similar to Equation 27 for Case 2, Equation 40 is appropriate for Case 3.

/ (40), the in vivo peak drug concentration in the absence of absorption, can be described by the slope of the in vitro concentration time profilein phase i as shown in Figure5(c).

/ (41)Combining and rearranging equations 39, 40 and 41 yields an expression for the time at which the peak drug concentration is reached in vivo () under the assumptions defining Case 3 (Equation 13 in text).

Case 3: Phase ii ( = to , peak to plateau)

In phase ii, the concentration of drug in solution in vivo in the absence of absorption () can be described by the slope and intercept of phase ii of the in vitro drug concentration vs.time profile (Figure 5(c)).

/ (42)where , the time in phase ii.

When absorption is taken into consideration, the concentration of drug in solution () can be determined by substituting from Equation 42 into Equation 27 and rearranging.

/ (43)Substituting (Equation 43) into Equation 7, integrating from 0 to and replacing with , yields the time profile for the amount of drug absorbed during phase ii.

/ (44)The total amount of drug absorbed during phase ii of Case 3is:

/ (45)At the plateau concentration () is reached (as shown in (Figure 5(c))and Equation 43 becomes:

/ (46)By combining and rearranging Equations45 and 46, an expression can be found for the time to reach the plateau concentration, (Equation 14 in text).

Case 3: Phase iii ( = to , the plateau region)

Making the same assumption for phase iii as for phase iii of Case 1 and 2, Equation 22 can be substituted into 7where = and integrated from to . The time profile for drug absorption in Case 3, phase iii becomes:

/ (47)The total drug absorbed during phase iii in Case 3 is:

/ (48)*Case 3: Total Amount of Drug Absorption (* = to )

There are two equations (9a and 9b) for calculating the total amount of drug absorbed for Case 3. When the time to peak () is smaller than the small intestinal transit time () drug absorption occurs during all three phases so that the sum of the amounts absorbed in each phase (Equations 39, 45 and 48) equals the total amount absorbed for Case 3 (Equations 12(a)). However, when ,phase i (Equation 39) completely dominates the absorption profile (Equation 12(b)).