Mathematical Models to Explore Potential Effects of Supersaturation and Precipitation on Oral Bioavailability of Poorly Soluble Drugs
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Abstract
Poorly soluble drugs are increasingly formulated into supersaturating drug delivery systems which may precipitate during oral delivery. The link between in vitro drug concentration profiles and oral bioavailability is under intense investigation. The objective of the present work was to develop closedform analytical solutions that relate in vitro concentration profiles to the amount of drug absorbed using several alternate assumptions and only six parameters. Three parameters define the key features of the in vitro drug concentrationtime profile. An additional three parameters focus on physiological parameters. Absorption models were developed based on alternate assumptions; the drug concentration in the intestinal fluid: (1) peaks at the same time and concentration as in vitro, (2) peaks at the same time as in vitro, or (3) reaches the same peak concentration as in vitro. The three assumptions provide very different calculated values of bioavailability. Using Case 2 assumptions, bioavailability enhancement was found to be less than proportional to in silico examples of dissolution enhancement. Case 3 assumptions lead to bioavailability enhancements that are more than proportional to dissolution enhancements. Using Case 1 predicts drug absorption amounts that fall in between Case 2 and 3. The equations developed based on the alternate assumptions can be used to quickly evaluate the potential improvement in bioavailability due to intentional alteration of the in vitro drug concentration vs. time curve by reformulation. These equations may be useful in making decisions as to whether reformulation is expected to provide sufficient bioavailability enhancement to justify the effort.
KEY WORDS
amorphous modeling oral absorption solubility supersaturationINTRODUCTION
The Biopharmaceutical Classification System (BCS) was designed to group compounds into one of four categories based on both solubility and permeation rate, in order to accelerate the drug development process (1). The majority of compounds currently in development are classified as BCS II, highly permeable and poorly soluble, requiring formulation strategies to increase solubility and dissolution rates (2).
One approach to increase solubility is to formulate the BCS II compounds in the amorphous state, where the amorphous solid produces a highly supersaturated solution. Once a maximum concentration is reached, precipitation can reduce the concentration to at/near crystalline solubility (3). It is often assumed that the higher the amorphous peak concentration, the higher the bioavailability. However, a higher degree of supersaturation does not always lead to increased absorption, indicating that this parameter alone is not a suitable predictor of bioavailability for potential amorphous formulation (4,5).
Over the past 25 years, mathematical models have been developed to estimate in vivo drug absorption from in vitro drug concentrationtime data (6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16). These biopharmaceutical models base their prediction of drug absorption on physicochemical properties of the drug (solubility, pKa, physical state), physiological properties (absorption rate constant, local pH, transit time), and dosage form (17). Current models range from simple equations to complex computer simulations. For instance, a simple model may use as few as four parameters to estimate drug absorption (e.g., the maximum absorbable dose (MAD) model considers only the absorption rate constant and solubility) (18,19), whereas computer models (e.g., GastroPlus™, Stella®, Intellipharm®, and Simcyp®) may utilize over 20 parameters (7,17,19, 20, 21, 22, 23, 24).
While there are several biopharmaceutical models in the literature that are reported to be capable of handling precipitating drug, most require a large number of parameters to reliably predict in vivo absorption (2,20). To date, there are no simple equations like the MAD in the literature that take into account the effect of precipitation in the intestinal lumen on bioavailability. It would be useful to have a model that has the capability of estimating the magnitude of the effect of formulation changes on oral drug absorption of amorphous compounds, based on a limited number of in vitro parameters in order to speed the screening of promising BCS II drugs during development (2).
The objective of the current work was to develop sparsely parameterized mathematical models relating in vitro drug concentrationtime profile (often referred to as a dissolution profile) of the amorphous form of a model drug to an estimate of in vivo drug concentrationtime profile. To obtain in vivo luminal concentrationtime profiles from in vitro dissolution profiles, precipitation was assumed to be dependent on time or dependentonconcentration. Closedform equations describing absorption were derived based on each set of assumptions.
MODEL DEVELOPMENT
Parameterization of In Vitro Drug ConcentrationTime Profiles
Each in vitro drug concentrationtime profile of an amorphous form differs in concentration at peak, duration of the peak, peak shape, and plateau concentration (Fig. 1). Previous investigators have modeled the supersaturation in vitro dissolution curve (26,27). However, our goal of obtaining a closedform analytical solution for the bioavailability of an amorphous formulation demands a more parsimonious parameterization of the in vitro curves. Here, we seek to capture the main features of each curve with a minimum number of parameters, allowing rapid characterization. Of course, the use of biological relevant medium and volumes will provide better input (28, 29, 30, 31, 32, 33). To be sure, there is some loss in accuracy when a minimum number of parameters are used to characterize a wide variety of peak shapes. The simplicity and timesavings gained from using few parameters on which to base an estimate of bioavailability is thought to outweigh any real or suspected accuracy obtained from describing the experimental dissolution curve in greater detail.
It would initially seem that characterizing the in vitro concentration vs. time profile using the area under the curve (AUC) would be the simplest and most direct approach. When there is negligible depletion of drug from intestinal fluids due to low absorption rate constant, as is the case for BCS IV compounds, the in vivo concentration vs. time profile mirrors in vitro. The AUC of the in vivo concentration vs. time profile can be determined from the in vitro curve when using biologically relevant medium and volumes. The total amount of drug absorbed is simply the product of in vivo AUC, intestinal fluid volume, and absorption rate constant. However, the main focus of the proposed model is to estimate the amount of drug absorbed for BCS II compounds. BCS II drugs have significant absorption rate constants which leads to significant depletion of drug from gastrointestinal fluid, so that the total amount of drug cannot be directly calculated from AUC without underestimating the amount of drug absorbed.
Definition of Model Terminology Using the Structure A_{b}c_{d}
Variable  Vitro/vivo  Condition  Case  

Concentration  C  In vitro = in glass  g  Excess  e  Simple  1 
Time  T  In vivo = in living  l  Maximum  m  Dependentontime precipitation  2 
Amount  X  Plateau  p  Dependentonconcentration precipitation  3  
Volume  V  Absorbed  a  
In solution/absorption occurring  s  
In solution and absorption NOT occurring  n  
Residence/end  r  
Initial  o 
A second approach is to approximate the peak as an isosceles triangle (Fig. 2b) where the maximum concentration (C _{g} m), the plateau concentration (C _{g} p), and the time to peak (T _{g} m) are identified directly from the in vitro drug concentration profile. Using this approach, the time to reach the plateau concentration (T _{g} p) can be calculated from the relationship between two similar triangles. The use of isosceles triangles provides simplicity in the development of the model and in the resulting equations used to estimate the bioavailability. However, this approach more severely underestimates the area under the experimental peak than does the first approach.
A third approach combines the advantages of the first two. The maximum drug concentration (C _{g} m) and the plateau drug concentration (C _{g} p) following the peak are identified directly from the in vitro drug concentration profile. However, in this approach, (T _{g} m) and (T _{g} p) are not determined directly from inspection of time points on the profile. Rather, the width of the peak is characterized by its full width at half maximum.
Using this heuristic approach, the area under the threeparameter approximation (C _{g} m, C _{g} p, and T _{HM}) of the dissolution curve differs from the area under the experimental dissolution profile by only 9% for the case of indomethacin, 12% for the case of hydrochlorothiazide, and 26% for the case of danazol (Fig. 1). Several other drug dissolution profiles (34) were evaluated and the difference in area under the curve is 14 ± 7%. It should be noted that the time to peak using the isosceles triangular approximation will not always equal the time to peak concentration from the in vitro drug concentrationtime profile. However, the simplicity in the derivation of bioavailability equations using the isosceles triangle to approximate the experimental data is thought to outweigh the loss in fidelity in the description of all aspects of the peak shape.
Indeed, there are a myriad of other ways to approximate the variety of shapes of in vitro concentrationtime profiles. For some curves, a peak drug concentration is reached quite rapidly and maintained for a period of time until very rapid precipitation occurs, resulting in a concentrationtime profile that may be better parameterized using a rectangular approximation as illustrated in Fig. 2d. However, with this approach another parameter, a “lag” time, or the time to reach in vitro peak concentration, C _{g} m, is often needed (Fig. 2e). For some profiles (e.g., hydrochlorothiazide and danazol in Fig. 1), a rectangular shape would overestimate the area under the curve. Using a rectangular approximation without the “lag” time (Fig. 2d) would allow peak approximation using just three parameters, as was achieved using the isosceles triangular approximation (Fig. 3). However, the area under the curve is often more severely overestimated than without the use of a “lag” time parameter.
Another approach would be to characterize in vitro profile by either the triangle or rectangular approximation, whichever fits best. However, this would eliminate the ability to use a single system to compare drug concentrationtime curves of a single drug from different formulations. Later in the text, the isosceles triangular approximation is compared to an equivalent rectangular approximation in order to identify any bias imposed by the use of the triangular approximation.
Relationship Between In Vitro Drug Concentration Profile and Drug Concentration in Intestinal Fluid
A onetank model (9) is used to calculate the amount of drug absorbed from the threeparameter approximation of the in vitro concentration profile. Originally, the onetank model was used to describe dissolutionlimited absorption, with no precipitation. Here, it is extended to include precipitation.
This process is more complicated in vivo (35). For the simplest case where the dose administered in vivo (X _{l} o)^{3} is introduced directly into the small intestines (by, for example, an entericcoated dosage form), many of the processes in vivo (Fig. 4b) and in vitro (Fig. 4a) are similar, where the amount of drug in solution (X _{l} s(t)) and amount of excess solid (X _{l} e(t)) vary with time and depend on the volume available for dissolution in the luminal fluid (V _{l}).
There are two major differences between the in vitro and in vivo systems. Due to depletion of dissolved drug via absorption, the concentration of drug in solution (X _{l} s(t)/V _{l}) in the small intestine will be lower than the concentration of drug during in vitro dissolution (X _{l} s(t)/V _{g}) (36). Furthermore, the in vitro fluid volume (V _{g}) is traditionally much greater than small intestinal fluid volume (V _{l}). Using USP Apparatus (type 1 or 2), the volume ranges from 250 mL to 1 L (32), while the volume of the small intestinal fluid has been reported to be only 45 to 320 mL (37). The lower in vivo volume can result in a significantly lower amount of drug in solution in vivo (22). This is particularly important for amorphous formulations where the degree of supersaturation and excess remaining solid can influence the time course of precipitation (3).
Alternatively, assuming drug precipitation is dependent on time, absorption would deplete the concentration of dissolved drug resulting in a peak drug concentration in vivo that is lower than in vitro. In this case, the bioavailability would be less than proportional to the area under the in vitro dissolution curve. Using the threeparameter isosceles triangular approximation of the dissolution curve, this is equivalent to assuming precipitation occurs at the same time in the small intestine as determined in vitro (Fig. 5b).
In contrast, assuming drug precipitation is dependentonconcentration, the drug concentration in vivo would eventually reach the same peak in vitro concentration. However, depletion due to absorption of drug from the small intestine would prolong the time to peak. The extended period of elevated drug concentration would result in greater absorption compared to that predicted from the in vitro dissolution area under the curve. Using the isosceles triangular approximation, this case is equivalent to in vivo precipitation occurring at the same concentration as observed in vitro (Fig. 5c).
It is not known with certainty whether the concentration profiles are the same in vitro and in vivo, or whether it is the in vivo peak drug concentration or the duration of the elevated concentration that corresponds better the in vitro dissolution profile. However, the drug concentration behavior in vivo is likely bracketed by assumptions (a) equivalent duration of the peak in vitro and in vivo (Fig. 5b) and (b) equal peak concentrations in vitro and in vivo (Fig. 5c). Potential bioavailability enhancement that could be achieved by formulation changes which alter the drug peak concentration and/or duration should be within these two extremes. Therefore, derivation of closedform analytical equations based on these assumptions allows us to easily quantify the potential effects of precipitation on bioavailability of supersaturating drug delivery systems.
Three Cases for In Vivo Dissolution Behavior
Depending on the drug properties and the conditions in the small intestine, three cases based on the above assumptions can be explored using the isosceles triangular approximation of the in vitro dissolution.
For Case 1, the drug concentration in small intestinal fluid (C _{l} s(t)) mirrors the in vitro drug concentration vs. time curve (C _{g} s(t)). Using the threeparameters characterizing the in vitro curve, the peak drug concentration (C _{l} m _{1}) in vivo occurs at the same time and concentration as seen in vitro (C _{g} m) (Fig. 5a). This scenario is limited to drugs with low absorption rate constants, such as is the case for BCS IV compounds.
For Case 2, the duration of the elevated drug concentration in vivo corresponds to the in vitro dissolution behavior. Using the isosceles triangle approximation, the peak drug concentration in vivo (C _{l} m _{2}) occurs at the same time (T _{l} m _{2} = T _{g} m) in vitro, but due to absorption, the peak drug concentration in vivo (C _{l} m _{2}) is lower than in vitro (C _{g} m) (Fig. 5b). This scenario is more consistent with precipitation being a dependentontime phenomenon and is referred to below as the dependentontime precipitation case.
For Case 3, precipitation in vivo (C _{l} m _{3}) occurs at the same concentration (C _{g} m) as seen in vitro, but due to absorption, the time to reach peak concentration in vivo (T _{l} m _{3}) occurs at a later time than in vitro (T _{g} m) (Fig. 5c). This scenario is more consistent with precipitation being a dependentonconcentration phenomenon and is referred to below as the dependentonconcentration precipitation case.
Development of Mathematical Equations Describing Drug Absorption for Precipitating Drugs
 1.
Drug dissolution and absorption occur in the small intestine only.
 2.
The small intestinal lumen can be considered a single mixing tank at a constant volume.
 3.
Drug absorption is a firstorder process with respect to concentration.
 4.
Once drug precipitation is complete, drug concentration remains constant.^{4}
 5.
There is excess solid drug remaining at the end of absorption.
 6.
Drug precipitation is nonpH dependent.

Phase i: The period between initial drug release and peak concentration

Phase ii: The time from peak drug concentration to the plateau concentration

Phase iii: The period over which the plateau concentration extends (where the drug in solution is in equilibrium with the solid drug)
The total amount of drug absorbed is the sum of drug absorbed during each phase. Equations describing the total amount of drug absorbed for each model assumption (i.e., Cases 1, 2, and 3) are presented below. Details of their deviation are found in Supplemental 1.
Case 1
Alphabetical Listing of Abbreviations
Term  Description  Term  Description 

C _{g} m  Maximum concentration reached in vitro  T _{l} m _{2}  Time maximum concentration or amount is reached in vivo for Case 2 = T _{g} m 
C _{g} m _{2rt}  Maximum concentration reached in vitro for Case 2rt  T _{l} m _{3}  Time maximum concentration or amount is reached in vivo for Case 3 
C _{g} m/C _{g} p  Supersaturation ratio in vitro  T _{l} p _{1}  Time plateau concentration or amount is reached in vivo for Case 1 = T _{g} p 
C _{g} p  Plateau concentration reached in vitro = solubility in vitro  T _{l} p _{2}  Time plateau concentration or amount is reached in vivo for Case 2 
C _{g} s  Drug concentration in solution in vitro  T _{l} p _{2rc}  Time plateau concentration or amount is reached in vivo for Case 2rc 
C _{l} m/C _{l} p  Supersaturation ratio in vivo  T _{l} p _{2rt}  Time plateau concentration or amount is reached in vivo for Case 2rt 
C _{l} m or C _{l} m _{1}  Maximum concentration reached in vivo for Case 1 = C _{g} m  T _{l} p _{3}  Time plateau concentration or amount is reached in vivo for Case 3 
C _{l} m _{2}  Maximum concentration reached in vivo for Case 2  T _{l} r  Small intestinal transit time or residence time in vivo 
C _{l} m _{3}  Maximum concentration reached in vivo for Case 3 = C _{g} m  V _{g}  In vitro dissolution volume 
C _{l} n  Drug concentration in vivo in the absence of absorption  V _{l}  Small intestinal fluid volume in vivo dissolution volume 
C _{l} p  Plateau concentration reached in vivo = C _{g} p  X _{g} e  Amount of excess solid drug in vitro 
C _{l} s  Drug concentration in solution in vivo  X _{g} o  Initial amount of solid drug in vitro 
k _{a}  Firstorder absorption rate constant  X _{g} s  Amount of drug in solution in vitro 
T _{g} m  Time maximum concentration or amount is reached in vitro  X _{l} a  Amount of drug absorbed in vivo 
T _{g} p  Time plateau concentration or amount is reached in vitro  X _{l} e  Amount of excess solid drug available for dissolution in vivo 
T _{g} r  End of experiment in vitro  X _{l} o  Initial amount of solid drug in vivo = dose 
T _{l} m _{1}  Time maximum concentration or amount is reached in vivo for Case 1 = T _{g} m  X _{l} s  Amount of drug in solution during in vitro dissolution 
Case 2
The dependentontime precipitation case assumes the peak duration in vivo corresponds to the in vitro time to peak in the absence of absorption. Using the threeparameter approximation of the in vitro dissolution time profile, this assumption translates to the time to peak in vivo equaling the time to peak in vitro (T _{l} m _{2} = T _{g} m), as shown in Fig. 5b. There are two equations (9a and 9b) for calculating the amount of drug absorbed (\( {\left.{X}_{\mathrm{l}}{a}_2\right}_0^{T_{\mathrm{l}}r} \)) using Case 2 assumptions depending on the supersaturation ratio (\( \frac{C_{\mathrm{g}}m}{C_{\mathrm{g}}p} \)).
With depletion due to absorption, the in vivo concentration will be lower than the concentration in solution in vitro (C _{g} s(t)), as shown in Fig. 5b. While the time to peak remains unchanged (T _{l} m _{2} = T _{g} m) in Case 2, we note that the peak drug concentration in vivo (C _{l} m _{2}) is lower than the peak concentration in vitro (C _{g} m) (Eq. 10). As in Case 1, the plateau concentration in vivo (C _{l} p) remains equal to that in vitro (C _{g} p), i.e., C _{g} p = C _{l} p. However, the time to reach the plateau in vivo (T _{l} p _{2}) is less than that observed in vitro (T _{g} p) (Eq. 11), again due to absorption. The time to reach plateau concentration (T _{l} p _{2}) is generally greater than T _{g} m. However, when the supersaturation ratio (C _{g} m/C _{g} p) drops below the product of k _{a} and T _{g} m, then T _{l} p _{2} < T _{g} m, eliminating phase ii. The total amount of drug absorbed is calculated from phases i and iii only (Eq. 9b).
Case 3
The dependentonconcentration precipitation case assumes drug concentration reaches the same peak value in vivo as in vitro (C _{l} m _{3} = C _{g} m). The total amount of drug absorbed during all three phases (\( {\left.{X}_{\mathrm{l}}{a}_3\right}_0^{T_{\mathrm{l}}r} \)) is determined by Eq. 12a or 12b, depending on the time to peak (T _{l} m _{3}) relative to small intestinal transit time (T _{l} r).
In the absence of absorption, as in Case 1, the concentration of drug in solution in vivo (C _{l} s(t)) would equal the concentration of drug in solution in vitro (C _{g} s(t)). However, as in Case 2, absorption results in a depletion in drug concentration in solution in vivo (C _{l} s(t)). Therefore, the peak drug concentration in vivo occurs at a later time (T _{l} m _{3}) than the time derived from the in vitro profile (T _{g} m), such that T _{l} m _{3} ≥ T _{g} m (Eq. 13). When the value of T _{l} m _{3} is greater than the small intestinal transit time (T _{l} r), phase i completely dominates the absorption profile (Eq. 12b).
In all three cases, the extent of drug absorption can be calculated from closedform analytical equations. Only six parameters are required to obtain solutions, three of which are determined from in vitro concentration vs. time profiles. From these equations, the potential bioavailability enhancement that could be gained from reformulation efforts to modify supersaturation peak duration and/or concentration may be quantitatively explored a priori.
METHODOLOGY
The amount of drug absorbed using the assumptions corresponding to each case described above was calculated from Eqs. 8, 9a, 9b, 12a, and 12b for three drug constructs in a patient described by average physiological parameters. The value for small intestinal fluid volume (V _{l}) is 70 mL and the transit time (T _{l} r) is 180 min, which represent median values from the literature (22,37,44, 45, 46). The plateau concentration (C _{g} p) was set to 10 μg/mL to reflect the low solubility values for many BCS II compounds, particularly those for which dissolution enhancement strategies, such as amorphization, are sought.
The values of the remaining three parameters—absorption rate constant (k _{a}), peak drug concentration reached during in vitro dissolution performance evaluation (C _{g} m), and the parameter reflecting drug concentration peak duration in vitro (T _{g} m)—were explored over wide ranges. The k _{a} values for BCS II and IV compounds were found to range from 0.0003 to 0.08 min^{−1}(47). The ranges for both C _{g} m and T _{g} m values were chosen based on experimental data (34). A 3 × 6 × 5 fullfactorial design for each case was utilized where k _{a} was evaluated at three levels, 0.001, 0.01, and 0.1 min^{−1}; C _{g} m ranged from 20 to 100 μg/mL in 20 μg/mL increments corresponding to maximum supersaturation ratio (C _{g} m/C _{g} p) from 2 to 10; and T _{g} m was evaluated in 10 min increments from 10 to 60 min.
RESULTS
The mathematical models (Eqs. 8, 9a, 9b, 12a, and 12b) were developed to quickly evaluate the potential improvement in bioavailability due to intentional alteration of the in vitro drug concentration vs. time curve by reformulation. These equations may be useful in making decisions as to whether reformulation is expected to provide sufficient bioavailability enhancement to justify the effort.
The models were applied to drug constructs with a range of absorption rate constant. As will be shown below, the relationship between in vitro dissolution profile and bioavailability is dependent on k _{a}. Therefore, the results for low, moderate, and high k _{a} values are presented in separate sections. Within each section, the bioavailability enhancement gained by prolonging T _{g} m and increasing C _{g} m/C _{g} p will be discussed in detail using the assumptions used in Cases 1, 2, and 3.
Drugs with a Low Absorption Rate Constant (k _{a} = 0.001 min^{−1})
For these poorly permeable compounds, an increase in peak duration (T _{g} m) and/or peak height relative to the plateau (C _{g} m/C _{g} p) generally results in an increase in the total amount of drug absorbed. However, at lower peak concentrations, C _{g} m/C _{g} p = 2, sustaining the duration of the peak (i.e., increasing T _{g} m from 10 to 60 min) does little to improve the amount of drug absorbed (i.e., less than 7% increase^{8}). Similarly, when the peak concentration cannot be sustained (i.e., T _{g} m = 10 min), there is only about a 40% improvement in the amount of drug absorbed when efforts are made to raise the in vitro peak drug concentration from 2 to 10 times the plateau (or drug solubility) over the short supersaturation period. These results are not surprising since the peak duration and extent of supersaturation are very low in each of these two scenarios. In contrast, when the in vitro peak drug concentration is high (e.g., 10 times the plateau concentration), increasing the peak duration, T _{g} m, from 10 to 60 min increases the amount of drug absorbed by 150% of the original value. Furthermore, when the peak duration is longer (e.g., T _{g} m = 60 min)^{9}, the bioavailability enhancement is increased by about 250% when the peak concentration is raised 5 times (i.e., C _{g} m/C _{g} p from 2 to 10).
Drugs with an Intermediate Absorption Rate Constant (k _{a} = 0.01 min^{−1})
Next, we consider drugs which have the same in vitro dissolution profiles as explored above, but now which have an order of magnitude greater absorption rate constant. Figure 6b, e, h shows about an order of magnitude increase in the amount of drug absorbed over Fig. 6a, d, and g, respectively. In fact, for Case 1, where the assumption is that the in vivo drug concentrationtime profile mirrors exactly that observed in vitro, increasing k _{a} by one order of magnitude will increase drug absorption by exactly one order of magnitude (Fig. 6a–c and Eq. 8). In contrast, when the drug concentrationtime profile in vivo does not exactly mirror that in vitro, but rather differs in the manner described by either Case 2 or 3, the amount of drug absorbed is not proportional to k _{a}.
In Case 2, where the in vivo peak is dependentontime and corresponds to in vitro time to peak, the amount absorbed is less than proportional to the absorption rate constant, whereas using the Case 3 assumptions, the amount of drug absorbed is more than proportional to k _{a}. This contrast is most clearly seen when the peak concentration and peak duration are both large (Fig. 6d vs. e and g vs. h). In this example, two drugs with the same in vitro dissolution profiles are characterized by T _{g} m = 60 min and C _{g} m/C _{g} p = 10. The drug with a k _{a} of 0.01 min^{−1} (vs. 0.001 min^{−1}) results in a 700% higher amount of drug absorbed using Case 2 assumptions (Fig. 6d vs. e).^{10} In contrast, using Case 3 assumptions, the extent of drug absorption is 1200% greater (Fig. 6g vs. h). In comparison, for Case 1, the increase in absorption is exactly one order magnitude greater or 900% (Fig. 6a vs. b). Therefore, the increase in the amount of drug absorbed one would expect for drugs with larger absorption rate constants (assuming the same in vitro dissolution profile) is highly dependent on the relationship between the in vitro and in vivo dissolution profiles (i.e., Case 2 vs. Case 3 assumptions). Until we can reliably extrapolate in vitro dissolution to in vivo, it is useful to quantitatively explore the effect of these two extremes in behavior on the potential range of amount of drug absorbed.
In contrast, we can explore the alternative assumption that the peak drug concentrations in vivo and in vitro reach equal values (dependentonconcentration precipitation—Case 3). In this case, drug depletion by absorption lengthens the time to reach in vivo peak concentration. Therefore, as the in vitro peak duration increases (i.e., higher T _{g} m), phase i dominates (Fig. 7g vs. h) since time to maximum concentration (T _{l} m _{3}) increases and eventually exceeds the intestinal transit time (T _{l} r) (Fig. 8c) (Eq. 12b).
Drugs with Large Absorption Rate Constant (k _{a} = 0.1 min^{−1})
Case 1 assumptions do not account for depletion of dissolved drug due to absorption. So, the trends in the amount of drug absorbed (Fig. 6c) are similar to those seen for the low (Fig. 6a) and intermediate (Fig. 6b) absorption rate constants, only an order of magnitude higher than for k _{a} = 0.01 min^{−1}. However, an absorption halflife corresponding to k _{a} of 0.1 min^{−1} is less than 10 min, which is short compared to the small intestinal transit time. Therefore, Case 1 is much less appropriate to use for high absorption rate constant drugs, due to depletion through absorption.
Using Case 2 and Case 3 assumptions, where the in vivo drug concentration profiles do not mirror those in vitro, the effect of enhancing the in vitro drug concentration profile has more interesting consequences on drug absorption. For the case in which the duration of the in vitro peak drug concentration corresponds to in vivo, Case 2, the amount of drug absorbed is fairly insensitive to alterations in the in vitro dissolution profile as characterized by T _{g} m and C _{g} m (Fig. 6f vs. d). This result can be traced to the reduction in the time to plateau, T _{l} p _{2}, by depletion due to absorption (Fig. 8a, b and Eq. 9b); little to no drug is absorbed in phase ii at the highest value of k _{a} (Fig. 7f vs. d). For drugs with very high absorption rate constants, formulation efforts to increase the duration of the peak would actually lead to a slight decrease in drug absorption, assuming dependentontime precipitation, Case 2.
The opposite is true for Case 3, which assumes the in vivo peak drug concentration reaches the in vitro value. Drugs with very high k _{a} (0.1 min^{−1}) have in vivo peak durations, T _{l} m _{3}, that exceed the small intestinal transit time (180 min) (Fig. 8c). Consequently, when absorption halflife approaches T _{g} m, T _{l} m _{3} approaches infinity and phase i completely dominates (Fig. 7i) the absorption profile. Therefore, the total amount of drug absorbed can be found from Eq. 12b. As in the case of drugs with intermediate absorption rate constants (k _{a} = 0.01 min^{−1}), we also find some counterintuitive consequences of altering the in vitro drug concentration profile for drugs with very high absorption rate constants (k _{a} = 0.1 min^{−1}). According to Eq. 12b, drug absorbed is proportional to C _{g} m/T _{g} m. Thus, a decrease in peak duration results in greater amount of drug absorbed. This counterintuitive result can be explained by examining the phases of absorption in more detail. For a given dissolution enhancement, C _{g} m/C _{g} p, a shorter peak duration in vitro translates to a shorter time to peak (using the isosceles triangular approximation) which provides more drug in solution sooner. When the absorption rate constant is very high, a large amount of drug is absorbed, maintaining the in vivo drug concentration below in vitro peak concentration that presumably “triggers” precipitation, in this case. The consequence is that the in vivo drug concentration remains supersaturated for the duration of the intestinal transit time.
Sensitivity of the Absorption Estimate to Geometry of Peak
The above results are based on the threeparameter isosceles triangular approximation of the in vitro drug concentration peak. To assess any bias imposed by this approximation, an alternative equivalent rectangular peak shape was explored. The rectangular peak was considered to represent the opposite extreme to the single time point at C _{g} m in the triangular approximation. Using a rectangular shape to approximate the peak, maximum drug concentration ranges from t = 0 to the end of phase ii (Supplemental 2). A comparison of triangular and rectangular peak approximations was performed for Case 2 since Case 2 generally includes all three phases unlike Case 3 which remains in phase i when the product of k _{a} and T _{g} m is large.
The amount of drug absorbed calculated from the triangular peak approximation differed from either rectangular approximations by −10 to 30% (Supplemental 4). The bias introduced by using the triangular peak approximation is small compared to the effect of maximum peak concentration, peak duration, and absorption rate constant shown in the previous section (Fig. 7). This analysis provides the confidence to use the threeparameter isosceles triangular approximation to quantitatively explore drug absorption for supersaturating drug delivery systems using Eqs. 8, 9a, 9b, 12a, or 12b, depending on the assumptions one wishes to make regarding the relationship of in vivo dissolution and in vitro dissolution behavior. More details can be found in Supplemental 3.
DISCUSSION
Reports show that measures of enhanced dissolution are not always predictive of enhanced bioavailability (5,25). Potential reasons for the discrepancies were explored quantitatively using absorption models that require limited data. The closedform equations developed here can also be used to determine the potential effects of key dissolution and physiological parameters on bioavailability for a promising new chemical entity with limited solubility.
The in vitro peak drug concentration and peak duration produced by supersaturating dosage forms are highly dependent on the drug, drug form, formulation components, and solution composition (5,28, 29, 30, 31, 32, 33,38,39,48, 49, 50). To bracket a range of bioavailability that could be reasonably expected from supersaturating dosage forms, two alternative assumptions were used: either (a) the duration of the peak drug concentration in vivo corresponds to the in vitro data or b) the drug concentration in vivo reaches the peak concentration observed during in vitro dissolution. When we assumed that the duration of the in vivo peak corresponds to the in vitro peak duration (i.e., Case 2), drug absorption was predicted to be less sensitive to alterations in the in vitro dissolution profile, especially for drugs with moderate to high k _{a} values (0.01–0.1 min^{−1}). It is particularly notable that when the absorption rate constant is high (k _{a} = 0.1 min^{−1}), formulation efforts to increase the in vitro drug concentration peak or duration are predicted to lead to a slight decrease in the amount of drug absorbed using Case 2 assumptions (Fig. 6f).
Contrast this result with Case 3, where the in vivo peak drug concentration is assumed to reach the in vitro peak value, despite depletion due to drug absorption. In this case, the amount of drug absorbed was very sensitive to alterations in the in vitro dissolution profile, particularly for compounds with high k _{a} values. When k _{a} is very large (k _{a} = 0.1 min^{−1}), formulation efforts to increase the drug peak concentration and/or decrease the peak duration were predicted to result in a substantial increase in drug absorption using Case 3 assumptions (Fig. 6i). In general, when the absorption rate constant of a drug is known or can be estimated, the current models may be useful to quantitatively explore whether or not it would be worth the effort to formulate for enhanced in vitro dissolution profiles.
In contrast to Cases 2 and 3, Case 1 predicts that the amount of drug absorbed is influenced by absorption rate constant, the peak concentration, and its duration in a straightforward manner. However, Case 1 only applies to a limited number of drugs which have both low solubility and very low absorption rate constants, such as BCS IV compounds. Case 1 is of limited use since most supersaturating drug delivery systems in development are formulated for BCS II compounds which have low solubility but higher absorption rate constant.
Several approaches to characterizing in vitro concentration vs. time curves were considered, including the simplest approach, the AUC approach. However, the AUC approach could not be utilized for Cases 2 and 3 without underestimating the amount of drug absorbed due to significant drug depletion. Therefore, an approximation or some estimate of a function that describes the course of the dissolved drug concentration was required. The isosceles triangulation approach was selected over others because it was a good representation of various experimental in vitro concentrations vs. time curve. Furthermore, this approach allowed the math to be significantly simplified, yielding closed analytical solutions.
For each of these cases, the in vitro concentrationtime curves are quickly characterize by the isosceles triangular approximation using just the three parameters: (1) peak drug concentration, (2) duration of the peak which is characterized by the full width at half maximum, and (3) the plateau drug concentration that often follows the peak. A comparison of the threeparameter triangular approximation with an equivalent rectangular peak showed that peak shape had much less influence on estimates of drug absorption than parameters such as absorption rate constant, in vitro peak drug concentration, and its duration. In short, for the purposes of rapidly calculating the potential effects of reformulation on bioavailability, the use of only three parameters to characterize the in vitro dissolution triangle profile provides a reasonable estimate of drug absorption without incurring large bias. Furthermore, this approach permits derivation of closed analytical solutions.
This analysis assumes the in vitro dissolution medium is carefully selected to mimic in vivo conditions, since three out of the six parameters needed for the model are determined from the in vitro drug concentrationtime profiles. Important factors to consider when performing dissolution experiments are temperature, hydrodynamics, and medium selection (28, 29, 30, 31, 32, 33). For example, the addition of bile salts and/or phospholipids has been shown to alter drug precipitation kinetics (33) by effecting the nucleation and/or crystal growth rate (28,51).
Two physiological parameters used in this model are small intestinal fluid volume, V _{l}, and small intestinal transit time, T _{l} r. An increase in V _{l} results in a proportional increase in the estimated drug absorbed (Eqs. 8, 9a, 9b, 12a, and 12b) in all cases. However, time dependence of the small intestinal fluid volume containing the drug will likely have more complex effects. Furthermore, variation in small intestinal transit time also leads to variation in the amount of drug absorbed, at least for cases in which there is undissolved drug remaining at the end of the absorption process as assumed in this work. Additional assumptions that limit the applicability of Eqs. 8, 9a, 9b, 12a, and 12b are described in the “MODEL DEVELOPMENT” section. Models may also be evaluated for nonenteric drug forms due to the slow introduction into the small intestine.
The advantage of the current approach is the small number of input parameters required to define key aspects affecting bioavailability. Due to the limited number of input variables, the models can be used to quantify variation in bioavailability based on known variations of small intestinal fluid volume and small intestinal transit time, as well as potential or experimentally determined variation in drug peak concentration and duration. Future studies will focus on using this model to evaluate variations in bioavailability contributed from both physiological and formulation input parameters which may potentially lead to clinically unacceptable high variation in the performance of the drug product.
CONCLUSION
Two alternative assumptions (i.e., equal drug peak concentrations or corresponding peak durations in vitro and in vivo) lead to what are expected to be two extremes in effect of reformulation on bioavailability of precipitating dosage forms. If we assume equivalent peak duration in vitro and in vivo, the model suggests that formulation improvements will have less impact on bioavailability than if we assume equal peak concentrations are reached. A simpler case, wherein both the peak drug concentration and peak duration are both assumed equal in vitro and in vivo, leads to a bioavailability estimate that lies between the extremes. The simpler case provides a more straightforward calculation, but its applicability may be limited to drugs with low absorption rate constant.
Until we can reliably extrapolate in vitro dissolution to in vivo, it is useful to quantitatively explore the effect of these two extremes in behavior on the potential range of amount of drug absorbed. The models developed here may be a useful tool to guide the thinking of formulators; potential outcomes of proposed strategies to either increase the peak drug concentration and/or the peak duration can be quantitatively evaluated using Eqs. 8, 9a, 9b, 12a, or 12b depending on assumptions.
Footnotes
 1.
The subscript g denotes in vitro (Latin for “in glass”). See Table I for a full description of the nomenclature.
 2.
Furthermore, it retains the benefits of using an isosceles triangle to approximate the peak. For example, the width of the base of the isosceles triangle which lies at C _{g} p is exactly twice T _{HM} as shown in Fig. 3. Additionally, the width of the base of the larger isosceles triangle is twice T _{g} m, the time to reach the apex of the isosceles triangle.
 3.
The subscript l denotes in vivo (Latin for “within the living”). See Table I for a full description of the nomenclature.
 4.
Drug precipitates to what is thought to be very small particles, perhaps in the nanometer range. These many minute drug particles have a very large surface area, making it less likely that dissolution is rate limiting.
 5.
Neglect for the purpose of this simple case, the reduction in drug concentration in the lumen due to absorption. This is handled in Case 2 and Case 3.
 6.
\( {\left.{X}_{\mathrm{l}}{a}_1\right}_0^{T_{\mathrm{l}}r} \) is also equal to the product of k _{a} and AUC of the in vivo concentration vs. time profile since the assumption is made that in vivo drug concentration vs. time curve mirrors the in vitro.
 7.
The subscript g denotes in vitro (Latin for “in glass”) and the subscript l denotes in vivo (Latin for “within the living”). A second subscript indicates the case (see Table II).
 8.
Percent change is expressed as a percent of the difference between new value and the original value relative to the original value. For example, if absorption increases from 5 to 10 mg, the percent increase is 100%.
 9.
T _{g} m is equal to the width of the peak at half maximum and not equal to the actual peak width at itsbase, which is about 2 times T _{g} m.
 10.
Nota bene the zaxis (total amount of drug absorbed) is different for each k _{a} value. For k _{a} = 0.01 min^{−1} (panels a, d, g), y = 0 to 0.6 mg; for k _{a} = 0.01 min^{−1} (panels b, e, h), y = 0 to 6 mg; and for k _{a} = 0.1 min^{−1} (panels c, f, i), y = 0 to 120 mg.
Notes
Acknowledgments
The authors wish to thank Jeff Browne, Kevin Johnson, and Jonathan Miller for their helpful discussions. M. Kleppe was supported in part by the Dane O. Kildsig Center for Pharmaceutical Processing Research (CPPR).
Supplementary material
References
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