Introduction

Drug-drug interaction (DDI) is an important aspect of drug development. When a drug is used concomitantly with other drug(s), one of the major concerns is the change of exposures, including the rate and extent of drug absorption, distribution, metabolism and elimination. To address these concerns, a common practice is to measure the exposure and report its increases (or decreases) when the drug is used in the presence of concomitant medication (COMED) compared to that when it is used alone.

In vivo DDI studies are generally designed in a crossover fashion to compare the exposures of the drug of interest with and without COMED. Most often, the differences of the exposures are based on the following three measures: (i) area under the plasma concentration versus time curve (AUC), a measure of the total exposure; (ii) maximum plasma concentration (Cmax), a measure of the extent of the exposure; and (iii) time to the Cmax (tmax), a measure of the rate of the exposure.

These exposure measures are usually obtained by non-compartmental analysis,[1] which is simple and accurate. Cmax and tmax are directly observed from the plasma concentration versus time curve. As long as the sampling times are appropriately selected, these two values can be adequately captured. If the sampling times are long enough, AUC is estimated from zero to the last measured timepoint (e.g. 24 hours after dosing) using the linear trapezoidal rule or elated methods,[1] such as log trapezoidal method. Usually, it is extrapolated from the last measured timepoint to infinity by incorporating the elimination rate constant, which describes the rate of drug removal from the body.[1]

When comparing the exposures between the presence and absence of COMED, a usual observation is the inconsistent changes between AUC and Cmax in a specific DDI study. For example, cases observed in the literature are presented in table I.

Table I
figure Tab1

Exposure changes due to drug-drug interactions reported in the literature

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As seen, not only the exposure folds vary, but the relative folds between AUC and Cmax for the same interaction differ as well. Roughly, the relative fold changes between AUC and Cmax can be classified into three categories. In the first category, AUC and Cmax have similar fold changes during DDI.[2,3] The second category includes the cases in which AUC has larger fold changes compared to Cmax.[4–8] In cases of the third category, Cmax has larger fold changes compared with AUC.[9–18]

It is interesting to note that rosuvastatin in three different studies with three different concomitant medications (gemfibrozil,[6] lopinavir/ritonavir[7] and ciclosporin,[8] respectively) showed similar pattern with Cmax having higher fold changes compared to AUC fold changes, although the absolute fold changes were different (2.21, 4.7 and 10.6 for Cmax, respectively). On the other hand, gemfibrozil as a COMED in another study[15] caused AUC and Cmax fold changes of repaglinide following another pattern: AUC was much higher than Cmax (7 vs 2). The difference was even larger in a study where salmeterol had 16-fold AUC in the presence of ketoconazole, whereas the Cmax fold change was only 1.4.[18] Also note that tmax change was positive (increase) in some situations and negative (decrease) in others.

In some cases, AUC and Cmax changes could be in the opposite directions. For example, full dose of amprenavir and nelfinavir resulted in decrease in amprenavir Cmax by 14%, but increase in AUC by 46%.[19] In another report, 14 healthy volunteers (7 male, 7 female) received single oral 0.5 mg doses of the anticholinergic agent scopolamine in a randomized crossover fashion with either 250 mL of water or fresh-squeezed, single-strength grapefruit juice. In the presence of grapefruit juice, scopolamine Cmax was decreased 11%, while AUC was increased 35%, and time to peak level (tmax) was extended from 23.5 minutes to 59.5 minutes (153% increase)[20] compared with water.

This difference of fold changes between AUC and Cmax could be of concern. For certain drug effects, the Cmax is an important consideration while under other circumstances, AUC is important. If a clinical outcome is most closely related to drug concentration (e.g. tachycardia with sympathomimetics), the exposure measure, Cmax, may be the most important to consider. Conversely, if the clinical outcome is related more to extent of exposure, AUC would be preferred[21] in the context that a long, low concentration exposure may be as important as shorter but higher concentration.

If we have a better understanding of this phenomenon, it will be greatly helpful to clinical interpretation of the observed DDI. Furthermore, if we can predict the relative change of AUC, Cmax and tmax based on the information available, it will be more valuable. This study intends to make such an attempt for didactical purposes. By assuming that the DDI was a result of changes of primary parameters of the drug of interest, the relative changes of AUC, Cmax and tmax were investigated by various simulations.

Methods

The important steps of this study included: setting up the model and specifications; implementation of the model by simulations; and data analyses and result interpretations.

The Assumptions

  • This study made the following assumptions.

  • DDI was considered as pharmacokinetic interaction and no pharmacodynamic interaction was involved.

  • DDI resulted in changes of one or more of the primary pharmacokinetic parameters of the drug of interest, such as absorption constant (ka), bioavailability (F), volume of distribution (Vd) and clearance (CL), which, in turn, altered the secondary parameters, such as AUC, Cmax and tmax. Based on this assumption, COMED was not specified in all the simulations performed regarding its dosage amount, dosing time, dosing frequency and dose titration scheme. However, the changes of the primary parameters of the drug of interest were considered to be the reflections of the combinations of all these factors.

  • The pharmacokinetics were assumed to follow a single dose, one-compartment model with first-order absorption and first-order elimination for oral administration route.

Pharmacokinetic Model

A pharmacokinetic model loosely based on midazolam in vivo characteristics was selected. The model was described by the formula in equation (1):

$$\rm C={F\bullet Dose\bullet k_a\over V-d(k_a-CL/V_d)}\bullet(e^{-CL/V_d\bullet t}-e^{-k_a\bullet t})$$
(1)

where C is the plasma drug concentration at t, the time elapsed after the dosage administration of the amount (Dose).

This model and associated parameters were for the drug of interest only. For COMED, no specific model and parameters were needed according to assumption (2). The CL, F, Vd, and ka of the drug of interest would be changed to reflect the effects of COMED (see Simulations and Glossary sections).

Simulations

The model parameters of CL and Vd were selected from a collection of drugs with different half-lives.[22] As shown in figure 1, drugs with various combination values of CL and Vd were located in several half-life zones.[22] One or two combinations (in case of two, one with higher CL and another with lower CL with similar half-lives) of CL and Vd values were selected from each zone.

Fig
figure 1

The selection of parameters of clearance (CL) and volume of distribution (Vd). CL (y-axis) and Vd (x-axis) vary widely. The diagonal lines show the combinations of CL and Vd with the same half-lives (in hours, labeled near the lines). The selected combinations of parameters are represented by different symbols.

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For each combination of CL and Vd, there were three levels of ka values (0.2, 1 and 5 h−1, respectively). F value was chosen as 0.2. There were a total of 24 cases as summarized in table II.

Table II
figure Tab2

Model parameters cases

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For each case, the model parameters and the simulation specifications varied as shown in table III for Case 2 as an example. Other cases used the same values except the second column (‘Value’). The ‘Value’ column was referred to the parameter values when the drug of interest was administered alone. When this drug was administered with COMED, these parameters would change. The possible values after change are listed in the third column (‘Values with COMED’). Twenty-five simulations were conducted for each of the four parameters (CL, F, ka and Vd) in each of the 24 cases. Therefore, there were a total of 2400 simulations conducted.

Table III
figure Tab3

Simulation model parameters

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The time values were selected from the interval between 0 to 580 hours for the first 15 cases. The rest of the cases used much longer intervals (58 000 hours for Cases 16–21 and 580 000 hours for the last three cases). These intervals were chosen with a consideration that most of the simulated profiles would be captured in order to get a relatively accurate AUC calculation by using trapezoidal method (zero to last timepoint). To get a smooth curve and relatively more accurate AUC, the interval between two adjacent timepoints was chosen to be 0.1 hours.

Simulations were conducted for each of the values of CL, F, Vd and ka using R program (version 2.8, R project group). For every scenario, concentration-time profile was generated and the Cmax, tmax and AUC were calculated.

R packages including ‘base’, ‘stats’, ‘graphics’ and ‘lattice’ were used for AUC, Cmax and tmax calculations, and for plotting the data. Several functions were defined, such as AUC to last timepoint (AUCt) and AUC to infinity (AUC∞) according to the trapezoid rule as shown in equations (2) and (3)

$$\matrix {\rm AUC_t=\mathop \int\limits_0\limits^t\ C\bullet dt\approx\Delta t\bullet(C_{t0}/2+C_{t1})\cr+\rm C_{t2}+\dots+C_t/2)}$$
(2)
$$\rm AUC_\infty={C_t\over (CL/V_d)}+AUC_t$$
(3)

where Ct0, Ct1, …, Ct refer to the concentrations at initial timepoint (0), timepoint 1 and so on, until timepoint t.

Cmax and tmax were identified by R using max() function. The values of AUC, Cmax and tmax were calculated for each scenario.

To investigate the situation where different parameters changed simultaneously, another set of simulations was conducted for Case 2, in which every combination of the four parameter fold changes were simulated as shown in table IV. In table IV, each parameter had five levels and there were 625 (54) possible combinations. Similar to the first set, simulations were conducted using the R program. For every scenario, the concentration-time profile was generated, and the Cmax, tmax and AUC were calculated.

Table IV
figure Tab4

Model parameters for the second set of simulations

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Glossary

To facilitate the discussion, it would be beneficial to define certain terms used in this study.

Standard Parameters. The standard setting for the simulation, the scenario where no COMED was used and the parameters of the drug of interest were not changed. For example of Case 2, the standard setting was given in the second column of table III: CL = 38 h/L, F = 0.2, ka = 1 h−1 and Vd = 260 L. The standard parameters for other cases can be found in table II.

Parameter Fold Changes. The ratios of the parameters in a specific simulation scenario to those of the standard setting when the parameters were not less than the standard parameters. Otherwise (i.e. the parameters in simulation were less than the standard parameters), they were referred to the negative ratios of the standard parameters to the parameters used in a simulation scenario. For example, the CL fold change was defined as the ratios of the CL used for a specific simulation to the standard CL (e.g. 38 L/h for Case 2) when CL was not less than the standard. If CL is less than the standard, the CL fold change was defined as the negative ratio of the standard CL (38 L/h for Case 2) to the CL used in the simulation (CLsim) as shown in equation (4)

$$\matrix {\rm When\ CL_{sim}\geq38,\cr\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \rm CL\ fold\ change=CL_{sim}/38\cr \rm When\ CL_{sim}<38,\cr\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \rm CL\ fold\ change=-38/CL_{sim}}$$
(4)

where CLsim was the value of CL in a specific simulation scenario.

Similarly, F fold change was defined as shown in equation (5)

$$\matrix {\rm When\ F_{sim}\geq0.2,\cr\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \rm F\ fold\ change=F_{sim}/0.2\cr \rm When\ F_{sim}<0.2,\cr\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \rm F\ fold\ change=-0.2/F_{sim}}$$
(5)

where Fsim was the value of F in a specific simulation scenario.

Vd fold change was defined as shown in equation (6)

$$\matrix {\rm When\ V_{d,sim}\geq260,\cr\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \rm V_d\ fold\ change=V_{d,sim}/260\cr \rm When\ V_{d,sim}<260,\cr\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \rm V_d\ fold\ change=-260/V_{d,sim}}$$
(6)

where Vd,sim was the value of Vd in a specific simulation scenario.

ka fold change was defined as shown in equation (7)

$$\matrix {\rm When\ k_{a,sim}\geq1,\cr\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \rm k_a\ fold\ change=k_{a,sim}/1\cr \rm When\ k_{a,sim}<1,\cr\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \rm k_a\ fold\ change=-1/k_{a,sim}}$$
(7)

where ka,sim was the value of ka in a specific simulation scenario.

Note that the standard parameters for Case 2 (CL = 38 L/h, F = 0.2, ka = 1 h−1 and Vd = 260 L) were used for illustration purpose. A set of specific standard parameters (see table II) were used in each specific case.

In this study, F fold changes were set at −5…,1, …, 5. The other parameter fold changes (CL fold change, Vd fold change and ka fold change) were set to −50, …, 1, …, and 50, for the first set of simulations (see table III) and −5, −2, 1, 2, 5 for the second set of simulations (see table IV).

Standard Exposures. The exposures resulted from the standard parameters (defined above), including standard AUC∞ (AUC∞,std), standard Cmax (Cmax,std) and standard tmax (tmax,std).

Exposure Changes. The ratios of the exposures resulted from a simulated scenario to the standard exposures (defined above). The AUC fold change and Cmax fold change were defined in a fashion similar to parameter fold changes as shown in equations (8) and (9)

$$\matrix {\rm When\ AUC_{\infty,sim}\geq AUC_{\infty,std},\cr\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \rm AUC\ fold\ change=AUC_{\infty,sim}/AUC_{\infty,std}\cr \rm When\ AUC_{\infty,sim}<AUC_{\infty,std},\cr\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \rm AUC\ fold\ change=-AUC_{\infty,std}/AUC_{\infty,sim}}$$
(8)

where AUC∞,sim was the value of AUC∞ in a specific simulation scenario.

$$\matrix {\rm When\ C_{\max,sim}\geq C_{\max,std},\cr\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \rm C_{\max}\ fold\ change=C_{\max,sim}/C_{\max,std}\cr \rm When\ C_{\max,sim}< C_{\max,std},\cr\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \rm C_{\max}\ fold\ change=-C_{\max,std}/C_{\max,sim}}$$
(9)

where Cmax.,sim was the value of Cmax in a specific simulation scenario.

Please note that by using fold changes, the increases were represented by the number >1 while the decreases were referred to negative values less than −1.

The tmax change was defined as the difference between (tmax)sim and tmax,std as shown in equation (10)

$$\rm t_{\max}\ change=t_{\max,sim}-t_{\max,std}$$
(10)

where tmax,sim was the value of tmax in a specific simulation scenario.

Fold Change Difference. The differences between AUC fold change and Cmax fold change as shown in equation (11)

$$\rm AUC\ fold\ change-C_{\max}\ fold\ change$$
(11)

Data Analyses

Parameter Fold Change, Exposure Fold Change and Fold Change Difference Calculation

During the simulations, the parameter fold changes (including CL fold change, F fold change, Vd fold change and ka fold change), the exposure fold changes (including AUC fold change, Cmax fold change and tmax change) and the fold change difference were calculated for each scenario. Upon the iteration of the simulations, a data sheet was accumulated to record the parameters (CL, F, Vd and ka) used, resulted AUC, Cmax and tmax, and calculated parameter fold changes, exposure fold changes and fold change differences. The data sheet was used for generating plots.

Plots Generation

Based on the results of the above calculations, the following plots were generated for the first set of simulations.

  • The concentration-time profiles for all scenarios in each case were plotted.

  • Exposure changes (including fold change difference, AUC fold change, Cmax fold change and tmax change) were plotted against each parameter fold changes (including F fold change, CL fold change, ka fold change and Vd fold change).

The following plots were generated for the second set of simulations.

  • The fold change difference was plotted against each of the parameter fold changes to investigate the influencing parameters in the separation of AUC∞ fold changes and Cmax fold changes.

  • The AUC fold change was plotted against each of the parameter fold changes.

  • The Cmax fold change was plotted against each of the parameter fold changes.

  • The tmax change was plotted against each of the parameter fold changes.

  • After the major influencing parameters were identified, the AUC fold change was examined against both CL fold change and F fold change in a 3-dimensional plot.

Results

The Simulated Profiles

For the first set of simulations, there were a total of 2400 concentration-time profiles accommodated in 96 pages. A representative page is shown in figure S1 in the Supplemental Digital Content 1, http://links.adisonline.com/DRZ/A1.

For the second set of simulations, the concentration-time profiles for each combination of different values of CL, F, Vd and ka were generated. There were a total of 625 profiles accommodated in 25 pages. A representative page including the standard profile (using Standard Parameters, with a circle in the figure) is shown in figure S2 in the Supplemental Digital Content.

The Effects of Parameter Fold Changes on Exposure Changes

For the first set of simulations, 96 plots for the exposure changes against parameter fold changes were generated. Figure 2 shows the plots of the exposure changes against CL fold changes for Cases 4–6 and Cases 10–12. Figure 3 shows the plots of the exposure changes against Vd fold changes for Cases 13–15 and Cases 19–21. Figure 4 shows the plots of the exposure changes against ka fold changes for Cases 2, 5 and 11. Figure 5 shows the exposure changes against F fold changes for all cases.

Fig. 2
figure 2

The plots of area under the plasma concentration vs time curve (AUC) and maximum plasma concentration (Cmax) fold changes, fold change difference and time to Cmax (tmax) change against clearance (CL) fold changes. The case number and corresponding standard parameters are labeled. Note that the y-axis labeled 1(0) indicates 1 for AUC fold change and Cmax fold change, while it is 0 for fold change difference and tmax. = bioavailability; k a = absorption rate constant; V d = volume of distribution.

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Fig. 3
figure 3

The plots of area under the concentration curve (AUC) and maximum concentration (Cmax) fold changes, fold change difference and time to Cmax (tmax) change against volume of distribution (Vd) fold changes. The case number and corresponding standard parameters are labeled. Note that the y-axis labeled 1(0) indicates 1 for AUC fold change and Cmax fold change while it is 0 for fold change difference and tmax change. = bioavailability; k a = absorption constant.

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Fig 4
figure 4

The plots of area under the plasma concentration vs time curve (AUC) and maximum plasma concentration (Cmax) fold changes, fold change difference, and time to Cmax (tmax) change against absorption rate constant (ka) fold changes. The case number and corresponding standard parameters are labeled. Note that the y-axis labeled 1(0) indicates 1 for AUC fold change and Cmax fold change while it is 0 for fold change difference and tmax change. = bioavailability; V d = volume of distribution.

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Fig 5
figure 5

The plots of area under the plasma concentration vs time curve (AUC) and maximum plasma concentration (Cmax) fold changes, fold change difference, and time to Cmax (tmax) change against bioavailability (F) fold changes. Note that the y-axis labeled 1(0) indicates 1 for AUC fold change and Cmax fold change, while it is 0 for fold change difference and tmax change. The diagonal lines for AUC fold changes and Cmax fold changes are overlapped, and the horizontal lines for fold change difference and tmax changes are also overlapped.

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The results from the second set were consistent with those from the first set of simulations. Figures S3–S6 in the Supplemental Digital Content show the effect of different parameter fold changes on AUC fold change, Cmax fold change, tmax Change, and fold change difference, respectively.

Area Under the Plasma Concentration versus Time Curve (AUC) Fold Change

For the first set of simulations, AUC fold change decreased with CL fold changes and increased with F fold changes, while it was not affected by Vd fold changes and ka fold changes as shown in figures 2–5.

For the second set, figure S3 in the Supplemental Digital Content indicated that the AUC fold change was not affected by ka and Vd evidenced by the fact that AUC fold change could be anywhere in the whole range regardless of ka fold change or Vd fold change. As a result, AUC fold change would be determined by CL and F fold changes as shown in figure S7 in the Supplemental Digital Content. When CL fold change decreased to the lowest (−5-fold in the second set of simulations) and F fold change increased to the highest (5-fold in the second set of simulations), the AUC fold change was the most pronounced.

These effects were expected and straightforward according to equation (12)

$$\rm AUC={F\times Dose\over CL}$$
(12)

Maximum Plasma Concentration (Cmax) Fold Change

Cmax fold change was more complex, which was the major factor accounting for the varied fold change difference.

The effect of F fold changes on Cmax fold change was the most obvious one. As shown in figure 5, Cmax fold change was proportional to F fold change. Since AUC fold change was also proportional to F fold change as shown in equation (12), if a DDI only caused F Change (no CL, Vd or ka changes), the AUC fold change and Cmax fold change would be exactly the same and the fold change difference would be zero (figure 5). Although Cmax fold change was in the same direction as the F fold change, the exact proportionality was not shown as indicated in the upper right panel of figure S4 in the Supplemental Digital Content. The reason was that the second set of simulation used the same standard parameters and the resulted standard exposures for all possible combinations. Therefore, the relationships between the parameter fold changes and the exposure changes were in a range, not a single line. Due to the same reason, unlike the one shown in figure 5, the relationship between the fold change difference and F fold change in the upper right panel in figure S6 in the Supplemental Digital Content was not a horizontal line.

The effect of CL fold change on Cmax fold change seemed generally small, especially for the drugs with long half-life and/or quick absorption. In figure 2, the difference in Cmax fold change among different panels can be appreciated. The lower three panels with longer half-life had minimal Cmax fold change, while for those in the upper row with shorter half-life, the Cmax fold change was significant. On the other hand, the panels on the left with smaller ka values showed larger Cmax fold change, while the panels on the right with larger ka values had smaller Cmax fold change, with the panel at lower right corner having an almost straight horizontal line indicating little effect on the Cmax fold change. Another interesting fact was that the Cmax fold change resulting from an increase of CL fold change was quite different from that resulting from a decrease of CL fold change. Taking the panel at upper left corner as an example, increase of CL fold change caused decrease of Cmax fold change with a steep slope whereas decreases of CL fold change induced a shallow increase of Cmax fold change. This difference was not so obvious in the upper left panel of figure S4 in the Supplemental Digital Content due to two reasons: (i) the second set of simulation was taking Case 2 in the first set as basis and therefore the range was not wide enough; and (ii) the multiple parameters were considered simultaneously, which may compensate for each other.

The effect of ka fold change on Cmax fold change was related to half-life as previously discussed and also as seen in figure 4. The standard parameter of ka was all the same (1 h−1) for the three panels in figure 4. The middle panel with shortest half-life produced the most prominent effect on Cmax fold change when ka fold change decreased, whereas the effect was much shallower for the right panel with longest half-life among the three cases. In addition, the different effects on Cmax fold change between the movement of ka fold change towards a positive direction (increase) and a negative direction (decrease) were observed. For the drugs with shorter half-life, the middle panel of figure 4, for example, the slope of the curve for the Cmax fold change against ka fold change in the negative direction was steep, while it was much shallower in the positive direction, and even leveled off in the other two panels. Since AUC fold change was not affected by ka fold change, the effect of ka fold change on the fold change difference was dependent on its effect on Cmax fold change. Roughly, the fold change difference curve was a mirror image of Cmax fold change, symmetric to the horizontal line at 1 (or 0). A general trend between ka fold change and Cmax fold change consistent with this observation was seen for multiple parameter analysis as shown in the lower left panel in figure S4 in the Supplemental Digital Content.

The curve representing the effect of Vd fold change on Cmax fold change was also related to half-life and ka fold change, especially the curve shape when the Vd fold change moved to the negative direction as shown in figure 3. When the half-life was shorter and ka was smaller, this part of the curve leveled off as shown in the lower left panel. Along with the increase of half-life and ka, the curve became steeper until it straightened up as shown in the upper right panel. Again, due to the stable AUC fold change, the curve for fold change difference was roughly a mirror image of the curve for Cmax fold change. When multiple parameters were considered simultaneously, the general trend between Vd fold change and Cmax fold change was consistent with this observation as shown in the lower right panel of figures S4 and S6 in the Supplemental Digital Content.

Time to Reach Cmax (tmax) Changes

Similar to Cmax fold change, tmax change was affected by multiple parameters, although F fold change did not have any effect as shown in figure 5 and the upper right panel in figure S5 in the Supplemental Digital Content. Both the effects of CL fold change and Vd fold change on tmax change were dependent on ka values with smaller ka having more significant effects as shown in figures 2 and 3. Although the effects of half-life could be appreciated in these two figures, the effects were more prominent when the effects of ka fold change on tmax change was considered. As shown in figure 4, for the drug with shorter half-life (the middle panel), ka fold change caused less tmax change compared to the tmax change for the drug with longer half-life (the right panel in figure S5 in the Supplemental Digital Content). For both cases, the negative ka fold change produced much larger tmax change compared to the positive ka fold change as seen from both figure 4 and the lower left panel in figure S5 in the Supplemental Digital Content. It seemed that there was a threshold for the decrease of tmax Change when the ka fold change going higher in the positive territory. The threshold value was also related to the half-life and the standard parameter of ka with smaller ka and longer half-life having more prominent threshold.

Discussion

Due to the wide range of the fold changes of the primary parameters, the length of the sampling time was of concern to capture the major part of the profiles for the purpose of accurate calculations. As shown in figures S1 and S2 in the Supplemental Digital Content, most of the profiles were well within the length of the sampling time. Visual inspections for other pages for both the first and second set of simulations were conducted to ensure it was the case for all scenarios. However, it was the author’s experience that the interval between adjacent sampling times was more important than the length of the sampling time. Because the accuracy of the calculations was of concern, the interval was set to 0.1 hour, which seemed to be adequate.

Instead of modeling for both the drug of interest and the COMED, the current approach examined the parameter changes of the drug of interest only. The purpose of the proposed approach is to get a general idea about the possible nature of the drug interaction by emphasizing the integration of all information available. The study provided visualizations for various exposure changes. The observations indicated that AUC was mainly influenced by CL and F, Cmax was affected by all four parameters, and tmax was mainly affected by CL and ka. F did little for tmax and ka was unrelated to AUC. From each plot alone, limited information was obtained. To get better interpretation, an integration approach should be taken.

The study was conducted with concentration on the individual factor considered independently (the first set of the simulations), although multiple parameters were taken into account (in the second set of simulations). Indeed, the multiple parameter consideration was more realistic and practical. For example, when the individual F fold change was considered independently, its effect on fold change difference showed a horizontal line (figure 5). However, when multiple parameters were taken into account simultaneously, the upper right panel in figure S6 in the Supplemental Digital Content provided a more realistic picture. Instead of a line, a broader range was given. More interestingly, the plot showed certain symmetry pattern. The symmetry of the effects of F fold change on the fold change difference in this case was caused by the symmetries of its effects on AUC fold changes (figure S3 in the Supplemental Digital Content) and Cmax fold changes (figure S4 in the Supplemental Digital Content). This symmetry had two possible consequences. First, at each level of F fold change, fold change difference had equal chances to go in either direction. For example, at level of 2 for F fold change, fold change difference ranged from about −10 to +10. The chances to go positive or negative were relatively equal dependent on the other parameters. Secondly, the increase or decrease of F fold change would have similar effects on fold change difference, with the centre point (F fold change = 1) having the smallest effect. For example, a 2-fold increase of F (F fold change = 2) would result in similar effects as it would for a 2-fold decrease of F (F fold change = −2). When F fold change was relatively small, the fold change difference would be smaller compared to the situation where F fold changes were larger (the upper right panel of figure S6 in the Supplemental Digital Content).

These consequences were important because when interpreting the data, one had to notice that a certain level of fold change difference observed could have been caused by either increased or decreased F fold changes.

With the importance of multiple parameter simultaneous consideration in mind, the following summary of the basic pattern of exposure changes according to the effects of parameter fold changes may be of help. Figure 6 was summarized from the 96 plots of the 24 cases studied. For the plot in each cell, the x-axis was the parameter fold changes and the y-axis was the exposure changes. The table can be read horizontally for each of the exposure changes affected by one of the parameter fold changes or vertically for the effects of each parameter fold changes on a specific exposure change. For example, reading horizontally for the effects of Vd fold change would result in the following.

Fig 6
figure 6

Basic pattern summary and pattern reasoning. The columns and the ordinate in each cell show the exposure (fold change difference, area under the plasma concentration vs time curve [AUC fold change, maximum plasma concentration [Cmax] fold change and time to Cmax [tmax] change), while the rows and the abscissa in each cell show the parameter fold changes as labeled. The intersection of the ordinate and the abscissa represents the situations when parameter fold changes are 1 (no change). The right side from this point represents the parameter fold changes more than 1 (increase), while the left side stands for the parameter fold changes less than −1 (decrease). The line indicates the exposure changes (fold change difference, AUC fold change, Cmax fold change and tmax change) when parameter fold changes increase (or decrease). Its slope signifies the extent of the changes. The shaded areas indicate the consistency of exposure changes induced by a specific parameter change, which provides the evidence for selecting the candidates for the example in the study by Hsu et al.[17]

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  • Decreased Vd fold change would cause a small decrease of fold change difference, no effect on AUC fold change, a small increase of Cmax fold change and a modest decrease of tmax change.

  • Increased Vd fold would cause an increase of fold change difference, no effect on AUC fold change, a decrease of Cmax fold change and a modest increase of tmax change.

To use this chart to interpret DDI, follow the steps below. For the purpose of illustration, the DDI of saquinavir with coadministration of ritonavir[17] is taken as an example.

  1. 1.

    Perform a drug interaction study and get the values of AUC, Cmax and tmax in the presence and the absence of COMED.

  2. 2.

    Calculate the AUC fold change, Cmax fold change and tmax change. Get the fold change difference. In the example, AUC fold change is 128.9, Cmax fold change is 32.5, tmax change is 0.8 h and the fold change difference is 96.4.

  3. 3.

    Find the candidates of parameter fold changes, which account for the exposure changes obtained from step 2. In the example, all the exposure changes are positive. For this reason, CL fold change could be a candidate because decreased CL fold change would make all exposure changes positive. However, Vd fold change may not qualify the candidacy due to the fact that although decreased Vd fold change could cause an increase of Cmax fold change, it would result in decreases of tmax change and fold change difference based on figure 6. On the other hand, although an increase of Vd fold change could make tmax change and fold change difference positive, it would induce a decrease of Cmax fold change. In addition, the increase of AUC fold change is so significant in the example, while figure 6 shows no change of AUC when Vd fold change is altered. For the similar reasons, ka fold change may be eliminated from the candidate list. It is noted that there is a significant increase of Cmax fold change (32.5). If reading the chart vertically for the column of Cmax fold change, significant increase of Cmax fold change only occurs when there is an increase of F fold change. Therefore, although the increase of F fold change alone may not produce increases of tmax change and fold change difference, its contribution could not be ignored due to the considerable increase of Cmax fold change. Therefore, two potential candidates are selected. The reasoning process can be found by the shaded areas in figure 6.

  4. 4.

    Consider all available information to confirm or reject the candidates. For the example, saquinavir is primarily eliminated by means of metabolism; only 3% of a 14C-labelled dose is recovered in urine after intravenous administration.[23] In addition, the reported systemic plasma CL of saquinavir is higher than hepatic plasma flow. The absolute oral bioavailability of saquinavir is about 3%. Cytochrome P450 (CYP) 3A, the major isoform in the metabolism of ritonavir and saquinavir is present in intestinal tissue,[24,25] in addition to the liver. Thus, it is likely that saquinavir undergoes first-pass metabolism by both intestinal and hepatic enzymes. Because of this, coupled with the fact that ritonavir has a low in vitro concentration that produced 50% inhibition (IC50), it would be expected that ritonavir would have a marked inhibitory effect on the first-pass metabolism of saquinavir. In contrast, because the plasma CL of saquinavir approaches or exceeds hepatic plasma flow rate, even though ritonavir has a low in vitro IC50, it is likely that the post-absorptive inhibition effect will not be large. The information listed here tends to confirm the candidates selected by figure 6. However, the apparent controversy between a large AUC fold change (128.9) and the relatively small effects of CL fold change implied by the statements that the post-absorptive inhibition effect will not be large needs more explanation. Since the effects of CL fold change on Cmax fold change are generally small (figure 2), one may assume that the effect observed for the Cmax fold change of saquinavir is primarily caused by F fold change (i.e. the suppression of the first-pass metabolism). Ritonavir would then yield a significant inhibition of the first-pass metabolism, resulting in a Cmax fold change of 32.5 and also an AUC fold change of 32.5. If the net effect on AUC is the product of first-pass and post-absorptive inhibition, then the latter corresponding inhibition can be estimated to be 4 expressed as the AUC fold change, which is relatively small and very close to the reported estimate.[17]

  1. 1.

    It is important to understand the nature of DDI in order to prevent its adverse consequences if any. When a DDI is metabolism based, it is critical to identify whether the DDI occurs at pre-systemic level or systemic level. One of the difficulties for such an interpretation is that the absolute bioavailability (F) of a drug is usually not readily available and furthermore, it is difficult to estimate the F changes during a DDI study. As a result, the proposed approach to perform a rough estimate seems attractive. The following potential applications of such a pattern recognition approach may be anticipated.

  2. 2.

    A general understanding and a rough estimate about the nature of the DDI observed may be obtained from a simple comparison.

  3. 3.

    Based on the understanding of the mechanism, the output can be predicted regarding which exposure index is more prominent when it is of concern. As mentioned earlier, for some drugs, AUC is the major concern of exposure and for others it is Cmax. Under certain circumstances, tmax is also of concern.

  4. 4.

    This approach would be helpful to understand any situation where AUC and Cmax have differences, such as providing an explanation for a failed bioequivalence study. In one scenario, the AUC meets the criteria but Cmax does not, while under another scenario, Cmax meets the criteria but AUC does not. The pattern recognition approach would suggest helpful hints to single out the possible mechanism in order to decide whether the differences between the test and reference are caused by the formulation differences or some other influential factors. By the same token, this pattern recognition approach would also be useful for interpretations of other similar studies, such as food effect study (with and without food), renal impairment and hepatic impairment studies (subject groups with different renal or hepatic functions), where the AUC and Cmax changes differ and tmax varies.

  5. 5.

    The approach would be beneficial for interpreting the exposure difference resulting from different genetic makeup. For example, poor metabolizers of CYP2D6 have a 10-fold higher AUC and a 5-fold higher peak concentration to a given dose of atomoxetine compared with extensive metabolizers.[14]

  6. 6.

    Manufacture parameters may affect the in vivo performance. The pattern recognition approach will help the understanding of the effects of the manufacture parameters.

  1. 1.

    It should be emphasized that this approach be integrated with other methodologies due to its limited abilities. The limitations for the current study are listed below.

  2. 2.

    A linear, one-compartment model with first-order elimination was assumed. The real world might be more complex. When metabolism-based DDI is considered, saturation is frequently observed. The drugs listed in table I might not all follow the assumptions. With emphasizing the general trend for common parameters related to drug absorption, bioavailability and elimination, caution should be exercised for assumption violation.

  3. 3.

    The first-order absorption was used (constant ka). It implied the unrealistic assumption that the maximum absorption rate was achieved instantaneously. Due to its popularity and the fact that in some cases it was sufficient to describe the process of drug input, the current study used it for easier interpretation.

  4. 4.

    Lag time, the delay between drug administration and the beginning of absorption, was not considered in this study. The lag time can be anywhere from a few minutes to many hours. It may be particularly important when a rapid onset of effect is desired.

  5. 5.

    Single dose was used in all the simulations. Multiple doses might be needed to show a significant interaction. However, considering that the purpose of this study was to compare the exposure between the subjects with COMED and those without, the single dose results could be extrapolated to the case of multiple doses with a linear assumption.

  6. 6.

    The results obtained from this approach could be inconclusive. However, a general trend is helpful, in principle. Furthermore, integration with other methodologies for interpretation is always advised.

Conclusion

With the assumptions that the primary parameters of a drug of interest are altered during a DDI, an approach, named DDI pattern recognition for didactical purposes, is proposed for interpreting the DDI results based on the exposure fold changes. DDI pattern recognition may provide an insight of the mechanistic nature of DDI and a general idea about the processes, which dominate a DDI.

This approach analyses DDI from a new angle. Due to the complexity of DDI, one angle is not enough. By emphasizing the integration of all available information and mechanistic interpretation from multiple angles, the new approach may play a significant role in interpreting DDI studies.