Target mediated drug disposition with drug–drug interaction, Part II: competitive and uncompetitive cases

Abstract

We present competitive and uncompetitive drug–drug interaction (DDI) with target mediated drug disposition (TMDD) equations and investigate their pharmacokinetic DDI properties. For application of TMDD models, quasi-equilibrium (QE) or quasi-steady state (QSS) approximations are necessary to reduce the number of parameters. To realize those approximations of DDI TMDD models, we derive an ordinary differential equation (ODE) representation formulated in free concentration and free receptor variables. This ODE formulation can be straightforward implemented in typical PKPD software without solving any non-linear equation system arising from the QE or QSS approximation of the rapid binding assumptions. This manuscript is the second in a series to introduce and investigate DDI TMDD models and to apply the QE or QSS approximation.

Appendices

Appendix 1: Derivation of the final QE and QSS approximation in free concentration variables

Competitive DDI

Step 1: Total concentration formulation

Similar to the single drug case [6] the key for the QE or QSS approximation is to reformulate Eqs. (1)–(5) in total drug and total receptor concentration variables. With

$$\begin{aligned} C_{totA}&= C_A + RC_A \end{aligned}$$

(51)

$$\begin{aligned} C_{totB}&= C_B + RC_{B} \end{aligned}$$

(52)

$$\begin{aligned} R_{tot}&= R + RC_A + RC_{B} \end{aligned}$$

(53)

we obtain

$$\begin{aligned} \frac{d}{dt} C_{totA}&= In_A(t)- k_{elA} C_A - k_{intA} RC_A \end{aligned}$$

(54)

$$\begin{aligned} \frac{d}{dt} C_{totB}&= In_B(t) - k_{elB} C_B - k_{intB} RC_B \end{aligned}$$

(55)

$$\begin{aligned} \frac{d}{dt} R_{tot}&= k_{syn} - k_{deg} R - k_{intA} RC_A - k_{intB} RC_B \end{aligned}$$

(56)

$$\begin{aligned} \frac{d}{dt} RC_A&= k_{onA} C_A \cdot R - (k_{offA} + k_{intA}) RC_A \end{aligned}$$

(57)

$$\begin{aligned} \frac{d}{dt} RC_B&= k_{onB} C_B \cdot R - (k_{offB} + k_{intB}) RC_B. \end{aligned}$$

(58)

The baseline initial values are

$$\begin{aligned} C_{totX} (0)=\, & {} C_{totX}^0 \;= \; C_X^0 + RC_X^0 \nonumber \\ R_{tot} (0)=\, & {} R_{tot}^0 \; = \; R^0 +RC_A^0 +RC_B^0 \nonumber \\ RC_X (0)=\, & {} RC_X^0 \end{aligned}$$

(59)

for $X \in \{ A,B \}$. The values $C_A^0, \, C_B^0, \, R^0, \, RC_A^0, \, RC_B^0$ in Eq. (59) are chosen according to Eqs. (6)–(8) and the input functions in Eqs. (54)–(55) according to Eq. (9). Substituting free variables in Eqs. (54)–(58) with total variables from Eqs. (51)–(53) we obtain

$$\begin{aligned} \frac{d}{dt} C_{totA}&= In_A(t)- k_{elA} (C_{totA}-RC_A) \nonumber \\&\quad - k_{intA} RC_A \end{aligned}$$

(60)

$$\begin{aligned} \frac{d}{dt} C_{totB}&= In_B(t) - k_{elB} (C_{totB}-RC_B) \nonumber \\&\quad - k_{intB} RC_B \end{aligned}$$

(61)

$$\begin{aligned} \frac{d}{dt} R_{tot}&= k_{syn} - k_{deg} (R_{tot} - RC_A - RC_B) \nonumber \\&\quad - k_{intA} RC_A - k_{intB} RC_B \end{aligned}$$

(62)

$$\begin{aligned} \frac{d}{dt} RC_A&= k_{onA} (C_{totA}-RC_A)(R_{tot}-RC_A-RC_B) \nonumber \\&\quad - (k_{offA} + k_{intA}) RC_A \end{aligned}$$

(63)

$$\begin{aligned} \frac{d}{dt} RC_B&= k_{onB} (C_{totB}-RC_B)(R_{tot}-RC_A-RC_B) \nonumber \\&\quad - (k_{offB} + k_{intB}) RC_B \, . \end{aligned}$$

(64)

In comparison to Eqs. (1)–(5), Eqs. (60)–(64) have the advantage that the parameters $k_{onX}$ and $k_{offX}$ appear in the equations of the complexes only.

Step 2: QE and QSS binding relations

We assume rapid binding between $C_A$ and R, as well as $C_B$ and R. Hence, QE or QSS approximation of the complexes $RC_A$ and $RC_{B}$ in Eqs. (57)–(58) provide the algebraic equations

$$\begin{aligned} 0&= (C_{totA}-RC_A)(R_{tot}-RC_A-RC_B) - K_{YA} RC_A \end{aligned}$$

(65)

$$\begin{aligned} 0&= (C_{totB}-RC_B)(R_{tot}-RC_A-RC_B) - K_{YB} RC_B \end{aligned}$$

(66)

for $Y \in \{ D, SS \}$ with Eq. (19) (see Appendices 2, 3). The differential algebraic equation (DAE) form in total variables is then given by Eqs. (60)–(62), (65)–(66).

Step 3: QE and QSS model equations

To avoid solving the coupled non-linear equation system Eqs. (65)–(66) numerically, we transform Eqs. (54)–(56), (65)–(66) back to the free variables. From Eqs. (65)–(66) we obtain the complexes

$$\begin{aligned} RC_X&= \frac{C_X \cdot R}{K_{XA}} \, . \end{aligned}$$

(67)

The next step is to differentiate Eq. (67) and to express $\frac{d}{dt} C_{totA}, \frac{d}{dt} C_{totB}, \frac{d}{dt} R_{tot}$ appearing at the left hand side of Eqs. (54)–(56) in terms of $C_A, C_B$ and R and their derivatives. Using Eqs. (51)–(53) we can calculate from Eqs. (54)–(56)

$$\begin{aligned} \frac{d}{dt} C_{totA}&= \frac{d}{dt} C_A + \frac{d}{dt} RC_A \nonumber \\&= \frac{d}{dt} C_A + \left( \frac{d}{dt} C_A \right) \frac{R}{K_{YA}} + \left( \frac{d}{dt} R \right) \frac{C_A}{K_{YA}}\nonumber \\&= In_A(t) - k_{elA} C_A - k_{intA} \frac{C_A R}{K_{YA}} \end{aligned}$$

(68)

$$\begin{aligned} \frac{d}{dt} C_{totB}&= \frac{d}{dt} C_B + \frac{d}{dt} RC_B \nonumber \\&= \frac{d}{dt} C_B + \left( \frac{d}{dt} C_B \right) \frac{R}{K_{YB}} + \left( \frac{d}{dt} R \right) \frac{C_B}{K_{YB}} \nonumber \\&= In_B(t) - k_{elB} C_B - k_{intB} \frac{C_B R}{K_{YB}} \end{aligned}$$

(69)

$$\begin{aligned} \frac{d}{dt} R_{tot}&= \frac{d}{dt} R + \frac{d}{dt} RC_A + \frac{d}{dt} RC_B \nonumber \\&= \frac{d}{dt} R + \left( \frac{d}{dt} C_A \right) \frac{R}{K_{YA}} + \left( \frac{d}{dt} R \right) \frac{C_A}{K_{YA}} \nonumber \\&\quad + \left( \frac{d}{dt} C_B \right) \frac{R}{K_{YB}} + \left( \frac{d}{dt} R \right) \frac{C_B}{K_{YB}} \nonumber \\&= k_{syn} - k_{deg} R - k_{intA} \frac{C_A R}{K_{YA}} - k_{intB}\frac{C_B R}{K_{YB}} \, . \end{aligned}$$

(70)

The equivalent matrix form reads

$$\begin{aligned} Q(C_A,C_B,R) \begin{pmatrix} \frac{d}{dt} C_A \\ \frac{d}{dt} C_B \\ \frac{d}{dt} R \end{pmatrix} = g_{Com}(C_A,C_B,R) \end{aligned}$$

(71)

with

$$\begin{aligned}&Q(C_A,C_B,R) \\&= \begin{pmatrix} 1 + \frac{R}{K_{YA}} &{} 0 &{} \frac{C_A}{K_{YA}} \\ 0 &{} 1 + \frac{R}{K_{YB}} &{} \frac{C_B}{K_{YB}} \\ \frac{R}{K_{YA}} &{} \frac{R}{K_{YB}} &{} 1 + \frac{C_A}{K_{YA}} + \frac{C_B}{K_{YB}} \end{pmatrix}\\&g_{Com}(C_A,C_B,R) \\&= \begin{pmatrix} In_A(t) - k_{elA} C_A - k_{intA} \frac{C_A R}{K_{YA}} \\ In_B(t) - k_{elB} C_B - k_{intB} \frac{C_B R}{K_{YB}} \\ k_{syn} - k_{deg} R - k_{intA} \frac{C_A R}{K_{YA}} - k_{intB}\frac{C_B R}{K_{YB}} \end{pmatrix} . \end{aligned}$$

Eq. (71) is equivalent to

$$\begin{aligned} \begin{pmatrix} \frac{d}{dt} C_A \\ \frac{d}{dt} C_B \\ \frac{d}{dt} R \end{pmatrix} = M_{Com}(C_A,C_B,R) g_{Com}(C_A,C_B,R) \end{aligned}$$

(72)

where

$$\begin{aligned} M_{Com}(C_A,C_B,R) = Q^{-1}(C_A,C_B,R) \, . \end{aligned}$$

$Q^{-1}$ denotes the inverse matrix of Q and the explicit representation of $M_{Com}$ is listed in Table 1.

Uncompetitive DDI

Step 1: Total concentration formulation

The total drug and receptor variables are

$$\begin{aligned} C_{totA}&= C_A + RC_A + RC_{AB} \end{aligned}$$

(73)

$$\begin{aligned} C_{totB}&= C_B + RC_{AB} \end{aligned}$$

(74)

$$\begin{aligned} R_{tot}&= R + RC_A + RC_{AB} \end{aligned}$$

(75)

and we obtain

$$\begin{aligned} \frac{d}{dt} C_{totA}&= In_A(t) - k_{elA} C_A - k_{intA} RC_A - k_{intAB} RC_{AB} \end{aligned}$$

(76)

$$\begin{aligned} \frac{d}{dt} C_{totB}&= In_B(t) - k_{elB} C_B - k_{intAB} RC_{AB} \end{aligned}$$

(77)

$$\begin{aligned} \frac{d}{dt} R_{tot}&= k_{syn} - k_{deg} R - k_{intA} RC_A - k_{intAB} RC_{AB} \end{aligned}$$

(78)

$$\begin{aligned} \frac{d}{dt} RC_A&= k_{onA} C_A \cdot R - k_{onAB} C_B \cdot RC_A + k_{offAB} RC_{AB} \nonumber \\&\quad - ( k_{offA} + k_{intA} ) RC_A \end{aligned}$$

(79)

$$\begin{aligned} \frac{d}{dt} RC_{AB}&= k_{onAB} C_B \cdot RC_A - ( k_{offAB} + k_{intAB} ) RC_{AB} . \end{aligned}$$

(80)

The baseline initial values are obtained by applying Eqs. (73)–(75) to the initial values Eqs. (28)–(31). This leads to

$$\begin{aligned} C_{totA} (0)=\, & {} C_{totA}^0 \; = \; C_A^0 + RC_A^0 + RC_{AB}^0 \\ C_{totB} (0)= & {} C_{totB}^0 \; = \; C_B^0 + RC_{AB}^0 \\ R_{tot} (0)= \,& {} R_{tot}^0 \; =\, \; R^0 + RC_A^0 + RC_{AB}^0 \end{aligned}$$

and the input functions Eqs. (32)–(33).

Again substituting the free variables in Eqs. (76)–(80) yields

$$\begin{aligned} \frac{d}{dt} C_{totA}&= In_A(t) - k_{elA} (C_{totA}-RC_A-RC_{AB}) \nonumber \\&\quad - k_{intA} RC_A - k_{intAB} RC_{AB} \end{aligned}$$

(81)

$$\begin{aligned} \frac{d}{dt} C_{totB}&= In_B(t) - k_{elB} (C_{totB}-RC_{AB}) \nonumber \\&\quad - k_{intAB} RC_{AB} \end{aligned}$$

(82)

$$\begin{aligned} \frac{d}{dt} R_{tot}&= k_{syn} - k_{deg} (R_{tot}-RC_A-RC_{AB}) \nonumber \\&\quad - k_{intA} RC_A - k_{intAB} RC_{AB} \end{aligned}$$

(83)

$$\begin{aligned} \frac{d}{dt} RC_A&= k_{onA} (C_{totA}-RC_A-RC_{AB}) \nonumber \\&\quad (R_{tot}-RC_A-RC_{AB}) \nonumber \\&\quad - k_{onAB} (C_{totB}-RC_{AB}) RC_A \nonumber \\&\quad + k_{offAB} RC_{AB} - ( k_{offA} + k_{intA} ) RC_A \end{aligned}$$

(84)

$$\begin{aligned} \frac{d}{dt} RC_{AB}&= k_{onAB} (C_{totB}-RC_{AB}) RC_A \nonumber \\&\quad - ( k_{offAB} + k_{intAB} ) RC_{AB} \, . \end{aligned}$$

(85)

Note that in the formulation Eqs. (81)–(85) the parameter $k_{onX}, \, k_{offX}$, intended for elimination show up in the equations of the complexes only.

Step 2: QE binding relations

In Appendix 2 it is shown that the QE approximation provides the algebraic equations

$$\begin{aligned} 0&= C_A R - K_{DA} RC_A \end{aligned}$$

(86)

$$\begin{aligned} 0&= C_B RC_A - K_{DAB} RC_{AB} \end{aligned}$$

(87)

and the resulting DAE consists of Eqs. (81)–(83), (86), (87).

Step 3: QE model equations

Using Eqs. (73)–(75) and Eqs. (76)–(78) we can compute

$$\begin{aligned} \frac{d}{dt} C_{totA}&= \frac{d}{dt} C_A + \frac{d}{dt} RC_A +\frac{d}{dt} RC_{AB} \nonumber \\&= In_A(t) - k_{elA} C_A - k_{intA} \frac{C_A R}{K_{DA}} \end{aligned}$$

(88)

$$\begin{aligned}&\quad -k_{intAB} \frac{C_A C_B R}{K_{DA} K_{DAB}} \nonumber \\ \frac{d}{dt} C_{totB}&= \frac{d}{dt} C_B + \frac{d}{dt} RC_{AB} \nonumber \\&= In_B(t) - k_{elB} C_B - k_{intAB} \frac{C_A C_B R}{K_{DAB}K_{DA}} \end{aligned}$$

(89)

$$\begin{aligned} \frac{d}{dt} R_{tot}&=\frac{d}{dt} R +\frac{d}{dt} RC_A +\frac{d}{dt} RC_{AB} \nonumber \\&= k_{syn} - k_{deg} R - k_{intA} \frac{C_A R}{K_{DA}} \nonumber \\&\quad - k_{intAB}\frac{C_A C_B R}{K_{DAB}K_{DA}} \, . \end{aligned}$$

(90)

In addition, from Eqs. (86)–(87) we obtain by differentiation

$$\begin{aligned} \frac{d}{dt} RC_A&= \frac{1}{K_{DA}} \left( \left( \frac{d}{dt}C_A \right) R + C_A \frac{d}{dt} R \right) \end{aligned}$$

(91)

$$\begin{aligned} \frac{d}{dt} RC_{AB}&= \frac{1}{K_{DA} K_{DAB}} \left( \left( \frac{d}{dt} C_A \right) C_B R \right. \nonumber \\&\quad \left. +\, C_A \left( \frac{d}{dt} C_B \right) R + C_A C_B \frac{d}{dt} R \right) . \end{aligned}$$

(92)

With Eqs. (88)–(92) the equivalent matrix form reads

$$\begin{aligned} P(C_A,C_B,R) \begin{pmatrix} \frac{d}{dt} C_A \\ \frac{d}{dt} C_B \\ \frac{d}{dt} R \end{pmatrix} = g_{Un}(C_A,C_B,R) \end{aligned}$$

(93)

with $P(C_A,C_B ,R) = I+\hat{P}(C_A,C_B,R)$,

$$\begin{aligned}&\hat{P}(C_A,C_B,R) \\&= \begin{pmatrix} \frac{R}{K_{DA}} +\frac{C_B R}{K_{DA}K_{DAB}} &{} \frac{C_A R}{K_{DA} K_{DAB}} &{} \frac{C_A}{K_{DA}} + \frac{C_A C_B}{K_{DA}K_{DAB}}\\ \frac{C_B R}{K_{DA}K_{DAB}} &{} \frac{C_A R}{K_{DA}K_{DAB}} &{} \frac{C_A C_B}{K_{DA}K_{DAB}} \\ \frac{R}{K_{DA}} + \frac{C_B R}{K_{DA} K_{DAB}} &{} \frac{C_A R}{K_{DA}K_{DAB}} &{} \frac{C_A}{K_{DA}} + \frac{C_A C_B}{K_{DA}K_{DAB}} \end{pmatrix} \end{aligned}$$

and

$$\begin{aligned}&g_{Un}(C_A,C_B,R) \\&= \begin{pmatrix} In_A(t) - k_{elA} C_A - k_{intA}\frac{C_A R}{K_{DA}}- k_{intAB} \frac{C_A C_B R}{K_{DA} K_{DAB}} \\ In_B(t) - k_{elB} C_B - k_{intAB} \frac{C_A C_B R}{K_{DA} K_{DAB}} \\ k_{syn} - k_{deg} R - k_{intA} \frac{C_A R}{K_{DA}} - k_{intAB} \frac{C_A C_B R}{K_{DA} K_{DAB}} \end{pmatrix} . \end{aligned}$$

Finally, Eq. (93) can be written as explicit ODE

$$\begin{aligned} \begin{pmatrix} \frac{d}{dt} C_A \\ \frac{d}{dt} C_B \\ \frac{d}{dt} R \end{pmatrix} = M_{Un}(C_A,C_B,R) g_{Un}(C_A,C_B,R) \end{aligned}$$

where

$$\begin{aligned} M_{Un}(C_A,C_B,R) = P^{-1}(C_A,C_B,R) \end{aligned}$$

is listed in Table 1.

Appendix 2: QE approximation

The QE approximation is based on the theory of Fenichel [14] which allows a specific selection of the rates to be accelerated.

Competitive

To justify the QE approximation we increase the binding rates $k_{onX},k_{offX}$, where $X \in \{A,B\}$, by replacing with $\frac{1}{\varepsilon } k_{onX}$, $\frac{1}{\varepsilon } k_{offX}$ with $\varepsilon > 0$ small in Eqs. (54)–(58). Since the new constants are much larger this can be regarded as rapid binding and we obtain

$$\begin{aligned} \frac{d}{dt} RC_A&= \frac{k_{onA}}{\varepsilon } C_A \cdot R - \left( \frac{k_{offA}}{\varepsilon } + k_{intA} \right) RC_A \end{aligned}$$

(94)

$$\begin{aligned} \frac{d}{dt} RC_B&= \frac{k_{onB}}{\varepsilon } C_B \cdot R - \left( \frac{k_{offB}}{\varepsilon } + k_{intB} \right) RC_B. \end{aligned}$$

(95)

Multiplying Eqs. (94)–(95) by $\varepsilon$ gives

$$\begin{aligned} \varepsilon \frac{d}{dt} RC_A&= k_{onA} C_A \cdot R - \left( k_{offA} + \varepsilon k_{intA} \right) RC_A \end{aligned}$$

(96)

$$\begin{aligned} \varepsilon \frac{d}{dt} RC_B&= k_{onB} C_B \cdot R - \left( k_{offB} + \varepsilon k_{intB} \right) RC_B. \end{aligned}$$

(97)

Taking the limit $\varepsilon \rightarrow 0$ in Eqs. (96)–(97) results in

$$\begin{aligned} 0&= k_{onA} C_A \cdot R - k_{offA} RC_A \end{aligned}$$

(98)

$$\begin{aligned} 0&= k_{onB} C_B \cdot R - k_{offB} RC_B. \end{aligned}$$

(99)

Dividing Eq. (98) by $k_{onA}$ and Eq. (99) by $k_{onB}$ gives the QE approximation of the complexes

$$\begin{aligned} 0&= C_A \cdot R - K_{DA} RC_A \end{aligned}$$

(100)

$$\begin{aligned} 0&= C_B \cdot R - K_{DB} RC_B \, . \end{aligned}$$

(101)

Uncompetitive

Accelerating the binding rates $k_{onX}$ and $k_{offX}$ with $X \in \{A,AB\}$ in Eqs. (76)–(80) gives

$$\begin{aligned} \frac{d}{dt} RC_A&= \frac{k_{onA}}{\varepsilon } C_A R - \frac{k_{onAB}}{\varepsilon } C_B RC_A + \frac{k_{offAB}}{\varepsilon } RC_{AB} \nonumber \\&\quad - \left( \frac{k_{offA}}{\varepsilon } + k_{intA} \right) RC_A \end{aligned}$$

(102)

$$\begin{aligned} \frac{d}{dt} RC_{AB}&= \frac{k_{onAB}}{\varepsilon } C_B RC_A - \left( \frac{k_{offAB}}{\varepsilon } + k_{intAB} \right) RC_{AB} . \end{aligned}$$

(103)

Multiplying Eqs. (102)–(103) by $\varepsilon$ leads to

$$\begin{aligned} \varepsilon \frac{d}{dt} RC_A&= k_{onA}C_A R - k_{onAB} C_B RC_A + k_{offAB} RC_{AB} \nonumber \\&\quad - ( k_{offA} + \varepsilon k_{intA} ) RC_A \end{aligned}$$

(104)

$$\begin{aligned} \varepsilon \frac{d}{dt} RC_{AB}&= k_{onAB} C_B RC_A - ( k_{offAB} + \varepsilon k_{intAB} ) RC_{AB} . \end{aligned}$$

(105)

Taking the limit $\varepsilon \rightarrow 0$ in Eqs. (104)–(105) results in

$$\begin{aligned} 0&= k_{onA}C_A R - k_{onAB} C_B RC_A + k_{offAB} RC_{AB} \nonumber \\&\quad - k_{offA} RC_A \end{aligned}$$

(106)

$$\begin{aligned} 0&= k_{onAB} C_B RC_A - k_{offAB} RC_{AB} \, . \end{aligned}$$

(107)

Substituting Eq. (107) in Eq. (106) leads to

$$\begin{aligned} 0&= k_{onA}C_A R - k_{offA} RC_A \end{aligned}$$

(108)

$$\begin{aligned} 0&= k_{onAB} C_B RC_A - k_{offAB} RC_{AB} \, . \end{aligned}$$

(109)

Dividing Eq. (108) with $k_{onA}$ and Eq. (109) with $k_{onAB}$ gives the QE approximation of the complexes

$$\begin{aligned} 0&= C_A R - K_{DA} RC_A \end{aligned}$$

(110)

$$\begin{aligned} 0&= C_B RC_A - K_{DAB} RC_{AB} \, . \end{aligned}$$

(111)

Appendix 3: QSS approximation

Following the classical singular perturbation theory [15] all complex related processes are assumed to be rapid, including the internalization from the complexes.

Competitive

Accelerating the rates with $\varepsilon$ small in Eqs. (54)–(58) yields

$$\begin{aligned} \frac{d}{dt} RC_A&= \frac{k_{onA}}{\varepsilon } C_A \cdot R - \left( \frac{k_{offA}}{\varepsilon } + \frac{k_{intA}}{\varepsilon } \right) RC_A \end{aligned}$$

(112)

$$\begin{aligned} \frac{d}{dt} RC_B&= \frac{k_{onB}}{\varepsilon } C_B \cdot R - \left( \frac{k_{offB}}{\varepsilon } + \frac{k_{intB}}{\varepsilon } \right) RC_B. \end{aligned}$$

(113)

Multiplying Eqs. (112)–(113) by $\varepsilon$ and taking the limit $\varepsilon \rightarrow 0$

$$\begin{aligned} 0&= k_{onA} C_A \cdot R - \left( k_{offA}+ k_{intA}\right) RC_A \end{aligned}$$

(114)

$$\begin{aligned} 0&= k_{onB} C_B \cdot R - \left( k_{offB} + k_{intB} \right) RC_B. \end{aligned}$$

(115)

Hence, the QSS approximation reads

$$\begin{aligned} 0&= C_A \cdot R - K_{SSA} RC_A \end{aligned}$$

(116)

$$\begin{aligned} 0&= C_B \cdot R - K_{SSB} RC_B. \end{aligned}$$

(117)

Uncompetitive

We obtain from Eqs. (76)–(80) with $\varepsilon$ small

$$\begin{aligned} \frac{d}{dt} RC_A&= \frac{k_{onA}}{\varepsilon } C_A \cdot R -\frac{k_{onAB}}{\varepsilon } C_B \cdot RC_A + \frac{k_{offAB}}{\varepsilon } RC_{AB} \end{aligned}$$

(118)

$$\begin{aligned}&\quad - \left( \frac{k_{offA}}{\varepsilon } + \frac{k_{intA}}{\varepsilon } \right) RC_A \end{aligned}$$

(119)

$$\begin{aligned} \frac{d}{dt} RC_{AB}&= \frac{k_{onAB}}{\varepsilon } C_B \cdot RC_A - \left( \frac{k_{offAB}}{\varepsilon } + \frac{k_{intAB}}{\varepsilon } \right) RC_{AB} \, . \end{aligned}$$

(120)

Multiplying these equations by $\varepsilon$ and then taking the limit $\varepsilon \rightarrow 0$ results in

$$\begin{aligned} 0&= k_{onA}C_A R - k_{onAB} C_B RC_A + k_{offAB} RC_{AB} \\ \nonumber &\quad - ( k_{offA} + k_{intA} ) RC_A \end{aligned}$$

(121)

$$\begin{aligned} 0&= k_{onAB} C_B RC_A - ( k_{offAB} + k_{intAB} ) RC_{AB} \, . \end{aligned}$$

(122)

Inserting Eq. (122) in Eq. (121) gives

$$\begin{aligned} 0&= k_{onA}C_A R - k_{intAB} RC_{AB} - ( k_{offA} + k_{intA} ) RC_A \end{aligned}$$

(123)

$$\begin{aligned} 0&= k_{onAB} C_B RC_A - ( k_{offAB} + k_{intAB} ) RC_{AB} \, . \end{aligned}$$

(124)

Dividing Eq. (123) by $k_{onA}$ and Eq. (124) by $k_{onAB}$ provides

$$\begin{aligned} 0&= C_A R - \frac{k_{intAB}}{k_{onA}} RC_{AB} - K_{SSA} RC_A \end{aligned}$$

(125)

$$\begin{aligned} 0&= C_B RC_A - K_{SSAB} RC_{AB} \, . \end{aligned}$$

(126)

Appendix 4: Baseline initial values for the uncompetitive TMDD model

According to Eqs. (26)–(27) the baseline conditions for the complexes with the concentrations $C_A^0, C_B^0 \ge 0$ are

$$\begin{aligned} \left( \begin{array}{c c} \frac{k_{intA} +k_{offA}}{k_{onA}} &{} \frac{k_{intAB}}{k_{onA}} \\ -C_B^0 &{} \frac{k_{intAB}+k_{offAB}}{k_{onAB}} \end{array} \right) \left( \begin{array}{c} RC_A \\ RC_{AB} \end{array} \right)= & {} \left( \begin{array}{c} C_A^0 R \\ 0 \end{array} \right) . \nonumber \\&\end{aligned}$$

(127)

Applying Cramer’s rule to Eq. (127) and using the definition from Eq. (19) yields the solution

$$\begin{aligned} RC_A^0= & {} \frac{C_A^0 R^0 K_{SSAB}}{K_{SSA}K_{SSAB} + C_B^0 \frac{k_{intAB}}{k_{onA}} } \end{aligned}$$

(128)

$$\begin{aligned} RC_{AB}^0= & {} \frac{C_A^0 C_B^0 R^0 }{K_{SSA}K_{SSAB} + C_B^0 \frac{k_{intAB}}{k_{onA}} } \, . \end{aligned}$$

(129)

Inserting Eqs. (128)–(129) into the baseline condition of the receptor equation (78) leads to

$$\begin{aligned} k_{syn}= & {} \left( k_{deg} + \frac{ k_{intA} C_A^0 K_{SSAB} + k_{intAB} C_A^0 C_B^0 }{ K_{SSAB} K_{SSA} + \frac{C_B^0 k_{intAB}}{k_{onA}} } \right) R, \end{aligned}$$

which is equivalent to

$$\begin{aligned} R^0= & {} \frac{ k_{syn}}{ k_{deg} + \frac{ k_{intA} C_A^0 K_{SSAB} + k_{intAB} C_A^0 C_B^0 }{ K_{SSAB} K_{SSA} + \frac{C_B^0 k_{intAB}}{k_{onA}} } }. \end{aligned}$$

The baseline concentrations of the input functions then follow from Eqs. (76)–(77).

Appendix 5: Source codes

The matrix representation applied in Eqs. (14)–(15) and Eqs. (43)–(44) is of the general form

$$\begin{aligned} \begin{pmatrix} H_1 \\ H_2 \\ H_3 \end{pmatrix} = \begin{pmatrix} M_{11} &{} M_{12} &{} M_{13} \\ M_{21} &{} M_{22} &{} M_{23} \\ M_{31} &{} M_{32} &{} M_{33} \end{pmatrix} \begin{pmatrix} G_1 \\ G_2 \\ G_3 \end{pmatrix} \, . \end{aligned}$$

Hence, performing matrix multiplication the right hand side of the differential equation reads

$$\begin{aligned} H_1&= M_{11} G_1 + M_{12} G_2 + M_{13} G_3 \\ H_2&= M_{21} G_1 + M_{22} G_2 + M_{23} G_3 \\ H_3&= M_{31} G_1 + M_{32} G_2 + M_{33} G_3 \end{aligned}$$

compare the lines 113–128 for the competitive and the lines 221–239 for the uncompetitive case. The variables $H_1$,...,$H_3$ correspond to DADT(1), ..., DADT(3) in NONMEM and XP(1), ..., XP(3) in ADAPT 5.

The lines of the code are numbered for referencing but are not part of the code implementation.

NONMEM control stream for competitive DDI TMDD

The $DES block of the control stream is presented. Additionally, the first lines of the data file is shown to present the IV infusion mechanism. The full control stream is available in the supplemental material.

101: $DES

102: EPSILON = 1e-4

103: ; Dose at T1 = 0

104: INA = 0

105: INB = 0

106: IF (T.GE.0.AND.T.LE.0+EPSILON) THEN

107: INA = 100*EPSILON**(−1)

108: INB = 100*EPSILON**(-1)

109: ENDIF

110: CA = A(1)/V

111: CB = A(2)/V

112: R = A(3)

113: DET = R**2+CA*KDB+CB*KDA+CA*R+CB*R+KDA*KDB+KDA*R+KDB*R

114: G1 = INA - KELA*CA - (KINTA*CA*R)/KDA

115: G2 = INB - KELB*CB - (KINTB*CB*R)/KDB

116: G3 = KSYN-KDEG*R-(KINTA*CA*R)/KDA-(KINTB*CB*R)/KDB

117: M11 = (1/DET)*(DET - R*(R+CB+KDB))

118: M12 = (1/DET)*(CA*R)

119: M13 = (1/DET)*(-CA*(R+KDB))

120: M21 = (1/DET)*(CB*R)

121: M22 = (1/DET)*(DET - R*(R+CA+KDA))

122: M23 = (1/DET)*(-CB*(R+KDA))

123: M31 = (1/DET)*(-R*(R+KDB))

124: M32 = (1/DET)*(-R*(R+KDA))

125: M33 = (1/DET)*(DET-CA*(R+KDB)-CB*(R+KDA))

126: DADT(1) = M11*G1 + M12*G2 + M13*G3

127: DADT(2) = M21*G1 + M22*G2 + M23*G3

128: DADT(3) = M31*G1 + M32*G2 + M33*G3

The first lines of the data file are:

150: #ID TIME TYPE DV MDV

151: 1 0 1 . 1

152: 1 0 2 . 1

153: 1 0.0001 1 . 1

154: 1 0.0001 2 . 1

155: 1 2 1 32.9432 0

156: 1 2 2 28.3621 0

ADAPT 5 source code for uncompetitive DDI TMDD

The subroutine DIFFEQ is presented. For full source code see supplemental material.

201: Subroutine DIFFEQ(T,X,XP)

202: Implicit None

203: Include ’globals.inc’

204: Include ’model.inc’

205: Real*8 T,X(MaxNDE),XP(MaxNDE)

206: Real*8 KELA,KDA,KINTA,KELB,KDAB,KINTAB,KSYN,KDEG

207: Real*8 CA,CB,RR,R0

208: Real*8 DET,M(3,3),G(3)

209: KELA = P(1)

210: KDA = P(2)

211: KINTA = P(3)

212: KELB = P(4)

213: KDAB = P(5)

214: KINTAB = P(6)

215: KSYN = P(7)

216: KDEG = P(8)

217: R0 = KSYN/KDEG

218: CA = X(1)

219: CB = X(2)

220: RR = X(3) + R0

221: DET = RR**2*CA+CA*RR*KDA+CB*RR*KDA+CA**2*RR+CA*CB*KDA

222: & +KDA**2*KDAB+KDA*KDAB*RR+CA*KDA*KDAB

223: G(1) = R(1)-KELA*CA-(KINTA*CA*RR)/KDA

224: & -KINTAB*((CA*CB*RR)/(KDA*KDAB))

225: G(2) = R(2)-KELB*CB-KINTAB*((CA*CB*RR)/(KDA*KDAB))

226: G(3) = KSYN-KDEG*RR-(KINTA*CA*RR)/KDA

227: & -KINTAB*((CA*CB*RR)/(KDA*KDAB))

228: M(1,1) = (1/DET)*(DET-RR*(CA*RR+CB*KDA+KDA*KDAB))

229: M(1,2) = (1/DET)*(-CA*RR*KDA)

230: M(1,3) = (1/DET)*(-CA*(CA*RR+CB*KDA+KDA*KDAB))

231: M(2,1) = (1/DET)*(-CB*RR*KDA)

232: M(2,2) = (1/DET)*(DET-CA*RR*(RR+CA+KDA))

233: M(2,3) = (1/DET)*(-KDA*CA*CB)

234: M(3,1) = (1/DET)*(-RR*(CB*KDA+CA*RR+KDA*KDAB))

235: M(3,2) = (1/DET)*(-CA*RR*KDA)

236: M(3,3) = (1/DET)*(DET-CA*(CA*RR+KDAB*KDA+CB*KDA))

237: XP(1) = M(1,1)*G(1)+M(1,2)*G(2)+M(1,3)*G(3)

238: XP(2) = M(2,1)*G(1)+M(2,2)*G(2)+M(2,3)*G(3)

239: XP(3) = M(3,1)*G(1)+M(3,2)*G(2)+M(3,3)*G(3)

240: Return

241: End

Fig. 1

Schematic of the competitive DDI TMDD model described by Eqs. (1)–(5) (panel a) and the uncompetitive DDI TMDD model described by Eqs. (23)–(27) (panel b)

Fig. 2

Properties of the original DDI TMDD models. Competitive: In panels a and b the single drug profiles of drugs A and B (blue dotted lines) are compared with the competitive model (black solid line) Eqs. (1)–(9) for $dose_A = 100$, $dose_B = 100$ and no baseline $C_A^0 = C_B^0 = 0$. In panels c and d, the effect of the administration of one drug only on a present concentration of the other drug is shown by two examples: (i) drugs A and B are in baseline $C_A^0 = C_B^0 = 1$, and one administration at time $t = 12$ of drug B with $dose_B = 100$ (red dashed lines) causes an increase of drug A concentration, (ii) drug A is administered with $dose_A = 100$ at $t = 0$ and drug B administered with $dose_B = 100$ at $t = 0$ and additionally at $t = 12$ (black solid lines), and causes an increase of drug A concentration. Uncompetitive: In panels e and f the single drug profiles of drugs A and B (blue dotted lines) are compared with the uncompetitive model (black solid lines) Eqs. (23)–(33) for $dose_A = 100$, $dose_B = 100$ and no baseline $C_A^0 = C_B^0 = 0$. In panels g and h drugs A and B are in baseline $C_A^0 = C_B^0 = 1$ with one administration at time $t=5$ of drug A with $dose_A = 100$ and non of drug B (red dashed line) and the other way around (black solid lines) (Color figure online)

Fig. 3

Visualization of the QE approximation. Competitive: In panels a and b concentration profiles from the original formulation (red dashed lines) Eqs. (1)–(9) and the approximation of the QE formulation Eqs. (14)–(18) (black solid lines) are shown for escalating doses of $dose_A = dose_B = 10, 100, 1000$ at $t=0$ and no baseline $C_A^0 = C_B^0 = 0$. In panels c and d the effect of one drug administration on the present concentration of the other drug is shown. The original (red dashed lines) and QE approximation (black solid lines) profiles with a baseline $C_A^0 = C_B^0 = 1$ are shown for a dose of $doseA = 1000$ at $t=0$. The $k_{onB}$ and $k_{offB}$ are multiplied by the factors 0.1, 1 and 10 in such a way that $K_{DB}$ stays the same to show the convergence of the original formulation towards the QE approximation. Uncompetitive: In panels e and f concentration profiles from the original formulation (red dashed lines) Eqs. (23)–(33) and the approximation of the QE formulation Eqs. (43)–(48) (black solid lines) are shown for escalating doses of $dose_A = dose_B = 10, 100, 1000$ at $t=0$ and no baseline $C_A^0 = C_B^0 = 0$. In panels g and h original (red dashed lines) and QE approximation (black solid lines) profiles with a baseline $C_A^0 = C_B^0 = 1$ are shown where for drug A is administered with a dose of $dose_A = 100$ at $t=24$. The $k_{onA}$ and $k_{offA}$ are multiplied by the factors 0.1, 1 and 10 in such a way that $K_{DA}$ stays the same (Color figure online)

Fig. 4

Visualization of plasma concentration versus time data fitting from the original formulation with the QE approximation: Fit (solid lines) of the QE approximation of the competitive Eqs. (14)–(18) in NONMEM (panels a and b) and the uncompetitive DDI TMDD model Eqs. (43)–(48) in ADAPT 5 (panels c and d) in ODE formulation with an IV short infusion. Data (crosses) were produced with the original formulations Eqs. (1)–(9) and Eqs. (23)–(33)

Table 1 Matrices $M_{Com}(C_A,C_B,R)$ and $M_{Un}(C_A,C_B,R)$ for QE approximation of DDI TMDD implementation

Table 2 Estimated model parameters of the QE approximation of the competitive and uncompetitive DDI TMDD models formulated as ODE in free variables