Target-mediated drug disposition with drug–drug interaction, Part I: single drug case in alternative formulations

Abstract

Target-mediated drug disposition (TMDD) describes drug binding with high affinity to a target such as a receptor. In application TMDD models are often over-parameterized and quasi-equilibrium (QE) or quasi-steady state (QSS) approximations are essential to reduce the number of parameters. However, implementation of such approximations becomes difficult for TMDD models with drug–drug interaction (DDI) mechanisms. Hence, alternative but equivalent formulations are necessary for QE or QSS approximations. To introduce and develop such formulations, the single drug case is reanalyzed. This work opens the route for straightforward implementation of QE or QSS approximations of DDI TMDD models. The manuscript is the first part to introduce DDI TMDD models with QE or QSS approximations.

Appendices

Appendix 1: Short infusion and IV bolus

By construction and the use of Eqs. (12, 43–44) the final system (46–51) is equivalent to

$$\begin{aligned} \frac{d}{dt} C + \frac{d}{dt} RC &= In(t) -k_{el} C - k_{int}RC \end{aligned}$$

(55)

$$\begin{aligned} \frac{d}{dt} R + \frac{d}{dt} RC &= k_{syn} - k_{deg}R -k_{int}RC \end{aligned}$$

(56)

$$\begin{aligned} 0 &= C \cdot R - K_Y RC . \end{aligned}$$

(57)

We assume that the short infusion with infusion rate $dose/(\varepsilon V)$ is given for $t \in [t_a, t_a + \varepsilon ]$. We then obtain by integrating Eqs. (55–56) over $[t_a, t_a + \varepsilon ]$

$$\begin{aligned} \int _{t_a}^{t_a +\varepsilon } \frac{d}{dt} C + \frac{d}{dt} RC \, dt= & {} \int _{t_a}^{t_a +\varepsilon } \frac{dose}{\varepsilon V} \, dt \\&- \int _{t_a}^{t_a +\varepsilon } k_{el}C + k_{int} RC \, dt \\ \int _{t_a}^{t_a +\varepsilon } \frac{d}{dt} R + \frac{d}{dt} RC \, dt= & {} \int _{t_a}^{t_a +\varepsilon } k_{syn} - k_{deg} R \\&\qquad - k_{int} RC \, dt \end{aligned}$$

which leads to

$$\begin{aligned} C(t_a + \varepsilon ) - C(t_a ) + RC(t_a + \varepsilon ) -RC(t_a ) \; = \;& \\ \quad \quad \frac{dose}{V} - \int _{t_a}^{t_a + \varepsilon } k_{el}C + k_{int}RC \, dt&\end{aligned}$$

(58)

$$\begin{aligned} R(t_a + \varepsilon ) - R(t_a ) + RC(t_a + \varepsilon ) -RC(t_a ) \; = \;& \\ \quad \quad \int _{t_a}^{t_a + \varepsilon } k_{syn} -k_{deg}R - k_{int}RC \, dt . \end{aligned}$$

(59)

Now letting $\varepsilon \rightarrow 0$ in Eqs. (58–59) and using Eq. (57) we get

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} C(t_a + \varepsilon ) + \lim _{\varepsilon \rightarrow 0} \frac{ C(t_a + \varepsilon ) R(t_a +\varepsilon )}{K_Y}=\,\,& {} C_{tot}(t_a ) \\&+ \frac{dose}{V} \, , \\ \lim _{\varepsilon \rightarrow 0} R(t_a + \varepsilon ) + \lim _{\varepsilon \rightarrow 0} \frac{ C(t_a + \varepsilon ) R(t_a +\varepsilon )}{K_Y}=\,\,& {} R_{tot} (t_a ). \end{aligned}$$

This means that $\lim _{\varepsilon \rightarrow 0} C(t_a + \varepsilon )$, $\lim _{\varepsilon \rightarrow 0} R(t_a + \varepsilon )$ satisfy Eqs. (53–54) and

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} C(t_a + \varepsilon ) &= C_{new} \\ \lim _{\varepsilon \rightarrow 0} R(t_a + \varepsilon ) &= R_{new} \end{aligned}$$

is shown.

Appendix 2: Source codes

The matrix representation applied in Eqs. (46–50) is of the general form

$$\begin{aligned} \begin{pmatrix} H_1 \\ H_2 \\ \end{pmatrix} = \begin{pmatrix} M_{11} &{} M_{12} \\ M_{21} &{} M_{22} \end{pmatrix} \begin{pmatrix} G_1 \\ G_2 \end{pmatrix} \, . \end{aligned}$$

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

$$\begin{aligned} H_1&= M_{11} G_1 + M_{12} G_2 \end{aligned}$$

(60)

$$\begin{aligned} H_2&= M_{21} G_1 + M_{22} G_2 \end{aligned}$$

(61)

To implement the matrix multiplication in Eq. (46), we apply Eqs. (60)-(61) where $H_1$,$H_2$ correspond to DADT(1), DADT(2) in NONMEM (lines 125-126) and XP(1), XP(2) in ADAPT 5 (lines 224–225).

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

NONMEM control stream for single drug TMDD with infusion coded in the control stream

The $PK and $DES blocks of the control stream are presented. Additionally, the first lines of the data file is shown to present the IV infusion mechanism. Full control stream is available in the supplemental material.

100: $PK

101: KEL = THETA(1)*EXP(ETA(1))

102: KD = THETA(2)*EXP(ETA(2))

103: KINT = THETA(3)*EXP(ETA(3))

104: KSYN = THETA(4)*EXP(ETA(4))

105: KDEG = THETA(5)*EXP(ETA(5))

106: V = THETA(6)*EXP(ETA(6))

107: A_0(1) = 0

108: A_0(2) = KSYN/KDEG

109: $DES

110: EPSILON = 1e-4

111: ; Dose at T1 = 0

112: IN = 0

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

114: IN = 100*EPSILON**(-1)

115: ENDIF

116: C = A(1)/V

117: R = A(2)

118: DET = KD+R+C

119: G1 = IN - KEL*C - (KINT*C*R)/KD

120: G2 = KSYN - KDEG*R - (KINT*C*R)/KD

121: M11 = (1/DET)*(KD+C)

122: M12 = (1/DET)*(-C)

123: M21 = (1/DET)*(-R)

124: M22 = (1/DET)*(KD+R)

125: DADT(1) = M11*G1 + M12*G2

126: DADT(2) = M21*G1 + M22*G2

The first lines of the data file are:

#ID,TIME,DV,MDV

127: 1,0,.,1

128: 1,0.0001,.,1

129: 1,1,77.8812,0

ADAPT 5 source code for single drug 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 kel,KD,kint,ksyn,kdeg

207: Real*8 C,RR,RR0

208: Real*8 Det,M(2,2),g(2)

209: kel = P(1)

210: KD = P(2)

211: kint = P(3)

212: ksyn = P(4)

213: kdeg = P(5)

214: RR0 = ksyn/kdeg

215: C = X(1)

216: RR = X(2) + RR0

217: Det = RR + C + KD

218: g(1) = R(1) - kel*C - (kint*C*RR)/KD

219: g(2) = ksyn - kdeg*RR - (kint*C*RR)/KD

220: M(1,1) = (1/Det)*(KD+C)

221: M(1,2) = (1/Det)*(-C)

222: M(2,1) = (1/Det)*(-RR)

223: M(2,2) = (1/Det)*(KD+RR)

224: XP(1) = M(1,1)*g(1) + M(1,2)*g(2)

225: XP(2) = M(2,1)*g(1) + M(2,2)*g(2)

226: Return

227: End

Fig. 1

Scheme of the necessary steps to construct a QE or QSS approximation of a TMDD system. The three systems in step 4 are equivalent

Fig. 2

Schematic of the single TMDD model described by Eqs. (1–3)

Fig. 3

Free concentration C(t) from the original TMDD model Eqs. (1–7) (dashed line) and the QE or QSS approximation Eqs. (46–50) (solid line) is shown. In panel a the QE and in panel b the QSS approximation without baseline for escalating doses (dose = 10, 100 and 1000) are presented. In panel c (QE) and in panel d (QSS) behavior for different baselines ($C^0$ = 0.01, 0.1, 1 and 10) with a fixed dose (dose = 100) are visualized. In panels e (QE) and f (QSS) the effect for increasing $k_{on} = 2.5$ and $k_{off} = 0.1$ (by multiplying the factors 0.1, 1, 10 to both parameters with equal ratio $K_D$ for QE) for a fixed dose (dose = 100) and baseline ($C^0$ = 1) are presented. In panel e (QE) the original system converges to the QE approximation and in panel f (QSS) the original systems and its QSS approximations converge for increasing $k_{on}$ and $k_{off}$ values

Fig. 4

Free concentration versus time data (crosses) produced with the original model Eqs. (1–7) and fitted with the QE approximation Eqs. (46–50) (solid line)

Table 1 The original parameters are defined in the "Methods" section