Dynamics of target-mediated drug disposition: characteristic profiles and parameter identification
Abstract
In this paper we present a mathematical analysis of the basic model for target mediated drug disposition (TMDD). Assuming high affinity of ligand to target, we give a qualitative characterisation of ligand versus time graphs for different dosing regimes and derive accurate analytic approximations of different phases in the temporal behaviour of the system. These approximations are used to estimate model parameters, give analytical approximations of such quantities as area under the ligand curve and clearance. We formulate conditions under which a suitably chosen Michaelis–Menten model provides a good approximation of the full TMDD-model over a specified time interval.
Keywords
Target Receptor Antibodies Drug-disposition Michaelis–Menten Quasi-steady-state Quasi-equilibrium Singular perturbationIntroduction
The interaction of ligand and target in the process of drug-disposition offers interesting examples of complex dynamics when target is synthesised and degrades and when both ligand and ligand–target complex are eliminated. In recent years such dynamics has received considerable attention because it is important in the context of data analysis, but also, more generally, in the context of system biology because this model serves as a module in more complex systems [1].
Based on conceptual ideas developed by Levy [2], the basic model for target mediated drug disposition (TMDD) was formulated by Mager and Jusko [3]. Earlier studies of ligand–target interactions go back to Michaelis and Menten [4]. We also mention ideas about receptor turnover developed by Sugiyama and Hanano [5]. Mager and Krzyzanski [6] showed how rapid binding of ligand to target leads to a simpler model, Gibiansky et al. [7] studied the related quasi-steady-state approximation to the model and Marathe et al. [8] conducted a numerical validation of the rapid binding approximation. Gibiansky et al. [9] also pointed out a relation with the classical indirect response model. For further background we refer to the books by Meibohm [10] and Crommelin et al. [11], and to the reviews by Lobo et al. [12] and Mager [13].
In practice, the Michaelis–Menten model is often used when ligand curves exhibit TMDD characteristics (see e.g. Bauer et al. [14]). Recently, Yan et al. [15] analysed the relationship between TMDD- and Michaelis–Menten type dynamics. We also mention the work by Krippendorff et al. [16] which studies an extended TMDD system which includes receptor trafficking in the cell.
The characteristic features of TMDD dynamics were first studied in [3] under the condition of a constant target pool, i.e., the total amount of target: free and bound, was assumed to be constant in time. Under the same assumption, a mathematical analysis of this model was offered by Peletier and Gabrielsson [17]. This assumption was made, in part for educational reasons, because it makes a transparent geometric description possible, in which qualitative and quantitative properties of the dynamics can be identified and illustrated. In a recent paper Ma [18] compared different approximate models under the same assumption of a constant target pool.
In the present paper we extend this analysis to the full TMDD model and do not make the assumption that the target pool is constant. This means, in particular, that we shall now also be enquiring as to how the target pool changes over time and how it is affected by the dynamics of its zeroth order synthesis and first order degeneration.
- 1.Properties of concentration profiles When the initial ligand concentration is larger than the endogenous receptor concentration, the dynamics of target-mediated drug disposition results in a characteristic ligand versus time profile. In Fig. 1 we show such a profile schematically.
One can distinguish four different phases in the dynamics of the system in which different processes are dominant: (A) a brief initial phase, (B) an apparent linear phase, (C) a transition phase and (D) a linear terminal phase. We obtain precise estimates for the duration of each of these phases and for each of them we obtain accurate analytical estimates for the concentration versus time graphs of the ligand, the receptor and the ligand–receptor complex.
On the basis of ligand concentration versus time curves we will develop instruments for extracting information about the target and the ligand–target complex versus time curves.
- 2.
Parameter identifiability We shed light on what we can predict when we have only measured (a) the ligand, (b) ligand and target, (c) ligand and complex, and (d) all of the above.
- 3.
Systems analysis Whilst focussing on concentration versus time curves we gain considerable qualitative understanding and quantitative estimates about the impact of the different parameters in the model and on quantities such as the area under the curve and time to steady state of the different compounds.
- 4.
Model comparison An important issue is the question as to how the full TMDD model compares with the simpler Michaelis–Menten model [7, 8, 15]. In this paper we point out how the full model and the reduced Michaelis–Menten model differ significantly in the initial second order phase and in the linear terminal phase, in that the terminal rate (λ_{z}) of ligand in the full model is much smaller than that in the Michaelis–Menten model.
(i) Through a bolus dose. Then k_{f} = 0. We denote the initial ligand concentration by L(0) = L_{0} = D/V_{c}, where D is the dose and V_{c} the volume of the central compartment.
(ii) Through a constant rate infusion In_{L}. Then k_{f} = In_{L}/V_{c} > 0 and L(0) = 0.
Anchoring our investigation in a case study in which ligand is administered through a series of bolus doses, we dissect the resulting time courses of the three compounds, L, R and RL and identify characteristic phases, Phases A–D shown in Fig. 1. We associate these phases with specific processes and show, using singular perturbation theory [19, 20, 21], that individual phases may be analysed through appropriately chosen simplified models, yielding accurate closed-form approximations. They offer tools which may be used to compute critical quantities such as residence time, and to verify whether different approximations to the full TMDD model, such as the rapid binding approximation [6] and the quasi-steady-state approximation [7, 18] are valid in the different phases. These issues are discussed at the conclusion of this paper.
Much of the mathematical analysis underpinning the results presented throughout the text is presented in a series of Appendices at the end of the paper.
Case study
Pre-selected parameter values
Symbol | Unit | Value |
---|---|---|
V_{t} | L/kg | 0.1 |
Cl_{d} | (L/kg)/h | 0.003 |
Cl_{(L)} | (L/kg)/h | 0.001 |
k_{on} | (mg/L)^{−1}/h | 0.091 |
k_{off} | 1/h | 0.001 |
k_{in} | (mg/L)/h | 0.11 |
k_{out} | 1/h | 0.0089 |
k_{e(RL)} | 1/h | 0.003 |
R_{0} | mg/L | 12 |
Data analysis
The purpose of this study is to demonstrate the possibility of fitting the eight-parameter model shown in Fig. 2, to three different sets of high quality data with increasing richness, and show how precision of the estimates of the model parameters increases when successively information about target (II) and target and complex (III) is added. We use this data set for two purposes: (i) for data analysis and (ii) for highlighting critical features of the temporal behaviour of the three compounds.
Simulated data from three sources (ligand, target and complex) were intentionally used. We have experienced that data of less quality gave biased and imprecise estimates as well as biased and imprecise predictions of ligand, target and complex.
Final parameter estimates and their relative standard deviation (CV%) on the basis of the three datasets
Symbol | Unit | I (L) | II (L & R) | III (L & R & RL) |
---|---|---|---|---|
V_{t} | L/kg | 0.101 (2) | 0.100 (2) | 0.100 (1) |
Cl_{d} | (L/kg)/h | 0.003 (4) | 0.003 (3) | 0.003 (3) |
Cl_{(L)} | (L/kg)/h | 0.001 (1) | 0.001 (1) | 0.001 (1) |
k_{on} | (mg/L)^{−1}/h | 0.099 (17) | 0.092 (2) | 0.096 (1) |
k_{off} | 1/h | 0.001 (27) | 0.001 (13) | 0.001 (3) |
k_{out} | 1/h | 0.009 (6) | 0.009 (2) | 0.009 (2) |
k_{e(RL)} | 1/h | 0.002 (27) | 0.002 (23) | 0.002 (2) |
R_{0} | mg/L | 12 (4) | 12 (1) | 12 (1) |
Dataset I is made up from simulated concentration–time profiles covering five orders of magnitude in concentration range and from 0 to 500 h. Dataset II contains the same simulated ligand (L) profiles as in dataset I as well as target (R) concentration–time profiles obtained at each dose level. Dataset III includes dataset II but is enriched by four simulated time-courses of the ligand–target complex (LR) as well.
The four doses are D = 1.5, 5, 15 and 45 mg/kg. The volume V_{c} of the central compartment being 0.05 L/kg, this yields the following initial ligand concentrations L_{0} = D/V_{c} = 30, 100, 300 and 900 mg/L.
Dataset I—which involves L—allows the prediction of robust ligand concentration–time profiles within the suggested concentration and time frame. We see that if only ligand data are available, the majority of parameters except for k_{on}, k_{off} and k_{e(RL)} are estimated with high precision. The latter three parameters are still highly dependent on information about the time courses of either target and/or complex. Since k_{kon}, k_{off} and k_{e(RL)} have low precision (high CV%) we would discourage the use of these parameters for the prediction of tentative target and complex concentrations.
Dataset II—which involves L and R—still gives good precision in all parameters except k_{off} and k_{e(RL)}, which will also be highly correlated. Since we also have experimental data of the target we encourage the use of this model for interpolation of target concentration–time courses, but not for concentration–time courses of the complex.
Dataset III—which includes L and R, as well as RL—gives high precision in all parameters. Since we also have measured the complex concentration–time course with high precision we obtained k_{off} and k_{e(RL)} values with high precision. We doubt that the practical experimental situation can get very much better than this latter case where we have simultaneous concentration–time courses of L, R and RL with little experimental error due to biology and bio-analytical methods. Dataset III is an ideal case; the true experimental situation seldom gets better.
We also doubt the practical value of regressing too elaborate models to data. Models that capture the overall trend nicely but result in parameters with low precision and biased estimates may be of little value.
The volume of the central compartment V_{c} ought to fall somewhere in the neighbourhood of the plasma water volume (0.05 L/kg) for large molecules in general and antibodies in particular. In our own experience of antibody projects this has been the case when data contained an acceptable granularity within the first couple of hours after the injection of the test compound. Therefore we assumed V_{c} to be a constant term (0.05 L/kg) in this analysis and not part of the list of parameters to be estimated. We think this increases the robustness of the estimation procedure and is biologically viable.
Critical features of the graphs
- (a)
Initially all the ligand graphs in Fig. 3 exhibit a rapid drop which increases in relative sense as the ligand dose decreases. Over this initial period, which we refer to as Phase A, (cf. Fig. 1), R(t) exhibits a steep drop that becomes deeper as the drug dose increases.
- (b)
After the brief initial adjustment period, the graphs for large doses reveal linear first order kinetics over a period of time (Phase B) that shrinks as the drug dose decreases. At the lowest dose the linear period has vanished and the graph exhibits nonlinear kinetics.
- (c)
For the larger doses, there is an upward shift of the linear phase that appears to be linearly related to the ligand dose; the slope of this linear phase appears to be dose-independent.
- (d)
The point of inflection in the log(L) versus time curve—the middle of Phase C—which we observe in the graphs for L_{0} = 100 and 300, moves to the right as the initial dose increases, but stays at the same level. This is clearly seen in Fig. 3 in which the baseline value R_{0} and the value of K_{d} and K_{m} are also shown.
- (e)
For the lower doses we see that the log(L) versus time curve eventually becomes linear again, with a slope that is markedly smaller than it was in the nonlinear Phase C that preceded it. This part of the graph corresponds to Phase D in Fig. 1.
Summarising, in the ligand graphs of Fig. 3 we see for the higher drug doses the different phases A–D that were pointed out in Fig. 1. In the following analysis we explain these features and quantify them in that, for instance, we present estimates for the upward shift referred to in (c) and the right-ward shift of the inflection point alluded to in (d).
Dynamics after a bolus administration
Simulations
Parameter estimates used for demonstrating the dynamics of L, R and RL after bolus and constant-rate infusion regimens of L
Symbol | Unit | Value |
---|---|---|
k_{e(L)} | 1/h | 0.0015 |
k_{on} | (mg/L)^{−1}/h | 0.091 |
k_{off} | 1/h | 0.001 |
k_{in} | (mg/L)/h | 0.11 |
k_{out} | 1/h | 0.0089 |
k_{e(RL)} | 1/h | 0.003 |
R_{0} | mg/L | 12 |
- (a)
A rapid initial adjustment (see the blow-up on the right).
- (b)
A first linear phase with a slope which is independent of the dose, and which shifts upwards as the drug dose increases.
- (c)
A transition phase which shifts to the right as the drug dose increases, but maintains its level.
- (d)
A final linear terminal phase with a slope λ_{z} that is again independent of the drug dose. For the parameter values of Table 3 we find that λ_{z} ≈ k_{e(RL)} = 0.003.
Alternatively, if k_{e(RL)} > k_{out}, then R_{*} < R_{0} and we show that the total target pool first decreases before it returns to the baseline value R_{0}.
(ii) As the drug dose increases, R(t) ≈ 0 for an increasing time interval and the graphs of RL(t) and R_{tot}(t) trace—for the same increasing time interval—a common curve \(\Upgamma\) in the (t, R_{tot})-plane (cf. Fig. 9). This curve \(\Upgamma\) is monotonically increasing and tends to the limit R_{*} as \(t \to \infty. \) If k_{e(RL)} > k_{out}, we show that an analogous phenomenon occurs along a curve \(\Upgamma, \) which still tends to R_{*}, but is now decreasing.
(i) The three graphs exhibit a kink (a sharp angle) which shifts to the right (increasing time) as the drug dose increases.
(ii) R(t) − R_{0} tends to zero as \(t \to \infty\) in a bi-exponential manner, whilst RL(t) and R_{tot}(t) − R_{0} converge to zero in a mono-exponential way.
Low dose graphs
We conclude these simulations with a comparison of high-dose and low-dose graphs. We do this by adding simulations for initial ligand concentrations which are smaller than R_{0}. Specifically, we add the values L_{0} = 0.3, 1, 3 and 10 mg/L to the graphs shown in Figs. 5 and 6.
Mathematical analysis
Phase A
Phase B
Phase C
Phase D
When L(t) ≪ K_{d}, i.e., beyond T_{3}, the ligand concentration is so small that the dynamics is linear again.
The critical times T_{1}, T_{2} and T_{3} provide a natural division of the dynamics in four phases: A, B, C and D, as was done in Fig. 1. In Phase A (0 < t < T_{1}), ligand, receptor and complex reach quasi-equilibrium, in Phase B (T_{1} < t < T_{2}), the bulk of the ligand is eliminated from the system while most of the receptor is bound to ligand and in quasi-equilibrium. Phase C (T_{2} < t < T_{3}) is a nonlinear transitional phase in which L exhibits a steep drop, and finally, in Phase D (\(T_3<t<\infty\)) the three compounds converge linearly towards their baseline values.
Receptor graphs: Phase B
In Fig. 9 we see how for the different drug doses, the simulations of R_{tot}(t) follow the graph \(\Upgamma\) up till some time, when they suddenly depart from \(\Upgamma. \)
Remark
We recall from the approximation (17) that R(t) ≈ 0 for T_{1} < t < T_{2} and hence that RL(t) ≈ R_{tot} in this phase of the dynamics, as we see confirmed in Fig. 6.
Ligand graphs: Phase B
Remark
(i) the ratio μ of R_{0} and L_{0} and
(ii) the ratio κ of the direct elimination rates of receptor, k_{out}, and ligand, k_{e(L)}.
Ligand and receptor graphs: Phases C and D
In the simulations shown in Figs. 5 and 6 we see that in Phase C, L drops rapidly from O(10 × K_{d}) to O(0.1 × K_{d}), i.e., by a factor 100, whilst R_{tot} stays relatively close to R_{*} and changes by no more than a factor 1/7 ≈ 0.15. This suggests making the following assumption:
Assumption
R_{tot}(t) ≈ R_{*} or v(t) ≈ R_{*}/R_{0} in Phase C.
(i) the terminal slope λ_{z}^{TMDD} (k_{e(RL)});
(ii) the intercept of the asymptote of log{L(t)} of the ligand graph in the terminal Phase D with the vertical line {t = T_{2}}.
For completeness we also compute the terminal slope by means of a standard analysis of the full TMDD system. This is done in Appendix 7. It is found that for the parameter values in Table 3, the terminal slope λ_{z} of all the compounds, is given—to good approximation—by λ_{z}^{TMDD} = k_{e(RL)}. This confirms the limit in (36) and the exponent in (38).
Comparison with Michaelis–Menten kinetics
Thus, the TMDD-model and the Michaelis–Menten (MM)-model exhibit very different terminal slopes, unless one also includes a non-specific peripheral volume distribution term in the MM-model.
The reduced model mimics the concentration–time data for the two highest doses reasonably well, whereas the two lower doses display systematic deviations between observed and predicted data.
Since the reduced model has two parallel elimination pathways (linear and nonlinear) it has the intrinsic capacity of exhibiting linear first-order kinetics at low and at high concentrations. In the concentration-range in between it behaves nonlinearly. For higher concentrations the MM-route is saturated and the linear elimination pathway dominates so that the system behaves linearly.
However, the typical concentration–time pattern for ligand seen in a true TMDD system (cf. Figs. 5, 8, 10), cannot be fully described by the parallel linear- and MM-elimination model. The reduced model displays typical bi-exponential decline (which is expected from a two-compartment model) at lower concentrations. That is generally not the case with the full TMDD model.
A clear distinction of the two models occurs initially, immediately after dosing (Phase A), when the second-order reaction between ligand and circulating target forms the complex. This process cannot be captured by the reduced model, which may cause biased estimates (too large) of the central volume.
Symbol | Unit | Value | CV% |
---|---|---|---|
V_{c} | L/kg | 0.05 | – |
V_{t} | L/kg | 0.1 | 10 |
Cl_{d} | (L/kg)/h | 0.00307 | 20 |
Cl_{(L)} | (L/kg)/h | 0.00090 | 10 |
V_{max} | mg/h | 0.0146 | 40 |
K_{M} | mg/L | 3.68 | 50 |
Phases that can be explained by the two MM-models and the TMDD-model
Phase | MM-model (41) | MM-model (45) | TMDD-model (9) |
---|---|---|---|
A | − | − | + |
B | + | + | + |
C | +/− | + | + |
D | − | − | + |
In this table, a plus (+) means that the corresponding phase can be adequately explained, whilst a minus (−) means that it cannot.
Constant rate drug infusion
When the infusion lasts long enough, i.e., when t_{washout} is large enough, the concentrations will converge towards their steady state values L_{ss}, R_{ss} and RL_{ss}. Then, at washout, they will return to their pre-infusion values: L = 0, R = R_{0} and RL = 0.
Steady state concentrations of L, R and RL
The formula (48) for the ligand shows that L_{ss} will be smaller than expected from the ratio of ligand infusion rate-to-clearance (In_{L}/Cl_{(L)} = k_{f}/k_{e(L)}), due to the removal of ligand as part of the complex RL_{ss}. The same reasoning may be used to explain why the circulating target concentration R_{ss} given by (49) is smaller than the baseline concentration R_{0} = k_{in}/k_{out}. Due to the removal of target by means of the complex, the target concentration R_{ss} will drop further as the infusion rate increases and RL_{ss} increases accordingly (cf. Eq. (49)).
Using the parameter values in Table 3, we obtain 1/k_{e(L)} = 667, k_{in}/k_{e(L)} = 73, R_{*} = k_{in}/k_{e(RL)} = 36.7 and log(K_{m}) + log(R_{*}) = 0.45. We see that these values are confirmed by the numerically obtained graphs shown in Fig. 12.
We note that the expressions (47)–(49) show that the steady state concentrations do not depend on the on- and off rates k_{on} and k_{off} individually, but only as part of the constant K_{m}.
Simulations
The washout dynamics is very similar to the dynamics after a bolus dose, as described before: Phase A (0,T_{1}) now covers the infusion period so that T_{1} coincides here with the time of washout. Phases B–D are plainly evident in the post-washout dynamics.
The dynamics of the receptor R and the receptor–ligand complex RL are shown in Figs. 14 and 15. As in Phase A in the bolus administration, the pre-dose receptor pool (R_{0}) quickly binds to the ligand. We see that the speed of receptor depletion increases with increasing infusion rate k_{f}, consistent with the half-life estimate (15) after a bolus administration.
After washout, when k_{f} is large, R(t) ≈ 0 and RL(t) ≈ R_{*} for a while before they abruptly return to their baseline values. It is evident from Fig. 15 that, as in Fig. 7, initially the slope of log(R_{0} − R) is steeper than that of log(RL). This is in agreement with the analysis presented in Appendix 5, where it is shown that over that period the half-lives of R(t) − R_{0} and RL(t) are, respectively, O(1/k_{out}) and O(1/k_{e(RL}).
Discussion and conclusion
When ligand is administered through a bolus dose, L_{0} > R_{0}, and the conditions in (58) are satisfied, four phases can be distinguished in the ligand elimination graph: a brief initial Phase A, a slow linear Phase B, a rapid nonlinear Phase C and then again a slow linear terminal Phase D (cf. Fig. 1). Thanks to accurate analytical approximations for these four phases as shown in the Eqs. (14) for Phase A, (20), (25)–(27) for Phase B, (36) for Phase C and (38) and (40) for Phase D, we may extract information about the model parameters.
Information contained in the four phases
Phase | TMDD-model (9) |
---|---|
A | R_{0} and k_{on} |
B | k_{e(L)}, k_{in} (k_{out}) |
C | K_{d} (k_{off}) |
D | k_{e(RL)} (K_{m}) |
It should be noted though that estimating R_{0} may be difficult, since often there are no data for the first phase because it is over very quickly.
The four phases identified in the ligand elimination graph after a bolus administration are reflected in the structure of the receptor versus time graphs (receptor, receptor–ligand complex, and total amount of receptor). During Phase A the receptor pool is quickly depleted, and it remains so during Phase B. Then, during the PhasesC and D it climbs back to the terminal baseline level R_{0}.
- 1.The rapid binding model [6, 18], in which it is assumed thatwhere K_{d} is defined in (7). In Phase B, which is characterised by R(t) ≈ 0, we have dR/dt ≈ 0 so that, according to the second equation of the system (9), we have approximately$$ L \cdot R = K_d RL $$(59)which disagrees with (59).$$ \Updelta \mathop{=}\limits^{{\rm def}} L \cdot R - K_d RL =\frac{k_{\rm in}}{k_{\rm on}} $$(60)
In contrast, in Phases C and D the identity (59) is satisfied according to the results established in Appendix 5 (cf. (102)), and Appendix 7 where it was shown that λ_{z} = k_{e(RL)}.
In Fig. 17 we show how the quantity \(\Updelta = L \cdot R - K_d RL\) varies with time and how \(\Updelta\) rapidly jumps from k_{in}/k_{on} down to zero at the transition of Phases B and C.
- 2.The quasi-steady-state model [7, 18] in which it is assumed thatwhere K_{m} is defined in (8). Evidently, this assumption is not valid during Phases C and D, but it is during that part of Phase B in which RL(t) ≈ R_{*}. In that interval dRL/dt ≈ 0 (cf. Fig. 6) and hence, by the third equation of the system (1), condition (61) is approximately satisfied.$$ L \cdot R = K_m RL $$(61)
Anchored on the data of the Case Study, the analysis in this paper is based on the Assumptions A, B and C (or C^{*}). The question arises whether the characteristic features of the ligand elimination curves, such as shown in Fig. 1, are still present when these assumptions are not met.
In general, the behaviour of nonlinear systems such as (1) is very sensitive to the values of the parameters and initial data involved. However, a number of features of the ligand versus time graphs is quite robust in that they may survive if e.g. Assumption A is not satisfied and K_{d} and R_{0} are comparable. Thus, the estimate (15) for T_{1} suggests that the initial Phase A will remain short relative to typical times over which the other processes develop when K_{d}/L_{0} is small. We refer to [17] and [18] for a detailed analysis of this situation.
The approximate expressions for L and R_{tot} in Phase B (Eqs. (25) and (20)) are still valid provided that R(t) ≈ 0. This will still be the case when Assumption A is replaced by K_{d} ≪ L_{0} [17, 18].
In contrast, the analysis of the dynamics in Phase C that is carried out in Appendix 5 depends critically on Assumption A. It will be interesting to study the dynamics beyond Phase B when Assumption A does not hold, as it will be interesting to see how the value of α, β and γ affects the dynamics.
We have selected a set of data (ligand and circulating target and complex) with low experimental variability, concentration–time courses at four ligand doses given as bolus injections, and well-spaced data in time that captures the necessary phases and shapes of a typical TMDD system. Based on this approach and the mathematical/analytical analysis, we can draw conclusions about the identifiability of the model parameters and appropriate system. When data are less precise and information rich, or, when target and/or complex are less accessible, the a priori expectations of parameter accuracy and precision will be lower.
Footnotes
Notes
Acknowledgments
The constructive criticisms of the manuscript of the reviewers improved the quality of the text, and are highly appreciated.
Open Access
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