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Lumping of physiologically-based pharmacokinetic models and a mechanistic derivation of classical compartmental models

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Abstract

In drug discovery and development, classical compartment models and physiologically based pharmacokinetic (PBPK) models are successfully used to analyze and predict the pharmacokinetics of drugs. So far, however, both approaches are used exclusively or in parallel, with little to no cross-fertilization. An approach that directly links classical compartment and PBPK models is highly desirable. We derived a new mechanistic lumping approach for reducing the complexity of PBPK models and establishing a direct link to classical compartment models. The proposed method has several advantages over existing methods: Perfusion and permeability rate limited models can be lumped; the lumped model allows for predicting the original organ concentrations; and the volume of distribution at steady state is preserved by the lumping method. To inform classical compartmental model development, we introduced the concept of a minimal lumped model that allows for prediction of the venous plasma concentration with as few compartments as possible. The minimal lumped parameter values may serve as initial values for any subsequent parameter estimation process. Applying our lumping method to 25 diverse drugs, we identified characteristic features of lumped models for moderate-to-strong bases, weak bases and acids. We observed that for acids with high protein binding, the lumped model comprised only a single compartment. The proposed lumping approach established for the first time a direct derivation of simple compartment models from PBPK models and enables a mechanistic interpretation of classical compartment models.

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Acknowledgements

The authors kindly acknowledge comments on the manuscript by Charlotte Kloft (Clinical Pharmacy, Martin-Luther-Universität Halle-Wittenberg/ Germany), Steve Kirkland (Hamilton Institute, NUIM/Ireland), Andreas Reichel (Bayer Schering Pharma) and Olaf Lichtenberger (Abbott). S.P. acknowledges financial support from the Graduate Research Training Program PharMetrX: Pharmacometrics and Computational Disease Modeling, Martin-Luther-Universität Halle-Wittenberg and Freie Universität Berlin, Germany (http://www.pharmacometrics.de).

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Correspondence to Wilhelm Huisinga.

Appendices

Appendix A: Derivation of the relation between the lumped and the original concentrations

The relation between the concentrations of the lumped compartment and the comprised original compartments is given by Eq. 22:

$$ C_{\rm L} = \frac{1}{V_{\rm L}} \sum_{\rm tis} V_{\rm tis} C_{\rm tis}. $$
(54)

The idea is to establish a link between the concentration C tis of one of the original organs (or tissues) and the concentration C L of the associated lumped compartment. Let us arbitrarily choose one organ (named ‘ref’). Using the lumping criteria, we have C tis/K tis = C ref/(K ref(1 − E ref)) and thus C tis = C ref K tis/(K ref(1 − E ref)) for all tissue/organs lumped into ‘L’. In combination with the above equation for C L this yielded

$$ C_{\rm L} = \frac{1}{V_{\rm L}} \sum_{\rm tis} V_{\rm tis} \frac{K_{\rm tis}}{K_{\rm ref} (1-E_{\rm ref})} C_{\rm ref}, $$
(55)

or equivalently

$$ C_{\rm L} = \frac{1}{V_{\rm L}} \sum_{\rm tis} V_{\rm tis} K_{\rm tis} \cdot \frac{C_{\rm ref}}{K_{\rm ref}(1-E_{\rm ref})}. $$
(56)

Using the definition of K L in Eq. 23 and 24 we obtain the desired relation:

$$ \frac{C_{\rm L}}{K_{\rm L}} = \frac{C_{\rm ref}}{K_{\rm ref}(1-E_{\rm ref})}. $$
(57)

Since the tissue/organ ‘ref’ was arbitrarily chosen, the above relation holds for every original compartment ‘tis’ that is part of the lumped compartment ’L’, being eliminating (in which case E ref > 0) or non-eliminating (in which case E ref = 0).

Appendix B: General derivation of lumped ODEs

In the following, the subscript tis refers to all organs of the PBPK model excluding the liver, i.e., adi, bra, bon, gut, hea, kid, lun, mus, ski, spl.

To derive general equations for the rate of change of the lumped concentrations C L it is advantageous to bring the original ODEs of the generic PBPK in a different but equivalent form, where elimination is associated with the venous compartment and all other compartments have the same structural form. In the generic PBPK model (see Section ‘Material and Methods’), the liver is assumed to be the only eliminating organ with ODE (see Eq. 3)

$$ V_{\rm liv}\frac{\hbox{d}}{\hbox{dt}} C_{\rm liv} = Q_{\rm liv} \cdot \left(C_{\rm in}- \frac{C_{\rm liv}}{K_{\rm liv}}\right) - \hbox{CL}_{\rm int} C_{\rm liv}. $$
(58)

Defining Rhep = CLint K liv/Q liv and noting that 1 + Rhep = 1/(1 − Ehep), where Ehep is the hepatic extraction ratio defined in Eq. 16, we obtain

$$ V_{\rm liv}\frac{\hbox{d}}{\hbox{dt}} C_{\rm liv} = Q_{\rm liv}\cdot\left(C_{\rm in}- \frac{C_{\rm liv}}{K_{\rm liv}}\right) - Q_{\rm liv} \hbox{R}_{\rm hep} \frac{C_{\rm liv}}{K_{\rm liv}} $$
(59)
$$ = Q_{\rm liv}\cdot\left(C_{\rm in}- (1+\hbox{R}_{\rm hep})\frac{C_{\rm liv}}{K_{\rm liv}}\right) $$
(60)
$$ = Q_{\rm liv}\cdot\left(C_{\rm in}- \frac{C_{\rm liv}}{K_{\rm liv}(1-\hbox{E}_{\rm hep})}\right). $$
(61)

Now, the inflowing concentration of the vein is given by (see Eq. 7)

$$ C_{\rm in} = \frac{1}{Q_{\rm co}} \sum_{\rm tis} Q_{\rm tis} \frac{C_{\rm tis}}{K_{\rm tis}}, $$
(62)

which can be rewritten as

$$ \begin{aligned} C_{\rm in} & = \frac{1}{Q_{\rm co}} \left(\sum_{{\rm tis}\neq{\rm liv}} Q_{\rm tis} \frac{C_{\rm tis}}{K_{\rm tis}}\right.\\ & \quad \left. + Q_{\rm liv}(1-\hbox{E}_{\rm hep})\frac{C_{\rm liv}}{K_{\rm liv}(1-\hbox{E}_{\rm hep})}\right) \end{aligned} $$
(63)
$$ \begin{aligned} &= \frac{1}{Q_{\rm co}} \left(\sum_{{\rm tis}\neq{\rm liv}} Q_{\rm tis}\frac{C_{\rm tis}}{K_{\rm tis}} + Q_{\rm liv} \frac{C_{\rm liv}}{K_{\rm liv}(1-\hbox{E}_{\rm hep})}\right)\\ &\quad - \frac{1}{Q_{\rm co}} Q_{\rm liv}\hbox{E}_{\rm hep}\frac{C_{\rm liv}}{K_{\rm liv}(1-\hbox{E}_{\rm hep})}. \end{aligned} $$
(64)

Let us define

$$ \widehat{K}_{\rm tis} = K_{\rm tis}(1-E_{\rm tis}), $$
(65)

where E tis denotes the tissue elimination ratio. In our case, it is E liv = Ehep and E tis = 0 otherwise. Moreover, formally define \(\widehat{K}_{\rm ven} = \widehat{K}_{\rm art} =1\). Then, we finally obtain an equivalent formulation of the whole-body PBPK model. For all organs, tissues and other spaces except vein, it is

$$ V_{\rm tis}\frac{\hbox{d}}{\hbox{dt}} C_{\rm tis} = Q_{\rm tis}\cdot \left(C_{\rm in} - \frac{C_{\rm tis}}{\widehat{K}_{\rm tis}}\right), $$
(66)

while for the vein it is

$$ V_{\rm ven} \frac{\hbox{d}}{\hbox{dt}} C_{\rm ven} = Q_{\rm co} \cdot \left(C_{\rm in} - \frac{C_{\rm ven}}{\widehat{K}_{\rm ven}}\right) - \hbox{CL}_{\rm blood} {{C_{\rm liv}}\over {\widehat{K}_{\rm liv}}}, $$
(67)

where we exploit CLblood = Q livEhep based on Eq. 67.

We determined the equation for the rate of change of the lumped concentration C L by differentiating Eq. 22, yielding:

$$ V_{\rm L} \frac{\hbox{d}}{\hbox{dt}} C_{\rm L} = \sum_{\rm tis} V_{\rm tis} \frac{\hbox{d}}{\hbox{dt}} C_{\rm tis}. $$
(68)

The right hand side V tisd/dt C tis is defined in Eqs. 66 and 67. The right hand side of contain the concentrations of the original PBPK model C tis, which can be determined using Eq. 57.

We obtained a very simple form of equations for the mechanistically lumped model, when spleen and gut were lumped together into the same compartment as the liver (hence they were part of the lumped ‘Liv’ compartment), and when lung and artery were lumped together into the same compartment as the vein (hence, they were part of the lumped ‘cen’ compartment). In this case, the influent concentrations will be identical for all compartments different from the central compartment. The rate of change for the concentration of the pre-lumped compartments spl-gut-liv (‘sgl’) and ven-lun-art (‘vla’) are

$$ V_{\rm sgl} \frac{\hbox{d}}{\hbox{dt}} C_{\rm sgl} = Q_{\rm liv} \left(C_{\rm art} - \frac{C_{\rm liv}}{\widehat{K}_{\rm liv}} \right) $$
(69)
$$ = Q_{\rm liv} \left(C_{\rm art} - \frac{C_{\rm sgl}}{\widehat{K}_{\rm sgl}} \right), $$
(70)

and

$$ V_{\rm vla} \frac{\hbox{d}}{\hbox{dt}} C_{\rm vla} = Q_{\rm co} \left(C_{\rm in} - \frac{C_{\rm art}}{\widehat{K}_{\rm art}} \right)- \hbox{CL}_{\rm blood} \frac{C_{\rm liv}}{\widehat{K}_{\rm liv}} $$
(71)
$$ = Q_{\rm co} \left(C_{\rm in} - \frac{C_{\rm vla}} {\widehat{K}_{\rm vla}} \right) - \hbox{CL}_{\rm blood} \frac{C_{\rm liv}}{\widehat{K}_{\rm liv}} $$
(72)

where C in is the concentration flowing into the vein.

Now, we have for any lumped compartment excluding the central compartment:

$$ V_{\rm L} \frac{\hbox{d}}{\hbox{dt}} C_{\rm L} = \sum_{\rm tis} V_{\rm tis} \frac{\hbox{d}}{\hbox{dt}} C_{\rm tis} $$
(73)
$$ = \sum_{\rm tis} Q_{\rm tis} C_{\rm art} - \sum_{\rm tis} Q_{\rm tis} \frac{C_{\rm tis}}{\widehat{K}_{\rm tis}}, $$
(74)

where the sum is taken over all tissues that are lumped into ‘L’. Our above assumption (resulting in Eq. 69) ensures that the inflowing concentration is the same and identical to C art for all tissue/organs. Exploiting the lumping condition \({C_{\rm tis}}/{\widehat{K}}={C_{\rm L}}/{K_{\rm L}}\) yields

$$ V_{\rm L} \frac{\hbox{d}}{\hbox{dt}} C_{\rm L} = \left(\sum_{\rm tis} Q_{\rm tis}\right) C_{\rm in} -\left(\sum_{\rm tis} Q_{\rm tis}\right) \frac{C_{\rm L}}{K_{\rm L}}. $$
(75)

Finally, using Q L = ∑tis Q tis and C artC cen/K cen results in

$$ V_{\rm L} \frac{\hbox{d}}{\hbox{dt}} C_{\rm L} = Q_{\rm L} \left( \frac{C_{\rm cen}}{K_{\rm cen}} - \frac{C_{\rm L}}{K_{\rm L}}\right). $$
(76)

For the central compartment, we obtain analogously

$$ V_{\rm cen} \frac{\hbox{d}}{\hbox{dt}} C_{\rm cen} = Q_{\rm cen} \left(C_{\rm in} - {{C_{\rm cen}}\over {\widehat{K}_{\rm cen}}} \right)- \hbox{CL}_{\rm blood} {{C_{\rm Liv}}\over {K_{\rm Liv}}}, $$
(77)

where we exploited the fact that \(C_{\rm liv}/\widehat{K}_{\rm liv} = C_{\rm Liv}/K_{\rm Liv}\).

The above equations do not take into account any dosing. Again, as in the whole-body PBPK case, the corresponding ODEs of the lumped compartments comprising vein and liver have to be amended correspondingly. For an i.v. infusion r iv (see Eq. 9 for the definition), it is

$$ V_{\rm cen} \frac{\hbox{d}}{\hbox{dt}} C_{\rm cen} = Q_{\rm cen} \left(C_{\rm in} - \frac{C_{\rm cen}}{\widehat{K}_{\rm cen}} \right) - \hbox{CL}_{\rm blood} \frac{C_{\rm Liv}}{K_{\rm Liv}} + r_{\rm iv}, $$
(78)

while for a p.o. administration \(r_{{\rm po}({\rm F}_{{\rm F}\cdot {\rm G}})}\) (see Eq. 9 for the definition) it is

$$ V_{\rm Liv} \frac{\hbox{d}}{\hbox{dt}} C_{\rm Liv} = Q_{\rm Liv} \left( \frac{C_{\rm cen}}{K_{\rm cen}} - \frac{C_{\rm Liv}}{K_{\rm Liv}}\right) + r_{{\rm po}({\rm F}_{{\rm F}\cdot {\rm G}})}. $$
(79)

If ‘cen’ and ‘Liv’ are identical, i.e., the liver is lumped into the central compartment, then it is C cen/K cenC Liv/K Liv and thus

$$V_{\rm cen} \frac{\hbox{d}}{\hbox{dt}} C_{\rm cen} = Q_{\rm cen} \left(C_{\rm in} - \frac{C_{\rm cen}}{\widehat{K}_{\rm cen}}\right) - \hbox{CL}_{\rm blood} \frac{C_{\rm cen}}{K_{\rm cen}}+ r_{{\rm iv,po}({\rm F}_{\rm bio})}. $$
(80)

Note that in this case, the p.o. administration model has to account for the hepatic extraction, i.e., the absorption model with F bio = (1 − Ehep)F F·G (see Eq. 13) is used rather then the model with F F·G (see Eq. 10).

Appendix C: Lumping of permeability rate-limited tissue model

The rates of change of the vascular concentration C vas and the tissue concentration C tis corresponding to a permeability limited tissue model are given by:

$$ V_{\rm vas}\frac{\hbox{d}}{\hbox{dt}} C_{\rm vas} = Q_{\rm tis}\cdot (C_{\rm in} - C_{\rm vas}) - \hbox{PS}_{\rm tis} \left(C_{\rm vas} -\frac{C_{\rm tis}}{K_{\rm tis}}\right) $$
(81)
$$ V_{\rm tis} \frac{\hbox{d}}{\hbox{dt}} C_{\rm tis} = \hbox{PS}_{\rm tis} \cdot \left(C_{\rm vas} - \frac{C_{\rm tis}}{K_{\rm tis}}\right). $$
(82)

The amount of drug that can transfer from the vascular to the tissue part is limited by the maximal amount of drug, entering the tissue, i.e, Q tis C in. This has to be taken into account, when lumping the tissue space together with other compartments. The following idea is similar to the approach to determine the blood clearance from the intrinsic clearance. It is based on a quasi-steady state assumption on C vas (Eq. 81) yielding

$$ 0 = Q_{\rm tis}\cdot (C_{\rm in} - C_{\rm vas}) - \hbox{PS}_{\rm tis} \left(C_{\rm vas} -\frac{C_{\rm tis}}{K_{\rm tis}}\right) $$
(83)

or

$$ C_{\rm vas} = \frac{Q_{\rm tis}}{\hbox{PS}_{\rm tis} +Q_{\rm tis}} C_{\rm in} + \frac{\hbox{PS}_{\rm tis}} {\hbox{PS}_{\rm tis}+Q_{\rm tis}}\cdot\frac{C_{\rm tis}}{K_{\rm tis}}. $$
(84)

Thus, in steady state, the vascular concentration is a weighted sum of the influent concentration C in and the concentration leaving the tissue compartment C tis/K tis. For permeability-rate limited organs, we would usually expect

$$ \frac{Q_{\rm tis}}{\hbox{PS}_{\rm tis}+Q_{\rm tis}} > \frac{\hbox{PS}_{\rm tis}}{\hbox{PS}_{\rm tis}+Q_{\rm tis}}, $$
(85)

and as a consequence, we lump the vascular compartment with volume V vas together with the blood compartment and approximate C vas = C blood.

Inserting this formula for C vas into Eq. 82 for the tissue concentration yields

$$ V_{\rm tis}\frac{\hbox{d}}{\hbox{dt}} C_{\rm tis} = \hbox{PS}_{\rm tis} \cdot \left(\frac{Q_{\rm tis} C_{\rm in} + \hbox{PS}_{\rm tis} \frac{C_{\rm tis}}{K_{\rm tis}}}{\hbox{PS}_{\rm tis}+Q_{\rm tis}} - \frac{C_{\rm tis}}{K_{\rm tis}}\right) $$
(86)

and finally

$$ V_{\rm tis}\frac{\hbox{d}}{\hbox{dt}} C_{\rm tis} = \frac{\hbox{PS}_{\rm tis}\cdot Q_{\rm tis}}{\hbox{PS}_{\rm tis}+Q_{\rm tis}} \cdot \left(C_{\rm in} - \frac{C_{\rm tis}}{K_{\rm tis}}\right). $$
(87)

Hence, when lumping the tissue part of a permeability-rate limited tissue model, the term

$$ \frac{\hbox{PS}_{\rm tis}\cdot Q_{\rm tis}}{\hbox{PS}_{\rm tis}+Q_{\rm tis}} $$
(88)

should take the role of the tissue blood flow Q tis. It is bounded by both Q tis and PStis, as one would expect.

Appendix D: Automated determination of the number of lumped compartments of the mechanistically lumped model, and its composition

To compare the mechanistically lumped PK compartment models for a number of different drugs, we used an automated detection algorithm to determine the number of lumped compartment and its composition. The input were the normalized concentration-time profiles as predicted by the detailed whole-body PBPK model (cf. Eq. 19):

$$ c_{\rm tis}(t) =\frac{C_{\rm tis}(t)}{K_{\rm tis}(1-E_{\rm tis})}. $$
(89)

We determined the similarity matrix M = (M ij ) with entries

$$ M_{i,j} = \frac{\langle c_{{\rm tis}(i)}, c_{{\rm tis}(j)} \rangle} {\langle c_{{\rm tis}(i)}, c_{{\rm tis}(i)} \rangle}, $$
(90)

where 〈·, ·〉 denotes the Euclidian scalar product. In our setting, if C tis is given as a vector at different time points c tis(t 1), ..., c tis(t M ) for some M > 0, then

$$ \langle c_{{\rm tis}(i)}, c_{{\rm tis}(j)} \rangle \approx \sum_{k=1}^M c_{{\rm tis}(i)}(t_k)\cdot c_{{\rm tis}(j)}(t_k) \Updelta t, $$
(91)

where we, for simplicity, assume that the time points are equally spaced with distance \(\Updelta t\).

Next, we determined the eigenvector v corresponding to the maximal eigenvalue of M. This eigenvector has an entry corresponding to each organ, tissue or other space of the whole-body PBPK model. We normalized the eigenvector

$$ w(\hbox{tis}) = \frac{v(\hbox{tis})}{v(\hbox{ven})} $$
(92)

such that the normalized eigenvector satisfied w(ven) = 1. See Fig. 15 for the eigenvector corresponding to our model compound Lidocaine (for sake of illustration, we ordered the entries in increasing order).

Fig. 15
figure 15

Normalized eigenvector w corresponding to the maximal eigenvalue of the similarity matrix M based on the PBPK predictions of an 60 min i.v. infusion of 400 mg Lidocaine

We then considered the smallest entry of w (in the example corresponding to the muscle tissue) and lumped all organs that satisfied:

$$ w(\hbox{tis}) < w(\hbox{mus})+\Updelta w, $$
(93)

with \(\Updelta w=0.045\). For our model compound Lidocaine, there was no such organ. Hence, the muscle tissue comprised a single lumped compartment. We proceeded with the next tissue (adipose in our example):

$$ w(\hbox{tis}) < w(\hbox{adi})+\Updelta w. $$
(94)

In this case, w(bon) was the only organ to satisfy the above inequality so that adipose and bone were lumped together. We then proceeded with skin etc. The value of \(\Updelta w\) was chosen so that the automated lumping procedure gave the same results as the manually chosen lumping for Lidocaine. The smaller the value of \(\Updelta w\) the more similar the concentration-time profiles have to be for two organs/tissues to be lumped together.

For Caffeine, Diazepam and Amobarbital the predicted mechanistically lumped model was not sufficient to predict all concentration-time profiles of the 13-compartment whole-body PBPK model. In this case, we manually added a lumped compartment, which resolved the problem and increased the number of compartments by 1.

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Pilari, S., Huisinga, W. Lumping of physiologically-based pharmacokinetic models and a mechanistic derivation of classical compartmental models. J Pharmacokinet Pharmacodyn 37, 365–405 (2010). https://doi.org/10.1007/s10928-010-9165-1

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