Appendix 1: Physiologically based pharmacokinetic models

Physiologically based pharmacokinetic (PBPK) models are ordinary differential equation models describing how a substance, e.g. a drug, is absorbed, distributed, metabolized, and excreted in an organism. For the reader not familiar with PBPK models we provide a brief overview over the basic concepts, building blocks and equations forming a PBPK model. For more details we refer to [45].

In PBPK models physiological organs and tissues are represented by compartments. The transport of substance via the blood is modeled by balance equations of the form

$$\begin \frac=\frac\left( C_-\frac\right) , \end$$

(5)

where \(C_i\) denotes the compound concentration in the compartment i, \(V_i\) its volume, \(Q_i\) the blood flow, \(P_i\) the partition coefficient between blood and tissue, and \(C_\) the compound concentration in arterial blood, which is governed by

$$\begin \frac}=-\sum _\left( C_-\frac\right) }, \end$$

(6)

To describe dissolution, absorption, metabolism and excretion, as well as additional distribution mechanism the Eqs. 5 and 6 need to be extended. For example, dissolution and absorption in a single GIT compartment is described by the following equations:

$$\begin \frac&=\frac\left( C_-\frac\right) +K_a C_, \end$$

(7)

$$\begin \frac}&=-K_aC_+\frac}, \end$$

(8)

$$\begin \frac}&=K\left( C_0-C_\right) ^\left( C_s-C_\right) , \end$$

(9)

Equation 7 describes concentration in the GIT tissue \(C_g\), which is sourced by a linear absorption process from the GIT lumen. Equation 8 is describes the compound concentration in the GIT lumen \(C_\), which is sourced by the dissolved compound \(C_\). Equation 9 is the Noyse-Withney equation describing the dissolution of the compound in the GIT lumen, with K being a compound dependent constant, \(C_0\) is the total amount of compound administered divided by the administered volume and \(C_s\) is the solubility, i.e. the compound concentration the GIT lumen at (thermal) equilibrium. Metabolism is described by the Michaelis–Menten-Kinetics, which for \(C\ll K_m\) can be linearized:

$$\begin \frac=-V_\frac= -\frac} C+O\left( \left( \frac\right) ^2\right) , \end$$

(10)

The constants \(V_\) and \(K_M\) depend on the compound and the metabolizing enzyme and control the speed and saturation of metabolism. We assume a single generic metabolizing enzyme, hence in our hybrid model hepatic clearance is fully characterized by the rate \(\frac}\).

An active P-gp like transport via membrane proteins, assuming a constant protein concentration, follows also a Michaelis–Menten-Kinetics

$$\begin \frac&=-V_\frac \end$$

(11)

$$\begin \frac&=V_\frac. \end$$

(12)

As for the metabolism, the constants \(V_\) and \(K_M\) control the speed and saturation of the transport are compound and are transport protein dependent. For our purpose it is sufficient to set \(K_M=1\, \mathrm }\), i.e. use the OSP default value, hence the transport is parametrized by its maximal velocity \(V_\).

Appendix 2: Validation of property constraintsFig. 9

Distribution of the molecule properties in the test set. The vertical line show bounds for the properties to lie within in the validity range of the surrogate. All molecule properties lie within in their bounds

In Fig. 9 the distribution of predicted molecule properties of the test set are shown together with the maximal and minimal values in the surrogate training data set. All predicted molecule properties lie within in the surrogates training range, confirming the effectiveness of the penalized loss described in  “Training strategy” section. Note that for \(V_\) and FU we used heavy tailed distributions for generating the surrogate training data, resulting in the large range shown in Fig. 9. For the FU this results in unphysiological values \(>1\), for which the equations of the PBPK model are still defined. But in practice the property net does not predict a \(FU >1\). Furthermore, to increase the flexibility of our clearance model we increased the maximal allowed value for the GFR fraction from 1 to 5.25.

Appendix 3: A posteriori surrogate validationFig. 10

Simulation vs surrogate predictions for the predicted properties of the compounds in our test set for \(\mathrm }\; \)(left), \(\mathrm }\; \)(center) and \(\mathrm }\; \)(right). The accuracy is a bit smaller compared to the estimate on the simulation test set, but still significantly better than the accuracy of the hybrid model, hence the accuracy of the surrogate is sufficient

Fig. 11

Hybrid model test set predictions using the full PBPK model instead of the surrogate predictions for the predicted properties of the compounds. The accuracy for the three end-points \(\mathrm }\; \)(left), \(\mathrm }\; \)(center) and \(\mathrm }\; \)(right) is similar to the accuracy when using the surrogate. Demonstrating the accuracy of the surrogate model

We can validate the surrogate model a posteriori by predicting the training targets of our hybrid model using the PBPK model instead of the surrogate. Figure 10 shows the predictions obtained using the PBPK model vs those obtained using the surrogate. The accuracy is not as good as expected from the analysis in “Surrogate” section, but still accurate enough to be used in the hybrid model, the mfce of the surrogate (\(1.2-1.4\)) is clearly better than the mfce of the hybrid model (\(mfce\gtrsim 1.6\)). Additionally, Fig. 11 shows the predictions using the full PBPK vs the observed values. These predictions are almost as accurate as those using the surrogate model. A maximal difference of 0.24 in the mfce can be observed, and no additional features are visible. This highlights again the accuracy of the used surrogate model.

Appendix 4: Charge state dependence of model performanceFig. 12

Dependence of the hybrid models accuracy on the compounds charge state at \(pH=7.4\), i.e the pH value of blood. Shown are the three endpoints \(\mathrm }\; \)(left), \(\mathrm }\; \)(center) and \(\mathrm }\; \)(right) for the case male rat and solution. For neutral compounds the predictions are most accurate, followed by positively and negatively charged compounds. For zwitterions the accuracy is significantly worse, but here the number of compounds is too low for a reliable estimate of the accuracy

We check for a potential dependence of the model accuracy on the charge state in Fig. 12. We evaluate the performance for male rats when a solution is used. As charge states can reliably be predicted, we use predicted charge states at the pH of blood (\(pH=7.4\)). We observe the best performance neutral compounds, and a worse performance for positively and negatively charged compounds. But, in all three cases we achieve \(mfce<3\), so predictions are accurate enough to guide decisions. For zwitterions the mfce for \(\mathrm }\; \)and \(\mathrm }\; \)is larger than 3, but here only very few compounds are in our test set.