## Abstract

An important aspect in the development of small molecules as drugs or agrochemicals is their systemic availability after intravenous and oral administration. The prediction of the systemic availability from the chemical structure of a potential candidate is highly desirable, as it allows to focus the drug or agrochemical development on compounds with a favorable kinetic profile. However, such predictions are challenging as the availability is the result of the complex interplay between molecular properties, biology and physiology and training data is rare. In this work we improve the hybrid model developed earlier (Schneckener in J Chem Inf Model 59:4893–4905, 2019). We reduce the median fold change error for the total oral exposure from 2.85 to 2.35 and for intravenous administration from 1.95 to 1.62. This is achieved by training on a larger data set, improving the neural network architecture as well as the parametrization of mechanistic model. Further, we extend our approach to predict additional endpoints and to handle different covariates, like sex and dosage form. In contrast to a pure machine learning model, our model is able to predict new end points on which it has not been trained. We demonstrate this feature by predicting the exposure over the first 24 h, while the model has only been trained on the total exposure.

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Even though the name Pharmacokinetics implies that the field is only concerned with pharmaceutical substances, the field is concerned with all types of xenobiotic substances, see https://en.wikipedia.org/wiki/Pharmacokinetics.

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## Appendices

### 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

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 \(C_{art}\) the compound concentration in arterial blood, which is governed by

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:

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 \(C_{lum}\), which is sourced by the dissolved compound \(C_{dis}\). 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:

The constants \(V_{max}\) 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 \(\frac{V_{max}}{K_M}\).

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

As for the metabolism, the constants \(V_{max}\) 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 \(K_M=1\, \mathrm {\frac{\mu mol}{L}}\), i.e. use the OSP default value, hence the transport is parametrized by its maximal velocity \(V_{max}\).

### Appendix 2: Validation of property constraints

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 \(V_{max}\) 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 validation

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 performance

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 \(\mathrm {AUC_{PO}}\; \)and \(\mathrm {C_{max,PO}}\; \)is larger than 3, but here only very few compounds are in our test set.

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Führer, F., Gruber, A., Diedam, H. *et al.* A deep neural network: mechanistic hybrid model to predict pharmacokinetics in rat.
*J Comput Aided Mol Des* **38**, 7 (2024). https://doi.org/10.1007/s10822-023-00547-9

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DOI: https://doi.org/10.1007/s10822-023-00547-9