Software Supported Modelling in Pharmacokinetics

Software Supported Modelling in Pharmacokinetics Regina Telgmann1 , Max von Kleist2 and Wilhelm Huisinga2 1 CiT GmbH, Oldenburger Str. 200, D-26180 Rastede, Germany r.telgmann@cit-wulkow.de 2 Freie Universität Berlin, Department of Mathematics and Informatics, Arnimallee 6, D-14195 Berlin, Germany {kleist,huisinga}@mi.fu-berlin.de Abstract. A powerful new software concept to physiologically based pharmacokinetic (PBPK) modelling of drug disposition is presented. It links the inherent modular understanding in pharmacology with orthogonal design principles from software engineering. This concept allows for flexible and user-friendly design of pharmacokinetic whole body models, data analysis, hypotheses testing or extrapolation. The typical structure of physiologically-based pharmacokinetic models is introduced. The resulting requirements from a modelling and software engineering point of view and its realizations in the software tool MEDICI-PK are described. Finally, an example in the context of drug-drug interaction studies is given, that demonstrates the advantage of defining a whole-body pharmacokinetic model in terms of the underlying physiological processes quite impressively: A system of 162 ODEs is automatically compiled based on the specification of 7 local physiological processes only. 1 Introduction Pharmacokinetics is the study of the time course of drug and metabolite levels in different fluids, tissues, and excreta of the body [12]. This includes the investigation and understanding of the processes of absorption, distribution, metabolism, and excretion (ADME). The pharmacokinetic profile of a drug strongly influences its delivery to biological targets, thereby affecting its efficacy and potential side effects. Following studies in the late 1990s indicating that poor pharmacokinetics and toxicity were important causes of costly late-stage failures in drug development, it has been widely perceived that these areas need to be considered as early as possible in the drug discovery process [1]. Today’s combinatorial chemistry and high throughput screening methods anlarged the space of drug candidates significantly, creating actual needs for in silico pharmacokinetic analysis to support the drug development pipeline. The pharmacokinetics of a compound are 2 Regina Telgmann et al. typically understood, analyzed and interpreted in the context of their underlying ADME processes. However, there is no unique mathematical model for any of these processes; usually a number of different models with different underlying assumptions, parameterization and applicability are concurrent. To efficiently support in silico modelling and simulation in pharmacokinetics, we propose to inherit the inherent modular structure, which is based on the physiological processes, to the software tool. We describe the concepts of modularity and orthogonality as fundamental principles for the design of a virtual lab in pharmacokinetics. The above mentioned design principles have recently been realized successfully in the software tool MEDICI-PK, that is especially designed to fit the needs in pharmacokinetic modelling. Our approach is illustrated by an example from drug-drug interaction studies. 2 Mathematical Modelling in Physiologically Based Pharmacokinetics (PBPK) Fig. 1. Organ structure of a physiologically based pharmacokinetic model. A physiologically based pharmacokinetic (PBPK) whole body model is a special type of compartmental model, in which the compartments represent anatomical volumes, such as organs or tissues. The compartments are connected in an anatomically meaningful way, to simulate drug exchange via the blood flow. The conceptional representation of a 15 organ PBPK model is shown in Fig. 1. Each compartment is further subdivided into the four phases erythrocytes, plasma, interstitium and cellular space (see Fig. 1). Many physiological processes in pharmacokinetics are accessible for a mechanistic description at this resolution. Typically, following processes are modelled: – Convection of drug molecules by the blood flow Lecture Notes in Bioinformatics (LNBI) 2006 Vol. No. – – – – 3 Binding to macromolecules in plasma and interstitial space Distribution into tissue Diffusion or active transport across the cellular membrane Metabolism or interaction with metabolic networks or signalling pathways etc. There are many levels of mathematical description for a certain biological process (e.g., each of the physiological processes stated in the above list). To give an example, the process of protein binding (complex formation) between a compound and some macromolecule can be explicitly modelled in terms of the corresponding differential equations derived from the law of mass action. However, often it is assumed that the binding process is fast in comparison to other processes and therefore in dynamical equilibrium (quasi-stationarity). This results in some algebraic equation, often still accounting for saturation effects of the binding process. A further simplification finally results in a linear algebraic equation that is not capable of accounting for saturation effects, however it may be directly parameterized in terms of a frequently generated in vitro parameter. Each mathematical model has its range of applicability and typically requires different knowledge about the process and in particular different input parameters. In broad terms, a chosen model will be a compromise between detailed mechanistic description and required ”quality” of the input parameters. Typically, the knowledge and the quality of parameters increases along the drug discovery and development process so that adaptation of the model to the current knowledge and parameter quality is possible (and should be aimed for). The characterization of the PBPK model already suggests a modular description of the whole body model, especially in drug discovery. In mathematical terms, a PBPK model constitutes a set of differential/algebraic equations describing the underlying processes. The current status of software development in pharmacokinetics is dominated by either a purely equation based approach— contradicting user-friendliness—or implementing a static model—contradicting flexibility [10]. Instead, the requirements on user-friendliness and flexibility can be fulfilled by the use of sophisticated modular software concepts and structures, as outlined in the next section. 3 Modular Software Design To support the specification of a whole-body pharmacokinetic model, a variety of physiological processes (mentioned above) and a collection of different mathematical models to describe these processes have to be regarded. In practice, identical processes in different compartments will often be described by identical mathematical models; –however, from a software engineering point of view it is important to encapsulate the mathematical descriptions into modular parts. These modular parts (”models”), collected in a model library, are defined only once and can be re-used inside the whole PK model wherever suitable. This 4 Regina Telgmann et al. prevents re-definition and rewriting and supports the even treatment of same processes wherever wanted. The models are specified in terms of concentrations and parameters, but (a prerequisite for this approach) are independent of a specific numerical value of the parameters. Using the ”same” model in a different context requires evaluating different parameter values for the same mathematical expressions, depending on the context where the model is used. A given topology (like the 15 organs example) linked together with a selection of models from the library for each compartment therefore builds a description of the PBPK model which is still independent of any specific numerical values. It may be evaluated for any selection of parameter values. The parameters to which a models (might) refer can be classified as compound dependent, species/individual dependent, dependent on both or on none. This classification suggests the introduction of corresponding software objects building another orthogonal structure. For the full specification of a PBPK model, parameters and compound- as well as species-specific values of these parameters have finally to be linked—this is realized by the simulation object. Model definition Organs (o) + Phases Parameter definition Physiological process library type (binding, diffusion, metabolisms, etc.) -description, formulas - general parameter dependencies (o,c,i) Topology templates Compounds (c): - definitions - parameters (o) Individuals (i): - definitions - parameters (o) Mix parameters (c+i) Simulation scenario Full body model: - topology - selected processes Simulation object - full body model - individual - compounds - dosing strategies Simulation results Fig. 2. Schematic illustration of the orthogonal approach to software supported pharmacokinetic modelling. Lecture Notes in Bioinformatics (LNBI) 2006 Vol. No. 5 Because the models address the concentration of a compound by fixed terms (e.g. ”Comp1”) it is necessary to map the actual selection of a compound to this expression. Models which consider only one compound need no mapping since this is internally handled. Multi-compound models that consider more than one compound (e.g, needed in metabolism models for interactions between different compounds) necessitate an explicit mapping of the terms addressing the different compound concentrations. For instance, the simple metabolism model dC (1) = k1 C (1) − k2 C (1) C (2) dt dC (2) = −k1 C (2) + k2 C (1) C (2) V dt will be evaluated for two compounds A and B, only if the mapping between compounds (A,B) and concentration terms (C (1) , C (2) ) in the above equations is performed, e.g., A to C (1) and B to C (2) . As soon as all missing mappings are defined, the PBPK model is completed. When starting the simulation, the resulting differential equation system is automatically generated, including the assignment of all compound-specific parameter values and species-specific physiological parameter values to the respective processes. An illustrative example from drug-drug interaction studies is given in the next section that illuminates the advantage of defining a whole-body PBPK model in terms of the underlying (local) physiologically processes quite impressively. V 4 Examples Some drugs are administered as so-called pro-drugs that are metabolized into active compounds by liver enzymes. One example is Oseltamivir, better known as Tamiflu [8]. Tamiflu is the main antiflu medicine recommended by the World Health Organization (WHO) [2]. In anticipation of a flu pandemic, the WHO suggests that countries should stockpile enough Tamiflu to allow the treatment of at least a quarter of their population. At present time, however, the supplies of Tamiflu are enough to cover about 2% of the world population only. Recently, Hill et al. [7] highlighted a way to effectively double the supplies of Tamiflu: When administered with a second drug, called probenicid, Tamiflu excretion into the urine is stopped. As a result, only half of the normal doses of Tamiflu are needed. This ”wartime tactic” could be used to double power of scarce resources of Tamiflu in case of a flu pandemic [2]. Here, in silico modelling and simulation could help to better understand and possibly further optimize the co-administration effects. Motivated by the above example, we want to illustrate how the previously introduced software concepts, –realized in MEDICI-PK, can be used to efficiently model the phenomena of pro-drug administration and drug-drug interaction. Our aim is not to reproduce clinical data, but rather to demonstrate the power of our orthogonal and modular approach to pharmacokinetic modelling. 6 Regina Telgmann et al. Table 1. Brief overview over the performed simulations. description interactions dosing independent pharmacokinetics - A: 2[mg/kg B: 2[mg/kg C: 2[mg/kg conversion of pro-drug A to A→B A: 2[mg/kg active metabolite B C: 2[mg/kg conversion of pro-drug A to B A → B↓⊢ C A: 2[mg/kg C: 2[mg/kg competition for active tubular excretion between B and C body weight] i.v. body weight] i.v. body weight] i.v. body weight] i.v. body weight] i.v. body weight] i.v. body weight] i.v. The starting point is the definition of the building blocks in our PBPK model. This is done in terms of the relevant physiological processes like: (a) i.v. absorption, (b) linear protein binding, (c) passive diffusion, (d) tissue distribution (according to [6]), (e) saturable metabolism, (f) renal excretion. Each of the processes (modules) is defined in a ’model basis’ by a corresponding mathematical equation. For instance, the processes of saturable metabolism is defined by vmeta = meta Vmax Cu , meta + C Km u (1) meta meta with maximum reaction velocity Vmax and Michaelis-Menten constant Km . The concentration of unbound drug is denoted by Cu . The process of excretion is specified by vexcr = QGF Cu + ren Vmax Cu − Qre−abs. Cu . ren + C Km u (2) ren denotes the maximum velocity of the saturable active tubuThe parameter Vmax ren lar excretion process with Michaelis-Menten constant Km . QGF and Qre−abs. are the glomerular filtration rate and the passive reabsorption rate respectively. The renal excretion has been modelled as a function of three processes: (i) passive glomerular filtration (efflux), (ii) active tubular secretion (efflux) and (iii) passive reabsorption. Metabolic clearance in the kidney has been neglected. In total, seven local processes have been defined to model the whole body pharmacokinetics of the three compounds. The overall PBPK model is then defined by the full body template that links the local physiological process modules on the organ level. For efficiancy, it is possibe to define a generic organ structure, which is taken as a default for the initialization of the entire list of organ models. Subsequent individual modifications are possible in order to model organ specific processes, like e.g., excretion by the kidneys. Next, we specify the physiological parameters of the considered species, in our case a 250 g weighting male rat. These values are later needed to fill the model parameters. Finally, we specify the compound-specific Lecture Notes in Bioinformatics (LNBI) 2006 Vol. No. 7 data. Motivated by the Tamiflu example, we consider three compounds named A, B, and C; a pro-drug, an active metabolite and a competitive inhibitor for the secretion (of compound B). II. I. 1 1 0.8 0.8 plasma 0.6 0.4 0.4 0.2 0.2 0 0 1 2 3 4 plasma 0.6 5 0 0 1 2 3 4 5 III. 1 4 0.8 plasma 0.6 concentration [ µg/ml] compound A compound B compound C 3 2 0.4 lung 1 0.2 0 0 1 2 3 4 5 0 4 4 3 3 2 0 1 2 3 5 4 5 2 liver kidney 1 0 4 1 0 1 2 3 time [h] 4 5 0 0 1 2 3 time [h] Fig. 3. Simulation results as concentration vs. time profiles in venous plasma for (I.) independent pharmacokinetics (top left) and (II.) conversion of pro-drug A to B (top right). The simulation results for (III.) competition for tubular excretion of compounds B and C are shown in the middle and bottom panels. The middle left and right panels shows the concentration vs. time profiles in the venous plasma and interstitial space of the lung, while the bottom panels show the profiles in the cellular space of the liver (bottom left) and kidney. At this stage, the three constituents are completely independent. The PBPK model is specified in terms of parameters, however, no actual numerical values are assigned in the model. Only if we map the specific numerical values of the parameters (corresponding to the compound and the species of interest), we obtain a fully specified and ready to simulate so-called ’simulation object’. The advantage of this orthogonal specification and data management is a large flexibility. 8 Regina Telgmann et al. The same PBPK model can be used for different species and compounds, while the same species database can be used in studies of different models etc. We now demonstrate how to user-friendly and efficiently set up a model for three compounds interacting in a way motivated by the Tamiflu example in three steps. An overview over the necessary modelling steps to be performed is given in Table 1. Independent pharmacokinetics. To start with, the pharmacokinetics of the three compounds A, B and C are simulated independently. This is easily performed by creating a simulation object, which links the full body model, the species data (rat) and the respective compound data. As a consequence, MEDICI-PK automatically generates a set of model equations for each compound. In this example we have identical models for the three compounds. The resulting pharmacokinetic profiles are shown in Fig. 3 (top left panel) for an intravenous administration of 2 mg/(kg bodyweight) of each compound. Compounds A, B and C show very different pharmacokinetic profiles. This is due to their distinctive distribution characteristics in the various tissues and due to their different elimination characteristics. While compound C is eliminated in an almost constant fashion, compound A and B are eliminated in an exponential fashion. Plasma levels of compound B are substantially higher than plasma levels of compound A and C respectively. This is because compound B is mainly distributed in the plasma, with significantly lower concentrations in the interstitium and cellular space. Conversion of pro-drug A to B. We next demonstrate how to link the pharmacokinetics of compound A and B. In physiological terms, we want to model the conversion of compound A into compound B in the liver. Given the full body template from the first simulation scenario, this requires only a single change, namely the adaptation of the metabolism model chosen for compound B in the liver. We define the so-called multi-compound metabolism model (1) vmeta = − (1) (2) (1) Vmax Cu (1) Km + Cu + C2 Cu Vmax (2) (2) Km + Cu and subsequently map B to C (1) and A to C (2) in the simulation object. Assuming no i.v. administration of compound B, the simulation results are shown in Fig. 3 (top right panel) for intravenous administration of 2 mg/(kg body weight) of compound A and C. While the simulation of non-interacting compounds based on the same PBPK model can be solved by successive simulation of a single compound at a time, the consideration of (dynamic) interactions requires to establish a joint model for the interacting compounds. In MEDICI-PK, this is automatically generated exploiting the described software concept. This will become even more obvious in the next case. Lecture Notes in Bioinformatics (LNBI) 2006 Vol. No. 9 Competition for active tubular excretion. Finally, we want to include the drug-drug interactions between compound B and C. In physiological terms, compound C will compete with compound B for active excretion. Given the full body template from the second simulation scenario, this again requires only a single change. We modify eq. (2) to include the competition by (1) (1) Km (1 (2) (2) Cu (1) ) Km + Cu(2) ) + Km (2) (2) Vmax Cu (2) vCLren = QGF Cu(2) + (1) Vmax Cu (1) vCLren = QGF Cu(1) + Km (1 + (1) (1) Cu (2) + Cu − Qre−abs. (1) Cu(1) − Qre−abs. (2) Cu(2) As with the case of Oseltamivir-Probenicid competitive inhibition, uni-directed inhibition can be achieved by greatly diverging Km values (factor 104 ) for compounds B and C. After mapping B to C (1) and C to C (2) in the simulation object, the simulation is performed; the results are shown in Fig. 3 (middle and bottom panels) for intravenous administration of 2 mg/(kg body weight) of compound A and C. This example nicely illustrates the phenomenon of extended drug exposition of compound B (active metabolite) as a result of a drug-drug interaction. In our physiological context (Fig. 1), a total of 162 ordinary differential equations is necessary to simulate the pharmacokinetics of the three compounds, including drug-drug interactions; – on the basis of the presented concepts, MEDICIPK generates these equations from seven user-defined local physiological models only! 5 Conclusion & Outlook Considerable progress has been made in the development of in silico models to predict and understand the pharmacokinetics of new compounds, in particular in early drug discovery. As a result, modelling and simulation is possible prior to any in vivo experiments, solely based on in vitro data. We present the principles and concepts of a software design that efficiently allows to build up PBPK models in terms of the underlying physiological processes, combining user-friendliness and flexibility. These principles and concepts are the basis of the software tool Medici-PK that has been used to illustrate our approach. We believe that the combination of in vitro experiments and in silico modeling has the potential to drastically increase the insight and knowledge about relevant physiological and pharmacological processes in drug discovery. In anticipation of modelling not only the distribution of the drug in the body, but also its effect (disease modelling), one future challenge will be the combination of pharmacokinetics and effect related metabolic networks or signalling pathways for a better understanding of the disease dynamics. An example would be the treatment-induced selective pressure on viral dynamics. The presented concepts, realized in Medici-PK, are powerful and flexible enough to also support these future tasks. 10 6 Regina Telgmann et al. Acknowledgements It is a pleasure to thank Michael Wulkow (CiT, Rastede) for fruitful and constructive discussions. 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