Software Supported Modelling in Pharmacokinetics

Transcription

1 Software Spported Modelling in Pharmacokinetics Regina Telgmann, Max von Kleist 2 and Wilhelm Hisinga 2 CiT GmbH, Oldenbrger Str. 2, D-268 Rastede, Germany r.telgmann@cit-wlkow.de 2 Freie Universität Berlin, Department of Mathematics and Informatics, Arnimallee 6, D-495 Berlin, Germany {kleist,hisinga}@mi.f-berlin.de Abstract. A powerfl new software concept to physiologically based pharmacokinetic (PBPK) modelling of drg disposition is presented. It links the inherent modlar nderstanding in pharmacology with orthogonal design principles from software engineering. This concept allows for flexible and ser-friendly design of pharmacokinetic whole body models, data analysis, hypotheses testing or extrapolation. The typical strctre of physiologically-based pharmacokinetic models is introdced. The reslting reqirements 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 drg-drg interaction stdies is given, that demonstrates the advantage of defining a whole-body pharmacokinetic model in terms of the nderlying physiological processes qite impressively: A system of 62 ODEs is atomatically compiled based on the specification of 7 local physiological processes only. Introdction Pharmacokinetics is the stdy of the time corse of drg and metabolite levels in different flids, tisses, and excreta of the body [2]. This incldes the investigation and nderstanding of the processes of absorption, distribtion, metabolism, and excretion (ADME). The pharmacokinetic profile of a drg strongly inflences its delivery to biological targets, thereby affecting its efficacy and potential side effects. Following stdies in the late 99s indicating that poor pharmacokinetics and toxicity were important cases of costly late-stage failres in drg development, it has been widely perceived that these areas need to be considered as early as possible in the drg discovery process []. Today s combinatorial chemistry and high throghpt screening methods anlarged the space of drg candidates significantly, creating actal needs for in silico pharmacokinetic analysis to spport the drg development pipeline. The pharmacokinetics of a compond are

2 2 Regina Telgmann et al. typically nderstood, analyzed and interpreted in the context of their nderlying ADME processes. However, there is no niqe mathematical model for any of these processes; sally a nmber of different models with different nderlying assmptions, parameterization and applicability are concrrent. To efficiently spport in silico modelling and simlation in pharmacokinetics, we propose to inherit the inherent modlar strctre, which is based on the physiological processes, to the software tool. We describe the concepts of modlarity and orthogonality as fndamental principles for the design of a virtal lab in pharmacokinetics. The above mentioned design principles have recently been realized sccessflly in the software tool MEDICI-PK, that is especially designed to fit the needs in pharmacokinetic modelling. Or approach is illstrated by an example from drg-drg interaction stdies. 2 Mathematical Modelling in Physiologically Based Pharmacokinetics (PBPK) Fig.. Organ strctre 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 volmes, sch as organs or tisses. The compartments are connected in an anatomically meaningfl way, to simlate drg exchange via the blood flow. The conceptional representation of a 5 organ PBPK model is shown in Fig.. Each compartment is frther sbdivided into the for phases erythrocytes, plasma, interstitim and celllar space (see Fig. ). Many physiological processes in pharmacokinetics are accessible for a mechanistic description at this resoltion. Typically, following processes are modelled: Convection of drg molecles by the blood flow

3 Lectre Notes in Bioinformatics (LNBI) 26 Vol. No. 3 Binding to macromolecles in plasma and interstitial space Distribtion into tisse Diffsion or active transport across the celllar 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 compond and some macromolecle can be explicitly modelled in terms of the corresponding differential eqations derived from the law of mass action. However, often it is assmed that the binding process is fast in comparison to other processes and therefore in dynamical eqilibrim (qasi-stationarity). This reslts in some algebraic eqation, often still acconting for satration effects of the binding process. A frther simplification finally reslts in a linear algebraic eqation that is not capable of acconting for satration effects, however it may be directly parameterized in terms of a freqently generated in vitro parameter. Each mathematical model has its range of applicability and typically reqires different knowledge abot the process and in particlar different inpt parameters. In broad terms, a chosen model will be a compromise between detailed mechanistic description and reqired qality of the inpt parameters. Typically, the knowledge and the qality of parameters increases along the drg discovery and development process so that adaptation of the model to the crrent knowledge and parameter qality is possible (and shold be aimed for). The characterization of the PBPK model already sggests a modlar description of the whole body model, especially in drg discovery. In mathematical terms, a PBPK model constittes a set of differential/algebraic eqations describing the nderlying processes. The crrent stats of software development in pharmacokinetics is dominated by either a prely eqation based approach contradicting ser-friendliness or implementing a static model contradicting flexibility []. Instead, the reqirements on ser-friendliness and flexibility can be flfilled by the se of sophisticated modlar software concepts and strctres, as otlined in the next section. 3 Modlar Software Design To spport 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 encapslate the mathematical descriptions into modlar parts. These modlar parts ( models ), collected in a model library, are defined only once and can be re-sed inside the whole PK model wherever sitable. This

4 4 Regina Telgmann et al. prevents re-definition and rewriting and spports the even treatment of same processes wherever wanted. The models are specified in terms of concentrations and parameters, bt (a prereqisite for this approach) are independent of a specific nmerical vale of the parameters. Using the same model in a different context reqires evalating different parameter vales for the same mathematical expressions, depending on the context where the model is sed. A given topology (like the 5 organs example) linked together with a selection of models from the library for each compartment therefore bilds a description of the PBPK model which is still independent of any specific nmerical vales. It may be evalated for any selection of parameter vales. The parameters to which a models (might) refer can be classified as compond dependent, species/individal dependent, dependent on both or on none. This classification sggests the introdction of corresponding software objects bilding another orthogonal strctre. For the fll specification of a PBPK model, parameters and compond- as well as species-specific vales of these parameters have finally to be linked this is realized by the simlation object. Model definition Organs (o) + Phases Topology templates Physiological process library type (binding, diffsion, metabolisms, etc.) -description, formlas - general parameter dependencies (o,c,i) Parameter definition Componds (c): - definitions - parameters (o) Mix parameters (c+i) Individals (i): - definitions - parameters (o) Fll body model: - topology - selected processes Simlation object - fll body model - individal - componds - dosing strategies Simlation scenario Simlation reslts Fig. 2. Schematic illstration of the orthogonal approach to software spported pharmacokinetic modelling.

5 Lectre Notes in Bioinformatics (LNBI) 26 Vol. No. 5 Becase the models address the concentration of a compond by fixed terms (e.g. Comp ) it is necessary to map the actal selection of a compond to this expression. Models which consider only one compond need no mapping since this is internally handled. Mlti-compond models that consider more than one compond (e.g, needed in metabolism models for interactions between different componds) necessitate an explicit mapping of the terms addressing the different compond concentrations. For instance, the simple metabolism model V dc() dt V dc(2) = k C () k 2 C () C (2) = k C (2) + k 2 C () C (2) dt will be evalated for two componds A and B, only if the mapping between componds (A,B) and concentration terms (C (), C (2) ) in the above eqations is performed, e.g., A to C () and B to C (2). As soon as all missing mappings are defined, the PBPK model is completed. When starting the simlation, the reslting differential eqation system is atomatically generated, inclding the assignment of all compond-specific parameter vales and species-specific physiological parameter vales to the respective processes. An illstrative example from drg-drg interaction stdies is given in the next section that illminates the advantage of defining a whole-body PBPK model in terms of the nderlying (local) physiologically processes qite impressively. 4 Examples Some drgs are administered as so-called pro-drgs that are metabolized into active componds by liver enzymes. One example is Oseltamivir, better known as Tamifl [8]. Tamifl is the main antifl medicine recommended by the World Health Organization (WHO) [2]. In anticipation of a fl pandemic, the WHO sggests that contries shold stockpile enogh Tamifl to allow the treatment of at least a qarter of their poplation. At present time, however, the spplies of Tamifl are enogh to cover abot 2% of the world poplation only. Recently, Hill et al. [7] highlighted a way to effectively doble the spplies of Tamifl: When administered with a second drg, called probenicid, Tamifl excretion into the rine is stopped. As a reslt, only half of the normal doses of Tamifl are needed. This wartime tactic cold be sed to doble power of scarce resorces of Tamifl in case of a fl pandemic [2]. Here, in silico modelling and simlation cold help to better nderstand and possibly frther optimize the co-administration effects. Motivated by the above example, we want to illstrate how the previosly introdced software concepts, realized in MEDICI-PK, can be sed to efficiently model the phenomena of pro-drg administration and drg-drg interaction. Or aim is not to reprodce clinical data, bt rather to demonstrate the power of or orthogonal and modlar approach to pharmacokinetic modelling.

6 6 Regina Telgmann et al. Table. Brief overview over the performed simlations. description interactions dosing independent pharmacokinetics - A: 2[mg/kg body weight] i.v. B: 2[mg/kg body weight] i.v. C: 2[mg/kg body weight] i.v. conversion of pro-drg A to A B A: 2[mg/kg body weight] i.v. active metabolite B C: 2[mg/kg body weight] i.v. conversion of pro-drg A to B A B C A: 2[mg/kg body weight] i.v. competition for active tblar C: 2[mg/kg body weight] i.v. excretion between B and C The starting point is the definition of the bilding blocks in or PBPK model. This is done in terms of the relevant physiological processes like: (a) i.v. absorption, (b) linear protein binding, (c) passive diffsion, (d) tisse distribtion (according to [6]), (e) satrable metabolism, (f) renal excretion. Each of the processes (modles) is defined in a model basis by a corresponding mathematical eqation. For instance, the processes of satrable metabolism is defined by v meta = with maximm reaction velocity V meta max V meta max C K meta m + C, () and Michaelis-Menten constant K meta m. The concentration of nbond drg is denoted by C. The process of excretion is specified by v excr = Q GF C + V ren maxc K ren m + C Q re abs. C. (2) The parameter Vmax ren denotes the maximm velocity of the satrable active tblar excretion process with Michaelis-Menten constant Km ren. Q GF and Q re abs. are the glomerlar filtration rate and the passive reabsorption rate respectively. The renal excretion has been modelled as a fnction of three processes: (i) passive glomerlar filtration (efflx), (ii) active tblar secretion (efflx) 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 componds. The overall PBPK model is then defined by the fll body template that links the local physiological process modles on the organ level. For efficiancy, it is possibe to define a generic organ strctre, which is taken as a defalt for the initialization of the entire list of organ models. Sbseqent individal 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 or case a 25 g weighting male rat. These vales are later needed to fill the model parameters. Finally, we specify the compond-specific

7 Lectre Notes in Bioinformatics (LNBI) 26 Vol. No. 7 data. Motivated by the Tamifl example, we consider three componds named A, B, and C; a pro-drg, an active metabolite and a competitive inhibitor for the secretion (of compond B). I. II plasma.6 plasma concentration [ µg/ml] III. plasma compond A compond B compond C 4 3 lng 2 liver 2 kidney time [h] time [h] Fig. 3. Simlation reslts as concentration vs. time profiles in venos plasma for (I.) independent pharmacokinetics (top left) and (II.) conversion of pro-drg A to B (top right). The simlation reslts for (III.) competition for tblar excretion of componds 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 venos plasma and interstitial space of the lng, while the bottom panels show the profiles in the celllar space of the liver (bottom left) and kidney. At this stage, the three constitents are completely independent. The PBPK model is specified in terms of parameters, however, no actal nmerical vales are assigned in the model. Only if we map the specific nmerical vales of the parameters (corresponding to the compond and the species of interest), we obtain a flly specified and ready to simlate so-called simlation object. The advantage of this orthogonal specification and data management is a large flexibility.

8 8 Regina Telgmann et al. The same PBPK model can be sed for different species and componds, while the same species database can be sed in stdies of different models etc. We now demonstrate how to ser-friendly and efficiently set p a model for three componds interacting in a way motivated by the Tamifl example in three steps. An overview over the necessary modelling steps to be performed is given in Table. Independent pharmacokinetics. To start with, the pharmacokinetics of the three componds A, B and C are simlated independently. This is easily performed by creating a simlation object, which links the fll body model, the species data (rat) and the respective compond data. As a conseqence, MEDICI-PK atomatically generates a set of model eqations for each compond. In this example we have identical models for the three componds. The reslting pharmacokinetic profiles are shown in Fig. 3 (top left panel) for an intravenos administration of 2 mg/(kg bodyweight) of each compond. Componds A, B and C show very different pharmacokinetic profiles. This is de to their distinctive distribtion characteristics in the varios tisses and de to their different elimination characteristics. While compond C is eliminated in an almost constant fashion, compond A and B are eliminated in an exponential fashion. Plasma levels of compond B are sbstantially higher than plasma levels of compond A and C respectively. This is becase compond B is mainly distribted in the plasma, with significantly lower concentrations in the interstitim and celllar space. Conversion of pro-drg A to B. We next demonstrate how to link the pharmacokinetics of compond A and B. In physiological terms, we want to model the conversion of compond A into compond B in the liver. Given the fll body template from the first simlation scenario, this reqires only a single change, namely the adaptation of the metabolism model chosen for compond B in the liver. We define the so-called mlti-compond metabolism model v meta = V maxc () () K m () + C () + V maxc C2 (2) K m (2) + C (2) and sbseqently map B to C () and A to C (2) in the simlation object. Assming no i.v. administration of compond B, the simlation reslts are shown in Fig. 3 (top right panel) for intravenos administration of 2 mg/(kg body weight) of compond A and C. While the simlation of non-interacting componds based on the same PBPK model can be solved by sccessive simlation of a single compond at a time, the consideration of (dynamic) interactions reqires to establish a joint model for the interacting componds. In MEDICI-PK, this is atomatically generated exploiting the described software concept. This will become even more obvios in the next case.

9 Lectre Notes in Bioinformatics (LNBI) 26 Vol. No. 9 Competition for active tblar excretion. Finally, we want to inclde the drg-drg interactions between compond B and C. In physiological terms, compond C will compete with compond B for active excretion. Given the fll body template from the second simlation scenario, this again reqires only a single change. We modify eq. (2) to inclde the competition by v () CL ren = Q GF C () + v (2) CL ren = Q GF C (2) + K () V () maxc () m ( + C(2) K (2) K m (2) V maxc (2) (2) m ( + C() K m () ) + C () ) + C (2) Q re abs. () C () Q re abs. (2) C (2) As with the case of Oseltamivir-Probenicid competitive inhibition, ni-directed inhibition can be achieved by greatly diverging K m vales (factor 4 ) for componds B and C. After mapping B to C () and C to C (2) in the simlation object, the simlation is performed; the reslts are shown in Fig. 3 (middle and bottom panels) for intravenos administration of 2 mg/(kg body weight) of compond A and C. This example nicely illstrates the phenomenon of extended drg exposition of compond B (active metabolite) as a reslt of a drg-drg interaction. In or physiological context (Fig. ), a total of 62 ordinary differential eqations is necessary to simlate the pharmacokinetics of the three componds, inclding drg-drg interactions; on the basis of the presented concepts, MEDICI- PK generates these eqations from seven ser-defined local physiological models only! 5 Conclsion & Otlook Considerable progress has been made in the development of in silico models to predict and nderstand the pharmacokinetics of new componds, in particlar in early drg discovery. As a reslt, modelling and simlation 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 bild p PBPK models in terms of the nderlying physiological processes, combining ser-friendliness and flexibility. These principles and concepts are the basis of the software tool Medici-PK that has been sed to illstrate or approach. We believe that the combination of in vitro experiments and in silico modeling has the potential to drastically increase the insight and knowledge abot relevant physiological and pharmacological processes in drg discovery. In anticipation of modelling not only the distribtion of the drg in the body, bt also its effect (disease modelling), one ftre challenge will be the combination of pharmacokinetics and effect related metabolic networks or signalling pathways for a better nderstanding of the disease dynamics. An example wold be the treatment-indced selective pressre on viral dynamics. The presented concepts, realized in Medici-PK, are powerfl and flexible enogh to also spport these ftre tasks.

10 Regina Telgmann et al. 6 Acknowledgements It is a pleasre to thank Michael Wlkow (CiT, Rastede) for fritfl and constrctive discssions. M.v.K. and W.H. acknowledge financial spport by the DFG Research Center MATHEON Mathematics for key technologies: Modelling, simlation, and optimization of real-world processes, Berlin. References. van de Waterbeemd, H. and Gifford, E. ADMET in silico modelling: towards prediction paradise? Natre Reviews Drg Discovery(23) Vol.2: Btler, D. Wartime tactic dobles power of scarce bird-fl drg. Natre (25) Vol Polin, P. and Theil, F. P., A priori prediction of tisse:plasma partition coefficients of drgs to facilitate the se of physiologically-based pharmacokinetic models in drg discovery., Jornal of Pharmacetical Sciences (2) Vol.89: Polin P., Schoenlein K and Theil F. P. Prediction of adipose tisse: plasma partition coefficients for strctrally nrelated drgs. Jornal of Pharmacetical Sciences (2) Vol.9: Polin, P. and Theil, F. P., Prediction of pharmacokinetics prior to in vivo stdies. I.Mechanism-based prediction of volme of distribtion. Jornal of Pharmacetical Sciences (22) Vol.9: Polin, P. and Theil, F. P., Prediction of pharmacokinetics prior to in vivo stdies. II. Generic physiologically based pharmacokinetic models of drg disposition. Jornal of Pharmacetical Sciences (22) Vol.9: Hill, G. Cihlar, T. OO, C. Ho, E.S. Prior, K. Wiltshire, H. Barrett, J. Li, B. and Ward, P., The Anti-Inflenza Drg Oseltamivir Exhibits Low Potential To Indce Pharmacokinetic Drg Interactions Via Renal Secretion Correlation Of In Vivo And In Vitro Stdies. Drg Metabolism and Disposition (22) Vol.3: He, G., Massarella, J. and Ward, P. Clinical Pharmacokinetics of the Prodrg Oseltamivir and its Active Metabolite Ro Clinical Pharmacokintics (999) 37: Hisinga, W. Telgmann, R. and Wlkow, M. The Virtal Lab Approach to Pharmacokinetics: Design Principles and Concept. Drg Discovery Today (26) (accepted). Rowland, M. Balant, L. and Peck C. Physiologically Based Pharmacokinetics in Drg Development and Reglatory Science: A Workshop Report (Georgetown University, Washington, DC, May 29-3, 22) AAPS Pharmacetical Science (24); 6 () -2. von Kleist, M. and Hisinga, W. Hierachical Approach to Physiologically Based Pharmacokinetics: Refining Models with Increasing Knowledge. in preparation 2. Gibaldi, M. and Perrier, D. Pharmacokinetics 2nd ed. Marcel Dekker (982), New York