Softsuppmodell16may06.dvi

Software Supported Modelling in Regina Telgmann1, Max von Kleist2 and Wilhelm Huisinga2 Oldenburger Str. 200, D-26180 Rastede, Germany 2 Freie Universit¨at Berlin, Department of Mathematics and Informatics, D-14195 Berlin, Germany Abstract. A powerful new software concept to physiologically basedpharmacokinetic (PBPK) modelling of drug disposition is presented. Itlinks the inherent modular understanding in pharmacology with orthog-onal design principles from software engineering. This concept allows forflexible and user-friendly design of pharmacokinetic whole body models,data analysis, hypotheses testing or extrapolation. The typical structureof physiologically-based pharmacokinetic models is introduced. The re-sulting requirements from a modelling and software engineering point ofview and its realizations in the software tool MEDICI-PK are described.
Finally, an example in the context of drug-drug interaction studies isgiven, that demonstrates the advantage of defining a whole-body phar-macokinetic model in terms of the underlying physiological processesquite impressively: A system of 162 ODEs is automatically compiledbased on the specification of 7 local physiological processes only.
Pharmacokinetics is the study of the time course of drug and metabolite levels indifferent fluids, tissues, and excreta of the body [12]. This includes the investiga-tion and understanding of the processes of absorption, distribution, metabolism,and excretion (ADME). The pharmacokinetic profile of a drug strongly influencesits delivery to biological targets, thereby affecting its efficacy and potential sideeffects. Following studies in the late 1990s indicating that poor pharmacokineticsand toxicity were important causes of costly late-stage failures in drug develop-ment, it has been widely perceived that these areas need to be considered as earlyas possible in the drug discovery process [1]. Today's combinatorial chemistryand high throughput screening methods anlarged the space of drug candidatessignificantly, creating actual needs for in silico pharmacokinetic analysis to sup-port the drug development pipeline. The pharmacokinetics of a compound are Regina Telgmann et al.
typically understood, analyzed and interpreted in the context of their underlyingADME processes. However, there is no unique mathematical model for any ofthese processes; usually a number of different models with different underlyingassumptions, parameterization and applicability are concurrent.
To efficiently support in silico modelling and simulation in pharmacokinet- ics, we propose to inherit the inherent modular structure, which is based on thephysiological processes, to the software tool. We describe the concepts of modu-larity and orthogonality as fundamental principles for the design of a virtual labin pharmacokinetics. The above mentioned design principles have recently beenrealized successfully in the software tool MEDICI-PK, that is especially designedto fit the needs in pharmacokinetic modelling. Our approach is illustrated by anexample from drug-drug interaction studies.
Mathematical Modelling in Physiologically BasedPharmacokinetics (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 representanatomical volumes, such as organs or tissues. The compartments are connectedin an anatomically meaningful way, to simulate drug exchange via the bloodflow. The conceptional representation of a 15 organ PBPK model is shown inFig. 1. Each compartment is further subdivided into the four phases erythro-cytes, plasma, interstitium and cellular space (see Fig. 1). Many physiologicalprocesses in pharmacokinetics are accessible for a mechanistic description at thisresolution. Typically, following processes are modelled: – Convection of drug molecules by the blood flow Lecture Notes in Bioinformatics (LNBI) 2006 Vol. No.
– 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 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 ex-ample, the process of protein binding (complex formation) between a compoundand some macromolecule can be explicitly modelled in terms of the correspond-ing differential equations derived from the law of mass action. However, oftenit is assumed that the binding process is fast in comparison to other processesand therefore in dynamical equilibrium (quasi-stationarity). This results in somealgebraic equation, often still accounting for saturation effects of the binding pro-cess. A further simplification finally results in a linear algebraic equation thatis not capable of accounting for saturation effects, however it may be directlyparameterized 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 param-eters. In broad terms, a chosen model will be a compromise between detailedmechanistic description and required "quality" of the input parameters. Typ-ically, the knowledge and the quality of parameters increases along the drugdiscovery and development process so that adaptation of the model to the cur-rent knowledge and parameter quality is possible (and should be aimed for).
The characterization of the PBPK model already suggests a modular de- scription of the whole body model, especially in drug discovery. In mathemat-ical terms, a PBPK model constitutes a set of differential/algebraic equationsdescribing the underlying processes. The current status of software developmentin pharmacokinetics is dominated by either a purely equation based approach—contradicting user-friendliness—or implementing a static model—contradictingflexibility [10]. Instead, the requirements on user-friendliness and flexibility canbe fulfilled by the use of sophisticated modular software concepts and structures,as outlined in the next section.
Modular Software Design To support the specification of a whole-body pharmacokinetic model, a varietyof physiological processes (mentioned above) and a collection of different math-ematical models to describe these processes have to be regarded. In practice,identical processes in different compartments will often be described by identi-cal mathematical models; –however, from a software engineering point of view itis important to encapsulate the mathematical descriptions into modular parts.
These modular parts ("models"), collected in a model library, are defined onlyonce and can be re-used inside the whole PK model wherever suitable. This Regina Telgmann et al.
prevents re-definition and rewriting and supports the even treatment of sameprocesses 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 ofthe parameters. Using the "same" model in a different context requires evaluat-ing different parameter values for the same mathematical expressions, dependingon the context where the model is used. A given topology (like the 15 organsexample) linked together with a selection of models from the library for eachcompartment therefore builds a description of the PBPK model which is stillindependent of any specific numerical values. It may be evaluated for any selec-tion of parameter values. The parameters to which a models (might) refer can beclassified as compound dependent, species/individual dependent, dependent onboth or on none. This classification suggests the introduction of correspondingsoftware objects building another orthogonal structure. For the full specificationof a PBPK model, parameters and compound- as well as species-specific valuesof these parameters have finally to be linked—this is realized by the simulationobject.
Physiological process
Organs (o) + Phases
type (binding, diffusion,
- parameters (o)
- parameters (o)
-description, formulas
- general parameter
dependencies (o,c,i)
Topology
Mix parameters (c+i)
Full body model:
- topology
- selected
processes
Simulation object
- full body model
- individual
- compounds
- dosing strategies
Fig. 2. Schematic illustration of the orthogonal approach to software supported phar-macokinetic modelling.
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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 thisexpression. Models which consider only one compound need no mapping sincethis is internally handled. Multi-compound models that consider more than onecompound (e.g, needed in metabolism models for interactions between differentcompounds) necessitate an explicit mapping of the terms addressing the differentcompound concentrations. For instance, the simple metabolism model −k1C (2) + k2C (1)C (2) will be evaluated for two compounds A and B, only if the mapping betweencompounds (A,B) and concentration terms (C(1), C(2)) in the above equationsis performed, e.g., A to C(1) and B to C(2) . As soon as all missing mappingsare defined, the PBPK model is completed.
When starting the simulation, the resulting differential equation system is au- tomatically generated, including the assignment of all compound-specific param-eter values and species-specific physiological parameter values to the respectiveprocesses. An illustrative example from drug-drug interaction studies is given inthe next section that illuminates the advantage of defining a whole-body PBPKmodel in terms of the underlying (local) physiologically processes quite impres-sively.
Some drugs are administered as so-called pro-drugs that are metabolized intoactive compounds by liver enzymes. One example is Oseltamivir, better knownas Tamiflu [8]. Tamiflu is the main antiflu medicine recommended by the WorldHealth Organization (WHO) [2]. In anticipation of a flu pandemic, the WHOsuggests that countries should stockpile enough Tamiflu to allow the treatmentof at least a quarter of their population. At present time, however, the suppliesof 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 intothe urine is stopped. As a result, only half of the normal doses of Tamiflu areneeded. This "wartime tactic" could be used to double power of scarce resourcesof 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 aboveexample, we want to illustrate how the previously introduced software concepts,–realized in MEDICI-PK, can be used to efficiently model the phenomena ofpro-drug administration and drug-drug interaction. Our aim is not to reproduceclinical data, but rather to demonstrate the power of our orthogonal and mod-ular approach to pharmacokinetic modelling.
Regina Telgmann et al.
Table 1. Brief overview over the performed simulations.
interactions dosing 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-drug A to A: 2[mg/kg body weight] i.v.
active metabolite B C: 2[mg/kg body weight] i.v.
conversion of pro-drug A to B A → B↓ C A: 2[mg/kg body weight] i.v.
competition for active tubular C: 2[mg/kg body weight] i.v.
excretion between B and C 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. ab-sorption, (b) linear protein binding, (c) passive diffusion, (d) tissue distribution(according to [6]), (e) saturable metabolism, (f) renal excretion. Each of the pro-cesses (modules) is defined in a 'model basis' by a corresponding mathematicalequation. For instance, the processes of saturable metabolism is defined by with maximum reaction velocity V meta and Michaelis-Menten constant Kmeta The concentration of unbound drug is denoted by C . The process of excretion The parameter V ren max denotes the maximum velocity of the saturable active tubu- lar excretion process with Michaelis-Menten constant Kren m . 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) pas-sive glomerular filtration (efflux), (ii) active tubular secretion (efflux) and (iii)passive reabsorption. Metabolic clearance in the kidney has been neglected. Intotal, seven local processes have been defined to model the whole body pharma-cokinetics 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 defaultfor the initialization of the entire list of organ models. Subsequent individualmodifications are possible in order to model organ specific processes, like e.g.,excretion by the kidneys. Next, we specify the physiological parameters of theconsidered species, in our case a 250 g weighting male rat. These values are laterneeded to fill the model parameters. Finally, we specify the compound-specific Lecture Notes in Bioinformatics (LNBI) 2006 Vol. No.
data. Motivated by the Tamiflu example, we consider three compounds namedA, B, and C; a pro-drug, an active metabolite and a competitive inhibitor forthe secretion (of compound B).
compound Acompound B 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 (topright). The simulation results for (III.) competition for tubular excretion of compoundsB and C are shown in the middle and bottom panels. The middle left and right panelsshows the concentration vs. time profiles in the venous plasma and interstitial spaceof 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 valuesare assigned in the model. Only if we map the specific numerical values of the pa-rameters (corresponding to the compound and the species of interest), we obtaina fully specified and ready to simulate so-called 'simulation object'. The advan-tage of this orthogonal specification and data management is a large flexibility.
Regina Telgmann et al.
The same PBPK model can be used for different species and compounds, whilethe same species database can be used in studies of different models etc. We nowdemonstrate how to user-friendly and efficiently set up a model for three com-pounds 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 Table1.
Independent pharmacokinetics. To start with, the pharmacokinetics of the three compounds A, B and C are simulated independently. This is easilyperformed 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 com-pound. In this example we have identical models for the three compounds. Theresulting pharmacokinetic profiles are shown in Fig. 3 (top left panel) for anintravenous administration of 2 mg/(kg bodyweight) of each compound. Com-pounds A, B and C show very different pharmacokinetic profiles. This is dueto their distinctive distribution characteristics in the various tissues and due totheir different elimination characteristics. While compound C is eliminated inan almost constant fashion, compound A and B are eliminated in an exponen-tial fashion. Plasma levels of compound B are substantially higher than plasmalevels of compound A and C respectively. This is because compound B is mainlydistributed in the plasma, with significantly lower concentrations in the intersti-tium 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 tomodel the conversion of compound A into compound B in the liver. Given thefull body template from the first simulation scenario, this requires only a singlechange, namely the adaptation of the metabolism model chosen for compoundB in the liver. We define the so-called multi-compound metabolism model and subsequently map B to C(1) and A to C(2) in the simulation object. Assum-ing no i.v. administration of compound B, the simulation results are shown inFig. 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 modelfor the interacting compounds. In MEDICI-PK, this is automatically generatedexploiting the described software concept. This will become even more obviousin the next case.
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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 fullbody template from the second simulation scenario, this again requires only asingle change. We modify eq. (2) to include the competition by − Qre−abs.
− Qre−abs.
As with the case of Oseltamivir-Probenicid competitive inhibition, uni-directedinhibition can be achieved by greatly diverging Km values (factor 104) for com-pounds 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 bottompanels) for intravenous administration of 2 mg/(kg body weight) of compound Aand C. This example nicely illustrates the phenomenon of extended drug expo-sition 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 equationsis necessary to simulate the pharmacokinetics of the three compounds, includ-ing drug-drug interactions; – on the basis of the presented concepts, MEDICI-PK generates these equations from seven user-defined local physiological modelsonly! Conclusion & Outlook Considerable progress has been made in the development of in silico models topredict and understand the pharmacokinetics of new compounds, in particularin early drug discovery. As a result, modelling and simulation is possible prior toany in vivo experiments, solely based on in vitro data. We present the principlesand concepts of a software design that efficiently allows to build up PBPK modelsin terms of the underlying physiological processes, combining user-friendlinessand flexibility. These principles and concepts are the basis of the software toolMedici-PK that has been used to illustrate our approach.
We believe that the combination of in vitro experiments and in silico mod- eling has the potential to drastically increase the insight and knowledge aboutrelevant physiological and pharmacological processes in drug discovery. In antic-ipation of modelling not only the distribution of the drug in the body, but alsoits effect (disease modelling), one future challenge will be the combination ofpharmacokinetics and effect related metabolic networks or signalling pathwaysfor a better understanding of the disease dynamics. An example would be thetreatment-induced selective pressure on viral dynamics. The presented concepts,realized in Medici-PK, are powerful and flexible enough to also support thesefuture tasks.
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It is a pleasure to thank Michael Wulkow (CiT, Rastede) for fruitful and con-structive discussions. M.v.K. and W.H. acknowledge financial support by theDFG Research Center MATHEON "Mathematics for key technologies: Mod-elling, simulation, and optimization of real-world processes", Berlin.
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