Toxicokinetic models and their applications

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1 Toxicokinetic models and their applications L. Zivkovic Toxicological Centre, Podgorica, Montenegro, Abstract. In this paper toxicokinetics was examined. Chemical distribution, absorption and elimination in the living organism were investigated. The aim of this paper kinetic of toxic substance during exposure in a biological system was studied. Several toxicokinetic models using physiological and biochemical parameters were derived. Physiologically based toxicokinetic models and similar pharmacokinetic models were represented. Physiological changes during chronic exposure to a xenobiotic are introduced. This paper contributes toxicokinetics investigation with the aim to pediction the contamined biological system behavior. Keywords: Chemicals, kinetics, toxic effect, chronic, living organism. 1. Introduction The toxic response to chemical exposure depends particularly on the magnitude, duration, frequency, and route of exposure. These will determine the amount of material to which an organism is exposed, the exposure dose, and hence, the amount of material to which an organism is exposed, the exposure dose[1]-[3]. The latter determines the amount of material available for distribution and formation of toxic metabolites and hence, the likelihood of inducing a toxic effect. Opposing absorption and metabolite accumulation is elimination. All of these factors define the disposition of a xenobiotic. The modelling and mathematical description of the time course of the disposition of a potentially toxic xenobiotic in the organism are termed toxicokinetics[4]-[6]. Most of the methodologies and principles applied in toxicokinetics were first used to model the kinetics of chemicals used as drugs, pharmacokinetics. The objective of both toxicokinetics and pharmacokinetics is to quantitate the dynamic time course of xenobiotic absorption, distribution, biotransformation, and elimination processes in living organisms. Both the entire processes of disposition and individual steps such as elimination may be characterized. 2. Pharmacokinetic models A common way to describe the kinetic of drugs is to represent the body as a number of interconnected compartments, which may or may not have an anatomical or physological reality. These compartments represent all those tissues and organs, and fluids in the body that are kinetically indistinguishable from each other. Model construction process is begin by conceptualization, decomposition and specifying of components of the system being modeled. Discrete event formalism provides a means of specifying a entity called system. Basically, a system has a time base, inputs, states, outputs and functions for determining next states and outputs given current states and inputs. The insight provided by the discrete event system formalism is in the simple way that characterized how discrete event simulation specify discrete event system parameters. Having this abstraction, it is possible to design new simulation language with sound semantics that is easier to understand. Discrete event simulation scheme, an implementation of the discrete event formalism supports building models in a hierarchical, modular manner [7]. 2.1 Basic model In discrete event system formalism one must specify basic model from which larger ones are built, and how these models are connected together in hierarchical fashion. In this formalism basic models are defined by the structure, M =< Y S, Y, if, of, tf, 2 1 (1) where Y 1 is the set of external input event types, S is the sequential state set, Y 2 is the set of external event types generated as output, if is the internal transition function dictating state transitions due to internal (external) events, of is the output function generating external events at the output, and tf is the time advance function. To specify modular discrete event models requires that adopt a different view than that fostered by traditional simulation languages. As with modular specifications in general, a model should be viewed as processing inputs and outputs ports, through which all interaction with the environment is mediated. In the discrete event case, event determine values appearing on such ports. > ISBN:

2 More specifically, when external events, arising outside the model, are received on its input ports, the model description must determine how it responds to them. Also, internal events arising within the model change its state, as well as manifest themselves as events on the output ports to be transmitted to other model components. This basic model contains the following information: 1)The set of input ports through which external events are received. 2)The set of output ports through which external events are sent. 3)The set of state variables and parameters. 4)The time advance function which controls the timing of internal transitions. 5)The internal transition function which specifies how the system will transit after the time given by the time advance function has elapsed. 6)The external transition function which specifies how the system changes state when an input is received, the next state is computed on the bases of the present state, the input port and value of the external event, and the time that can elapsed in the current state. 7) The output function which generates an external output just before an internal transition takes place. 2.2 Multicomponent model Basic models may be coupled in the discrete event system formalism to form a multicomponent model which is defined by the structure. DN =< D, M i, I i, Z ij, Select > (2) where DN is a set of component names for each i in D, M i is a component basic model, I i is a set the influences of i and for each j in I i, Z ij is a function the i to j output translation. Select is a function the tie-breaking selector. Multicomponent models are implemented in discrete event simulation scheme as coupled models. A coupled models tell how to couple (connect ) several component models together to form a new model. This later model can itself employed as a component in a larger coupled model, thus giving to hierarchical construction. A coupled model contains the following information: the set of component, for each components its influences, the set of input ports through which external events are received, the set of output ports through which external event are sent. The coupling specification consists of: 1)The external input coupling which connects the input ports of the coupled model to one or more of the input ports of the components.2)the external outputs coupling which connects the output ports of components to output ports of the coupled model.3)the internal coupling which connects output ports of components to input ports of other components. 4)The select function which embodies the rules of employed to choose which of the imminent components is allowed to carry out is next event. A multicomponent model DN can be expressed as an equivalent basic model in the discrete event simulation formalism. Such a basic model can itself be employed in a larger multicomponent model. The model construction process is being by conceptualizing decomposition and specialization of components of the system being modeled The medicine activities description As a case study medicine influence effect in the human body was considered. L (medicine) + R (receptor) (LR ) stimulus effect Medicine transfer is consisted of resorbtion, distribution, metabolisms and eliminating process(fig. 1). Resorbtion means transport medicine from environment into blood circulation. The libosoluble medicine, which do not dissociates resorbing very quickly after oral application and the others applications. In opposite hydrosoluble medicine and which dissociating, they resorbtion slow. Process resorbtion is carried out mostly by diffusion through membrane and than filtration through the membrane pores between cells. One medicines group by active transport resorbs, over specific carrier. After resorbtion medicine distributes non uniformly in tissue and organs. Medicine distribution depends of libosolubilty, ability protein plasma connection, and ability tissue connection. Metabolism is medicine biotransformation in liver by enzymatic systems which it has decomposed and metabolite formed. These metabolite are soluble in water and so eliminating urine. The most important are oxidizes, esterase, transferases, flavin enzyme etc.. Individual dose can be minimal, maximal therapy dose as well as toxically dose. Dose increasing make numerous undesired effects of medicine. Medicine application can be in the liquid form, powder, tablet, drug and the others forms. A compartment might be represented by a cluster of cells within an organ, the blood or the entire body taken together. Recently, so called physiologically based pharmacokinetic and toxicokinetic models have been developed. These models permit the modelling of the time course of the concentration of a chemical in a tissue based on physiological considerations and may be particulary useful for the interspecies extrapolations necessary in risk assessment. ISBN:

3 Connected medicine o =total dose. The amount of the xenobiotic present in the organism at a certain time after observed in then: RESORPTION Free medicine RECEPTORS METABOLISM FILL OUT Figure 1. A medicine transport scheme 3.Toxicokinetic models The simplest toxicokinetic model depicts the body as a single homogenous unit within which a xenobiotic is uniformly distributed at all times. The toxicokinetics of a xenobiotic may be, analyzed by a one compartment model if the plasma concentrations determined after a single dose decrease exponentially. The plot of the logarithm of the plasma concentration versus time yields a straight line. In this model elimination of chemical from the body occurs by first order rate. dc dt = (3) k c e where c is chemical concentration in the body at time t, and k e is the rate constant for first order elimination. In compartment analysis, k e is often referred to as an apparent first order rate constant, to emphasize that the underlying processes may in reality only approximate first order kinetics. For example, the xenobiotic might actually be eliminated by active biliary secretion for which zero order saturation kinetics would be observed under suitable conditions. If the xenobiotic is not present in the organism and a known amount o of the xenobiotic is rapidly administered, the total amount of the xenobiotic initially in the body will be approximately c ket ( t) = c e o (4) The first order rate constant of elimination can be determined from the slope of the plant of the log plasma concentration versus time and can be applied to estimate the half-life of elimination for the xenobiotic. Within seven half-lifes, a xenobiotic is almoust completely eliminated (99%), although complete elimination will never theorethically be achieved. Important characteristics of the first order elimination of a xenobiotic according to the one compartment model are:1) The half-life of the xenobiotic is independent of dose, the semilogarithmic plot of the plasma concentration of the xenobiotic versus time yields a straight line. 2) The concentration of the xenobiotic in plasma decreases by a conctant fraction per unit of time. enobiotics responsible for toxic effects are usually not injected into blood, so intake is not instantaneous compared to distribution and elimination will determine the time course and a large period is observed before maximal concentration in plasma. If intake also approximates a first order absorption process then the rates of absorption and elimination will determine the time course of the plasma concentration (Fig.2). The rate of change in the concentration of a xenobiotic in a compartment model with first order absorption and elimination can be described by equation (5). dc dt = k c k c (5) a e where c is concentration, denoted xenobiotic, k a is the rate constant of absorption, k e is the rate constant of elimination. The rate constant of elimination can be determined experimentally by treating the elimination phase as if it occurred after the intravenous injection. This is done by extrapolation of the terminal straight line of the semilog plot of the ordinate at a putative o. By determining the plasma concentration of the xenobiotic at time points shortly after administration, the rate constant for absorption may be determined by plotting the difference between the early plasma concentration and the extrapolated portion of the elimination curve. ISBN:

4 0 k a k e Fig.2 Macroscopic model of absorption and elimination If a xenobiotic does not distribute and equilibrate throughout the body rapidly, a two compartment model provides a better description of the kinetics of disposition. In this case, the semilogarithmic plot of plasma concentration versus time does not yield a straight line. In two compartments models, the main or central compartment is assumed to represent the blood and highly perfused organs and tissues such as the liver, heart, and kidneys that are in rapid distribution equilibrium with the blood, whereas the second or peripheral compartment corresponds to poorly perfused tissues such as muscles and fat. This model is depicted in Fig. 4. o k 21 Conc. c E P k 12 C k 10 B Excretion A Fig.4 Two compartment model scheme The time course of the plasma concentration of the xenobiotic can be described by two overlaying monoexponential terms of the type: time t A-absorption, E-elimination, B-blood Fig.3 Concentration of a xenobiotic in absorption, elimination and blood versus time The rate constant of elimination can be determined experimentally by treating the elimination phase as if it occurred after the intravenous injection. This is done by extrapolation of the terminal straight line of the semilog plot of the ordinate at a putative o. By determining the plasma concentration of the xenobiotic at time points shortly after administration, the rate constant for absorption may be determined by plotting the difference between the early plasma concentration and the extrapolated portion of the elimination curve. ke t k ' t c ( t) Ae + Be = (6) where A and B are proportionality constant and k and k are rate constants. Thus, the toxicokinetics of a two compartment model describe a biexponential decay in the amount of xenobioptic in the body. Analogous to a one compartment model, a distribution equilibrium is assumed to exist within each compartment that can be expressed in terms of blood concentration. According to the equation (6), a plot of the logarithm of blood concentration versus time will yield a biphasic decay from which the constant A,B, k, and k can be estimated either graphically or by nonlinear regression analysis. Once these experimental constants have been determined, the toxicokinetic parameters can be calculated. ISBN:

5 o Conc. c P to physical growth or induction of biotransformation and changes in excretion rates, can be introduced (Fig. 6). Q alv Q t c inh Alveolar space Lung/blood Q alv Calv Q f C c vf Fat c vm Lean tissues (muscles) Q m time t C-central compartment, P-peripheral compartment, o - initial dose. c vt Well perfused tissues Q t Fig.5 Concentration of a xenobiotic in compartment versus time Liver Q l The numerical values of these parameters can aid in assessing the relative importance of tissue distribution and elimination to the disposition of the xenobiotic. c vl Metabolic excretion 4. Physiologically based toxicokinetic model Although compartmental analysis models are convenient to use and provide useful description of the overall time course of xenobiotic disposition, the practical limitations of curve fiting to experimental data generally restrict such models to one or two compartments. Alternatively, a compartmental model can often be developed from physiological pharmacokinetic or toxicokinetic model is a mathematical description of the disposition of a xenobiotic in the organism or in part of it, specific organ. Such a model is constructed by using physiological and biochemical parameters such as blood flow rates, tissue and organ sizes, binding, and biotransformation rates[5]-[6]. These are generally more complex and require the specification of many parameters, thus physiological compartmental models are still simplified representations of bilogical systems. In addition, the determination of physical and biochemical parameters is often both difficult and inaccurate. However, the physiological framework provides several advantages. The physical definition of compartments and transfer rates facilitates the incorporation of existing knowledge about the quantitative behavior of biological systems into the model. Physiological changes with time during chronic exposure to a xenobiotic, such as those due Fig. 6 Scheme of a physiologically based toxicokinetic model Most important, however, a reasonable basis exists for extrapolating the kinetics of a xenobiotic to predict its disposition following various types and patterns of exposure and most important, to extrapolate from experimental data obtained in one species of experimental animal to other animal species and humans. Physiologically based models use the concepts of material balance and flow limited transport under the following definitions blood serves to distribute a xenobiotic from the site of absorption to other parts of the body. In the normal sequence of events, a chemical entering the bloodstream is often distributed rapidly within the blood, and its blood concentration can be considered essentially uniform. The chemical enters and leaves the compartment with the blood flow diffuses or is transported from blood to tissue and brack. The chemical may also undergo a variety of physical interactions such as binding to macromolecules, the results is a partitioning between ISBN:

6 tissue and blood, depending on the chemical s particular affinity for each medium. Diffusion or transport directly between adjacent comparments, enzymatic biotransformations, and excretion may also occur. The net results of all of these changes is expressed by a material balance differential equation, which is simply a a mathematical statement of the conservation of chemical. Frequently, these material balance equation can be greately simplified because many of terms required may not apply to a particular compartment. 10. Conclusions The diffusional transfer rate, enzymatic biotransformation and excretion toxicitied organism were examined. Fundamentals xenobiotic transfer rate relations at the toxication were derived. Toxicokinetics investigation taking into account only elimination and elimination and absorption was performed. Also, physologically based toxicokinetic model was studied. Different toxicokinetic models of dynamic transition are defined.. [1] J.Savkovic-Stevanovic,Reliability and safety analysis of the process plant, Petroleum and Coal, 56 (4) 62-68,2010. [2] J. Savković-Stevanović,Waste water transport system safety, EG2009-Environment and Geology, WSEAS, ID , pp.71-76, University of Brasov, Romania, Sept.24-26, [3] S.Končar-Djurdjević, Life Environment Protection, Rad, Belgrade,1971. [4] J.Savković-Stevanović.,L. Živković, Risk analysis and life environment protection, 2nd International Congress on Engineering, Ecology and Materials in theprocess Industries, Jahorina, Serbian Republic, Bosnia and Hercegovina, 9-11, March,2011 [5] M.E. Andersen et a., Toxicol. App. Pharmacol. 87, ,1987. [6]Q.Chen B.N.Ames, Toxicologit, 13, 110, [7] J.Savković-Stevanović, Discrete events simulation at the drug distribution in the human body, Comput. Chem. Eng.3 (2)44-51,2007. Acknowledgement The author wishes to express their gratitude to the Fund of Serbia for Financial Support Notation A- absorption a-constant B-blood b-constant c - component concentration, mole/cm 3 DN -domain names I i -set the influences of i and for each j in i if-internal transition function dictating state transitions due to internal (external) event k-specific constant rate, s -1 M i -- basic model of- output function generating external events at the output tf- time transition function S- sequential state set Y 1 - set of external input event types Y 2 - set of external event types generated as output Z ij - function the i to j output translation. Subscript a-absorption C-central e-elimination o-initial P-peripheral -xenobiotic References ISBN: