Matematical and Computational Applications Article Nonlinear Elimination of Drugs in One-Compartment Parmacokinetic Models: Nonstandard Finite Difference Approac for Various Routes of Administration Oluwaseun Egbelowo ID Scool of Computer Science and Applied Matematics, University of te Witwatersrand, Joannesburg, Private Bag 3, Wits 25, Sout Africa; oluwaseun@aims.edu.g Received: 25 April 28; Accepted: 22 May 28; Publised: 23 May 28 Abstract: Te motivation for tis study is to introduce and motivate te use of nonstandard finite difference NSFD scemes, capable of solving one-compartment parmacokinetic models. Tese models are modeled by bot linear and nonlinear ordinary differential equations. Exact finite difference scemes, wic are a special NSFD, are provided for te linear models wile we apply te NSFD rules, based on Mickens idea of transferring nonlinear models into discrete scemes. Te metod used was compared wit oter establised metods to verify its efficiency and accuracy. One-compartment parmacokinetic models are considered for different routes of administration: I.V. bolus injection, I.V. bolus infusion and extravascular administration. Keywords: parmacokinetics; intravenous bolus injection; intravenous bolus infusion; extravascular; nonstandard finite difference; Micaelis-Menten elimination. Introduction Parmacokinetics modeling is te matematical representation of te beaviour of a drug in te body or an area of te body, created to describe te parmacologic or pysiologic kinetics caracteristics. Parmacokinetics is te study of te basic processes tat determine te duration and intensity of a drug effect witin an organism. Tese models can assist in simulating te biological processes involved in te kinetic beaviour of a drug after it as been introduced into te body, leading to a better understanding of its dynamic effects. Matematical modeling is currently a common tool used in te study of pysiological and biocemical systems. It can be developed from non-compartmental representations to large scale multi-compartment models. One of te early uses of compartment models was reported by Widmark []. He used a compartment model to describe te distribution of alcool in te body. Suc compartment models ave proven to be a great advantage wen screening drugs used by umans at any instant in time. In te case of compartment models, mass-balance equations are used to represent eac compartment. Te number of compartments in te model depends on te rate of drug distribution to different parts of te body. Most studies use one- or two-compartment models. Wen a drug is eliminated, te drug concentration in te systemic circulation and in all tissues decline at te same rate because of te rapid distribution equilibrium. Drugs tat follow tis beaviour follow te one-compartment parmacokinetic model, wile in two-compartment models, te movement of te administered drug is distributed instantaneously to some tissues and slowly to oter tissues. However, if te distribution of te drug appened at tree different rates, a tree-compartment model would be applicable. Our focus is on a one-compartment parmacokinetic model, specifically aimed at different models of drug elimination. Mat. Comput. Appl. 28, 23, 27; doi:.339/mca23227 www.mdpi.com/journal/mca
Mat. Comput. Appl. 28, 23, 27 2 of 2 After a drug is released from its dosage form, te drug is absorbed into te surrounding tissue and/or te body. As commented on by Sargel et al. [2], te distribution and elimination of te drug in te body varies for eac patient but can be caracterized using matematical models and statistics. Being able to caracterise drug distribution and elimination is an important prerequisite to be able to determine or modify te dosing regimens of individuals and groups of patients. Among te tree main types of parmacokinetic PK models: compartment, pysiologic and non-compartmental models, compartmentally based models are known to be a very simple and useful tool in parmacokinetic. In essence, a compartment model provides a simple way of grouping all te tissues into one or more compartments were drugs move to and from te central or plasma compartment. In tis manner, we are able to model te transport processes between interconnected volumes, suc as te movement of drugs and ormones in te uman body. Compartment models assume tat tere is rapid and perfect mixing, so tat te drug concentration remains te same in eac compartment. Te complex transport processes are approximated by assuming tat te flow rates between te compartments are proportional to te concentration difference in te compartments. Compartment models play a significant role in understanding te dynamics of drug concentration in te body. In practice, PK models seldom consider all te rate processes ongoing in te body, Sargel et al. [3]. Due to te complexity of te models wic incorporate suc information, simplifying assumptions are often made so tat solutions may be obtained. Traditional PK models, being simplified matematical expressions, are based on te assumption of a linear relationsip between te dose of a drug and its concentration see Beňová et al. [4]. In a linear model, tese rate coefficients called k are assumed to be constant. However, suc assumptions regarding te linearity of te model do not necessarily describe te actual pysical processes as accurately as a non-linear relationsip may. In fact, te non-linearities seen in suc models are related to drug absorption, distribution, metabolism and excretion and te parmacokinetic of drug action. Since in most cases, tese compartment models are described by autonomous linear or nonlinear ordinary differential equations, we coose to consider te latter in tis work, focusing on tree regimes of excretion. In te case wen te nonlinearity or kinetics in te system are complex or wen te number of compartments in te model becomes large, suc as in te work of Sarma [5], exact solutions are not obtainable and ence we turn to numerical metods. Some of te well known standard numerical scemes produce unnecessary oscillations, introduce extraneous or spurious solutions, and converge to fixed-point solutions different from te corresponding derivative [6]. Hence, one observes te occurrence of numerical instabilities. Te nonstandard finite difference NSFD sceme was developed by Mickens as an alternative metod providing an approximate solution to a wide range of differential equations and catering for te numerical instabilities tat occur wen using standard metods. NSFD metods ave been well reported in recent years, mainly because tey are efficient and and preserve qualitative properties, see for example Villatoro [7], Roeger [8], Ibijola & Obayomi [9], Manning & Margrave [], Mickens [ 6] and Sunday [7] wic give te relevant background materials on tis topic. Some of tese autors deal wit te exact finite difference sceme wic is a special NSFD metod. A standard finite difference SFD sceme is said to be exact for a particular differential equation if its local truncated error is exactly zero for its general solution Gander & Meyer-Spasce [8] and Mickens [9]. In oter words, wen te analytical solution of a differential equation can be matced exactly wit its corresponding SFD equation, ten te exact finite difference solution exists. If exact general solutions of a differential equation are explicitly known, ten an exact finite difference sceme exists. Te idea of an exact finite difference sceme was first conceived by Mickens [9,2] wo as sown tat an exact explicit sceme is easily obtained from te knowledge of its analytical solution. Terefore, exact finite difference scemes are designed in suc a fasion tat te difference equation as te same general solution as te corresponding differential equation. In te situation were exact solutions exist, te solution can be re-structured in suc a way to obtain te exact finite difference sceme. However, in te case were exact solutions are not possible, te rules proposed by Mickens [9] will be deployed. In suc situations, te metod is referred to as te NSFD metod as
Mat. Comput. Appl. 28, 23, 27 3 of 2 discussed by Anguelov & Lubuma [2]. Te consequence of tese rules is tat wile te sceme may not be exact, qualitative properties of te corresponding differential equations for all step-sizes are preserved and tus elementary numerical instabilities tat can arise are eliminated. Our researc is aimed at te well-known one-compartment model. We aim to investigate tree forms of drug elimination from te body. In a one-compartment model, te body is assumed to be a single compartment and te drug absorbed acieves instantaneous distribution trougout te body, metabolizing between tissues. Te drug output is caracterized by an elimination rate. In tis study, tree dose regimen models are considered: One-compartment model I.V. bolus injection, One-compartment model I.V. bolus infusion, One-compartment model Extravascular administration. We consider tese cases to illustrate te effectiveness of te NSFD metod for te solution of nonlinear differential equations of tis nature. Te work conducted ere is done wit te aim of introducing a numerical metod wic may act as an effective tool to be employed in future researc for te solution of models wic are non-linear and/or describe multiple compartments. As suc, we propose and illustrate te use of a numerical metod of solution, namely te NSFD metod, capable of efficiently obtaining solutions wic are not only accurate but maintain te underlying dynamics of te system of equations. Tis coice of metod impacts on weter we are able to consider non-compartment models; te NSFD metod is not amenable to te simulation of non-compartment models as it provides a meta-analysis of te inter-compartment dynamics, wereas non-compartment models are unable to describe tese meta-dynamics and instead conduct parameter estimation of te entire system as a wole troug te use of experimental data. Te advantage of te NSFD metod is te ability to predict te concentration-time profile of a drug wen tere are alterations in te dosing regimen tis would not be possible were one to consider non-compartment analysis. Anoter advantage of te NSFD metod is tat it preserves significant properties of te analogous models and consequently gives reliable numerical results even wen analytical solutions are not possible. Te standard approaces to multi-compartment models assume linear dynamics over te duration of eac time step, wereas te NSFD metod assumes exponential dynamics. Hence, in te case of a linear model, te NSFD metod recovers te model dynamics exactly. Tis paper illustrates te ability of te NSFD metod to solve a one-compartment PK model wit various modes of elimination, in a stable and robust fasion, wit te ability to be extended to non-linear and/or multi-compartment models. Te variables of importance and teir meaning are give below: C: Drug concentration in te central compartment. V max : Te maximum rate of cange of concentration. K m : Te Micaelis-Menten constant. k a : Te absorption rate constant for oral administration. k el : Elimination rate of te drug leaving te central compartment. V : Te apparent volume of distribution. 2. Metods Wile te implementation of te NSFD metod is te focus of tis researc, we employ te Runge-Kutta as a means of comparison. Tis section provides an overview of NSFD and Runge-Kutta. 2.. NSFD Modeling Fundamental Principles NSFD metods provide numerical solutions to differential equations by constructing discrete models. Tey preserve te significant properties of teir continuous analogues and consequently give reliable numerical results. Te following rules were given by Mickens in [9] for constructing an NSFD sceme:
Mat. Comput. Appl. 28, 23, 27 4 of 2 Rule Rule 2 Rule 3 Rule 4 Rule 5 Te orders of te discrete representation of te derivative must be equal to te orders of te corresponding derivatives appearing in te differential equations. Denominator functions for te discrete representations for derivatives must, in general, be expressed in terms of more complicated functions of te step-sizes tan tose conventionally used. Nonlinear terms must, in general, be modeled by nonlocal discrete representations. All te special conditions tat correspond to eiter te differential equation and/or its solutions sould also correspond to te difference equation and/or its solutions. Te discrete sceme sould not introduce extraneous or spurious solutions. Remark. Exact finite difference is a special NSFD. 2.2. Runge-Kutta Metod In a similar fasion wit te finite difference sceme, we introduce te concept of te Runge-Kutta metod from Taylor s teorem, were is te step size between te values of te independent variable x. Consider x = f t, x. Ten, te Taylor s series expansion of Equation is given by Differentiating Equation, we ave xt + = xt + x t + 2 2! x t + O 3. 2 x t is given in Equation, terefore Equation 3 becomes x t = f t t, x + f x t, xx t. 3 x t = f t t, x + f x t, x f t, x. 4 Substituting Equations and 4 into Equation 2, we ave xt + = xt + f t, x + 2 2 f tt, x + f x t, x f t, x + O 3. 5 Wit some manipulations, we ave xt + = xt + 2 f t, x + 2 f t +, x + f t, x + O3. 6 From Equation 6, if k = f t n, x n, 7 k 2 = f t n +, x n + k, 8 ten classical second order Runge-Kutta metod is given as x n+ = x n + 2 k + 2 k 2. 9 Te approximation given by Equation 9 as a local truncation error O 3. Tis second order Runge-Kutta metod is also known as Heun s metod. Te most widely used metod is te fourt-order Runge-Kutta metod wic can be developed in a similar fasion to te second order Runge-Kutta. Te local truncated error of te fourt-order
Mat. Comput. Appl. 28, 23, 27 5 of 2 Runge-Kutta metod is O 5. Equation can be solved using te classical fourt-order Runge-Kutta as follows: were x n+ = x n + 6 k + 2k 2 + 2k 3 + k 4, k = f t n, x n, k 2 = f t n + 2, x n + k, 2 2 k 3 = f t n + 2, x n + k 2, 3 2 k 4 = f t n + 2, x n + k 3. 4 Te approximation given by Equation subjected to k i, i =, 2, 3, 4 as an error of O 5 and tus is deemed te most accurate of all te approximations provided. 3. Results 3.. I.V. Bolus Injection Wen drugs are administered by an I.V. bolus injection, te entire dose administered enters te bloodstream directly and is able to produce parmacological effects. Tis is followed by te distribution of te drug troug te circulatory system to all tissues in te body. Hence, we assume tat a drug given by an I.V. bolus injection is rapidly mixed. Naturally, if you inject it directly into te bloodstream, te drug is immediately found in te bloodstream and does not ave to be absorbed. Te concentration at time t = corresponds to te dose given in tis manner and ence, it describes te I.V. bolus injection route of administration. We consider tis mode of administration alongside two different elimination processes: drugs eliminated by linear parmacokinetic and 2 drugs eliminated by nonlinear processes. We solve te differential equation arising from tese elimination processes, employing te NSFD metod as a means of comparison wit te SFD metod. Te case were te drugs are given via an I.V. bolus injection, distributed as a two-compartment model and ten eliminated only by linear parmacokinetic, is presented in Egbelowo et al. [22]. In te manuscript, we did not provide te results in te case of a one-compartment model, wic is of interest ere. Wen considering te SFD metod for te one-compartment model tat describes te distribution and elimination after an IV bolus dose, i.e., = k el C k, 5 we notice te following interesting dynamics: i if < k el <, C k monotonically tends to, ii if k el =, C k = for k, iii if < k el < 2, C k tends to wit an oscillating amplitude via an alternating sign at eac step, iv if k el = 2, C k oscillates wit a constant amplitude C, and v if k el > 2, C k oscillates wit an increasing amplitude, were is te step-size, k el is te elimination rate of te drug, and C k represents te concentration of te drug at time t k. Results indicate tat te SFD sceme developed as te same qualitative beaviour as te analytical solution of te parmacokinetic model if < k el. 6
Mat. Comput. Appl. 28, 23, 27 6 of 2 In turn, te corresponding NSFD sceme constructed gives accurate results witout te requirement 6 as needed by te SFD sceme. 3... I.V. Bolus Injection: Nonlinear Parmacokinetic Elimination Te equation tat describes te elimination of a drug tat is distributed in te body as a one-compartment and is eliminated by nonlinear parmacokinetic after an I.V. bolus injection is given as per Sargel et al. [23]. From te compartment diagram in Figure, we ave te differential equation dc dt = V maxc K m + C V max >, K m >, 7 subject to C = C, 8 were V max is te maximum elimination rate and K m is te Micaelis constant. Micaelis-Menten kinetics are also referred to as te capacity-limited metabolism, saturable metabolism, or mixed-order kinetics as discussed in Beňová et al. [4]. C V max, K m Figure. Sceme of I.V. bolus injection wit Micaelis-Menten elimination. Te solution given as Ct to te differential Equation 7 as no closed form expression. Te explicit closed-form solution of te one-compartment I.V. bolus injection model tat follows Micaelis-Menten kinetics given in Equation 7 is, owever, expressible in terms of te Lambert W-function as Ct = K m W C e C Km Vmaxt Km. 9 K m Definition. Te Lambert W-function is defined to be a multivalued single valued in te case of PK applications inverse of te function x xe x satisfying [24] y = Wx, 2 suc tat x = ye y. 2 Equation 2 can ten be written as x = Wxe Wx, 22 were W is Lambert s function. Equation 9 satisfies te transcendental equation 22. Te exact finite difference sceme of 9 is given as C k+ = K m W C ke C k Km Vmax Km K m A SFD sceme for te Micaelis-Menten Equation 7 migt take te form. 23 = V maxc k. 24
Mat. Comput. Appl. 28, 23, 27 7 of 2 We provide NSFD scemes for Equation 7 for te case were Equation 7 is discretized using tree different discretization metods, as done in te works of Mickens [25] and Capwanya et al. [26,27] namely, semi-implicit forward-euler, implicit forward-euler and explicit forward-euler. Case : Semi-Implicit Forward-Euler Te NSFD sceme ere is obtained from te semi-implicit forward-euler discretization as = V maxc k+. 25 Tis can be rewritten as We ten use te fact tat C k+ = K m C k + C 2 k K m + V max + C k. 26 + V max K m = e Vmax Km + OV 2 max 2 /K 2 m, 27 allowing us to make te replacements + V max K m = e Vmax Km, 28 wic implies K me Vmax Km V max. 29 Terefore, te denominator function for te semi-implicit forward-euler discretization is given as φ, V max, K m = K me Vmax Km V max. 3 Te NSFD semi-implicit discretization of Equation 7 is φ = V maxc k+. 3 Te NSFD sceme 3 is compared wit te corresponding SFD sceme 25. Te sceme given by Equation 3 as te same qualitative beaviour as te original differential equation for all step sizes. Case 2: Implicit Forward-Euler Wen Equation 7 is discretized using an implicit discretization, we obtain = V maxc k+ +, 32 and following te same process as described by Equations 25 29, we obtain te same denominator function given in Equation 3. Terefore, te NSFD sceme for Equation 7, wen discretized using an implicit forward-euler approximation, is φ = V maxc k+ +. 33 Case 3: Explicit Forward-Euler Te NSFD sceme, in tis case, is obtained from an explicit forward-euler discretization as
Mat. Comput. Appl. 28, 23, 27 8 of 2 = V maxc k. 34 Te denominator function is given by Similarly, te NSFD discretization of Equation 7 is φ 2 = K m e Vmax Km V max. 35 φ 2 = V maxc k. 36 From te tree cases, we ave tat te denominator function obtained depends on te discretization metod used. Upon employing te Lambert function, te resultant sceme for Equation 7 as exibited by Equation 23, is φ 2 = K m W C k C k e Km Vmax Km K m C k φ 2. 37 Tis sceme will serve as a means of comparison wit te results obtained from te tree cases above. 3..2. I.V. Bolus Injection: Mixed Drug Elimination Anoter possible route of drug elimination is mixed drug elimination. In tis elimination process, as depicted in Figure 2, drugs are eliminated by nonlinear processes. Terefore, te equation tat best describes a drug tat is eliminated by Micaelis-Menten kinetics after an I.V. bolus injection is given by dc dt = k elc V maxc K m + C, 38 were k el is te first-order rate constant representing te sum of all first-order elimination processes. Te second term of te Equation 38 represents te saturable process. Te SFD sceme of Equation 38 is given by = k el C k+ V maxc k+. 39 Implementing te NSFD sceme as before, we obtain a denominator function given by wic provides te following NSFD sceme for Equation 38 φ 3 = K me kel Km+Vmax Km K m, 4 k el K m + V max φ 3 = k el C k+ V max C k+. 4
Mat. Comput. Appl. 28, 23, 27 9 of 2 C V max, K m k el Figure 2. Scematic representation of I.V. bolus injection wit mixed drug elimination. 3.2. I.V. Bolus Infusion I.V. bolus infusion is te process of infusing a drug at a constant rate. Te drug input is constant and equal to te rate of infusion of te drug. On starting te infusion, tere is no drug in te body and terefore no elimination. Te concentration of te drug in te body ten rises, but as te drug concentration increases, so does te rate of elimination. Tus, te rate of elimination will keep rising until it matces te rate of infusion. Te concentration of te drug in te body is ten constant and is said to ave reaced a steady state. A similar approac is used for te I.V bolus injection process, and we consider tis mode of administration alongside two different elimination processes as before: drugs eliminated by linear parmacokinetic and 2 drugs eliminated by nonlinear processes. In te case wen te drug is given via I.V. bolus infusion and te drug is excreted in a linear way, we observe similar dynamics as we did for te case were an I.V. bolus injection is te means of administration see Egbelowo et al. [22] were bot cases was considered for a two-compartment model. Te one-compartment model tat describes te distribution and elimination after an IV infusion dose is given by = R k el C k. 42 Te solution obtained via te SFD sceme gives te following results: i ii iii iv v if < k el <, C k monotonically tends to R k el, if k el =, C k = k R for k, el if < k el < 2, C k tends to k R wit an oscillating amplitude via an alternating sign at eac step, el if k el = 2, C k oscillates wit a constant amplitude 2R k, and el if k el > 2, C k oscillates wit an increasing amplitude. is te step-size, k el is te elimination rate of te drug, C k represents te concentration of te drug at time t k, R = R V is te flow rate of te drug, and R is te infusion rate per unit time. From tese results, we conclude tat tis model will ave numerical instabilities for all cases except for cases i and ii. Maintaining te requirements given by cases i and ii, te same qualitative beaviour is observed as te original differential equation. Te NSFD sceme constructed in turn gave accurate results for all te cases given above. 3.2.. I.V. Bolus Infusion: Nonlinear Parmacokinetic Elimination In Figure 3, te drug is administered by constant infusion and is eliminated by nonlinear parmacokinetic processes. Te equation tat describes te rate of cange of te plasma concentration, as depicted in Figure 3, is given by dc dt = R V maxc K m + C V max >, K m >, 43 subject to C =. 44
Mat. Comput. Appl. 28, 23, 27 of 2 All te parameters in Equation 43 are defined as for model 7. Solving Equation 43 using te concept of te W-Lambert function, we ave Ct = K mv max W explt+mt K m V max RK m, 45 R V max were Lt = Mt = R2 t K m V max Q = K m V max log R 2 Q K m V max R V max 2 + 2RQ K m R V max 2, V maxq tv K m R V max 2 max K m + 2Rt R K m 2 V max R V max + RK m e R Vmax RK m, R R V max. 46 Equation 45 can be written in te form φ = K m V max W expl +M KmVmax R V max φ RK m C k, 47 were L = L and M = M. Te steady-state concentration of Equation 43 is determined by te following equation C ss = K mr V max R. 48 Te NSFD sceme for Equation 43 is structured as φ = R V maxc k+, 49 were φ is defined as before. Comparing Equation 49 wit te SFD sceme given by = R V maxc k, 5 sows tat te NSFD sceme is dynamically consistent wit te original differential equation for any step size. R C V max, K m Figure 3. Scematic representation of IV infusion wit Micaelis-Menten elimination. 3.2.2. I.V. Bolus Infusion: Mixed Drug Elimination Figure 4 describes te rate of cange in te plasma drug concentration for a drug tat is given by I.V. infusion and eliminated by nonlinear parmacokinetic. Tis is an extension of Figure 3, wic leads to Equation 5 dc dt = R k elc V maxc C =. 5 K m + C All te parameters are defined as done for te model given by Equation 7. Te steady-state concentration of Equation 5 can be determined by
Mat. Comput. Appl. 28, 23, 27 of 2 C ss2 = R k elk m V max + R k el K m V max 2 + 4Rk el K m 2k el. 52 Te NSFD sceme of 5 is given by φ 3 = R k el C k+ V maxc k+, 53 were φ 3 is given in Equation 4. Te SFD sceme of te relevant equation is = R k el C k V maxc k. 54 R C V max, K m k el Figure 4. Scematic representation of I.V. infusion wit mixed drug elimination. 3.3. Extravasular Administration A drug administered via te extravascular route of administration undergoes te process of absorption before it gets to te systemic circulation. Tis type of drug delivery is complicated by te variable at te site of absorption. Te level of absorption of a drug from te gastrointestinal tract GIT depends on te anatomy and pysiology of te absorption site, pysiocemical properties of te drug, and pysiocemical properties of te dosage form. Initially, te entire drug is in te site of absorption and none as yet reaced te systemic circulation [28]. Most drugs administered extravascularly act systemically. In suc cases, systemic absorption is a prerequisite for efficacy. Tis section describes te extravascular route of administration. We consider tis mode of administration via two different elimination processes: drugs eliminated by linear parmacokinetic 2 drugs eliminated by nonlinear processes. 3.3.. Extravasular Administration: Linear Parmacokinetic Elimination Wen a drug is administered troug extravascular administration and eliminated by a linear process as sown in Figure 5, we apply Equations 55 and 56 in order to model te process. Equation 55 describes te drug at te site of absorption before it reaces te systemic circulation, wile Equation 56 describes te concentration of te drug at te systemic circulation. D is te amount of drug in te GIT at any time t, k a is te first-order absorption rate constant, C is te plasma concentration of te drug in te body and k el is te elimination rate. Tus, te disappearance rate of te drug from te GIT is given by also termed te equation for drug in GIT, dd dt = k ad, D = D. 55 Te rate of cange of te amount of drug in te body is given by dc dt = k ad k el C, C =, 56
Mat. Comput. Appl. 28, 23, 27 2 of 2 were C is te amount of te drug available at te absorption site. Te time course of te te amount of drug tat follows te oral route of administration is given by Ct = k a D e kelt e k at, k a = k V k a k el el, 57 were D is te dose of te administered drug, k a is te constant of te absorption, and k el is te rate of elimination. At t = at later time intervals te above equation reduces to i.e., wen e k at Ct = k a D V k a k el e k elt. 58 Te exact finite difference sceme of te model is derived from te analytical solution. Since Equations 55 and 56 can be solved simultaneously, we proceed as dd dt Te particular solution of Equation 59 is = k a D dc dt = k a D k el C, 59 D = Dt, C = Ct. 6 D =, C =. 6 Considering te system of Equations 59 in matrix form, te corresponding matrix is M = k a. k a k el Te matrix as eigenvalues λ if detm λi = or λ 2 tramλ + detm =. Terefore, te eigenvalue equation to be λ 2 + k a + k el λ + k a k el =, 62 wic provide te eigenvalues λ = k a, λ 2 = k el. 63 v Suppose v = represents te eigenvectors corresponding to te eigenvalues, v 2 ten M λiv =. Hence, we ave tat k a λ,2 v =, k a k el λ,2 v 2 wic gives k a λ,2 v =, k a v + k el λ,2 v 2 =. 64 Hence, v = giving te general solution of te system as v = v 2 kel +λ,2 k a,
Mat. Comput. Appl. 28, 23, 27 3 of 2 kel + λ Dt = A e λt + B k a kel + λ 2 k a e λ 2t, 65 Ct = Ae λ t + Be λ 2t. 66 To calculate A and B, we use te initial values given in Equation 6 so tat Equation 66 simplifies to From Equation 65, we ave tat Ct = Ae λ t + Be λ 2t = C, 67 A = C e λ t Be t λ 2 λ. 68 kel + λ Dt = A e λ t + B k a kel + λ 2 k a e λ 2t = D. 69 Substituting Equation 68 into Equation 69 and after some algebraic manipulations, we obtain B = ka λ 2 λ D e λ 2t Substituting Equation 7 into 68, we obtain kel + λ A = C e λ ka t D λ 2 e λ t + λ λ 2 λ C e λ 2t. 7 kel + λ λ 2 λ C e λ t. 7 Substituting Equations 7 and 7 into Equation 66 wit some manipulations we obtain [ Ct = p D k ] [ el + λ 2 C e λ t t + p D λ ] + k el C e λ 2t t. 72 k a k a were p = ka λ 2 λ. Te exact finite difference sceme of Equation 59 is obtained by making te following transformations in Equation 72 t t k = k, t t k+ = k +, D D k, Dt D k+, 73 Tus, [ C k+ = p C C k, Ct C k+, D k λ 2 + k el C k k a ] [ e λ + p D k λ + k el C k k a ] e λ 2. 74 λ C k+ 2 e λ λ C e λ 2 e k = k a D λ 2 λ k λ2 e λ k el C k, 75 λ 2 λ giving te exact finite difference sceme D k+ ψd k φ = k a D k C k+ ψc k φ = k a D k k el C k, 76
Mat. Comput. Appl. 28, 23, 27 4 of 2 were ψ = k ae k el k el e k a k a k el, φ = e kel e ka k a k el. 77 Te exact finite difference result obtained in system 76 will be compared wit te SFD sceme D k+ D k = k a D k C k+ C k = k a D k k el C k. 78 D k a C k el Figure 5. One-compartment parmacokinetic model for first-order drug absorption and first-order elimination. 3.3.2. Extravascular Administration: Mixed Drug Elimination In te situation wen te drug is administered by te extravascular mode of administration and eliminated by parallel patways, Equation 79 is applied to describe Figure 6. Consider dd dt = k a D 79 dc dt = k a D k el C V maxc K m +C, wit initial conditions D = Dt and C = Ct. Te NSFD sceme of Equation 79 is D k+ ϕd k φ C k+ ϕc k φ = k a D k and may be compared to te SFD sceme of te form D k+ D k C k+ C k = k a D k k el C k V maxc k K m +C k, = k a D k = k a D k k el C k V maxc k K m +C k. 8 8 D k a C V max, K m k el Figure 6. Scematic presentation of extravascular administration wit bot linear and Micealis-Menten elimination.
Mat. Comput. Appl. 28, 23, 27 5 of 2 4. Numerical Simulations and Discussion One-compartment models wit different routes of administration I.V. bolus injection, I.V. bolus infusion and extravasular are considered for simulations. In order to perform a useful comparison, tese metods were tested under similar conditions corresponding to te intended practical application. Te purpose of te tests was to compare te accuracy and stability of te various numerical scemes employed. Tis is done for varying step-sizes and te results are examined in te figures and numerical results presented in te next few sections. Te numerical calculations are carried out in MATHEMATICA, and te results are ten processed in MATLAB to generate visual representations. 4.. I.V. Bolus Injection: Simulations 4... Results Describing Nonlinear Parmacokinetic Elimination Ti section presents te results of te case wen te drug is administered by I.V. bolus injection and eliminated by Micaelis-Menten elimination. NSFD scemes 3, 33 and 36, and te SFD scemes 25, 32 and 34 respectively, are compared wit te analytical solution 9 in Figure 7. Te analytical solution was obtained troug te use of te W-Lambert function..9.8 Concentration Profile Analytical Semi implicit forward Euler NSFD wit Semi implicit forward Euler.9.8 Concentration Profile Analytical Implicit forward Euler NSFD wit Implicit forward Euler.7.6.5.4.3.7.6.5.4.3.2.2.. 5 5 2 25 3 a 5 5 2 25 3 b.9.8 Concentration Profile Analytical Explicit forward Euler NSFD wit Explicit forward Euler.7.6.5.4.3.2. 5 5 2 25 3 c Figure 7. a NSFD sceme 3 in case plotted against te analytical solution 9 and te corresponding SFD sceme 25; b NSFD sceme 33 in case 2 plotted against te analytical solution 9 and te corresponding SFD sceme 32 and c NSFD sceme 36 in case 3 plotted against te analytical solution 9 and te corresponding SFD sceme 35.
Mat. Comput. Appl. 28, 23, 27 6 of 2 4..2. Results Describing Mixed Drug Elimination In tis section, simulations of te equation tat describe a drug tat is eliminated by mixed drug elimination after an I.V. bolus injection are provided. Figure 8 sows a comparison between te SFD sceme in Equation 39, te NSFD sceme in Equation 4 and MATLAB built-in function ODE45. 2.8.6 Concentration Profile ODE45 SFD Mixed drug elimination NSFD Mixed drug elimination.4.2.8.6.4.2 2 3 4 5 6 7 8 Figure 8. Trajectory representation of te one-compartment I.V. bolus injection model tat follows mixed drug elimination. Te SFD sceme 39 and NSFD sceme 4 plotted against ODE45 of 38. 4.2. I.V. Bolus Infusion: Simulations 4.2.. Results Describing Nonlinear Parmacokinetic Elimination Te results for te case wen te drug is administered by I.V. bolus infusion and eliminated by Micaelis-Menten elimination is presented. Te NSFD sceme 49 and SFD sceme 5 are compared wit MATLAB built-in, ODE45. From Figure 9 we see tat regardless of te step-size, te NSFD sceme 49 converges to te steady state. Table gives te simulation results of te I.V. bolus infusion case were nonlinear parmacokinetic elimination is present. Table. Te absolute error results of Equations 43 for C wit parameters values R =.5, K m = 4, and V max = 2. Absolute Error for C N Error in Sceme 5 SFD Error in Sceme 49 NSFD 2 4..27.8 4 2..338.3 8..5.7 6.5.5.2 32.25.24. 64.25.2. 28.62.6. 256.3.3. 52.6.. 24.8..
Mat. Comput. Appl. 28, 23, 27 7 of 2 2.5 Concentration Profile 2.5 Concentration Profile 2 2.5.5.5 ODE45 SFD Nonlinear elimination NSFD Nonlinear elimination Analytical C ss.5 ODE45 SFD Nonlinear elimination NSFD Nonlinear elimination Analytical C ss 2 4 6 8 2 4 6 5 5 2 a b Figure 9. NSFD 5 sceme, SFD sceme 49 and ODE45 are compared analytical solution 45. C ss is te steady state Equation 48. Te concentration of te drug wen administered via I.V. bolus infusion and eliminated by nonlinear parmacokinetic processes for a =.563 and b = 6.456. 4.2.2. Results Describing Mixed Drug Elimination Figure sow te simulation results of te NSFD sceme 53 and SFD sceme 54 in comparison to te results obtained via te in-built function ODE45. 8 Concentration Profile 8 Concentration Profile 7 7 6 6 5 4 3 5 4 3 2 ODE45 SFD Mixed drug elimination NSFD Mixed drug elimination C ss2 2 4 6 8 2 2 ODE45 SFD Mixed drug elimination NSFD Mixed drug elimination C ss2 5 5 2 25 3 35 4 a Figure. NSFD sceme 53 and SFD sceme 54 is comparison wit ODE45. C ss2 is te steady state Equation 52. Te concentration of te drug wen administered via I.V. bolus infusion and eliminated by mixed drug processes for a =.75 and b = 3.3333. 4.3. Extravasular Administration: Simulations 4.3.. Results Describing Linear Parmacokinetic Elimination Results of te one compartment parmacokinetic model administered by an extravascular mode of administration and following linear elimination are presented ere. Te NSFD sceme 76 is compared to te SFD sceme 78. Te scemes obtained from te model using different metods are tested under similar conditions. Simulations are provided for =.5 and = in Figure. b
Mat. Comput. Appl. 28, 23, 27 8 of 2 9 8 7 6 5 4 3 2 Concentration Profile wit =.5 ODE45 SFD Linear elimination NSFD Linear elimination Analytical 6 4 2 8 6 4 Concentration Profile wit = ODE45 SFD Linear elimination NSFD Linear elimination Analytical 2 2 4 6 8 2 4 6 8 a Figure. NSFD sceme 76, SFD sceme 78 and ODE45 is compared wit exact solution 57. Plasma concentration-time curve for a drug given as a single oral dose wit linear parmacokinetic elimination a =.5 and b =. b Table 2 sows numerical results for te exact finite difference sceme for extravascular administration in comparison wit standard metods Euler, Heun and Runge-Kutta and te analytical solution of te model. Te numerical results for te exact finite difference sceme are te same as te analytical solution for any value of t. Table 2. Te numerical results for te extravascular administration model. Numerical Results Euler Heun Runge-Kutta Exact FD Exact........5.3763.3823.3823.3823.2.27525.254.2554.2554.2554.3.3795.3524.35374.35374.35374.4.46592.4358.4366.4366.4366.5.53723.5432.5597.5597.5597.6.59573.5623.56373.56374.56374.7.64337.6983.654.654.654.8.688.6492.658.658.658.9.7245.682.68275.68275.68275..7365.7686.7842.7842.7842 4.3.2. Results Describing Mixed Drug Elimination Figure 2 sows te simulation results for te model describing extravascular administration along wit mixed drug elimination processes.
Mat. Comput. Appl. 28, 23, 27 9 of 2.9.8 Concentration Profile ODE45 SFD Mixed drug elimination NSFD Mixed drug elimination.7.6.5.4.3.2. 2 4 6 8 Figure 2. Comparison of metods for one-compartment extravascular administration tat follows mixed drug elimination i.e., NSFD sceme 8 and SFD sceme 8 is compared wit ODE45. 5. Conclusions In tis work, we presented one-compartment parmacokinetic models wit different routes of administration. We presented numerical results via a variety of scemes for eac of te developed models, paying attention particularly to te efficiency of te NSFD metod in comparison to standard metods. From te results obtained, we observe tat te stability of te NSFD sceme is independent of te cosen step-size for te linear cases. Tis is not te case wit standard metods suc as Euler and Heun metods. Wit te later metods, te step-size must be cosen in a reasonable domain, oterwise numerical instabilities will occur. Te numerical simulations conducted verify tat NSFD scemes are efficient and accurate for te solution of problems modelling parmacokinetic processes. Importantly, as pointed out troug test cases in tis work, te NSFD metod is able to generate numerical scemes tat are dynamically consistent wit te original differential equations. Acknowledgments: Oluwaseun Egbelowo tanks te Council for Scientific and Industrial Researc CSIR Sout Africa for a bursary and te Deutscer Akademiscer Austauscdienst DAAD for a scolarsip. Oluwaseun Egbelowo is particularly grateful to Caris Harley Wits and Byron Jacobs Wits for considerably elping in improving te manuscript. Conflicts of Interest: Te autor declares no conflict of interest. Abbreviations Te following abbreviations are used in tis manuscript: PK I.V. NSFD SFD Exact FD GIT parmacokinetic intravenous nonstandard finite difference standard finite difference exact finite difference gastrointestinal tract
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