A Comparative Analysis of Numerical Methods for Mathematical Modelling of Intravascular Drug Concentrations Using a Two-Compartment Pharmacokinetic Model

Abstract

Pharmacokinetic modelling is extensively used in understanding drug behavior, distribution and optimizing dosing regimens. This study presents a two-compartment pharmacokinetic model developed using three numerical approaches that includes the Euler method, fourth-order Runge–Kutta method, and Adams–Bashforth–Moulton method. The model incorporates key parameters including elimination, transfer rate constants, and compartment volumes. The numerical approaches are used to simulate the concentration of drug profiles, which are then compared to the exact solution. The results reveal that with an average error of 1.54%, the fourth-order Runge–Kutta technique provides optimized results compared to other methods when the overall average error is taken into account, which shows that the Runge–Kutta method is better in terms of accuracy and consistency for drug concentration estimates in the two-compartment model. This mathematical model may be used to optimize dosing procedures by providing a less complex method. Along with that, it also monitors therapeutic medication levels, which provides accurate analysis for drug distribution and elimination kinetics.

1. Introduction

The role of modern drug delivery systems in clinical pharmacology and personalized medicine is increasing with improvements in health and allied facilities [1,2]. Different mathematical techniques and methods are adopted to determine the optimized solution for solving the differential equations of different systems [3,4]. Specifically, when dealing with infectious diseases, various methods and techniques are tested in order to optimize the drug delivery process [5,6]. Pharmacokinetics is the science of the kinetics of drug absorption and disposition. These studies involve both experimental and theoretical methodologies. The subsequent methodology involves the establishment of pharmacokinetic models that aid in the estimation of drug disposition after administration [7]. Pharmacokinetic modeling plays a vital role in understanding the behavior of drugs in the human body and optimizing dosing regimens for therapeutic efficacy [8]. It provides useful insights into drug kinetics and enables the estimation of drug concentrations at different periods by measuring the mechanisms of drug absorption, distribution, metabolism, excretion, and toxicity (ADMET) [9,10]. These models are designed to apprehend the pharmacokinetics of oral, intravenous, intramuscular, subcutaneous, transdermal, topical, and inhalation routes of drug administration. These models enable easy interpretation of the volume of distribution and drug clearance, which in turn can be substantial in the calculation of the half-life of the drug [11]. This information is important for setting proper dosage levels, dosing intervals, and monitoring therapeutic medication levels in patients [12].

Pharmacokinetic models are divided into three different categories that include compartmental, non-compartmental, and physiological models [13], with the former being an extensively used model for the pharmacokinetic depiction of drugs. Among mammillary, catenary, open, and closed models of the compartmental approach, an open model for the intravascular route is described in this paper. This classification is based on the arrangement of compartments that includes a space where drugs can be distributed and eliminated, which could be in parallel or series. For capturing the dynamic interactions between drug concentrations in different parts of the body in complex pharmacokinetic systems involving numerous compartments, mathematical modeling becomes of vital importance [14]. Various studies and research have been performed in order to devise an efficient mechanism for drug delivery systems. The two-compartment model framework for representing drug distribution between a central compartment, which includes highly perfused organs, and a peripheral compartment, in which the drug distributes slowly, is discussed [15]. This model includes the drug’s transfer between compartments as well as its elimination from the system. Most medications, through an intravenous infusion, enter the systemic circulation and take a certain amount of time to distribute throughout the body. The concentration of drugs in plasma will drop more quickly during the distributive phase than it will during the post-distributive phase, which includes the complex phase, which involves interaction between drugs, tissues, and organs. In the central compartment, the concentration of the drug decreases during the distributive phase more rapidly as compared to the post-distributive phase. On the other hand, in the peripheral compartment, the drug concentration declines exponentially with respect to time [16]. The serum levels of such intravenous infusions could be estimated using this two-compartment model [17].

The pharmacokinetics of sisomicin were investigated in four healthy volunteers following intramuscular and intravenous injection [18]. The compartment model for pharmacokinetics was aligned with the elimination rate of the antibiotic, which is typified by the slow and fast disposition rate constants of 0.072 and 0.004 min⁻¹, correspondingly [19]. For the assessment of plasma concentration after levodopa infusion as per the loading and maintenance dose protocol, the two-compartment model was employed. Blood samples were obtained for the determination of kinetic rates of levodopa as it is transported between two compartments and eliminated from the vascular space [20]. A non-standard finite difference technique was used to address the two models, which are IV infusion and IV bolus injection two-compartment pharmacokinetic models. The authenticity of the nonstandard finite difference (NSFD) method is addressed through its evaluation against other methods. It was found out that this technique provides consistent results in resolving the troubles regarding pharmacokinetic modelling [21]. Another study was conducted on human beings to investigate the ADMET of pentobarbital that was administered by extravascular and intravascular routes by employing the two-compartment open model [22]. The effectiveness of the two-compartment model is proved with the help of two tests conducted in vivo. One of them was performed to compare the different routes of administrations of trobicin, and the other was conducted on five subjects in which d-lysergic acid diethylamide was administered through IV route [23]. With the purpose of investigating the pharmacokinetics of mizolastine, which was the latest H1 antagonist, through intravenous and oral routes of administration, non-compartmental and two-compartmental models were utilized. Later, one with zero-order absorption was proven to be the optimistic approach for the analysis of the fundamental kinetics of the drug [24].

Numerical approaches are essential tools for solving differential equations that characterize pharmacokinetic models and simulating drug concentration profiles [25]. To approximate the solutions of these equations and offer estimates of drug concentrations over time, multiple numerical approaches have been developed. In this study, three widely used numerical approaches, the Euler method [26], fourth-order Runge–Kutta method [27], and Adams–Bashforth–Moulton method [28], were compared. The Euler method is a fundamental and less computational method that approximates the solution by applying the system’s derivative iteratively at each time step. Its precision, however, is restricted by its first-order approximation, which can result in major inaccuracies, especially in complicated pharmacokinetic systems [29].

The fourth-order Runge–Kutta method is a higher-order numerical method that increases accuracy by computing many intermediate stages. It outperforms the Euler method in terms of stability and precision and is commonly used in pharmacokinetic modeling [30].The Adams–Bashforth–Moulton method associates an explicit Adams–Bashforth predictor with an implicit Adams–Moulton corrector. It balances accuracy and computational efficiency and is suitable for modeling drug concentrations in complex systems [31]. In this study, we analyzed and examined the performance of numerical approaches, the Euler method, fourth-order Runge–Kutta method, and Adams–Bashforth–Moulton method, in modeling drug concentration profiles in a two-compartment pharmacokinetic system. To verify the models’ applicability to practical situations, key factors such as elimination and transfer rate constants, as well as compartment volumes, were considered [32]. The fractional differential equation model was also utilized, which accounts for local dependencies. The Caputo derivative was used to define the dynamics of the system in the fractional differential equation-based two-compartment model [33]. In other research, different methods of using the fractional differential equation related to a two-compartment model are discussed [34].

This study was conducted to evaluate the accuracy, precision, and computing efficiency of different numerical approaches by analyzing and comparing the numerical results produced using the Euler method, fourth-order Runge–Kutta method, and Adams–Bashforth–Moulton method, with a precise solution. The results of this research study will provide sensitive information that will assist in the selection of appropriate numerical and mathematical approaches for pharmacokinetic modeling. Furthermore, the findings will help to optimize dosing techniques and improve drug-concentration prediction in two-compartment systems [35].

2. Materials and Methods

2.1. Two Compartmental Pharmacokinetic Model

The two-compartment pharmacokinetic model employed in this study is a mathematical representation of how a drug operates and is delivered in the human body. The model assumes a single intravenous (IV) bolus administration, with the drug introduced directly into the central compartment at time t = 0. To effectively describe the behavior of the drugs, the model includes the following parameters. The elimination rate constants k₁₀ and k₁₂ represent the rates at which the drug is eliminated from the central compartment and transported from the central to the peripheral compartment, respectively. The rate at which the drug moves from the peripheral to the central compartment is represented by the transfer rate constant (k₂₁). The unit of all the rate constants is per hour (1/h). Furthermore, compartment volumes V₁ and V₂ are provided to account for distribution volume (in liters) in the central and peripheral compartments, respectively. The amount of drug in each compartment is represented in Equations (1) and (2) [36],

$$A_c\left(t\right)=C_c\left(t\right)\cdot V_1$$

(1)

$$A_p\left(t\right)=C_p\left(t\right)\cdot V_2$$

(2)

The system of ordinary differential equations (ODEs) for the amount of drug in each compartment is given by Equations (3) and (4) [37],

$${\displaystyle \frac{d{A}_c}{dt}}=-\left(k_{12}+k_{10}\right)A_c+k_{21}{A}_p$$

(3)

$${\displaystyle \frac{d{A}_p}{dt}}=k_{12}{A}_c{-k}_{21}{A}_p$$

(4)

Alternatively, these equations can be expressed in terms of concentrations by dividing the amount of drug by its compartment volume. A system of ordinary differential equations determines the drug concentrations in each compartment, indicated as C_c and C_p for the central and peripheral compartments (in mg/L), respectively. Equations (5) and (6), as follow, illustrate how drug concentrations fluctuate over time in the human body [38]:

$${\displaystyle \frac{d{C}_c}{dt}}=-\left(k_{12}+k_{10}\right)C_c+k_{21}{C}_p$$

(5)

$${\displaystyle \frac{d{C}_p}{dt}}=k_{12}{C}_c{-k}_{21}{C}_p$$

(6)

These equations depict the rates of change in drug concentration in the central and peripheral compartments. The elimination, transfer, and distribution processes are taken into consideration.

In Equation (5), ${\displaystyle \frac{d{C}_c}{dt}}$ is the rate of change in drug concentration in the central compartment and is determined by three terms. The first term, $-k_{12}{C}_c$, denotes the drug transfer from the central to the peripheral compartment. The second term, $-k_{10}{C}_c$, represents the elimination process from the central compartment, where k₁₀ represents the elimination rate constant. The negative sign specifies that the drug concentration falls over time. The third term, $k_{21}{C}_p$, signifies the drug transfer from the peripheral to the central compartment and also depicts an increase in drug concentration in the central compartment.

In Equation (6), ${\displaystyle \frac{d{C}_p}{dt}}$ defines the rate of change in drug concentration in the peripheral compartment and also emphasizes the dynamics of the peripheral compartment. The first term, $k_{12}{C}_c$, represents the drug transfer from the central to the peripheral compartment, and the second term, ${-k}_{21}{C}_p$, represents the drug transfer from the peripheral to the central compartment.

These equations, together, capture the relationship between drug elimination, inter-compartmental transfer, and the resulting changes in drug concentration over time in both central and peripheral compartments. Differential equations, solved by the Euler method, fourth-order Runge–Kutta method, and Adams–Bashforth–Moulton method, are used to simulate the drug concentration and analyze the drug behavior in a two-compartment pharmacokinetic system. This provides a better understanding of the dynamics of drug distribution and elimination, and how medications are dispersed throughout the body and eliminated over time.

2.2. Numerical Methods

To simulate drug-concentration profiles and solve systems of ordinary differential equations of two-compartment models, three numerical methods were used, including the Euler method, fourth-order Runge–Kutta method, and Adams–Bashforth–Moulton method. These methods are widely utilized in finding numerical solutions for initial value problems due to their reliability and computational efficiency. Especially, the Euler method offers a simple first-order approach, the Runge–Kutta method provides higher accuracy through intermediate slope evaluations, and the Adams–Bashforth–Moulton method balances efficiency and precision in multistep integration processes.

2.2.1. Euler Method

The Euler method is a simple numerical method that is used to approximate the solution of ordinary differential equations. In this method, time interval is divided into small steps and approximating the derivative of each step based on the slope of the tangent line. The concentration of the drug at each time step is evaluated using Equations (7) and (8):

$$C_c\left(t+∆t\right)=C_c\left(t\right)+∆t\left(-\left(k_{12}+k_{10}\right)C_c+k_{21}{C}_p\right)$$

(7)

$$C_p\left(t+∆t\right)=C_p\left(t\right)+∆t\left(k_{12}{C}_c{-k}_{21}{C}_p\right)$$

(8)

2.2.2. Fourth-Order Runge–Kutta Method

The fourth-order Runge–Kutta method is a numerical approach for solving ordinary differential equations accurately. It gives a more accurate approximation than the Euler method. This method computes multiple intermediate values to estimate the derivative at various points in the time step. The fourth-order Runge–Kutta method is used in Equations (9)–(12), as follow, to calculate the drug concentration in the central compartment at each time step.

$$m_1=∆t\left(-\left(k_{12}+k_{10}\right)C_c+k_{21}{C}_p\right)$$

(9)

$$m_2=∆t\left(-\left(k_{12}+k_{10}\right)\left(C_c+{\displaystyle \frac{k_1}{2}}\right)+k_{21}\left(C_p+{\displaystyle \frac{k_1}{2}}\right)\right)$$

(10)

$$m_3=∆t\left(-\left(k_{12}+k_{10}\right)\left(C_c+{\displaystyle \frac{k_2}{2}}\right)+k_{21}\left(C_p+{\displaystyle \frac{k_2}{2}}\right)\right)$$

(11)

$$m_4=∆t\left(-\left(k_{12}+k_{10}\right)\left(C_c+k_3\right)+k_{21}\left(C_p+k_3\right)\right)$$

(12)

The drug concentration in the central compartment can be simulated as in Equation (13),

$$C_c\left(t+∆t\right)=C_c\left(t\right)+\left({\displaystyle \frac{\left(m_1+2m_2+2m_3+m_4\right)}{6}}\right)$$

(13)

For the peripheral compartment, Equations (14)–(18) can be used to evaluate the drug concentration,

$$l_1=∆t\left(k_{12}{C}_c-k_{21}{C}_p\right)$$

(14)

$$l_2=∆t\left(k_{12}\left(C_c+{\displaystyle \frac{l_1}{2}}\right)-k_{21}\left(C_p+{\displaystyle \frac{l_1}{2}}\right)\right)$$

(15)

$$l_3=∆t\left(k_{12}\left(C_c+{\displaystyle \frac{l_2}{2}}\right)-k_{21}\left(C_p+{\displaystyle \frac{l_2}{2}}\right)\right)$$

(16)

$$l_4=∆t\left(k_{12}\left(C_c+l_3\right)-k_{21}\left(C_p+l_3\right)\right)$$

(17)

The drug concentration in the peripheral compartment can be simulated as,

$$C_p\left(t+∆t\right)=C_p\left(t\right)+\left({\displaystyle \frac{\left(l_1+2l_2+2l_3+l_4\right)}{6}}\right)$$

(18)

2.2.3. Adams–Bashforth–Moulton Method

Another numerical technique for approximating the solution of ordinary differential equations is Adams–Bashforth–Moulton method. This is a prediction correction method that combines the Adams–Bashforth prediction method and the Adams–Moulton correction method. The following procedures based on predictor and corrector steps are used to compute the drug concentration at each time step. The predictor formula approaches a solution at the next time step based on the current and previous time steps. The general form of the predictor formula equations is depicted as Equations (19) and (20):

$$C_{c,pred}=C_{c,i-1}+h\left({\displaystyle \frac{55f_{1,i-1}-59f_{1,i-2}+37f_{1,i-3}-9f_{1,i-4}}{24}}\right)$$

(19)

$$C_{p,pred}=C_{p,i-1}+h\left({\displaystyle \frac{55f_{2,i-1}-59f_{2,i-2}+37f_{2,i-3}-9f_{2,i-4}}{24}}\right)$$

(20)

where $C_{c,i-1}\mathrm{a}\mathrm{n}\mathrm{d}{C}_{p,i-1}$ represent drug concentration in the central and peripheral compartments at the previous time step, and ${f_{1,i-1},f}_{1,i-2},f_{1,i-3},f_{1,i-4},{f_{2,i-1},f}_{2,i-2},f_{2,i-3},\mathrm{a}\mathrm{n}\mathrm{d}{f}_{2,i-4}$ are the corresponding derivative values.

The corrector step is then performed to improve the estimations, as depicted in Equations (21) and (22),

$$C_{c,i}=C_{c,i-1}+h\left({\displaystyle \frac{9f_{1,pred}+19f_{1,i-1}-5f_{1,i-2}+f_{1,i-3}}{24}}\right)$$

(21)

$$C_{p,i}=C_{p,i-1}+h\left({\displaystyle \frac{9f_{2,pred}+19f_{2,i-1}-5f_{2,i-2}+f_{2,i-3}}{24}}\right)$$

(22)

where $C_{c,i}\mathrm{a}\mathrm{n}\mathrm{d}{C}_{p,i}$ are the updated concentrations at the current time step; $f_{1,pred}$ and $f_{2,pred}$ are the predicted derivative values obtained at the predictor step.

2.3. Analytical Approach

The analytical solution of the two-compartment pharmacokinetic model provides a mathematical framework to describe the distribution and elimination of a drug within the body. This model considers the central compartment and the peripheral compartment, with drug transfer and elimination governed by first-order kinetics. The dynamics of these concentrations over time are represented by the system of Equations (1) and (2). To solve this system analytically, the system is written in matrix vector form:

$${\displaystyle \frac{dC}{dt}}=A·C\left(t\right)$$

where $C\left(t\right)=\left[\begin{array}{c}{C}_c\left(t\right)\\ C_p\left(t\right)\end{array}\right]$ and $A=\left[\begin{array}{cc}-\left(k_{12}+k_{10}\right)& k_{21}\\ k_{12}& -k_{21}\end{array}\right]$.

The eigenvalues of matrix A are determined by solving the characteristic equation:

$$\det \left(A-\mathsf{\lambda }\mathrm{I}\right)=0$$

This leads to a quadratic equation in λ, and solving it gives two eigenvalues. These values are then used to form the general solution, which includes exponential terms based on how the system behaves over time. The direction and shape of the solution are guided by the corresponding eigenvectors. To get the final expressions, initial values of the drug concentrations are used, which are explained in the Section 3.

3. Results

To determine the analytical solution, consider the constant of the rate of drug transfer from the central compartment to the peripheral compartment as $0.5/\mathrm{h}$, and the elimination rate constant from the central compartment as $0.05/\mathrm{h}$. The transfer rate constant at which the drug moves from the peripheral compartment to the central compartment is $0.25/\mathrm{h}$. The initial condition of the system is considered as $C_c\left(0\right)=0,C_p\left(0\right)=200.$ Then the system of equation will be represented as,

$${\displaystyle \frac{d{C}_c}{dt}}=-0.55C_c+0.25C_p$$

(23)

$${\displaystyle \frac{d{C}_p}{dt}}=0.5C_c-0.25C_p$$

(24)

The matrix form of the system is:

$$C^{\prime }=AC$$

$$C^{\prime }=\left[\begin{array}{c}{\displaystyle \frac{d{C}_c}{dt}}\\ {\displaystyle \frac{d{C}_p}{dt}}\end{array}\right],A=\left[\begin{array}{cc}-0.55& 0.25\\ 0.5& -0.25\end{array}\right],C=\left[\begin{array}{c}{C}_c\\ C_p\end{array}\right]$$

The eigenvalues of matrix A can be determined by solving the equation $\det \left(A-\mathsf{\lambda }\mathrm{I}\right)=0$, which leads to a quadratic equation.

$${\mathsf{\lambda }}^2+0.8\mathsf{\lambda }+0.0125=0$$

The eigenvectors for each eigenvalue ${\mathsf{\lambda }}_1=-0.0159$ and ${\mathsf{\lambda }}_2=-0.78405$ are $\left[\begin{array}{c}0.4681\\ 1\end{array}\right]$ and $\left[\begin{array}{c}-1.0681\\ 1\end{array}\right]$, respectively.

The general solution of the given system is:

$$C\left(t\right)=a{e}^{-0.0159t}\left[\begin{array}{c}0.4681\\ 1\end{array}\right]+b{e}^{-0.78405t}\left[\begin{array}{c}-1.0681\\ 1\end{array}\right]$$

(25)

To find out the particular solution, it is necessary to determine the values of the constants ‘a’ and ‘b’ by applying the initial conditions,

$$\left[\begin{array}{c}0\\ 200\end{array}\right]=a\left[\begin{array}{c}0.4681\\ 1\end{array}\right]+b\left[\begin{array}{c}-1.0681\\ 1\end{array}\right]$$

(26)

The following two equations are obtained,

$$0.4681a-1.0681b=0$$

$$a+b=200$$

The particular solution of the given system is:

$$C\left(t\right)=139.05e^{-0.0159t}\left[\begin{array}{c}0.4681\\ 1\end{array}\right]+60.94e^{-0.78405t}\left[\begin{array}{c}-1.0681\\ 1\end{array}\right]$$

(27)

$$C\left(t\right)=e^{-0.0159t}\left[\begin{array}{c}65.089\\ 139.05\end{array}\right]+e^{-0.78405t}\left[\begin{array}{c}-65.09\\ 60.94\end{array}\right]$$

(28)

Separately, the particular solution can be written as:

$$C_c=65.089e^{-0.0159t}-65.09e^{-0.78405t}$$

(29)

$$C_p=139.05e^{-0.0159t}-60.94e^{-0.78405t}$$

(30)

To simulate drug-concentration profiles in the two-compartment pharmacokinetic model, three numerical methods (Euler, fourth-order Runge–Kutta, and Adams–Bashforth–Moulton) were used in computational software. To test the performance of each method, the simulated results were compared with the accurate solution.

The simulation results show that in the two-compartment model, the fourth-order Runge–Kutta approach consistently provided accurate estimates for drug concentrations in both compartments, as depicted in

Table 1. Drug concentrations in the central compartment.

Time	Euler’s Method	Fourth-Order Runge–Kutta Method	Adams–Bashforth–Moulton Method	Analytical Solution
0	0.0000	0.0000	0.0000	−0.0013
1	35.2940	34.3460	34.5668	34.3452
2	50.3344	49.4835	49.6841	49.4845
3	56.4320	55.8598	55.9964	55.8631
4	58.5859	58.2448	58.3272	58.2505
5	59.0053	58.8158	58.8621	58.8239
6	58.6666	58.5668	58.5915	58.5773
7	58.0010	57.9515	57.9639	57.9643
8	57.1992	57.1770	57.1826	57.1920
9	56.3452	56.3375	56.3395	56.3547
10	55.4760	55.4760	55.4761	55.4953

and

Table 2. Drug concentrations in the peripheral compartment.

Time	Euler’s Method	Fourth-Order Runge–Kutta Method	Adams–Bashforth–Moulton Method	Analytical Solution
0	200.0000	200.0000	200.0000	199.9900
1	163.7910	164.6810	164.4737	164.6789
2	146.5941	147.3956	147.2066	147.4002
3	137.8190	138.3619	138.2323	138.3723
4	132.7846	133.1135	133.0340	133.1295
5	129.4219	129.6111	129.5648	129.6323
6	126.8160	126.9233	126.8967	126.9496
7	124.5624	124.6247	124.6091	124.6559
8	122.4818	122.5206	122.5108	122.5566
9	120.4950	120.5221	120.5152	120.5626
10	118.5664	118.5882	118.5827	118.6331

. The drug concentration profiles acquired using this approach were identical to the profiles obtained from the analytical solution, exhibiting minimal deviation. This shows that the fourth-order Runge–Kutta approach can accurately capture the dynamics of drug distribution and elimination in the two-compartment system, as depicted in Figure 1.

In Figure 1, the Euler method shows significant deviations from the exact solution, especially in the scenario with rapidly changing drug concentrations. This method was less accurate and stable compared to the fourth-order Runge–Kutta method. The greater accuracy and simplicity of the Euler method come with increasing errors and limitations in handling complex reaction kinetics and rapid changes in drug concentration. A predictor correction method, the Adams–Bashforth–Moulton method, showed intermediate performance between the Euler method and fourth-order Runge–Kutta method. It gave more accurate results than the Euler method, but still had more deviations from the exact solution than the fourth-order Runge–Kutta method.

Error Analysis

Three numerical approaches were used to determine the errors for drug concentration in the central compartment, including Euler, fourth-order Runge–Kutta, and Adams Basforth Moulton. The errors represent the difference in drug concentrations predicted by each approach, and the exact solution.

Table 3. Error of drug concentrations in the central compartment.

Time	Euler’s Method	Fourth-Order Runge–Kutta Method	Adams–Bashforth–Moulton Method
0	0.0100	0.0100	0.0100
1	0.8879	0.0021	0.2052
2	0.8061	0.0046	0.1936
3	0.5533	0.0104	0.1400
4	0.3449	0.0160	0.0955
5	0.2104	0.0212	0.0675
6	0.1336	0.0263	0.0529
7	0.0935	0.0312	0.0468
8	0.0748	0.0360	0.0458
9	0.0676	0.0405	0.0474
10	0.0667	0.0449	0.0504

illustrates the absolute errors for drug concentration in the central compartment at various time intervals. All three approaches demonstrate some degree of error, with the fourth-order Runge–Kutta method having the lowest error when compared to the Euler method and Adams–Basforth–Moulton method. This suggests that the fourth-order Runge–Kutta technique predicts drug concentrations more accurately in the central compartment. As the simulation progresses, the error decreases, showing that the numerical approaches converge to the exact solution over time. However, at the end of the process, there are some errors and deviation in all approaches, as shown in Figure 2.

Table 4. Error of drug concentrations in the peripheral compartment.

Time	Euler’s Method	Fourth-Order Runge–Kutta Method	Adams–Bashforth–Moulton Method
0	0.0013	0.0013	0.0013
1	0.9488	0.0008	0.2216
2	0.8499	0.0010	0.1996
3	0.5689	0.0033	0.1333
4	0.3354	0.0057	0.0767
5	0.1814	0.0081	0.0382
6	0.0893	0.0105	0.0142
7	0.0367	0.0128	0.0004
8	0.0072	0.0150	0.0094
9	0.0095	0.0172	0.0152
10	0.0193	0.0193	0.0192

shows the absolute errors for drug concentration in the peripheral compartment at various time points. In terms of the best approach, a comparison of errors shows that the fourth-order Runge–Kutta method consistently produces the lowest errors among the three ways for calculating drug concentration. As a result, the fourth-order Runge–Kutta method is the most reliable and accurate solution for simulating drug concentrations and optimizing dosing techniques in this specific pharmacokinetic system, not only for the drug concentration in the central compartment but also for the drug concentration in the peripheral compartment. The errors tend to decrease as time increases, indicating convergence towards the exact solution, as shown in Figure 3.

Table 5. Overall average of errors of drug concentrations in both compartments.

Method	Central Compartment	Peripheral Compartment	Overall Average	Overall Average (%)
Euler Method	0.2771	0.2953	0.2862	28.62
Fourth-Order Runge–Kutta Method	0.0086	0.0221	0.0154	1.54
Adams–Bashforth–Moulton Method	0.0663	0.0868	0.0766	7.66

shows the average errors for the central and peripheral compartments, as well as the overall average error, calculated using three different numerical methods that include the Euler method, fourth-order Runge–Kutta method, and Adams–Bashforth–Moulton method. With an average error of 0.0154, the fourth-order Runge–Kutta method shows better results than the other methods when the overall average error is taken into account. The total average error for the Adams–Bashforth–Moulton method is 0.0766, while the overall average error for the Euler method is 0.2862.

Based on these findings, the fourth-order Runge–Kutta method delivers the most correct and reliable predictions for drug concentrations in both compartments and also in the two-compartment pharmacokinetic system. The performance comparison with reference to error between these analytical methods and exact solution are depicted in Figure 4.

4. Discussion

The two-compartment pharmacokinetic model was used to evaluate the operation of drugs in the body. To estimate the process of drug function, different parameters were used in the model. The rate at which the drug was eliminated from the central compartment was k₁₀ = 0.05 per hour, while the rate at which the drug was transported from the central to the peripheral compartment was k₁₂ = 0.5 per hour. The transfer rate constant, k₂₁ = 0.25 per hour, is the rate at which the drug transfers from the peripheral to the central compartment. The initial condition of the drug concentration in the central compartment at t = 0 is 0, and the drug concentration in the peripheral compartment at t = 0 is 200. By taking all these values of the different parameters, the differential equations of the model are solved analytically to get an exact solution.

Several studies have been conducted in order to determine the approximation results of the mathematical model of the drug concentration in the two-compartment model. In this study, three different numerical methods are compared with the exact solution of the two-compartment model. The simplest method to get the approximate solution of the ordinary differential equations is the Euler method. In this method, the concentration of the drug in the central compartment and peripheral compartment is calculated at each time step of 0.1. The results obtained from this method are highly deviated from the exact solution.

Another approximation method used to estimate the drug concentration is fourth-order Runge–Kutta method. This method calculates multiple intermediate values to approximate the drug concentration at each time step. For a single iteration, intermediate values have to be calculated, making it a lengthy process. But the results obtained from this method are very close to the exact solution for both compartments. One more approximation method is the Adams–Bashforth–Moulton method, which is used to calculate drug concentration in the central compartment and peripheral compartment. This method consists of two steps, predictor and corrector steps, that are based on the current and previous time steps. The values obtained from the predictor step are used in the corrector step to approximate the drug concentration of each iteration. This method gives good approximation results but still has slightly more deviations from the exact solution.

Our comparative examination of the traditional numerical methods adds to the existing researches in sophisticated modelling techniques, which include fractional-order compartmental models employing the Grünwald–Letnikov scheme to capture anomalous diffusion behavior more effectively [39]. Ref. [34] also shows that fractional-order compartmental models can better describe anomalous pharmacokinetic behavior, though with added complexity due to mass balance difficulties [34]. Recent research on nonstandard finite difference schemes shows that they are effective in solving one-compartment pharmacokinetic models for a variety of delivery methods, including IV bolus and extravascular dosage [40]. These findings emphasize the need to use robust numerical approaches for pharmacokinetic modelling.

After comparison of these three approximation methods with the exact solution, the fourth-order Runge–Kutta method showed better approximation than the other two methods. The drug concentration values in both compartments that were obtained from the fourth-order Runge–Kutta method were more accurate and stable and very close to the exact solution. While each numerical method evaluated in this study offers advantages, they also come with specific limitations. The Euler method, though simple and computationally efficient, demonstrates lower accuracy, making it less suitable for scenarios requiring precise drug concentration predictions. The fourth-order Runge–Kutta method provides high accuracy but at the cost of increased computational load, which may not be ideal for real-time applications. The Adams–Bashforth–Moulton method, requiring multiple previous values to start and maintain stability, can be limited in certain practical implementations. These understandings support practitioners in selecting the most appropriate method for clinical modeling tools, depending on the required balance between accuracy, computational resources, and implementation complexity.

From a pharmacological perspective, the accurate prediction of drug concentration is essential in maintaining the therapeutic level. The findings suggest that using higher-order numerical methods can enhance the reliability of pharmacokinetic simulations, and higher-order numerical methods are particularly useful when exact solutions are not available. This supports better dose optimization and safer treatment planning in clinical and pharmaceutical research.

5. Conclusions

This study examined three numerical methods systems, including the Euler method, fourth-order Runge–Kutta method, and Adams–Bashforth–Moulton method, to model drug concentration in a two-compartment model. The fourth-order Runge–Kutta method produced the highest accurate estimations, closely approaching the precise answer. The Euler method and Adams–Bashforth–Moulton method also deliver accurate results but have greater error. The results reveal that with an average error of 1.54%, the fourth-order Runge–Kutta method provides more optimized results than the other methods. When the overall average error is taken into account, it shows that the Runge–Kutta method is better in terms of accuracy and consistency for drug concentration estimates in the two-compartment model. The findings demonstrate the need to select a suitable numerical method based on the accuracy and computational efficiency. For enhanced modeling accuracy, future studies can investigate advanced techniques and incorporate experimental data. Overall, this study contributes to the understanding of drug concentration dynamics and assists in the selection of appropriate numerical methods for pharmacokinetic studies and drug development.