The concentration of digoxin after intravenous and oral administration studied by a two-compartment model

The present paper is designed to compare the distribution of digoxin administered through an Intravenous (i.v.) and Oral (p.o.) in central and peripheral compartments. The model under consideration is denoted by a systemof three non-linear ordinary differential equations. The Eigenvalue and the Laplace transformmethods were used to solve the system of equations. In this model, digoxin was administered to three subjects through i.v. and three subjects through p.o. then, the serum concentrations were measured over a period of 6 h. The transfer coefficients were obtained from digoxin concentrations using themethodof residuals and the variationof digoxin concentration–time curves plotted using MATLAB. In this model, we consider excretion is only from the peripheral compartment. ARTICLE HISTORY Received 4 October 2018 Accepted 17 June 2019


Introduction
Explanation of the simplest pharmacokinetic model in accordance with data observed for human drug adaptation is a necessary step for a full understanding of distribution, elimination, activity and drug toxicity. This is becoming increasingly important with drugs that have a low therapeutic index and serious side effects such as the cardiac glycoside digoxin. In view of its narrow therapeutic window, it is important to study the drug concentration in men at various times after oral (p.o.) and intravenous (i.v.) administration. Doherty, Flanigan, and Dalrymple (1972) distinguish the pharmacokinetics of digoxin after oral administration in seven normal patients of tritium-labelled digoxin. The mean half-life from digoxin concentration was 32.7 h., while urinary elimination rates capitulated the mean half-life of 35.6 h., approximately similar values of creatinine and digoxin renal clearances were identified. Gitfrich, Bodem, and Chidsey (1974) inspect the urinary excretion of tritiated digoxin after administering intravenously and reported an average half-life value of 50.8 h. in six chronic disease patients with normal renal function. They found that the renal clearance of digoxin slightly exceeded the creatinine clearances. The steady-state volume of distribution of digoxin in normal subjects has been calculated by Reuning, Sams, and Notari (1973) from digoxin concentration data reported in three studies. J. J Vallner, Stewart, Kotzan, Kirsten, and Honigberg (1981) examined the elimination half-life of hydromorphone in six subjects was 2.64 ± 0.88 h and the drug distributed in a high volume of 1.22 l/kg. Also, the drug was very quickly but partially absorbed after oral administration. Kannan, Singh, and Przekwas (2018) has shown using his Q3D-compartment multiscale models, the effect of the solubility on the regional presence of the ICS and its absorption across the different lung tissues to the vein/artery. In particular, it was found that highly soluble drugs like Budesonide (solubility coefficient ∼ 45 μg/ml) dissolves very fast, then diffuses and then is cleared to the GUT region. On the other hand, poorly soluble drugs like FP (solubility coefficient ∼ 0.5-1.5 μg/ml) being nearly insoluble, are present in chunks even after 1-2 h after the inhalation, and thus there is hardly any diffusion along the lung axis. Consequently, we see a secondary spike in the vein concentration at around 1.25 h time-frame, when the FP chunk (and the dissolved FP molecular drug, sticking to it) entering the GUT and to the vein. Similarly, the deposition of the ICS drug on the airway walls is equally important. For instance, Kannan et al. (2017), (Kannan, Guo, & Przekwas, 2016) provides the factors affecting the deposition on the airways: (i) drug density, (ii) the inhaler geometry, (iii) the inhaled flow rate, (iv) the vorticity in the airways (causes increased deposition for smaller particles), (v) the cartilaginous ridges in the trachea and the generation-1 airways (causes increased deposition for smaller particles) and (vi) the enhanced deposition at the airway bifurcations, thereby restricting the drug to reach the lower airways and the alveoli. In cases like the FP/Budesonide drug, a significant portion of the inhaled corticosteroids (ICS) (from the dry powder inhalers) go and deposit on the alveoli (due to the factors mentioned earlier). All of the above contribute to the efficacy of the inhaled drug and the clearance effects (almost none in the alveoli).
In this present study, we formulated two-compartmental model as shown in Figure 1 which is denoted by a system of three non-linear ordinary differential equations (ODEs). The Eigenvalue and the Laplace transform methods were used to solve the system of equations. In this model, digoxin was administered to three subjects through i.v. and three subjects through p.o. then, the serum concentrations were measured respectively over a period of 6 h. The transfer coefficients were obtained from digoxin concentrations using the method of residuals and the variation of digoxin concentration-time curves plotted using MATLAB In detail, we compare the distribution of digoxin in men administered   i.v.). In this study, we considered digoxin as a drug and method of residuals used for identifying the rate constants from the clinical data. Digoxin: Digoxin is a cardiac glycoside indicated for the treatment of mild-to-moderate heart failure. Most frequently, it is used for atrial fibrillation, atrial flutter and heart failure. Digoxin is taken by mouth or by injection into a vein. Digoxin is extracted from the leaves of Digitalis lanata. The term 'digitalis' is used to designate the whole group of glycosides. The glycosides are composed of 2 portions: a sugar and a cardenolide. Digoxin is described chemically as (3β, 5β, 12β)-3-[O-2, 6-dideoxy-β-D-ribohexopyranosyl-(1→4)-O-2, 6-dideoxy-β-D-ribo-hexopyranosyl-(1→4)-2, 6-dideoxy-β-D-ribo-hexopyranosyl) oxy]-12, 14-dihydroxy-card-20(22)-enolide. Its molecular formula is C 41 H 64 O 14 , its molecular weight is 780.95, and its structural formula is:

Method of Residuals:
The method of residuals is a commonly employed technique for resolving a curve into various exponential terms. This method allows the separation of the mono-exponential constituents of a bi-exponential plot of plasma concentration against time and therefore, it is a useful procedure for fitting a curve to the experimental data of a drug.
As mentioned in this paper, a bi-exponential equation is derived, which is in the form of Here the constants λ 1 and λ 2 are rate constants for the distribution phase and elimination phase, respectively. The constants A and B are intercepts on the y-axis for each exponential segment of the curve in the above equation. Clearly, for the data given the decline in the initial distribution phase is more rapid than the elimination phase. The rapid distribution phase is confirmed with the constant λ 1 being larger than the rate constants λ 2 . Therefor at a time following the attainment of distribution equilibrium, the term Ae −λ 1 t will approach to zero, while Be −λ 2 t will still have a finite value. At this later time, the above equation will reduce to Here the rate constant can be obtained from the slope (−(λ 2 /2.3)) of a straight line representing the terminal exponential phase or λ 2 phase is extrapolated to the y-axis; the y intercept is equal to B. values from the extrapolated line are then subtracted from the original experimental data points and a straight line is obtained. This line represents the rapidly distributed λ 1 phase or the residual plasma concentration against time. A number of pharmacokinetic parameters derived by using rate constants λ 1 and λ 2 and y intercepts A and B.

Assumptions and model building
A two-compartment model consists of two physiological meaningful parts (Kwon, 2001). The first central compartment x 1 is identified with the blood and organs were heavily supplied with blood like liver or kidney. The second peripheral compartment x 2 describes a tissue or the part of the body which is not heavily supplied with blood such as skin and fat tissue. Only, in the case of p. o. administration x 3 describes the absorption compartment in which the absorption takes place. The central and peripheral compartments are connected with each other in both directions and therefore, a distribution between the central and peripheral compartment takes place. The main assumption in pharmacokinetics is that the drug is completely eliminated from the body through the central compartment. In most cases, the metabolism takes place in the liver and the excretion via the kidneys.
We consider two different types of drug administration, first, the drug is directly administered by intravenous bolus injection (i.v.) into the blood. It is assumed that the drug is immediately completely distributed in the blood. Second, the drug is orally (p. o.) administered by a tablet. Hence, absorption through the stomach takes place. Therefore, the distribution is not immediate further, only a part of the amount of drug will reach the blood circulation called bio-availability. A schematic overview of the two-compartment model is presented in Figure 1.
To shorten the notations, we consider i. v. and p. o. administration at once. The general form of a two-compartment model describing either i. v. or p. o. drug administration is given by the set of ODEs as follows: Equation (1) describes the central compartment, Equation (2) the peripheral compartment and Equation (3) the absorption in the case of p. o. administration. Note that the third absorption compartment for p. o. does not count for the nomenclature of the model. Table 3 describes the parameters and variables used in this model. Also, Table 4 describes the particular setting for Equations (1)-(3) to present either p.o. or i.v administration.

Case I: Oral administration (p. o.)
In matrix notation of (1)-(3), we have  the digoxin concentration in the central compartment at a time t x 2 (t) the digoxin concentration in the peripheral compartment at a time t x 3 (t) the digoxin concentration in the absorption compartment at a time t 0 < f ≤ 1 Fraction parameter regulating the amount of drug which effectively reaches the blood in case of p. o. administration. We set f = 1 without loss of generality for our mathematical consideration.
The Laplace transform (Doetsch 1976;Widder, 1966) is an integral transform where the linear operator L{f (t)} transform a function f (t) with t ∈ R from the time domain to a function F(s) with s ∈ C in an image domain. The advantage of this transformation is that differentiation and integration in the time domain correspond to simple algebraic operations in the image domain.

Case II: Intravenous administration (i. v.)
Since the third compartment does not exist for the intravenous administration, the matrix notation of Equations (1) and (2) from Equation (4), we have.
Making use of inverse Laplace Transform, Applying Heaviside's Theorem to Equations (19) and (20) to obtain the solution

Results -oral administration
Once the pharmacokinetic parameters in Table 5 are determined using the method of residuals for an individual, the amount of drug remaining in the central and peripheral  compartments is calculated using Equations (12) and (13). The pharmacokinetic data for digoxin are calculated in Subject 1, Subject 2 and Subject 3. Simulated and actual digoxin concentrations in the central compartment of Subject 1, Subject 2 and Subject 3 are tabulated in Table 6 and plotted for Subject 1 in Figure 2   found in all the three subjects and the same is illustrated in Figure 2(a). and Figure 3(a). In Subject 1 it takes 0.2 h to reach the high concentration of 9.3435 ng/ml whereas, in Subject 2 and Subject 3 it takes 0.8 h to reach the high concentration of 2.7466 ng/ml and 1.8467 ng/ml, respectively. However, within 1.5 h the digoxin concentration falls to 1.8801 ng/ml and 1.3948 ng/ml for Subject 1 and Subject 2, respectively and also within 2.1 h the digoxin concentration falls to 1.5208 ng/ml for Subject 3. The rate of disintegration of digoxin is very slow after 1.5 hours in both Subject 1 and Subject 2, whereas in Subject 3 it is slow down after 2.1 h. It takes almost 6-7 h to reduce from 1.8801 ng/ml to 0.5645 ng/ml in Subject 1 and 1.3948 ng/ml to 0.2921 ng/ml in Subject 2 also, in Subject 3 it reduces from 1.5208 to 0.5701 ng/ml. In the peripheral compartment of three subjects, the digoxin concentration quickly raises in the span of 1-1.5 h, then slowly decreases in the next 4.5 h. More clearly, the digoxin concentration of Subject 1 in the peripheral compartment (Figure 2(b)) increases from 1.8881 to 11.6384 ng/ml in the span of 1.5 h and takes 4.8 h to reduce 7.1440 ng/ml. In Subject 2, the dioxin concentration in the peripheral

Results -intravenous administration
The pharmacokinetic parameters in Table 7 are determined using the method of residuals for an individual, the amount of drug remaining in the central and peripheral compartments is calculated using Equations (20) and (21). The pharmacokinetic data for  digoxin are calculated in subjects C.J. and T.F. Simulated and actual digoxin concentrations in the central compartment of subjects C.J. and T.F are given in Table 8 and plotted for C. J. in Figure 6(a) (Central Compartment), Figure 6(b) (Peripheral Compartment) and Figure 9(a) (Actual and Simulated data in Central Compartment), also plotted for T.F. in Figure 7(a) (Central Compartment), Figure 7(b) (Peripheral Compartment) and Figure 9(b) (Actual and Simulated data in Central Compartment). When an intravenous dose of digoxin is administered, the digoxin enters directly into the central compartment.
It can be seen that digoxin concentration of the central compartment declines rapidly during the initial distribution phase, while digoxin amount in the peripheral compartment equilibrates more slowly in C.J., T.F. and W.S. The digoxin concentration in the central compartment of C.J. (Figure 6(a)), T.F. (Figure 7(a)) and W.S. (Figure 8(a)) decreases rapidly within 1 h from 22.58to 8.26 ng/ml, 26.50to 6.88 ng/ml and 21.56 to 5.48 ng/ml, respectively. In the next 6 hours, the concentration of digoxin slowly decreases to 0.12 ng/ml,  0.51 ng/ml and 0.75 ng/ml in C.J., T.F. and W.S, respectively. However, the digoxin concentration of the peripheral compartment in C.J. (Figure 6(b)) increases rapidly from 2.58 ng/ml and reaches the peak 8.98 ng/ml within 0.3 h then it starts decreasing and reaches to 3.38 ng/ml in 2 h and it equilibrates. It takes 6 h to reach 0.28 ng/ml. The digoxin concentration of the peripheral compartment in T.F. (Figure 7(b)) increases quickly from 1.83 ng /ml and reaches the highest point 6.64 ng/ml in 0.3 h and declines to 2.06 ng/ml in 2 h then slowly decrease to 0.10 ng/ml in 6 h. Also, the digoxin concentration of the peripheral compartment in W.S. (Figure 8(b)) increases quickly from 1.21 ng/ml and reaches the highest peak of 13.45 ng/ml in 1 h then slowly decreases to 3.25 ng/ml in 6 h.

Conclusion
In this present study, we develop a two-compartment model and consider digoxin drug administration through oral (p.o.) and intravenous (i.v) to four volunteers. Kramer et al considered a two-compartmental model with excretion from the central compartment and describes the time course of drug concentration in the central compartment model after an intravenous injection, whereas the authors considered excretion from the peripheral compartment for both the oral and intravenous injection. The solution curves for clinical data and model were drawn using MATLAB. In this two-way cross-over study it was noted that the digoxin concentration in the central compartment after oral administration, our model-simulated data almost coincide with the available clinical data (Table 1) with RMSE 0.1697 (Subject 1), 0.0974 (Subject 2) and 0.1282 (Subject 3) which we can see from Figure 4 to Figure 10 of each subject. Also, after the intravenous administration, the data simulated by our model almost coincide with the available clinical data (Table 2) with RMSE 0.4993 (C.J.), 0.4950 (T.F) and 0.9543 (W.S.) which we can see from Figure 13 to Figure 19 of each subject. Eventually, our model predicts the measures of the drug concentration in the peripheral compartment for any given time in both p.o. administration and i.v. administration. Also, we were unable to demonstrate any significant relationship between digoxin doses of 500 μg and 1 mg administered through oral and intravenous respectively and the parameters k 12 , k 21 , k 31 and k 20 . This supports the observation that the pharmacokinetics of digoxin when given through oral and intravenous, are independent of dose. Also, there is no significant relation between pharmacokinetic parameters and RMSE of each subject in both the administrations. When we are using a non-linear compartment model in pharmacokinetics, the general approach is to use ODEs. Furthermore, stochastic differential equations (SDEs) are achieved by adding a Wiener noise component. This extra noise permits handling of auto-correlated residuals originating from natural variation or a systematic model error. Auto-correlated residuals are often partly ignored in pharmacokinetic modelling although it violates the hypothesis for many standard statistical tests (Klim, Mortensen, Kristensen, Overgaard, & Madsen, 2009). Since a curved pattern occurs in the residual plots for the deterministic minimal model a stochastic approach could decrease this curve pattern and give more accurate results.