The Presence of Phases and the Inability of the Classical Compartment Models to Provide Pharmacokinetic Parameters of Physiological Significance for Lipophilic Drugs

The first biphasic open one-compartment pharmacokinetic model is described. Its analytical solutions to drug concentration were developed from parameters of an open two-compartment pharmacokinetic model. The model is used to explain the unusually large compartment volumes and apparent volumes of distribution of lipophilic drugs, as well as to identify which of the pharmacokinetic parameters of the classical compartment models are biologically relevant.


Introduction
In our recent work, we have defined the apparent volume of distribution anew and we were able to use this concept to fully explain the classical pharmacokinetic compartment models for hydrophilic drugs. In addition, we have proven that the calculated apparent volume of distribution reflects the extent of the true distribution of hydrophilic drugs in the body and stated that lipophilic drugs that do not follow a one-compartment model are unlikely to be fully described by the theory of the classical pharmacokinetic compartment models [1] [2] [3]. Lipophilic drugs exhibit an apparent volume of distribution that is much larger than the volume of total body fluids. If their distribution/binding to tissues outside the blood circulation is very rapid and comparable to the kinetics of mixing of the drug in the plasma, the central compartment of the pharmacokinetic model will be composed of more than one phase. In this paper, firstly, we have processed clinical data of the drug Prazosin, clearly, a drug that follows an open two-compartment pharmacokinetic model, using an open biphasic one-compartment model [4]- [9]. The intent is to develop the model and derive explicit solutions to drug concentration in two immiscible phases at equilibrium within a single compartment. Such a model could help us better understand the disposition of drugs in the body that has an apparent volume of distribution larger than the volume of the total body fluids. Secondly, we have used published clinical data of Doxazosin, Digoxin and Pindolol to demonstrate the inability of the classical compartment models to extract pharmacokinetic parameters of physiological significance for lipophilic drugs.

Methods
Doxazosin, Prazosin, Digoxin and Pindolol plasma drug concentrations were extracted from published work using the graph reader tool provided by the graphreader.com. Pharmacokinetic compartment modeling of the clinical data was conducted using methods described elsewhere [3] [9].

Results and Discussion
It was demonstrated in our recent work that the apparent volume of distribution of Prazosin as determined from an open two-compartment model is the true distribution volume of the drug in the body [9]. It is well established that the elimination half-life of the drug regardless of the mode of drug administration is about 2.5 h whereas after an intravenous bolus injection the drug is distributed to the peripheral compartment with an average half-life of about 10 minutes. Figure 1 displays patient HA's plasma Prazosin concentration as a function of time as reported in the work of Grahnen [4]. If the initial three blood samples after 5, 10 and 20 minutes were not collected, the drug disposition would have followed a one-compartment pharmacokinetic model, and we would have calculated from the parameters given by the best fit equation (C 0 = 10.6 ng/mL) and the intravenous dose (500 μg), an apparent volume of distribution also known as Figure 1. Plasma drug concentration as a function of time for patient HA [4]. The drug was administered by an 0.5 mg rapid intravenous push and data from 1 to 10 hours were fitted using an open one-compartment pharmacokinetic model.  21 , is complete in the terminal or β elimination phase. In that pharmacokinetic phase, the two compartments behave as a single one and the established pseudoequilibrium in plasma drug concentration is controlled only by the rate of drug elimination. The hypothesis of the first part of this work is the potential modeling of the β elimination phase or period of a two-compartment model drug using a biphasic one-compartment pharmacokinetic model.
In the system of Figure 2 (right), we have a single compartment composed of two immiscible phases at equilibrium. Ideally, all the drug is added directly to the blood which is part of the chemical phase 1 and drug distribution in the one-compartment model from the blood to all other body tissues (phase 2) is instantaneous. The distribution of Prazosin in the body, however, being a two-compartment model drug, doesn't reach completion until its disposition enters the terminal elimination phase. As with the one-compartment pharmacokinetic model, an important condition of the biphasic one-compartment model is the instantaneous drug distributional equilibrium between the two phases upon any drug addition into the system. Thus, to establish the pseudoequilibrium conditions of the two-compartment model from time zero in our biphasic single compartment system we have to rapidly introduce part of the dose in phase 2.  1, 2, 1a, 1b, k 1b1a , k 1a1b , k 1a2 , k 21a and k 1a0 are the central compartment, the peripheral compartment, the plasma or extracellular fluid which is part of the central compartment, other tissues which are also kinetically part of the central compartment, in contact and immiscible with the phase 1a, the transfer rate constant of drug from phase 1b to phase 1a within the central compartment, the transfer rate constant of the drug from phase a to phase b of the central compartment, the transfer rate constant of the drug from compartment 1 phase 1a to compartment 2, the transfer rate constant of drug from compartment 2 to phase 1a, and the elimination rate constant of the drug from phase 1a of the central compartment, respectively. (Right) Biphasic open one-compartment pharmacokinetic model. Equilibrium between the two phases in the system upon rapid addition of the solute in the two phases is instantaneous. The solute distribution rate constants k 12 and k 21 have the same values as originally calculated from the two-compartment model of prazosin [9]. k 10 is the elimination rate constant of the drug from phase 1. ; The analytical solutions to the drug amount in the two phases as a function of time are, Our next task is to choose the value of 0 2 x . As it can be seen from Figure  It must be noted that these calculations were carried out using the average pharmacokinetic parameters of four patients (not just patient HA) [9]. Under the pseudoequilibrium conditions, the amount of drug in each compartment varies but the fractional concentration stays constant. The fractional ratio in the two compartments at pseudoequilibrium conditions is equal to 2.71. Using the pharmacokinetic parameters described in our recent work, 1.03 μg of prazosin, 0.278 μg x 1 and 0.752 μg x 2 , are removed from the system per minute during the pseudoequilibrium conditions [9]. Thus, the biphasic one-compartment model requires 365.177 μg to be introduced into phase 2 at time zero ( 0 2 x ) and the remaining from the 500 μg total dose to be added in phase 1 ( 0 1 134.823 μg x = ). The mono-exponential decay pattern of the simulations shown in Figure  3(b) verifies that the biphasic single compartment system is always at equilibrium. The simulations shown in Figure 3(c) were conducted at x 2 /x 1 fractional ratios of 9, 2.71 and 0.41. It is quite interesting to note that the second part of Equation (3)   were calculated from the clinical data of four patients (not only patient HA) using Equation (7) and Equation (8) of the published article of Savva and coworkers [9]. Thin and thick continuous lines are the drug quantities x 1 and x 2 , respectively, that were calculated using the biphasic one-compartment model explicit solutions to drug amount (Equations (3) and (4)). (c): Thick and thin continuous lines are x 2 and x 1 simulations that were carried out using Equation (4) and Equation (3) [11]. In agreement with our estimates, the same group of investigators has calculated an elimination t 1/2,z of 562 minutes and an average value of V d,ss of 110 L in twelve elderly volunteers [12].
Despite the perfect fit of the raw data to the two-compartment model, an estimated value of the volume of the central compartment so close to the volume of the total body water suggests that the model lacks physiological significance. It is possible that the more lipophilic than prazosin, quinazoline derivative, partitions into tissues other than plasma as soon as it is injected into the bloodstream. In this case, the central compartment is at least biphasic (Figure 2, left). One of the possibilities is that the drug enters the system via an intravenous push into phase 1a of the central compartment, it is distributed in phase 1b within the central compartment and to the peripheral compartment and it is eliminated out of the system from the central compartment. The drug transfer rate constants k 1b1a and k 1a1b in the two phases 1a and 1b of the central compartment are extremely high and practically unmeasurable in the time domain of sample collection as suggested by the excellent goodness of fit of plasma drug concentrations by the two-compartment model (Figure 4). Phase 1b is not in contact and is not related in any way to the peripheral compartment. The drug can be distributed to phase 1b and to the peripheral compartment only from phase 1a. Drug distribution upon its entry from phase 1a into the tissues that make up phase 1b within the central compartment is considered to be instantaneous. Contrary to that, the rate of drug distribution from phase 1a into the tissues comprising the peripheral compartment, and vice versa, takes place at a slower rate and is measurable in the system. The chemical equilibrium constant of the two consecutive reversible reactions (R1) is related to the partition coefficients of the drug in the three phases at equilibrium. The drug dose (D) is added into phase 1a and instantly the drug diffuses and reaches equilibrium in phase 1b.
Initial conditions: 1a,eq 2,eq 1b 1a 2 ; ; Partition coefficients: 1b 2b 1b,eq 1b 1b,1a 1a,eq 1b 2a The chemical equilibrium constant, K of the process is, 1b 2b 2,1a 1b The apparent volume of distribution with respect to the phase 1a of the central compartment V d,1a , which plasma is definitely part of it, and like the partition coefficient is also an equilibrium quantity, can be described at distribution equilibrium with phase 1b and the peripheral compartment, using the mass balance equations (Equation (8)).
s,eq d,1a 1a,eq V C At zero time, all the drug is in the central compartment and has reached equilibrium in both phases 1a and 1b. No drug has yet been eliminated out of the system or distributed into the peripheral compartment. Thus, at t = 0 we can consider the central compartment as being a closed and isolated system where all the drug is located. This unique situation can be described by Equation (8b) and Equation (8c).
Hence, according to the multiphasic compartment model, the volume 34.6 L which is determined from the intercept of the line derived after application of the method of residuals, is not the real volume of the central compartment V 1 . It is rather the apparent volume of distribution of doxazosin with respect to phase 1a when phase 1a is at equilibrium with phase 1b within the central compartment. d,1a V ′ suggests that at time zero it would require 34.6 L of plasma to accommodate 1 mg of Doxazosin in the absence of the peripheral compartment. As soon as the drug starts getting distributed in the peripheral compartment, the apparent volume of distribution of Doxazosin with respect to the plasma changes and becomes equal to V d,1a (Equation (8a)) which can only be estimated at the time of momentary distribution equilibrium between the three phases, if only we knew how much drug is initially distributed in the two phases within the central compartment. The problem is that the analytical solution to drug concentration in the central compartment will always be a function of the unknown initial quantity of the drug in phase 1b. It is also important to understand that since the system undergoes two totally different conditions, d,1a V ′ is only valid at zero time and is totally unrelated to V d,1a .
Even if the system in Figure 2 is set up as a three-compartment kinetic model, it may not be feasible to collect plasma drug concentrations at earlier times to distinguish kinetically compartment 1a from compartment 1b. Kramer et al., have studied the pharmacokinetics of Digoxin using a two-and a three-compartment model. They have reported an average value of V 1 in a two-and a three-compartment model equal to 58 L and 46 L, respectively [13]. We have carried out our own analysis on the clinical data of patient TF in an open three-compartment mammillary pharmacokinetic model [13] [14] [15] [16]. The initial value problem for the system of differential equations was set up with all the drug being in the central compartment at time zero. After applying the method of residuals twice, C 0 was calculated to be 26.28 ng/mL and 1 38 L V = . These volumes cannot be the real volumes of the central compartment. They are rather the digoxin apparent volumes of distribution d,1a V ′ at zero time. This initial rapid distribution phase was also reported by Doherty [17]. Although these investigators were collecting blood samples every 2 minutes, as early as 2 minutes after intravenous injection, it was not possible to resolve the digoxin distribution kinetics in the different tissues, hence different phases, that comprise the central compartment. Two more studies have reported a V 1 of 37 L while a third one reported a V 1 equal to 110 L [18] [19] [20]. The V d,ss of digoxin is reported to be larger than 350 L.
None of these volumes has physiological significance. Digoxin has an aqueous solubility of 65 μg/mL. It is not very lipophilic (logP = 1.26) and it is not freely permeable through plasma membranes (MW = 781). Yet, it is extensively distributed at high concentrations in heart, kidney and liver and at lower concentrations essentially in all human tissues, apart from the brain, at very early times after an intravenous push of the drug [24]. It appears that as soon as the drug is injected into the blood it is instantly distributed to other tissues of vital organs. x x x = + in a two-compartment model after an intravenous bolus administration is, V 1 and k 10 in Equation (9) are constants, C 1 is the only parameter that changes with time.
By arbitrarily defining Clearance as the volume of plasma cleared of drug per unit time, 10 1 CL k V = ⋅ (10) and Nagashima, Gibaldi and coworkers have shown that [26] [27], 1 10 x k f β = (11) and 1 1 x Therefore, We have verified the validity of Equation (11) and Equation (12)  Hicks and coworkers have conducted clinical studies of pindolol administered by an intravenous push to four normal subjects. They have published the mean plasma levels of pindolol but did not carry out pharmacokinetic analysis [28].
We have extracted the published mean plasma levels of pindolol using methods described elsewhere and determined the pharmacokinetic parameters of Pindolol using an open two-compartment model (Table 1 and Figure 5) [9].
The average value of total clearance of Pindolol using Equation (10) was calculated to be 443.00 mL/min, which in agreement with that calculated by Gugler

Conclusion
The

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.