PBPK program and its assumptions

PKQuest [1] was used for all the analysis in this paper. The details of the implementation of the program and a complete derivation of the pharmacokinetic equations is described in [1]. It is assumed that each organ can be modeled as a single well-stirred compartment. This is clearly an approximation and effects such a diffusion gradients, countercurrent exchange and heterogeneous organ blood flows [7–9] will produce small deviations from this assumption.

The free water solute concentration (c, moles/liter) plays a central role in these calculations because the rate of exchange across the capillary depends on differences in the value of c between the capillary and tissue (see eq. 1). For the blood or any organ, the free concentration can be related to the total (measured) concentration (C_tot, moles/Kg) by the relation C_tot = c wfract/fw where wfract (liters/Kg) is the water fraction in the organ and fw is the fraction of the solute that is free in the water phase. The parameter fw_i, the fraction free in organ i (including vascular compartments), characterizes the fraction free (and bound) and the corresponding tissue/blood partition coefficients. Solute binding is a ubiquitous aspect of pharmacokinetics and PKQuest has provided a number of different procedures for handling it. For example, the propranolol pharmacokinetics [1] were described in terms of "ktiss" (tissue/blood partition coefficients) and "freepl" (the fraction free in plasma). In the application of PKQuest to the volatile solutes [3], fw was described in terms of "Kfwat" (oil/water partition coefficient) and "klip_i" (the fraction of lipid in each organ).

For the protein bound antibiotics it will be assumed that the value of fw_i for organ i can be described by the single site Scatchard type binding equation (see [1] for details):

where c_i is the free concentration in organ i, k is the protein binding constant and Pb_i is the concentration of protein binding sites in organ i. For the special case of linear binding (no saturation), kc_i<<1, eq. 3 reduces to

As is the case for other beta-lactam antibiotics [6, 10], it will also be assumed that albumin is the binding protein so that the protein binding site concentration in organ i (Pb_i) is proportional to the extravascular albumin concentration in organ i. In PKQuest, the default values for the tissue protein concentrations (cProt_i) relative to plasma have been preprogrammed. These values are based primarily on the measurements of Tsuji et. al. [6] for the rat. (The user can change these values by entering values for cProt_i.) Only the experimental value of Pb of plasma needs to be entered in PKQuest and the value of Pb_i for the different organs are then calculated using these stored default values.

For the general case (eq. 3), the PKQuest protein binding is characterized by 2 parameters: kProt (the binding constant, k) and cProt_vein (the concentration of binding sites in venous plasma, Pb). The special case of linear albumin binding is completely described by just the one parameter k*Pb (see eq. 4). In PKQuest, the linear case is turned on by entering fw [vein] (the fraction free in the vein plasma, which is independent of concentration) and setting kProt = -2 and the following procedure is used to find the default values of fw_i for each organ i. First, eq. 4 for i=vein is solved for the value of kPb_vein. Then, using the default values of albumin concentration in each organ (cProt [i]), the value of kPb_i is found, and, from eq. 4, the corresponding fw_i determined.

Experimental pharmacokinetic data

Three set of data from previous publications were used. The data for inulin was based on the experimental measurements of Odeh et. al. [11] in which the venous plasma inulin was measured following a 5 minute, constant intravenous infusion of 4 gm of inulin. The data from subject #1 was used (42 year old normal female, weight 87.3 Kg, height 173 cm). The data for dicloxacillin was obtained from the measurements of Lofgren et. al. [12] of the venous plasma time course following a 30 minute constant infusion of 2 gms dicloxacillin. The averaged data from 6 young males (average weight 75 Kg) was used. The data for ceftriaxone was obtained from the measurements of Stoeckel and colleaques [13, 14]. The data are for a single subject (M.Fl.) given 4 different IV doses of ceftriaxone – 0.15, 0.5, 1.5 and 3 gms as a 5 minute constant infusion. The experimental data points were obtained by using UN-SCAN-IT (Silk Scientific Corporation) to read the data from the published figures. All the figures shown in this paper are direct copies (in jpeg format) of standard PKQuest output.

Standard human parameters

Inulin

As described previously [1], the PBPK data unique to each solute is input to PKQuest in the form a short Maple http://www.maplesoft.com procedure that completely characterizes the pharmacokinetics:

inulin:=proc()

   defaultpar():

   Wtot:=87.3;

   standardhuman(Wtot);

   Fat:=0.28; # 26% Fat by weight based on 173 cm height.

   mecf:=1.0; # Inulin distributes only in default extracellular space

   fclear [muscle]:=0.354; # permeability value for muscle, all other organs set from it

   rclr:=0.1; #Renal water clearance of 0.10 liters/min

   cunit:="milligrams"; #unit used in printed and plotted output

   concunit [vein]:=4;# input data = venous plasma concentration

   ninput:=1; finput[1]:=table([organ=vein, type=1, rate=800, tbeg=0,

      tend=5, csteady=0, padjust=0]); # 5 minute constant infusion of 4000 mg

The experimental data was for the subject in fig. 2 of Odeh et. al. [11] (this female had a weight and height of 87.3 Kg and 173 cm, and a predicted 28% fat content). There are 3 parameters that specifically characterize inulin: mecf, rclr, and fclear [muscle]:

The parameter mecf determines the extravascular volume of distribution. The default value is mecf = -1, which indicates that the solute distributes in all the body water. If mecf is >0 then the solute distributes in organ i in a water volume equal to mecf*ecf [i] where ecf [i] is the default extracellular space for organ i. The user has the option to either adjust each of the individual values of ecf [i], or scale all the values by the factor mecf. The default ecf [i] values in "standardhuman" were initially set equal to the values directly determined in the rat by Tsuji et. al. [6] from steady state tissue inulin measurements. Then, the value of mecf that optimized the fit to this data of Odeh et. al. was determined, and these new values were chosen for "standardhuman". (The human values were 16% larger than the rat values). Since the default data was chosen to fit this data, setting mecf = 1 simply selects this optimized data.

The parameter rclr is the free water concentration renal clearance. It is equal to the volume of blood water cleared per min (liters/min). It differs slightly from the standard renal clearance which is equal to plasma volume/min (rclr = 0.94 × plasma clearance). The value of rclr of 0.1 liter/min corresponds to the value directly measured by Odeh et. al. for this subject.

The user has two different options for entering the values of fclear [i] (see eq. 1). If, as in this case, just the value of fclear [muscle] is input, then PKQuest will, by default, use the relation in eq. 2 to determine the value of PS_muscle; use the default ratios of PS_i/PS_muscle to determine the value of PS_i for the other organs, and then use eq. 2 to find the corresponding flcear [i] for the other organs. The default ratios PS_i/PS_muscle assume, for example, that the liver and kidney have a high permeability and are flow limited, while, because of the blood brain barrier, the brain capillary permeability is zero. The user can also individually input the value of fclear [i] for any organ i. If an organ's value for fclear is not set, then PKQuest will use the value set from the above calculation based on the muscle value. The parameter fclear [muscle] was adjusted to find the best fit to the inulin data using the Powell minimization feature of PKQuest [1]. The optimal value of fclear_muscle was 0.354.

Figure 1 shows the PKQuest output for this inulin data (all the figures in this paper are copied directly from the standard output of PKQuest). There is good agreement between the predicted venous plasma concentration (solid line) and the experimental values (squares). PKQuest also routinely outputs the experimental pharmacokinetic clearance and volume of distribution determined from the first and second moments of the plasma concentration extrapolated exponentially to infinite time [15]. These experimentally derived values are in good agreement with the model values: Plasma clearance: 0.104 liters/min (experimental) versus 0.1 (model); Volume of distribution: 10.7 liters (experimental) versus 10.7 (model). PKQuest also outputs the values of fclear [i] and the corresponding value of PS_i determined from eq. 2. The fclear [muscle] of 0.354 corresponds to a PS of muscle of 0.6 ml/min/100 gm.

Dicloxacillin

Approximately 97% of human serum dicloxacillin is protein bound (fw_B = 0.03) [16], making it a likely candidate to show some permeability limitation. Another advantage of dicloxacillin is that this binding is linear, showing no evidence of saturation at concentrations up to 230 mg/liter [6, 16], simplifying the PBPK. Since the binding does not saturate, it can be described by eq. 4 and the binding is characterized by the single parameter fw [vein] (see Methods).

The complete Maple procedure for dicloxacillin is:

dicloxacillin:=proc()

   defaultpar():

   Wtot:=75.0;

   standardhuman(Wtot);

   Fat:=0.16;

   concunit [vein]:=4;

   mecf:=1.0;

   kProt:=-2; #this turns on use of cProt [i] to determine fw [i]

   fw [vein]:=0.03;

   rclr:=0.12; # renal filtraton clearance

   Tclr [kidney]:=3.0;# Renal tubule secretion clearance

   fclear [muscle]:=0.26;

   ninput:=1; finput[1]:=table([organ=vein, type=1, rate=2000.0/30, tbeg=0,

      tend=30, csteady=0, padjust=0]); # constant 30 minute infusion of 2 gms

The data of Lofgren et. al. [12] for the venous plasma time course of normal young males given a 30 minute constant infusion of 2 gms dicloxacillin was used as input. The average body weight for this group was 75 Kg, and the standard fat content for young males (16%, [17]) was assumed. The parameter mecf is set = 1 because dicloxacillin, like inulin, distributes only in the extracellular space as shown by its negligible distribution into red cells and its low blood/tissue partition coefficient [6]. This means that dicloxacillin has the same volume of distributions in the different organs as was used above for inulin. Dicloxacillin is cleared primarily by the kidney in these subjects [12] and there is a large component of renal tubular secretion that is blocked by probenecid [18] determined by the parameter Tclr [kidney]. (Other studies have indicated that the liver can account for 40% or more of the total clearance [16]. PKQuest calculations (not shown) based on the assumption that the clearance was divided equally between the kidney and liver yield almost identical results). The value of renal filtration (rclr) was set to the normal value of 0.12 liters/min. The capillary permeability limitation is characterized by the single parameter fclear [muscle] (from which the permeabililty of the other organs is calculated by PKQuest). The above values of Tclr [kidney] = 3.0 (liters/min) and fclear [muscle]:= 0.26 (dimensionless) were obtained by simultaneously optimizing both values using the Powell minimization procedure in PKQuest [1] to find the values that provided the best fit to the experimental data.

The left hand panel in figure 2 shows a comparison of the PKQuest model predictions (solid line) for this set of input parameters with the experimental data of Lofgren et. al. [12]. Again, the agreement is quite good. This model fit used a permeability limitation (fclear) for muscle of 0.26. This means that the venous blood leaving the muscle is 26% equilibrated with the tissue. Using eq. 2, this corresponds to a PS_muscle of 13 ml/min/100 gm.

The right hand panel in figure 2 shows the best fit that can be found for this same data, assuming that dicloxacillin is flow limited (fclear = 1). The fit is clearly worse, especially at the early time points where a permeability limitation should be important. Part of the better fit may have resulted from adjusting 2 parameters (fclear [muscle] and Tclr [kidney]) for fig. 2 (left), while, in fig. 2 (right), only one parameter was adjusted (Tclr [kidney]). However, the fact that the fit in fig. 2 (right) could not be significantly improved by trying to adjust other parameters, such as muscle blood flow, extracellular space and/or plasma protein binding, suggests that this permeability limitation is real.

Ceftriaxone

In addition to having a high degree of protein binding, Ceftriaxone has the additional complication that this binding is non-linear and saturates at high concentrations. Stoeckel and colleagues have carried out detailed investigations of the pharmacokinetics of ceftriaxone in humans and the influence of this non-linear binding [13, 14]. They showed that the non-linear kinetics of ceftriaxone arise solely from this non-linear binding by demonstrating that the kinetics became linear if the free drug concentrations are used in the pharmacokinetic calculations. In this section, PKQuest will be used to develop a PBPK model for this data. The model includes both non-linear binding and a permeability limitation, the first time that both these factors have been included in a PBPK.

Because the protein binding saturates, the general Scatchard binding equation (eq. 3) must be used. This requires a value for the protein binding constant (k) and the concentration of binding sites in each organ (Pb_i). As described above (Methods), these constants are determined by two PKQuest parameters: kProt (the association constant for protein binding) and cProt [vein] (the plasma concentration of binding sites). PKQuest then uses the default information about the tissue albumin concentration in each organ relative to the plasma albumin to determine the concentration of binding sites in each organ (Pb_i). The data of Stoeckel et. al. [13, 14] are ideal for this analysis because they provide information about the venous blood levels for the same subject (M.Fl.) given 4 different IV doses of ceftriaxone – 0.15, 0.5, 1.5 and 3 gms. It is important to have these different inputs in order to test if the PBPK can accurately model the non-linear kinetics. Since all the doses were given to the same subject, an identical set of PBPK parameters is used for all the doses:

ceftriaxone:=proc()

   defaultpar():

   Wtot:=70.0; #Body weight

   standardhuman(Wtot);

   Fat:=0.18;# Fat body weight fraction for 70 Kg, 183 cm man.

   cunit:="micromoles";

   concunit [vein]:=4;# input = venous plasma concentration

   mecf:=0.85; # distribute only in extracellular space – adjusted to fit data

   fclear [muscle]:=0.15; # determinse muscle capillary permeability

   rclr:=0.11; # Normal renal clearance

   Tclr [kidney]:=0.035; # Renal tubule secretion

   Tclr [liver]:=0.06; # Liver metabolism

   cProt [vein]:=806; # total conc. protein binding site in plasma (micromoles/liter)

   kProt:=0.0315; # Association constant (liters/micromoles)

Subject M.Fl. had a weight of 70 Kg and height of 183 cm (K. Stoeckel, personal communication). The values of cProt [vein] and kProt used above are the values determined by equilibrium dialysis for the subject M.Fl. [13]. Ceftriaxone is eliminated by both renal secretion and liver metabolism. The normal value of 0.ll liters/min was used for renal clearance, and the tubule secretion (Tclr [kidney]) and liver metabolism (Tclr [liver]) were adjusted to have a renal clearance of about 65% of the total clearance as was observed experimentally. The liver metabolism and tubule secretion were assumed to be linear because the kinetics were linear if corrections were made for the plasma binding. The value mecf has been set to 0.85 to slightly improve the fit to the data. This makes the extracellular volume of distribution 85% of the default value. It is not clear whether this has some physiological significance for this subject or if it is just a "fudge" factor that corrects for some other limitation in the PBPK. The two parameters fclear [muscle] and Tclr [liver] were simultaneously adjusted using the Powell minimization procedure in PKQuest to obtain the best fit to the experimental data. For this case, where the binding is non-linear, the value of fclear [muscle] that is input is interpreted as the value of fclear in the limit of zero concentration. As the concentration increases, the value of the fraction free (fw_B) will increase (eq. 3) and the corresponding value of fclear (eq. 2) will also increase (i.e. become less permeability limited). The optimized value of fclear [muscle] of 0.15 corresponds (see eq. 2) to a PS_muscle of 6 ml/min/100 gm.

Figure 3 shows the agreement between the model predictions of PKQuest and the experimental data for inputs of 0.15, 0.5, 1.5 and 3.0 grams. The same parameter set was used for each of these 4 fits. Only the input function was varied. Over the plasma concentration ranges seen at early times for each of the 4 doses, the fraction free in the blood and the fclear_muscle (fraction of solute that equilibrates with the tissue in one pass through the muscle capillary) varies as follows: At 0 μM: fraction free = 0.038, fclear = 0.15; At 50 μM: fraction free = 0.092, fclear = 0.33; At 150 μM: fraction free = 0.18, fclear = 0.55; At 450 μM: fraction free = 0.374, fclear = 0.8; At 800 μM: fraction free = 0.51; fclear = 0.89. The non-linearity in the kinetics is apparent both in the changes in the qualitative shape of the curve and in the absolute values of the plasma concentration. The agreement between the model and experiment for this 20 fold dose range and roughly 10 fold range of fraction free is surprisingly good, especially considering that there are so few adjustable parameters (see below).

Figure 4 show the corresponding fits for the case where ceftriaxone is assumed to be flow limited (fclear = 1) for the 0.15 and 3.0 gm dose. Comparing figures 3 and 4 it can be seen that the permeability limitation significantly influences the pharmacokinetics only for the lower doses because of the saturation of the protein binding. For the low dose (0.15 grams), the plasma concentration at early times is about 50 μM which will have a corresponding fclear of 0.33. This means that there will be only 33% equilibration during one pass through the capillary – a significant permeability limitation. However, for the high dose (3.0 grams), the early plasma concentration is about 800 μM which corresponds to a fclear of 0.89 which differs only sightly from the flow limited condition (fclear = 1) assumed for figure 4.