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 PSmuscle; use the default ratios of PSi/PSmuscle to determine the value of PSi for the other organs, and then use eq. 2 to find the corresponding flcear [i] for the other organs. The default ratios PSi/PSmuscle 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 fclearmuscle 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 PSi 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 (fwB = 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 PSmuscle 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 (Pbi). 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 (Pbi). 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 (fwB) 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 PSmuscle 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 fclearmuscle (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.