The Effect of Halothane on the Recirculatory Pharmacokinetics of Physiologic Markers 

The design of this pharmacokinetic study entailed 16 individual experiments. Four purpose-bred male coon-hounds, weighing 18–31.5 kg (27.6 +/- 6.4 kg, Table 1), were studied on four occasions each in this study approved by our institutional animal care and use committee. Approximately 1 month before being studied, a Vascular-Access-Port catheter (Access Technologies, Skokie, IL) was placed into a femoral artery of each dog and secured to the muscle fascia of the upper hind leg to facilitate frequent percutaneous arterial blood sampling. [20] The dogs were allowed to recover from surgery for at least 2 weeks before being studied for the first time; the interval between subsequent studies was no less than 2 weeks.

All dogs were studied while awake (0% halothane, control) and while anesthetized with halothane at end-tidal concentrations of 1%, 1.5%, and 2%, which correspond to 1.2, 1.7, and 2.3 minimum alveolar concentration, [21,22] the order of which was randomized using a repeated measures Latin square experimental design. The dogs were trained to lie in the left lateral decubitus position for the awake study.

On the day of an awake study, the dog was brought to our laboratory and positioned. When it was determined that the dog was cooperative, the neck was prepped and the skin overlying the right external jugular vein was infiltrated liberally with 2% 2-chloroprocaine. Using a modified Seldinger technique, an 8-French percutaneous sheath introducer was inserted into the external jugular vein. If the insertion could not be accomplished in a timely and humane manner, the study was abandoned and rescheduled.

When the dogs were studied while anesthetized, anesthesia was induced with methohexital (10–15 mg/kg given intravenously) via a foreleg vein, the trachea was intubated with a 9-mm tracheal tube, and the animal was placed in the left lateral decubitus position. Mechanical ventilation was instituted at a tidal volume of 20–25 ml/kg at a rate sufficient to maintain end-tidal carbon dioxide tension at 30 +/- 5 mmHg. Anesthesia was maintained with 1%, 1.5%, or 2% halothane in oxygen. End-tidal halothane concentrations were monitored with an Engstrom EMMA (Engstrom Medical AB, Bromma, Sweden) after its calibration with known standards. A heat and moisture exchanger was placed between the endotracheal tube and the EMMA in-line detector to minimize interference by water vapor.

For all studies, after an overnight fast during which the dog was allowed water ad libitum, the dog was brought to the laboratory and arterial access for blood sampling by roller pump or syringe was achieved by inserting a 20-gauge Huber needle percutaneously in the Vascular-Access-Port; this also allowed systemic arterial blood pressure to be monitored via a solid-state pressure transducer (Trantec; Baxter-Edwards, Irvine, CA). A flow-directed thermal dilution pulmonary artery catheter (Baxter-Edwards 93A-140–7F, with a 20-cm proximal port) was inserted through a sheath introducer in the right external jugular vein, positioned, and secured for determination of pulmonary arterial pressure and thermal dilution CO and to facilitate right atrial administration of the physiologic markers. The side arm of the sheath introducer was used for maintenance fluid administration and readministration of autologous blood. Hydration was maintained throughout the study by an infusion of 0.9% saline at a rate of 5–10 ml [center dot] kg sup -1 [center dot] h sup -1 to maintain a constant pulmonary artery diastolic pressure (+/- 2 mmHg). Once the study was well underway, a well-lubricated 7-French single-lumen, balloon-tipped catheter was inserted through the urethra into the bladder to facilitate urine ([sup 14 C]-inulin) collection. All dogs, whether awake or anesthetized, easily tolerated this procedure.

One hundred fifty milliliters of whole blood was removed from the dog through the arterial catheter and anticoagulated with 1,000 U heparin. This blood was immediately replaced with 600 ml 0.9% saline solution administered intravenously for 30 min. During the first 10 min of the study (from time t = 0 min to t = 10 min), this autologous blood was reinfused intravenously to replace the blood obtained during the period of frequent blood sampling.

The study was not begun until the dog was hemodynamically stable. This was defined as less than 10% variation of CO and pulmonary and systemic arterial blood pressures during a 30-min period when measured every 15 min. The dogs were hemodynamically stable approximately 1 h after removal and saline replacement of the 150 ml blood.

At the onset of the study (time t = 0 min), ICG (Cardio-Green; Hynson, Westcott, and Dunning, Baltimore, MD), 5 mg in 1 ml of diluent, [sup 14 C]-inulin (DuPont NEN, Boston, MA), 30 micro Ci in 1.5 ml of diluent, and antipyrine (Sigma Chemical Co., St. Louis, MO), 25 mg in 1 ml of diluent, were injected within 1 s through the proximal pulmonary artery catheter port. Arterial blood samples were collected every 0.05 min for the first minute and every 0.1 min for the next minute using a computer-controlled roller pump (Masterflex; Cole-Parmer, Chicago, IL) and chromatography fraction collector (model 203; Gilson, Middleton, WI). Subsequent arterial blood samples were drawn manually at 0.5-min intervals to 4 min, every 2 min to 20 min, every 10 min to 60 min, every 15 min to 120 min, and every 30 min to 360 min.

Plasma ICG concentrations of all samples obtained up to 20 min were measured on the study day using the high-performance liquid chromatography technique of Grasela et al. [23] as modified in our laboratory to provide sensitivity of 0.2 to 20.00 micro gram/ml with coefficients of variation of 5% or less. [8] Plasma ICG concentrations were converted to blood concentrations by multiplying them by one minus the hematocrit concentration because ICG does not partition into erythrocytes.

Plasma [sup 14 C]-inulin concentrations of all samples were determined by liquid scintillation counting, using an external standard method for quench correction. [24] Counts that were less than three times the background count were considered to be less than the lower limit of detection. The coefficient of variation of the assay was less than 3%. Plasma inulin concentrations were converted to blood concentrations by multiplying them by one minus the hematocrit, because inulin does not partition into erythrocytes.

Plasma antipyrine concentrations were measured in all samples using a modification of a high-performance liquid chromatography technique developed in our laboratory. [5,17] The antipyrine method is linear from plasma concentrations of 0.10 to 10.00 micro gram/ml with coefficients of variation of 5% or less. Plasma antipyrine concentrations were converted to blood concentrations using an in vivo technique. [5]

Arterial marker concentration versus time data before evidence of ICG recirculation (i.e., first-pass data) were weighted uniformly and fitted to the sum of two right-skewed gamma (Erlang) distribution functions using Tablecurve 2D (version 3.0; Jandel Scientific, San Rafael, CA) on a Pentium-based PC (Gateway 2000, North Sioux City, SD). [25] Because neither ICG nor inulin distribute beyond the intravascular space before recirculation, they were modeled simultaneously to improve the confidence in the model parameters of the central (first-pass) circulation. Antipyrine has measurable tissue distribution during this time and was modeled independently.

In subsequent pharmacokinetic analysis, the descriptions of the central circulation were incorporated as parallel linear chains, or delay elements, into independent recirculatory models for the individual markers using SAAM II (SAAM Institute, Seattle, WA) implemented on a Pentium-based PC. [25,26] The first-pass data were excluded from further data fitting; the results of the Erlang model of the central circulation were placed as fixed parameters into the recirculatory model, thereby reducing the number of parameters to be optimized. The data were weighted in proportion to the reciprocal of the estimated variance for each datum, assuming all data have estimated fractional standard deviations of 0.5. Possible model misspecification was examined by testing for random scatter about the calculated values [27] using the two-tailed one-sample runs test, with P < 0.05, corrected for multiple applications of the runs test, as the criterion for rejection of the null hypothesis. [28]

The pharmacokinetic modeling method was based on the approach described by Jacquez [29] for obtaining information from outflow concentration histories, the so-called inverse problem (fig. 1). Inulin and antipyrine distribution to extracellular fluid space and total body water space, respectively, were analyzed as the convolution of their intravascular behavior, determined by the pharmacokinetics of concomitantly administered ICG, and tissue distribution kinetics. [5]

The intravascular mixing model of ICG disposition has been modified from that which we described previously because of the improved descriptions of circulatory delays enabled by SAAM II. [9,5] The two lumped parallel pathways of the central circulation, described by the sum of two Erlang distribution functions, were incorporated into a full recirculatory model, including lumped parallel fast and slow peripheral circuits and Cl^E, using SAAM II. The antipyrine pulmonary tissue volume (V^T-P) is the difference between the antipyrine central volume (mean transit time [MTT]^antipyrine[center dot] CO) and the central intravascular volume codetermined by ICG and inulin (MTT^ICG, inulin [center dot] CO).

Peripheral drug distribution can be lumped into identifiable volumes and clearances: a fast nondistributive peripheral pathway (V^ND-V and Cl^ND-F), a slow nondistributive peripheral pathway (V^ND-S and Cl^ND-S), rapidly (fast) equilibrating tissues (V^T-F and Cl^T-F), and slowly equilibrating tissues (V^T-S and Cl^T-S). The fast and slow nondistributive peripheral pathways (delay elements) represent intravascular circuits in the ICG and inulin models; the single nondistributive peripheral pathway in the antipyrine model represents pathways with minimal apparent tissue distribution. In the inulin and antipyrine models, the parallel rapidly and slowly equilibrating tissues are the fast and slow compartments of traditional three-compartment pharmacokinetic models, respectively, and therefore the central circulation and nondistributive peripheral pathways are detailed representations of the ideal central volume of the three-compartment model. [9] Because of the direct correspondence between the recirculatory model and three-compartment models, Cl^Ewas modeled from the arterial (sampling) compartment to allow comparison of these results with previous ones.

The eight model variables determined from arterial blood ICG concentrations, the nine model variables determined from arterial blood antipyrine concentrations, and the 12 model variables determined from arterial blood inulin concentrations were determined to be both sensible and identifiable for our sampling schedule by the IDENT2 program of Jacquez and Perry. [30,5]

The effect of the order of treatment on observed pharmacokinetic variables was ruled out using a general linear model analysis of variance for a repeated-measures Latin square experimental design (NCSS 6.0.2 Statistical System for Windows; Number Cruncher Statistical Systems, Kaysville, UT). Some data failed either the Kolmogorov-Smirnov test for a normally distributed population or the Levene median test for equal variance (SigmaStat Statistical Software version 2.0; Jandel Scientific Software, San Rafeal, CA). Therefore, all variables were treated as ordinal data and compared across treatments using the Friedman repeated-measures analysis of variance on ranks. When the analysis of variance met the criterion for rejection of the null hypothesis, post hoc comparisons with control were made using Dunnett's test. The relation of the pharmacokinetic variables to the end-tidal halothane concentration was sought using the Spearman rank-order correlation (Sig-maStat) using the Bonferroni correction of the criterion for rejection of the null hypothesis. The criterion for rejection of the null hypothesis was P < 0.05.

Halothane anesthesia was associated with a decrease in hematocrit compared with the awake control (Table 1). There was a significant, halothane dose-related increase in heart rate and a decrease in blood pressure and CO with no change in systemic vascular resistance.

The blood ICG, inulin, and antipyrine concentration versus time relations were well characterized by the models from the moment of injection (Figure 2, Figure 3, Figure 4). The one-sample runs test confirmed that there was random scatter of the observed data about the calculated values, indicating that there were no systematic errors produced by the models. Our recirculatory model of drug disposition was able to describe the effect of halothane on CO and its distribution (the ICG model, Table 2) and the effect of reduced and redistributed CO on inulin (Table 3) and antipyrine (Table 4) disposition. Our model accounted for the twofold increase in the area under the first minutes of the antipyrine concentration versus time curve (AUC) produced by 2% halothane anesthesia (Figure 4, Table 5). The total volumes of distribution (Vss) of ICG, inulin, and antipyrine were unaffected by halothane anesthesia (Table 1, Table 2, Table 3, Table 4). The ICG Cl^Ewas not affected by halothane anesthesia (Table 1and Table 2;Figure 2[inset]), but the Cl^Eof both inulin and antipyrine were significantly decreased by halothane anesthesia, as reflected in the decreased slopes of their terminal drug concentration versus time relations (Table 1, Table 3, Table 4;Figure 3[inset], Figure 4[inset]).

The volumes of the central and peripheral circulations described by ICG disposition did not change with halothane anesthesia; more than one third of the blood volume was in the central circuit (V^c), approximately 10% was in the fast nondistributive circuit (V^ND-F), and just over half was in the slow nondistributive circuit (V^ND-S)(Figure 1, Table 2). Halothane did cause a progressive and dose-related decrease in flow through the central circuit (dye-dilution CO), up to a 45% reduction at 2% halothane. The slow nondistributive intercompartmental clearance of ICG (Cl^ND-S), but not its fast nondistributive intercompartmental clearance (Cl^ND-F), was significantly correlated with end-tidal halothane concentration and was significantly different from the baseline value at 2% end-tidal halothane. The systemic distribution of CO not represented by Cl^E(90–94% of CO) was nearly evenly divided between the peripheral intravascular circuits described by ICG in the awake dog and remained so at 1% and 1.5% halothane anesthesia, with flows through each circuit decreasing directly in proportion to the decrease in CO. At 2% halothane, relatively more of the CO flowed through the fast circuit (51%) than the slow circuit (40%).

The volumes of the recirculatory inulin pharmacokinetic model were also unaffected by halothane anesthesia; approximately 20% of the volume of inulin was in the central and peripheral nondistributive circuits whereas one quarter of the volume was the rapidly equilibrating (fast) tissue and the balance was the slowly equilibrating tissue (Table 3). As in the ICG model, Cl^ND-S, but not Cl^ND-F, was correlated with end-tidal halothane concentration; in contrast to the ICG model, Cl sub ND-S was significantly different from the baseline value at all end-tidal halothane concentrations. Approximately 80% of CO was nondistributive in the inulin models in the dogs under all experimental conditions. Although the nondistributive flows were nearly evenly divided between the fast and slow circuits during all levels of halothane anesthesia, in the awake animals 30% was Cl^ND-F and nearly 50% was Cl sub ND-S. Neither the fast nor the slow distributive (intercompartmental, tissue) clearance (Cl^T-F and Cl^T-S) of inulin was affected by halothane in a statistically significant way in halothane-anesthetized dogs. Inulin Cl^E(glomerular filtration rate) was correlated with end-tidal halothane concentration, and at 2% end-tidal halothane it was significantly less than the baseline value.

The recirculatory antipyrine pharmacokinetic model was most profoundly affected by halothane anesthesia (Table 4). The only peripheral nondistributive volume that could be identified in the antipyrine model, V^ND, [5] increased significantly and in a halothane dose-related manner from a miniscule volume of 0.05 l in the awake animals to 0.32 l in dogs anesthetized with 2% halothane. The other antipyrine compartmental volumes were largely unaffected by halothane anesthesia; less than 5% of the volume of antipyrine was in the central vascular circuit and pulmonary tissue (V^T-P), the latter of which had a volume of approximately 0.1 l. The antipyrine VT-F had approximately 17% of the distribution volume, while the bulk of the volume (79%) was in V sub T-S. Only 7.5% of CO was nondistributive (Cl^ND) in the antipyrine models in awake dogs, but as CO decreased with halothane anesthesia, nondistributive clearance (Cl^ND) actually increased significantly, both absolutely and relatively, and represented 33–45% of CO in halothane-anesthetized dogs. Antipyrine tissue distributive clearances (Cl sub T-F and Cl^T-S) not only decreased significantly in absolute terms in a halothane dose-dependent manner but, more importantly, it also decreased as a percentage of CO; although more than 85% of the CO was involved in drug distribution in the awake dogs, this decreased with halothane anesthesia to a minimum at which only 50% of CO was involved in drug distribution in dogs anesthetized with 2% halothane. As a result of the increase in nondistributive clearance and decrease in distributive clearance during halothane anesthesia, the AUC during the first 3 min after drug administration more than doubled (Figure 4, Table 5). Antipyrine Cl^Ewas correlated with end-tidal halothane concentration and was significantly less than the baseline value at all levels of end-tidal halothane.

The observed halothane-induced increase in heart rate, decrease in blood pressure, and decrease in CO, with no change in systemic vascular resistance (Table 1), is consistent with the cardiovascular effects of halothane reported by others. [1] The association of halothane anesthesia with a decrease in hematocrit has also been described. [31]

Cardiac outputs and halothane dose-related changes in CO determined by ICG and inulin (i.e., indicator) dilution (Table 2) are nearly identical to those determined by thermal dilution (Table 1), indicating that our sampling schedule was appropriate. These progressive decreases in CO with increasing halothane concentrations are associated with the expected increases in the first-pass AUCs (AUC^first-pass = dose/CO) not only for ICG and inulin but also for antipyrine (Figure 2, Figure 3, Figure 4). The first-pass curve is largely complete within the first half minute.

The AUC⁰-[infinity] of a complete drug concentration history can also increase significantly as a result of a decrease in the Cl^Eof the drug (AUC⁰-[infinity]= dose/Cl^E). This Cl^Edependent increase in AUC is readily apparent in the curves of several hours duration, as illustrated in the insets of Figure 3and Figure 4. However, simulations revealed that changes in Cl^Ecannot explain the increase in the antipyrine AUC after first-pass, long before Cl^Ebecomes the major determinant of the drug concentration versus time relation (Figure 4).

An important observation of the present study is that halothane anesthesia causes a significant increase in the AUC of antipyrine in the critical first minutes after drug administration due to an increase in the apparent flow of blood not involved in drug distribution to the tissues, despite a halothane dose-dependent decrease in CO. Using the blood drug concentrations of the recirculation (second-pass) peak, the recirculatory model describes an independent nondistributive blood flow, or clearance, which returns the lipophilic marker to the central circulation after minimal tissue equilibration. This pharmacokinetic shunt reflects the inhomogeneity of tissue blood flow, the presence of significant diffusion barriers, or both. Halothane-induced changes in CO and its distribution alters the balance of distributive and nondistributive blood flows to various tissues, which are not proportional to changes in CO. Because nondistributive blood flow quickly returns the lipophilic marker to the central circulation, the increase in nondistributive blood flow by halothane increases the arterial blood AUC during the early minutes after drug administration (Table 5). The increased arterial drug concentrations resulting from a larger nondistributive clearance increases drug exposure of the sites of action of drugs with a rapid onset of effect, for which antipyrine is a pharmacokinetic prototype, and would be expected to result in a more profound and prolonged effect of these drugs.

The study of regional blood flow during halothane anesthesia by Gelman et al. [32] suggests the basis of the increase in nondistributive blood flow during halothane anesthesia in dogs. These investigators found that during 1 and 2 minimum alveolar concentration halothane anesthesia, the percentage shunting of 15-micro meter radiolabeled microspheres was 2.5 and 4 times that during awake control, respectively. Although preportal and muscle blood flow decreased in these studies, renal blood flow was preserved and cerebral blood flow nearly doubled. Thus the substantial increase in nondistributive blood flow described in the present recirculatory antipyrine pharmacokinetic model may be due, at least in part, to the decrease in muscle tissue blood flow and the maintenance of renal blood flow (which is essentially nondistributive) described by Gelman et al. [32] Together, the doubling of the antipyrine AUC⁰-3 min due to increased nondistributive blood flow observed in the present study and the near doubling of cerebral blood flow reported by Gelman et al. [32] would provide mechanistic explanations for profound increases in the central effects of intravenous anesthetic agents administered during halothane anesthesia.

Physiologically based pharmacokinetic models describe measured blood and tissue drug concentration histories by proportioning CO, and hence drug distribution, among tissues or tissue groups with similar perfusion and drug solubility characteristics. [39] These models have proved useful in providing insight into factors affecting drug disposition [33] and predicting the effect of altered physiology on drug disposition. [34] However, such models have well-known limitations, including their requirements for large amounts of data (tissue drug concentrations and blood flows), which are often unavailable (especially in humans), their inability to characterize individuals, and their many assumptions that ignore interindividual variability and physiologic changes. [39] They are also limited by the many assumptions that must be made in interpreting tissue concentration data and adjusting the model for different conditions of body composition and blood flow. For example, because physiologic models generally assume that tissue blood flow is homogeneous, some models assume all blood leaving a tissue is in equilibrium with it, [34] whereas others interpret their tissue data as demonstrating diffusion barriers for flow-limited drug uptake. [35] When simulating drug disposition in the presence of altered physiology, physiologic models adjust regional blood flows in one of several ways. Regional blood flows may be changed in physiologic simulations in direct proportion to changes in CO, [36] or adjusted arbitrarily and independently, [34] or adjusted based on radioactive microsphere regional blood flow measurements [37,38] while maintaining the assumption of either complete tissue equilibration with exiting blood or diffusion barriers for flow-limited drug uptake.

Traditional multicompartmental pharmacokinetic models are mathematical constructs based on the characterization of the multiexponential plasma drug concentration versus time relation. [39] These models have been useful in characterizing drug disposition in subjects under various conditions and provide guidance for drug dosing, particularly when drugs are administered as multiple doses or by continuous infusion. However, because these models lack an anatomic and physiologic basis, they cannot be used to predict the effect of alterations in CO on drug disposition. In addition, neither the traditional compartmental model nor the physiologic model account for measured drug blood, or tissue, concentrations in the first minutes after drug administration when intravenous anesthetic drugs produce their maximum effect. [40]

The present recirculatory multicompartmental model of the disposition of physiologic markers based on frequent early arterial blood sampling retains the best aspects of the traditional multicompartmental model and the physiologically based model besides offering several significant advantages over both. The traditional multi-compartmental model and the recirculatory model describe data from individuals (including humans) collected under various conditions, such as physiologic changes produced by different levels of anesthesia. The physiologic model and the recirculatory model incorporate physiologic factors, such as CO and its distribution, in a description of drug disposition. However, only the recirculatory model can describe drug disposition from the moment of injection, including a description of pulmonary drug distribution (uptake), which an early version of this model [41] lacked. [42] Unlike the physiologic models, a recirculatory model neither assumes that all tissues are in equilibrium with the blood leaving it nor invokes the concept of diffusion barriers for flow-limited drug uptake. Instead, the recirculatory model uses the blood drug concentrations of the recirculatory peak to describe a nondistributive blood flow, or clearance, which returns blood to the central circulation after minimal tissues equilibration. This can be thought of as a pharmacokinetic shunt (second-pass peak, Figure 1, Figure 2, Figure 3, Figure 4).

The AUC is often used as a measure of drug exposure. In evaluating the effect of changes in AUC, the time course of drug effect must be considered. For example, the AUC⁰-[infinity] for cancer chemotherapeutic agents has been correlated with efficacy and toxicity because these drugs, which act irreversibly, produce effects that are a function of the product of drug concentration and time. [43] For rapidly and reversibly acting drugs, it may be more useful to gauge drug exposure of the effector site by arterial AUC during times that might correspond to peak drug effect.

The increase in AUC of antipyrine for at least the first 3 min after drug administration is due to increased nondistributive blood flow during halothane anesthesia (Table 5) and reflects an increased exposure of the brain to antipyrine. For rapidly acting drugs, such as the centrally acting intravenous anesthetics for which the lipophilic marker antipyrine is a prototype, this increase in nondistributive blood flow (and, as a result, AUC) would produce a more profound and longer-lasting drug effect due to exposure of potential sites of drug action to higher drug concentrations for a longer period of time. A more profound onset and longer duration of effect of the hydrophilic drugs for which inulin is a prototype (e.g., the relatively slow-onset neuromuscular blockers) is not likely to be seen during halothane anesthesia for distributional pharmacokinetic reasons because observed differences in the AUC of inulin did not correspond to the expected time course of drug (e.g., neuromuscular blocker) onset and duration of action (Table 5).

Traditional three-compartment pharmacokinetic models are constructed from triexponential equations describing the blood or plasma drug concentration versus time relation. The initial volume of distribution (V^t) is that volume in which a drug appears to mix instantaneously before being distributed throughout the rest of the total volume of distribution; the volume estimate of V¹is related to the time interval between drug administration and the collection of the first blood samples, because the drug will be more extensively distributed with the passage of time. From V¹, drug distributes throughout the rest of its volume by a process called intercompartmental clearance (Cl^I) and is irreversibly removed from the body by elimination clearance (Cl^E). The volume of distribution at steady state (V^ss) is the total volume of distribution and, as such, is the sum of V^Iand the rapidly (fast) and slowly equilibrating volumes of distribution (V^Fand V^S, respectively). Intercompartmental clearances between V^Iand both V sub F and V^S(Cl^F, and Cl^S, respectively) are volume-independent estimates of drug transfer that are directly determined by blood flow and transcapillary permeability.

The recirculatory pharmacokinetic model of ICG disposition based on frequent early arterial blood samples describes intravascular mixing and blood flows through the many circuits of the body by deriving estimates of not only blood volume and CO but also their systemic distribution (Figure 1). The many blood circuits lump into kinetically distinct pathways based on their blood volume to blood flow ratios (MTTs). [5,9,44,45] This ICG pharmacokinetic model is isomorphic with the lumped parameter recirculatory models describing the physiology of the systemic circulation, [46–48] which have a central volume and two parallel peripheral circuits representing the peripheral circulation. These parallel peripheral circulations are characterized by time constants that reflect a low capacitance, low volume (fast) circuit and a high capacitance, larger volume (slow) circuit, the latter of which presumably represents the splanchnic circulation. Both the physiologic models and our model distribute CO between the two parallel peripheral circuits, and the models can accommodate the redistribution of CO and blood volume. Physiologic models require estimates of blood flow to each significant tissue group. Recirculatory models contain estimates of the lumped behavior of the circulation, which are determined directly from easily acquired data. This multi-compartmental recirculatory model can be interpreted physiologically. [49]

Simultaneous time-density function analysis of an intravascular marker and a tissue water marker has been used to estimate extravascular lung water. [50,51] Blood flow to the lungs is assumed to be the CO determined by ICG and inulin kinetics, and the central (thoracic) blood volume is derived from the product of CO and the MTT for the markers between the injection and sampling sites. Similarly, the antipyrine central volume is the product of the composite mean transit time of the central pathways for antipyrine and the CO. [8] The apparent lung tissue volume (V^T-P) of antipyrine is the difference between the antipyrine central blood and tissue volume and central intravascular volume codetermined by ICG and inulin.

Characterization of the dispositions of inulin and antipyrine simultaneously with that of ICG allows characterization of their early disposition as the convolution of their intravascular behavior, determined by the pharmacokinetics of concomitantly administered ICG and tissue distribution pharmacokinetics. [5] For pharmacokinetic models in which drug distribution is known to occur from intravascular space, defined in the present recirculatory, models, Cl^Ioccurs by transcapillary exchange, which is determined by blood flow (Q) and diffusion (capillary permeability coefficient surface area product, P)[52,53]:Equation 1. The physicochemical properties of some substances, such as antipyrine, make them freely diffusible; for them, Cl^Iapproaches actual blood flow to tissue (i.e., Cl^I= Q when P much > Q because e sup -P/Q [righ arrow] 0). [52,53] For substances that are not freely diffusible, such as inulin, Cl^Iis related to the permeability coefficient surface area product (i.e., Cl^I, = P when P much < Q because the higher-order terms of the series expansion of Equation 1can be neglected). [52,53] The difference between CO and the sum of intercompartmental clearance to tissues for a given drug is the portion of CO not involved in flow-limited tissue distribution of that drug, or nondistributive flow, which returns (shunts) drug to the central circuit. Tissue volume to tissue distribution clearance ratios determine whether a compartment is a fast or slow nondistributive peripheral pathway or a rapidly or slowly equilibrating tissue. [5,45]

Our model also includes recirculation and time delays, as does the detailed model of the circulation developed by Jacques et al. [54] A time delay accounts for the noninstantaneous appearance of drugs in a compartment after being transferred from another compartment. A delay element is a mathematical description of the distribution of drug transit times which can be described by the average (MTT). It also has an associated apparent volume calculated as the product of the flow through the delay and its MTT. In the recirculatory model, the central delay represents the central blood and tissue volume between the sites of drug injection and blood sampling. The peripheral delay elements represent kinetically distinct peripheral nondistributive pathways. Because flow in a medium generates a distribution of time lags, [29] the time delays in the present model are represented as delay elements rather than as discrete time lags, which are more appropriate for characterizing ideal plug flow models.

We used the distributed delay elements of the SAAM II kinetic analysis software to describe the first-pass data in the recirculatory model. [25] These delay elements are composed of n compartments connected by identical rate constants k such that n/k is equal to the MTT for the delay. As n increases, the arterial drug appearance curve becomes narrower until at n =[infinity] the delay becomes a discrete lag (i.e., a spike). Normally the number of compartments, n, in a model cannot be adjusted during parameter optimization. Therefore, we fit the first-pass data using the closed form equations of this type of distributed delay in sequence and the TableCurve 2D software. First, the data, weighted uniformly, were fit to the sum of two gamma distribution functions, which allow n to be any positive value (i.e., it permits fractional compartments). Then n was rounded to the nearest integer value and fixed as a constant, and the data were fitted again using the sum of two Erlang distribution functions; the Erlang function is essentially the gamma distribution with n confined to integer values (i.e., it permits only whole compartments):Equation 2where A¹and A²are the proportional blood flows through the two parallel elements. Thus the number of compartments for the delay element, their rate constants, CO (ICG and inulin), and the proportion of flow through each parallel central pathway are estimated. These results were placed as fixed parameters into the recirculatory model.

The central circuits and the nondistributive peripheral compartments (shunts) can be estimated only by obtaining frequent early arterial blood samples. If a less-rigorous sampling schedule is used (i.e., if the sampling frequency is inadequate to define the first- and second-passes), then these data can only be modeled as the traditional V sub I[center dot] V^Fand V^Sso derived are similar (equivalent) to the peripheral tissue compartments in the recirculatory model.

The authors thank Mary Pat Janowski for technical assistance.