Abstract
Thirtythree paired indicator/nutrient dilution curves across the mammary glands of four cows were obtained after rapid injection of paraaminohippuric acid (PAH) plus glucose into the external iliac artery. For the measurement of extracellular volume and kinetics of nutrient uptake from indicator dilution curves, several models of solute dispersion and disappearance have been proposed. The CroneRenkin models of exchange in a single capillary assume negligible washout of solutes from the extracellular space and do not describe entire dilution curves. The Goresky models include a distribution of capillary transit times to generate whole system outflow profiles but require two indicators to parametize extracellular behavior. A compartmental capillary, convolution integration model is proposed that uses one indicator to account for the extracellular behavior of the nutrient after a paired indicator/nutrient injection. With the use of an iterative approach to least squares, unique solutions for nonexchanging vessel transit time _{μ} and its variance σ were obtained from all 33 PAH curves. The average of heterogeneous vascular transit times was approximated as 2σ = 8.5 s. The remainder of indicator dispersion was considered to be due to washout from a wellmixed compartment representing extracellular space that had an estimated volume of 5.5 liters or 24% of mammary gland weight. More than 99% of the variation in the time course of venous PAH concentration after rapid injection into the arterial supply of the mammary glands was explained in an unbiased manner by partitioning the organ into a heterogeneous nonexchanging vessel subsystem and a wellmixed compartmental capillary subsystem.
 indicator dilution
 mathematical models
 arteriovenous differences
the mammary glands of a lactating cow constitute a highly vascularized tissue that is perfused with 500–600 liters of blood for every kilogram of milk produced (9, 38). As in other tissues, blood flow appears to be regulated in accord with mammary metabolic activity (10, 34). Presumably, regulation of blood flow serves to modify concentrations of milk precursors in extracellular fluid bathing mammary epithelial cells and thereby influences precursor uptake and milk synthesis. Attempts to express hypotheses of milk synthesis in mathematical form (10, 23) have relied on characterizations of extracellular space taken from other tissues such as muscle. A technique particularly suited to the task of measuring extracellular volume and kinetics of nutrient uptake in intact organs is the indicator dilution (ID) technique. The ID technique has had a long history of development and implementation (11, 20, 31). It entails the rapid injection of a mixture of single or multiple indicators and target nutrients into the artery immediately at the entrance of an organ and the characterization of outflow dilution profiles from serial venous samples. The bolus injection of nutrients can elicit a dynamic concentration profile over a wide range for a short period of time without influencing the physiological status of the animal.
Indicators are included in the bolus injection to indicate what would happen to the target nutrient if there were no uptake. Intuitively, then, the difference between nutrient and indicator dilution curves represents uptake. How exactly the uptake rates can be ascertained from this difference is the subject of several mathematical models. The mathematics for the various methods range from simple algebra to partial differential equations with complex integral calculus. Algebraic models describe, with arguable precision, only a portion of the entire dilution curve. The simplifying assumptions that have been made to facilitate analytic solution of the partial differential equations are no longer necessary with the ready availability of computer simulation software for numerical integration. In this paper we examine the underlying assumptions of several ID models and synthesize from them a new simulation method to interpret paired indicator/nutrient dilution curves.
In the proposed compartmental capillary, convolution integration (CCCI) model, all capillaries are identical and consist of wellmixed compartments representing extracellular and intracellular spaces and the various fluxes between and without. The heterogeneity of nonexchanging vessel transit times is used to transform concentrations of indicator in the outflow of a single capillary to concentrations in the whole system outflow in a linear accumulative manner. Common extracellular indicators for indicator dilution studies are radiolabeled substances, such as sucrose, creatinine, inulin, or mannitol. Dyes, such as Evans blue (20) and indocyanine green (28), have been used without being radiolabeled. Unlabeled paraaminohippuric acid (PAH) has not been used in bolus indicator dilution studies but is often infused continuously as an indicator for blood flow measurements (43). Objectives of the current work were to evaluate the suitability of PAH as an extracellular indicator for the mammary glands and of the CCCI model to estimate parameters of the extracellular space in lactating bovine mammary glands. A companion paper describes testing of alternative capillary submodels for estimation of glucose transport and metabolism parameters (35).
MATERIALS AND METHODS
Unlabeled PAH sodium salt, dglucose, and Evans blue were purchased from SigmaAldrich Canada (Oakville, Ontario, Canada). All other chemicals and reagents were commercial products of analytic grade.
Either 0.3–0.5 g of PAH or 0.25 mg of Evans blue, as extracellular indicators, with 1.4–16.4 g glucose, were dissolved in 25 ml water at 50°C, and the solution was passed through a 0.22μm disposable syringe filter (Millipore Canada, Etobicoke, Ontario, Canada). Doses of each solution (5 ml) were stored at 4°C in capped syringes. Also 1 ml of each solution was diluted 100 times and stored at −20°C for chemical analysis.
All procedures were approved by the Animal Care and Use Committee of the University of Guelph according to the guidelines of the Canadian Council of Animal Care. Thirtythree PAH dilution curves across the mammary glands of four cows were obtained after rapid injection of PAH plus glucose into the external iliac artery. In three of the injections, Evans blue was included. Four Holstein cows ranging from 99 to 250 days in milk and producing 24.9 kg/day (SD 4.6) milk were surgically fitted with polyethylene catheters in both external iliac arteries and in both subcutaneous abdominal veins draining the udder (29). The catheters were flushed twice a day with saline and filled with 2 ml concentrated heparin (1,000 USP U/ml, Organon Teknika, Toronto, Canada). All injections were conducted between 10:00 AM and 3:00 PM of a single day for each cow. The venous sampling system consisted of a peristaltic pump delivering blood at an average of 0.48 g/s (SD 0.08) to a fraction collector programmed to collect 25 blood samples at 5s intervals into tubes containing 15% K_{3}EDTA. Immediately before injection into the artery, a blood sample was collected from the contralateral arterial catheter for measurement of background indicator concentration. The fraction collector was started simultaneously with rapid injection of 5 ml of the test solution into the arterial catheter. The approximate time over which the dose was administered, t_{dose}, was recorded. After sampling, arterial and venous catheters and the sampling system were flushed with heparinized saline (10 USP U/ml). Plasma was removed by centrifugation and stored at −20°C until analyzed.
For Evans blue, absorbance at 620 nm of plasma samples and test solutions was determined on a spectrophotometer (Spectronic 1201, Milton Roy, Rochester, NY). PAH was analyzed by a modification of the spectrophotometric procedure of Smith et al. (43) to use gravimetric rather than volumetric dispensation and a standard PAH solution (10 ppm) in each batch to calculate concentration.
The dose (q_{o}) of indicator injected was calculated from the difference in syringe weight before and after injection, and indicator concentration was analyzed in the test solution. The sampling rate, f (ml/s), was calculated as the difference in weight of the sample tray, before and after sampling, divided by the total sampling time (125 s). The sampling delay, t_{s}, was calculated as, V_{s}/f, where V_{s} is the volume of sampling tubing.
Each observation on the indicator dilution curve was corrected for baseline by subtracting the average of the first three observations. Recirculation of the dose was removed by monoexponential extrapolation of six observations of the downslope of the concentrationtime dilution curve ic(t). Area under the curve was calculated as (1) where b is the monoexponential downslope, Δt is the sampling interval, and n is the number of observations before the extrapolation starts.
External iliac plasma flow (ml/s) was calculated as (2) The average residence time (s) of indicator in extracellular space was calculated as (3)
RESULTS AND DISCUSSION
Recirculation time.
Recirculation time was measured by rapid injection of PAH into one arterial catheter and sampling from the contralateral venous catheter (Fig. 1A). A small peak between 22.5 and 42.5 s indicated that there was some lateral mixing of blood in the mammary glands. The early peak of 2.0 × 10^{−6}/ml was only about 1% of the peak value of venous outflow from an ipsilateral PAH injection (Fig. 1, B and C), so lateral mixing was negligible. Metcalf et al. (32) noted significant lateral crossover of arterial PAH in only two of nine cows studied. Recirculation commenced 40 s after first appearance time and reached a peak 5 s later (Fig. 1A). This recirculation time of 45 s agrees with the 40–50 s measured by Thivierge et al. (45) in secondlactation cows. Accordingly, the six observations after peak and before 45 s (excluding first appearance time) were used in subsequent experiments to obtain the monoexponential slope for extrapolation to infinity.
Evaluation of extracellular indicator.
Three paired Evans blue and PAH injections were conducted. Normalized for dose, the Evans blue curve displayed a higher peak and a smaller wash out tail than the PAH curve (Fig. 1B). The ratio of the mean transit times (excluding first appearance time) between Evans blue and PAH was 0.47 (SD 0.06; n = 3), indicating that there was a significant portion of Evans blue binding with albumin (36) and staying in the vascular space (25). The venous recovery of PAH was 99.0% when Evans blue was used as reference indicating that PAH was not removed by the mammary glands.
The ideal extracellular indicator for application of the CCCI model to parameterize nutrient uptake should freely and rapidly diffuse through the capillary wall and into the same extracellular space as the nutrient under study. Evans blue binds with albumin in the blood and did not freely diffuse into the extracellular compartment. The molecular mass of PAH is 216 Da, which is similar to that of glucose and the large amino acids. Also, the clearance of PAH by the kidney is very efficient so that PAH was almost completely cleared out of plasma by 15 min postinjection. PAH was deemed a good indicator for detecting properties of the extracellular space of the bovine mammary glands.
Paired indicator/nutrient dilution curves.
A typical paired PAH/glucose dilution curve is shown in Fig. 1C. After correction for recirculation, the two extrapolated curves cross at 110 s, indicating washout of the injected nutrient from a space inaccessible to PAH. There is currently no established method for estimating characteristics of uptake of the nutrient in a paired indicator/nutrient dilution experiment that considers washout. However, components of previous ID models could be amalgamated into a new construct that could serve in interpretation of Fig. 1C.
Previous indicator dilution models.
The single capillary where exchange of the nutrient occurs is the basic unit for most ID models. However, the whole system outflows that are measured reflect not only exchanges in capillaries but also dispersion along sampling tubes and nonexchanging blood vessels. Figure 2 illustrates how each model takes different account of the single capillary exchange and the influence of vasculature on whole organ outflow.
Several investigators (16, 17, 48) have used the classic CroneRenkin (14, 37) model of single capillary extraction to calculate the permeability of an organ’s vasculature to nutrients of interest. In these studies, a vascular indicator is injected simultaneously with the nutrient, the latter of which is an extracellular indicator that does not enter cells, such as [^{14}C]sucrose or [^{14}C]mannitol. The capillary in the CroneRenkin model is assumed to be a selectively permeable cylinder that exchanges with an extravascular space of constant radius along its length (26). It is assumed that there is no flux of extracellular indicator back into the capillary once it has left and that there is no diffusion occurring along the length of the capillary. The consequence of the assumptions is that both indicators exit from the single capillary at the same time, the extracellular one at a lower concentration due to extraction (Fig. 3A). To produce the characteristic whole tissue outflow curve, this venous labeled blood mixes with varying proportions of unlabeled blood from capillaries that the pulse dose has not reached yet or has passed already. Thus the extraction ratio of indicators should remain constant across time.
In practice, extraction is not constant and declines rapidly to below zero (15). After the pulse dose of indicators has passed through the capillary, extracellular fluid is loaded with indicator molecules that will wash back into the venous outflow that contains no reference vascular indicator and thereby artificially reduces extraction. In fact, washout does not wait for the dose to finish traversing the capillary but will commence immediately on appearance of test indicator in the extracellular space.
Yudilevich and colleagues (55, 56) adapted the CroneRenkin model to estimate rates of unidirectional nutrient uptake by organs. In their approach, the reference is an extracellular indicator, such as [^{14}C]mannitol and [^{14}C]sucrose, instead of a vascular indicator. In addition, the nutrients of interest differ from the reference indicator in that they are taken up into cells and metabolized. Again, the observed extraction of nutrient relative to extracellular indicator starts low, increases to a peak value, and then decreases and even becomes negative at the tail (8, 25, 56). The change in extraction is also observed in Fig. 1C. Arbitrarily, peak extraction is used to calculate parameters of nutrient utpake.
The CroneRenkin models do not explain how an input curve of a few seconds or less is transformed to a whole organ outflow dispersed over a couple of dozen seconds. Goresky (20) introduced two elements into the modeling that allowed entire dilution curves to be described. The first was a consideration of extravascular spaces of the capillary supply zone and the second was a distribution of capillary transit times.
The single capillary of Goresky (20) is similar to that of the CroneRenkin model except that time and distance are considered simultaneously to predict indicator concentration at the venous end of a twospace capillary. Appearance of an extravascular indicator at the venous end is delayed relative to that of the vascular indicator because of mixing between a nonconvective extravascular space and a convective vascular space but the shape of the outflow curve is unchanged from the input function because this mixing is assumed to be instantaneous. The contrast between the Goresky model and the CroneRenkin model in describing exit of extracellular indicator from a single capillary is shown in Fig. 3A.
The dispersion of an arterial pulse dose of indicators into its venous concentration profile is a matter of transit time heterogeneity (31). The dispersion model of Roberts and Rowland (39) is based on the hypothesis that a pulse dose of an extracellular indicator injected into the arterial supply of an organ will disperse, due to the heterogeneity of blood flow, into an inverse Gaussian distribution of concentrations in the collecting vein (Fig. 2). The single capillary is not explicitly represented. Exchange of injected nutrients between extracellular and intracellular spaces is represented with firstorder rate constants to yield an equation for the profile of nutrient concentrations in the organ outflow. The five unknown parameters of the equation, representing whole system extracellular volume, degree of dispersion, and rate constants for nutrient influx, efflux, and sequestration, can be estimated by nonlinear fits to a single nutrient dilution curve without the use of any indicators (52). However, in the Goresky (20) proposition that the heterogeneity of vascular indicator transit times was due to different lengths of capillaries, a vascular indicator provides the capillary transit time distribution function. The normalized indicator concentration flowing out of the whole organ is a convolution of the concentrations flowing out of each capillary and their respective transit times from the paired vascular indicator curve.
Goresky (20) assigned all of the heterogeneity of transit times to characteristics of capillaries, but later work (22, 40, 41) explored the consequences of assuming heterogeneity in nonexchanging vessels. Indicator dilution curves across the heart were best fit with a distribution of nonexchanging vessel transit times, setting pathway length as a function of capillary transit time (40). The majority of the heterogeneity, though, was attributed to capillaries. Audi et al. (1) described the distribution of capillary and nonexchanging vessel transit times as two separate curves (Fig. 2), the convolution of which yields the whole organ outflow.
Cobelli et al. (13) proposed a network of wellmixed compartments (Fig. 2) to account for the heterogeneity of transit times through an organ. Three parallel pairs of identical compartments between injection and sampling compartments were found to be adequate to fit the downslope of extracellular indicator and transportable substrate concentrations in the venous drainage of the human forearm. Each pathway from compartment 1 to 8 is essentially a series of firstorder delays as pulsedosed indicator molecules mix with unlabeled blood and washout, mix with the next compartment and washout, and so on. Uptake of nutrient is considered possible from the intermediate compartments only. The parallel pathways are a representation of the different routes indicator molecules may take as they proceed through an organ but the compartments in series have no direct physiological corollary. The single capillary and nonexchanging vessels are not represented explicitly.
Van der Ploeg et al. (47) used a single wellmixed compartment in series with a venous transit time delay (Fig. 2) to estimate O_{2} distribution volumes in goat hearts from a step change in blood flow. Assuming multiple identical compartments in series to approximate Krogh cylinder behavior, like in the CroneRenkin model, yielded gross overestimates of coronary volume (47). It was concluded that the Krogh cylinder was an inappropriate model of coronary O_{2} exchange. Distributing flow heterogeneously to nine parallel pathways, each consisting of a wellmixed compartment and transit time delay in series, yielded volume estimates that differed only 8–15% from those of the single pathway model.
A CCCI model.
Implementation of a Goresky model to study nutrient transport and metabolism requires that both a vascular and extracellular indicator be injected along with the nutrient of interest. Intuitively, it seems that an extracellular indicator alone should be able to account for the extracellular behavior of the nutrient in a paired indicator/nutrient injection as used in the Yudilevich et al. (56) adaptation of the CroneRenkin model. The CroneRenkin models, however, do not consider heterogeneity of transit times between injection and sampling sites and, consequently, cannot describe the entire time course of venous dilution. A new approach, therefore, is proposed here to model paired indicator/nutrient dilution curves by adapting ideas from previous models of nutrient exchange across organs. A split of the heterogeneity of transit times between exchanging and nonexchanging vessels as implemented by Audi et al. (2) is assumed. But instead of interpreting the capillary transit function from a cylinder model of the single capillary, the wellmixed compartment of van der Ploeg (47) will be used (Fig. 2). Audi et al. (2) indicated that capillary dispersion modeled by a single exponential decay was consistent with observations of the capillary transit time function in isolated rabbit lungs. Capillaries in many tissues form closeknitted networks (30, 34) that cause extensive mixing of the fluid elements and where multiple capillaries may be draining a common extracellular space (49). It has often been recognized that the Krogh cylinder is an inadequate representation of solute delivery to cells (6, 47). Considering the capillary and its surrounding fluid as a wellmixed compartment may be an appropriate simplification.
Goresky (20) introduced the use of the heterogeneity of transit times to integrate from single capillary to whole system outflows. The heterogeneous component of vascular indicator transit through rat liver averaged 10.9 (19), 8.3 (54), and 9.8 s (42). More than 9 s of mean capillary transit time seems too long to exchange nutrients and wastes in a flowlimited manner. Mean transit time through the capillary has been estimated to be ∼20% of the mean of whole organ transit (5, 27). As an alternative to assigning heterogeneity to capillary transit, then, our model considers the heterogeneity of vascular indicator transit times to arise exclusively in the nonexchanging vessels.
In the CCCI model, it is assumed that indicators and nutrients instantaneously mix with the extracellular space served by a single capillary. The extracellular space and intracellular space are treated as two wellmixed compartments (Fig. 4). A convolution integral is implemented to integrate from uniform single capillary outflows to whole system outflow according to the heterogeneity of transit times in nonexchanging vessels.
Assuming identical capillaries in the organ, a general description of the single compartmental capillary model is (4) and (5) where V_{e}, V_{c}, j_{1}, j_{2}, and j_{3} represent the whole system extracellular volume, intracellular volume, and influx, efflux and metabolism rates, respectively, and other variables are defined in Fig. 4.
The input dose function, c_{sc_a}(t), is based on a rectangular pulse (6) where q_{0} is the dose injected, t_{dose} is the duration of the injection, p(t) is a rectangular pulse equal to 1 between time 0 and t_{dose} and 0 otherwise, and c_{a}(0) is the background arterial concentration. If only the extracellular indicator is of interest, j_{1} = j_{2} = j_{3} = 0 so that Eq. 5 is ignored, Eq. 4 has the analytic solution, (7) Figure 3B compares indicator venous outflows from a single capillary simulated by the CroneRenkin model and the wellmixed compartmental model. The indicator reaches peak concentration at t_{dose}. How fast it is washed out of the capillary distribution space is positively related with F and negatively related with V_{e} (Eq. 7).
The Gaussian function is used to describe the density distribution of vascular transit times as (8) where t̄_{μ} represents the mean of transit times ascribed to the nonexchanging blood vessels before and after the capillary, which includes a common large vessel transit time, and σ represents SD of the mean. Both indicator and nutrient share the same nonexchanging vessel pathways, described by the normal distribution function f(t). The entire system outflow profile is the convolutionary accumulation of each single capillary outflow in these different delays as (9) Equations 4, 5, and 8 were solved numerically with a fourthorder RungeKutta algorithm using a step size of 0.0001 s. At each increment of simulated time, c_{sc_v} and the area under the f(t) function, Δf(n), were added into respective arrays for numerical solution of Eq. 9 with a step size of 0.001 s. Simulations of extracellular indicator and nutrient concentrations in the organ outflow, where j_{1} = k_{1}V_{e} c_{sc_v}, j_{2} = k_{2}V_{c} c_{sc_c}, and j_{3} = k_{3}V_{c} c_{sc_c} for the nutrient, are shown in Fig. 5. Extraction according to the CroneRenkin model started low, increased to a peak of 0.52, and declined to negative values past t = 75 s (Fig. 5). This pattern is typical of the paired indicator/nutrient experiment (8, 21, 28, 56). The steadystate solution of Eqs. 4 and 5 for j_{3} = k_{3}V_{c} c_{sc_c} gives an extraction (10) which equals 0.22 for the scenario shown in Fig. 5. Thus peak extraction calculated according to the paired indicator/nutrient method of Yudilevich et al. (56) is not applicable to the steadystate situation. Depending on the numerical value of parameters for the volume of and rate of exchange with the intracellular space, the peak in extraction can occur at different time points and at various degrees of offset from the true extraction value. The steadystate extraction of Eq. 10 is, in fact, equivalent to the cumulative extraction of nutrient c_{sc_v} relative to indicator ic_{sc_v} (11) used by Wåhlander et al. (48) to interpret ID curves.
Parameter estimation.
Twelve to fourteen observations from each PAH curve were used to estimate t̄_{μ} and σ with the CCCI model using an iterative LevenbergMarquardt algorithm to find lowest residual sums of squares. Observations on the tail part of the PAH curves were not used because of the Hamiltonian extrapolation. To save simulation time, the initial three observations of the original curves, which include the sampling delay t_{s} and part of the common large vessel transit time t_{0}, were excluded. Because a portion of blood in the external iliac artery into which indicators were injected does not flow to the mammary glands, F and V_{e} could not be directly estimated so the area iA and monoexponential downslope b were substituted for iq_{0}/F and F/V_{e}, respectively, in Eq.4. Preliminary fits in which t_{dose} was estimated along with t̄_{μ} and σ yielded a median t_{dose} of 1.7 s (SD 0.5). Due to consistency of the estimate and to increase the power of estimation of t̄_{μ} and σ, a constant t_{dose} of 1.5 s was used for all simulations. Fits of the CCCI model to the paired glucose dilution curves, using iA, b, t̄_{μ}, and σ from the PAH fits, are presented in a companion paper where four alternative capillary submodels were evaluated (35).
Fitted parameters of PAH distribution and statistics are listed in Table 1. An example of a fitted curve is given in Fig. 6. A major goal of this study was to evaluate the CCCI model for its ability to describe indicator dilution curves. The model would be rejected if it were not possible to obtain a unique set of precisely estimated parameters. Repeatability of parameter estimation was high, with coefficients of variation of 13 and 14% within cows for t̄_{μ} and σ, respectively. Thus the optimization algorithm was able to converge on final solutions and those solutions were unique and identifiable.
The CCCI model was evaluated with respect to its ability to account for the observed plasma PAH values in all experiments. All r^{2} in Table 1 were close to 1. In addition, there was no evidence of a relationship between residuals and time scale (Fig. 7A) indicating unbiased prediction error across time.
In the indicator dilution technique, there is the assumption that bolus injection does not disturb the physiological status of the target system. Accordingly, estimates of the physiological parameters of the CCCI model should not be related with dose administered. Indeed there were no significant relationships between leastsquares parameter estimates and PAH dose (Fig. 7, B and C).
Finally, the model would be rejected if estimated parameters were implausible in the physical or physiological sense. The average of σ across all cows was 4.24 s (Table 1). The part of t̄_{μ} which is the sampling delay t_{s} and the common large vessel transit time t_{0} could be approximated as t̄_{μ}−2σ, because with lateral symmetry of the normal distribution function, 96% of nonexchanging vessel transit times would have values within t̄_{μ} ± 2σ. Thus the mean of heterogeneous nonexchanging vessel transit times t̄_{μ} was ∼2σ = 8.5 s. This mean derived from an extracellular indicator should be equivalent to that of an indicator that does not enter the extravascular space. The mean transit time (excluding common large vessel transit time) of a vascular indicator was 5.35 s in dog livers (14) and 8 s in pig hearts (19). These values are similar to our estimate of 8.5 s in cow mammary glands.
Using iA, b, t̄_{μ}, and σ from the PAH curve, paired glucose dilution curves were fitted without bias with an average r^{2} of 1.0 and an average SE of parameter estimation less than 32% of the parameter value (35). The rate constant for intracellular sequestration of glucose from the injected dose was not different from that calculated from the background areteriovenous difference of basal glucose. This equality indicated not only linear kinetics of glucose sequestration but also that the parameters obtained using the CCCI model in nonsteady state apply to the steadystate situation.
Physiological interpretation of parameters.
Whereas, for the Goresky models, a vascular indicator is actually injected to define the vascular transit time function, the CCCI model is meant to fit vascular distribution parameters from an extracellular indicator curve. The presumed distribution of vascular transit times is Gaussian but dilution of a vascular indicator, e.g., labeled erythrocytes, across an organ is typically skewed to the right (20, 42) and, therefore, not Gaussian. Bassingthwaighte et al. (7) also presumed that transit time of a vascular indicator was symmetrically distributed about a mean but that mixing and washout within the vasculature produced the skewness. In fact, Bassingthwaighte et al. (7) fitted essentially the same three parameters (b, t̄_{μ}, and σ) of the convolution of a Gaussian distribution with an exponential decay from indicator dilution curves between arterial injection and sampling sites. In applying the same approach in the CCCI model to describe extracellular indicator dilution, the estimate of volume V_{e} that is obtained thus includes vascular mixing space and is not representative of the interstitial space alone. The identical extracellular distribution space is also presumed to be accessed by the nutrient in the injectate so that the parameter b = F/V_{e} is relevant to the nutrient dilution Eqs. 4 and 5.
The CCCI model treats outflow from the whole organ as the convolution of capillary and nonexchanging vessel subsystems in the manner of Audi et al. (2). In the current study, the mean transit time of extracellular indicator in nonexchanging vessels (t̄_{μ}) averaged 25% of the mean residence time in extracellular space (t̄_{b}). Similarly, Audi et al. (2) found that nonexchanging vessels accounted for 27% of the dispersion of pulsedosed extracellular indicators in isolated, perfused rabbit lung. However, all of the heterogeneity of vascular transit times, represented by t̄_{2σ} and σ, is assumed to occur in nonexchanging vessels rather than in capillaries as in Goresky (20) and Linehan and Dawson (28). With timelapse radiography, Clough et al. (12) estimated that only 5% of the variation in transit time of a radiopaque vascular indicator through a single lobe of the dog lung could be attributed to the parallel distribution of dose into pathways of different duration within the arterial tree. Assuming the same dispersion occurs in the venous tree, up to 10% of the variation in transit time was observed outside of capillaries. However, Clough et al. (12) arrived at the figure of 5% from indicator dispersion in arteries as small as 200 μm in diameter, whereas diameters of terminal arterioles and capillaries are in the 10 to 30μm range (18). The small arteries and veins between 200 and 10 μm in diameter, distributed in parallel, would be expected to contribute substantially to the dispersion of transit time because of the reduced velocity of flow through them. When mean transit time (excluding common large vessel transit time) of a vascular indicator through pig myocardium was 8 s, transit through the capillary was estimated at 1.5 s (27). Similarly, capillary transit time in the rabbit lung was estimated at 0.87 s when whole organ transit time was 4.4 s (4). In many tissues, capillary transit time has been estimated at <1 s (24, 44). Thus our measured t̄_{2σ} of 8.5 s for bovine mammary glands is not a plausible capillary transit time and is better ascribed to a range of delays in nonexchanging vessels, especially the small vessels.
The difference between the CCCI and Audi et al. (2) models is in the interpretation of the capillary transit time function. We assumed the function represents washout from a wellmixed compartment in the manner of van der Ploeg et al. (47) and Cobelli et al. (13). Each lobule of 1.5 × 0.9 × 0.8 mm in the bovine mammary gland contains ∼200 alveoli, and each alveolus of secretory cells is surrounded by a basketlike capillary network (30, 51, 53). The length of mammary capillaries has not been established because of the complexity of the capillary network. A lobule is supplied by several arterioles and drained by several venules (34). The diameter of capillaries is 3–5 μm, and the distance between the basal membrane of the capillary endothelium and that of the secretory cells is 0.8–2 μm (51). Capillaries adjacent to secretory epithelial cells display welldeveloped basal infoldings, thus increasing the surface area available for the exchange from blood to tissue. Length of these marginal folds and microvillous processes increased significantly during lactation in both the arteries and capillaries (3, 30). The short diffusion distance and mixing action of blood movement in the vascular phase would allow small and nonbinding substances to reach equilibration in extracellular space within seconds. The outflow dilution curves are thus dominated by postequilibration dynamics. Additionally, the compartmental capillary in the CCCI model could be taken to represent groups of capillary supply zones in an alveolus as apposed to a single capillary segment. The nature of capillary structure and drainage in the mammary glands may have uniquely contributed to the good fits of the CCCI model to outflow dilution curves. Whether such fits would be obtained from the nonlactating mammary glands or other organs remains to be determined.
With the CCCI model, a single flowlimited extracellular indicator is needed to obtain both t̄_{2σ} of nonexchanging vessels and t̄_{b} of extracellular space in the target system. These two parameters were estimated independently of each other; t̄_{b} from six observations on the downslope only and t̄_{2σ} from all 13 observations after b was known. To examine a potential correlation between these two parts of the indicator outflow, the mixed procedure of SAS Institute (48) was used to analyze the following statistical model (12) where t̄_{b} was the mean residence time of PAH in extracellular space (Eq. 3), t̄_{2σ} was calculated as 2σ, i = cow 1 to 4, and j was the injection number for each cow. It was found that the mean residence time t̄_{b} was significantly different between cows (P = 0.0002), but the regression of t̄_{b} on t̄_{2σ} was not significant (P = 0.34). Thus there was no evidence of a relationship between the independent estimations of monoexponential downslope b and the SD σ of t̄_{μ}. It might be expected that the two transit times would be correlated if, for example, a decrease in blood flow rate were responsible for a longer t̄_{2σ}, then t̄_{b} would also increase. However, the lack of correlation suggests capillary recruitment in mammary blood flow regulation. For instance, if 10% of small arteries closed, both the blood flow (F) and the volume of small blood vessels would drop 10% so that the mean of nonexchanging vessel transit times would not change. Extracellular volume would not be affected by the change of blood flow, but the mean residence time of extracellular indicator would increase 10%. Thus with capillary recruitment, transit times through the two parts of the system are not correlated.
The reciprocal of monoexponential downslope b is the ratio of PAH distribution space to mammary gland plasma flow. Given that mammary gland plasma flow is ∼580 l/kg milk produced (9, 38), the cows in this experiment were producing 24.7 kg/day on average, and b averaged 0.03 s^{−1} (Table 1), the PAH distribution space was about 5.5 liters. Udders of lactating cows from the same herd as in this experiment weighed ∼22 kg (46) so the extracellular space, as indicated by PAH dilution, was ∼24% of mammary gland weight. Although there is no reason to expect extracellular space of a single organ to be equal to that of the whole body, the estimate of 24% falls within the range of 20–26% of total body weight observed in cattle and horses (25, 33).
In conclusion, to create a model for measurement of nutrient transport and metabolism from paired indicator/nutrient dilution across an organ, pieces from several earlier models were stitched together. Using a simulation approach to solve equations of the CCCI model removes the need for some of the simplifying assumptions employed to obtain analytic solutions of previous models, such as an instantaneous t_{dose}. Using just one indicator to describe vascular and extracellular behavior is an approach intermediate to the two indicators used in the Goresky models and no indicators in the dispersion model. Essentially, the convolution of a capillary function with a nonexchanging vessel function to obtain the whole organ outflow profile is identical to how Audi et al. (2) accounted for transit time heterogeneity. The difference between the CCCI and Audi et al. (2) models is in the interpretation of the capillary transit time function. We assumed compartmental kinetics. There is no axial distribution of indicator or nutrient concentrations in the compartments, so for those tissues where such gradients influence rates of transport and metabolism, e.g., liver (21), the CCCI model may be found to be inappropriate. Representation of capillary dynamics with wellmixed compartments allows for easy consideration of washout or backflux of indicator and nutrient from intracellular and extracellular spaces. Such washout is apparent when the monoexponential downslope of the nutrient curve is less than that of the indicator dilution curve (as in Refs. 8, 42, 55, 56). The compartmental capillary also allows for easy incorporation, as a set of ordinary differential equations, of alternative submodels of nutrient exchange and transformation within the capillary supply region.
More than 99% of the variation in the time course of venous extracellular indicator concentration after rapid injection into the arterial supply of the mammary glands was explained in an unbiased manner by partitioning the organ into a heterogeneous nonexchanging vessel subsystem and a wellmixed compartmental capillary subsystem. The density distribution of the heterogeneity of nonexchanging vessel transit times can be described by a normal distribution function f(t). The parameters t̄_{μ} and σ of the normal distribution can be obtained by simulating the convolution of f(t) with the set of ordinary differential equations describing single capillary compartmental dynamics. The mean of heterogeneous nonexchanging vessel transit times was 8.5 s, which was 25% of the mean residence time of PAH in extracellular space in lactating bovine mammary glands.
GRANTS
This work was funded by a grantinaid from Agribrands International, a wholly owned subsidiary of Cargill, National Sciences and Engineering Research Council Canada, and the Ontario Ministry of Agriculture and Food.
Acknowledgments
Thanks to Donna Benschop, Scott Cieslar, Heather Copland, Torben Madsen, Norm Purdie, Lori Sheehan, Andrea Sotirakopoulos, Ying Wu, Changting Xiao, and Gao Yingxin for assistance in sample collection and analysis. Thanks also to Peter deVries and Jim van Dusen for animal care.
Footnotes

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 Copyright © 2005 the American Physiological Society