Abstract Local vascular drug delivery systems provide elevated concentrations in target arterial tissues, while minimizing systemic side effects. Drug can now be released to isolated arterial segments from the endovascular or perivascular aspects of the blood vessel, yet the forces that determine drug distribution and deposition for these different modes of delivery have not been rigorously investigated. This study examines mechanisms of transmural transport of a model vasoactive drug, heparin, and compares estimates of the distribution after administration from either aspect of the artery. We showed that (1) heparin traversed the arterial wall rapidly; (2) diffusion far outweighed convection in the control of transmural heparin transport in the normal artery, but after endothelial injury, convective forces rose to one quarter the magnitude of diffusive forces; (3) the endothelium posed a minimal diffusive barrier to heparin; and (4) the diffusive barrier imposed by the adventitia depended on its thickness. These findings strongly suggest that vasoregulatory compounds can be administered to target tissue by either perivascular or endovascular means with equal efficacy, because the forces governing transport of heparin from either aspect of the blood vessel wall are not significantly different. Furthermore, the differences in arterial transport properties between heparin and other macromolecules suggest that distribution and the optimal aspect of delivery will depend just as much on the physicochemical properties of the drug as the state of the blood vessel wall.
Mechanical relief of obstructive vascular disease has yielded substantial benefit. More than 400 000 balloon angioplasties and 460 000 coronary bypass operations are performed in the United States annually.1 2 Yet these procedures initiate accelerated obstructive arteriopathies of their own and may require complex adjunctive pharmacotherapy.2 The utility of drugs designed for this purpose is limited by short half-lives and significant systemic toxicity. Local vascular pharmacotherapy provides heightened concentrations of vasoactive agents in and around the blood vessel wall, without the adverse effects of systemic administration. A local delivery system might continuously provide drug to target tissues while minimizing drug concentrations elsewhere.2 3 Attempts at sustained local delivery of vasoactive compounds can be divided into two classes based on the site of delivery (Fig 1⇓): endovascular, administered from within the blood vessel,4 5 and perivascular, released from outside the artery.6 7 The choice between these two classes may be dictated by parameters beyond simple convenience, as the direction of administration may significantly alter tissue deposition. To date, there has been insufficient systematic experimental study of the mechanisms of delivery3 that might illustrate the conditions under which one mode might be preferred over the other. We attempted to address this concern by quantifying the physical forces that govern transarterial drug transport. We contrasted the convective forces that arise from the physiological transmural hydrostatic pressure gradient with the diffusive forces that arise from the drug concentration gradients in the arterial wall and assessed the potential barrier function of the endothelium and the adventitia.
Heparin was used as a model vasoactive compound because of its therapeutic potential in limiting the accelerated vascular disease that develops in response to mechanical interventional therapy.6 8 9 10 11 In addition, heparin closely resembles endogenous factors, such as heparan sulfate, that regulate many aspects of vascular biology. Knowledge of the local transport and distribution of these compounds may help to better understand the role they play in endogenous vascular repair and their potential as therapeutic agents.9 10 We found that heparin traverses the arterial wall rapidly, that diffusive forces are almost always dominant over convective forces, and that the endothelium poses a minor barrier to heparin, whereas the barrier of the adventitia depends on its thickness. These findings show that the forces governing transport of compounds from either aspect of the blood vessel wall are not significantly different. Furthermore, the arterial transport properties of heparin and other macromolecules differ, implying that the distribution will depend on the physicochemical properties of the drug as well as the physical structure of the blood vessel.
Materials and Methods
In Vitro Perfusion Experiments
Sprague-Dawley rats (320 to 360 g) were anesthetized with an intraperitoneal injection of ketamine (50 mg/kg) and xylazine (10 mg/kg). The abdominal aorta was exposed, cleaned of fat and excess fascia, and cannulated proximally just below the splenic vein, and distally just above the iliac bifurcation. Ligatures were placed around each cannula so that the intermediate segment of artery was isolated from the rest of the circulation. All branch vessels were ligated and severed. The cannulas were clamped to a rigid frame so that the length of the isolated artery was preserved at its in vivo dimensions. The artery was excised, and the length of the artery between the tips of cannula was measured under a dissecting microscope (0.99±0.03 cm). Leaks from the artery were assessed by connecting one cannula to an elevated (100-cm) reservoir and closing the other cannula. The artery was inspected under the microscope and discarded if any leak was noted.
The artery was placed in an in vitro perfusion apparatus (Fig 2⇓), simulating plasma flow through the artery. The perfusate flowed from an upper reservoir through the artery, emptied into a lower reservoir, and was pumped back to the upper reservoir, forming a well-mixed endovascular compartment (100 mL). The artery was immersed in an extravascular bath (4 mL), to which known concentrations of radiolabeled heparin were added, establishing a fixed transmural concentration gradient. Krebs-Henseleit buffer (Sigma Chemical Co) was used as the perfusate and extravascular bath. The transmural pressure gradient and the luminal volume flow rate were set by the height, or hydrostatic pressure head, of the upper reservoir (ΔH) and the downstream resistance to flow, which was adjusted through a throttle valve. An overflow line connected the upper and lower reservoirs directly, holding ΔH constant regardless of pump speed. The entire perfusion system was placed within a closed cabinet and maintained at 37°C and 100% relative humidity. Not shown are a stir bar in the extravascular bath, a thermally controlled water jacket surrounding the lower reservoir, and in the lower reservoir, a thermometer and a 95% O2/5% CO2 bubbler. The volume flow rate of perfusate was measured by counting the rate at which drops fell from an outflow needle. Drop volumes were determined before each experiment from the number of drops collected in a measured volume.
Heparin was administered perivascularly by immersing the artery in the extravascular bath of [3H]heparin (Du Pont NEN) and unlabeled heparin (Hepar Industries) in buffer. The total heparin concentration was 2.5 mg/mL, and the activity was 6 μCi/mL. The artery was perfused for 6 hours at 37°C. At 1-hour intervals, three 50-μL samples were taken from the lower reservoir, and one 50-μL sample was removed from the extravascular bath. The perfusate volume flow rate, temperature, and pH were monitored hourly.
At the end of the experiment, the perfusion system was drained, and its volume was recorded. Loss of the perfusate was attributed to steady evaporation, and calculations of drug concentration in perfusate samples were adjusted for this concentrating effect by multiplying the concentration of heparin by the perfusate volume at that time point over the initial volume. The extravascular bath was switched to a modified Bouin’s fixative (53% ethanol, 4% formaldehyde, 2.5% glutaraldehyde, 7% acetic acid, and 0.7% KCl), and the artery was perfused with fresh buffer for 3 hours. The artery was then immersion-fixed for an additional 40 hours without perfusing, after which it was removed, dehydrated, and processed for paraffin embedding. Serial 10-μm cross sections were taken from one cannula tip to the other and stained with Verhoeff’s elastin stain.
Computer-assisted morphometric analysis was performed on cross sections taken at 1-mm intervals along the arterial length. The IEL, EEL, and outer edge of the adventitia were traced with image analysis software (IPLab Spectrum, Signal Analytics). The length of the IEL and EEL and the area of the lumen, media, and adventitia were measured. The medial thickness of each cross section was calculated by dividing the medial area by the length of the IEL. The adventitial thickness of each cross section was calculated by dividing the adventitial area by the length of the EEL. Mean values for medial thickness, adventitial thickness, luminal area, and perimeter were calculated for each artery and used in subsequent calculations. The extravascular concentration was the average of the measurements at each time point. The transmural heparin mass transfer rate was defined as the time rate of change of heparin in the endovascular compartment and was calculated by a linear regression fit over the steady state portion of the data.
Nine rat aortas were perfused without a hydrostatic pressure head (ΔH=0 cm), setting the transmural pressure gradient (ΔP) to zero and establishing a scenario wherein all the measured mass transfer should have been governed solely by diffusion. The endothelium of four of these arteries were denuded with three passes of an inflated 2F embolectomy catheter (Baxter Diagnostics).12 Another 11 rat aortas were perfused with ΔH=100 cm, mimicking a physiological pressure gradient. Before each experiment, the pressure just downstream of the artery was measured with a diaphragm manometer (Omega Engineering, Inc). The flow rate was adjusted with the throttle valve over a range that resulted in a physiological pressure gradient of 99 to 103 cm H2O. During the subsequent perfusions, the flow rate remained within this range. Five of these arteries were also denuded of endothelium before cannulation.
To assess the integrity of the vessel wall after dissection and to exclude arteries from the analysis where trauma might lead to potential artifact, each artery was pressurized to 125 cm H2O by connecting an elevated bag of Ringer’s solution with the other cannula closed. The artery was examined for leaks under a dissecting microscope, and the bag was examined for flow for several minutes. In addition, in many arterial preparations minor injury led to a slow leak that overflowed the extravascular bath. The subsequent drop in extravascular heparin concentration allowed us to discard completed experiments in which injury led to artifactual transport measurements. In a separate experiment, the integrity of the endothelial monolayer of nondenuded excised arteries was confirmed by perfusing an artery with 4% albumin and Evans blue dye in buffer.13
Diffusion of Heparin in Aqueous Solutions
The diffusivity of [3H]heparin in buffer was measured by using a standard diffusion cell (Crown Glass) with a porous hydrophilic membrane (model GVW; mean pore size, 0.22 μm; Millipore) that separated two 3-mL chambers. [3H]Heparin was added to the source chamber, and an equal concentration of unlabeled heparin was added to the sink chamber to create isosmotic conditions. Each chamber was well mixed with magnetic stir bars and maintained at room temperature. Aliquots (10 μL) were taken from each chamber at 10-minute intervals for 90 minutes. The concentration gradient of [3H]heparin was large enough to be considered constant over the short time of the experiment and was approximated by the average concentration of heparin in the source chamber (ch*). The time rate of change of heparin concentration in the sink chamber (∂ch/∂t) was calculated by performing a linear regression over the steady state portion of the sink chamber measurements. From a mass balance for the sink chamber, the diffusivity of heparin in aqueous solutions (D) is as follows:
where vh is the volume of the sink chamber, Ao is the total open area of all of the pores, and lmem is the thickness of the membrane.
Diffusivities and Resistances of Heparin Within the Arterial Wall
The perfusion experiments performed with no hydrostatic head (ΔH=0 cm) had no transmural pressure gradient and therefore, no transmural hydraulic flux. Thus, once steady state was established, the mass transfer data reflected diffusion alone. The arterial wall was modeled as a series of concentric cylindrical tubes (Fig 3⇓), and the medial and adventitial thicknesses were approximated as the average of those measured from all the histological sections of an artery. Furthermore, the extravascular and endovascular compartments were well mixed, so that the only concentration gradient existed in the transmural direction. The transport was modeled as four resistors in series, one each for the adventitia (Radv), media (Rmed), endothelium (Rend), and the mass transfer boundary layer within the lumen flow (Rbl), which separate the potential or concentration gradient (cev−cp). Thus, by analogy to Ohm’s law, the potential difference for diffusive mass transfer is the product of the flux and the series sum of these resistances.
where j is the transmural heparin transfer rate, L is the length, and P is the perimeter of the lumen. The coefficient of Rend (bend) was 0 after a denuding injury and 1 with intact native arteries. The mass transfer was purely diffusive in these perfusions performed at ΔH=0 cm, defining the following resistances: and where ladv and lmed are the adventitial and medial thicknesses, and Dadv and Dmed are the diffusivity of heparin within the adventitia and media, respectively. The boundary layer resistance (Rbl) is defined in the “Appendix.” Rearrangement of Equation 2 allows the unknowns, Dmed, Dadv, and Rend, to be determined by multiple linear regression:
Balance Between Diffusion and Convection in Transmural Transport
The physiological hydrostatic pressure gradient gives rise to transmural convective currents. The ratio of the convective to diffusive forces of transmural transport of a given drug molecule is embodied in the Peclet number (Pe).13 14 15 A Pe much less than 1 implies that the transmural transport is purely diffusive. Conversely, a Pe much greater than 1 implies that the transmural transport is purely convective, and under these conditions, the oncoming hydraulic flow might potentially prevent drug in the perivascular space from entering the arterial wall. A Pe of unity implies that both diffusive and convective effects play a role in drug transport. The Pe for heparin in arterial media is calculated as follows:
where Umed is the heparin drift velocity in arterial media and is less than the hydraulic velocity because of steric and charge interactions in the arterial tissue.15 16 The degree of retardation may differ for diffusive and convective movements. The retardation coefficient for diffusive flux of heparin in arterial media (fDmed) is defined as the degree by which the diffusivity in arterial media is reduced from the diffusivity in aqueous solutions15 17 :
Similarly, a retardation coefficient for convective flux in arterial media (fCmed) can be defined as the degree by which the solute drift velocity is reduced from the transmural hydraulic velocity (U)17 : Thus,
The retardation coefficient for convection has not been explicitly measured for any solute in any model of arterial interstitium.16 The physical constraints that generate the diffusive and convective retardation coefficients can be similar15 ; however, the media does not necessarily have to retard convection. Thus, Pe can only be framed within limits by assuming at one extreme that the retardation for convection and diffusion are equivalent (fDmed=fCmed) and at the other fCmed equals 1, such that The transmural hydraulic velocity (U) was calculated from published correlations of hydraulic conductivity (see “Appendix”). Pe numbers were calculated for arteries over a range of medial thicknesses for normal and deendothelialized arteries.
The diffusivity of heparin in aqueous solutions at room temperature was measured to be 1.39×10−6 cm2/s (R2=.996). After correction from room temperature to 37°C using the Stokes-Einstein relation,18 the diffusivity was 1.45×10−6 cm2/s. The thicknesses of each arterial layer and the corresponding heparin flux normalized by the concentration gradient are shown in the Table⇓. The multiple linear regression (r=.920) of the data taken without a hydrostatic pressure gradient (ΔH=0 cm) showed the following: Dmed=7.73×10−8 cm2/s (P=.03), Dadv=1.21×10−7 cm2/s (P=.07 ), and Rend=25 100 s/cm (P=.004). The diffusive resistances of these three arterial layers are calculated over a range of thicknesses (Fig 4⇓).
The estimations of Pe are shown for a range of medial thicknesses for both native and deendothelialized arteries (Fig 5⇓). Because the degree to which the arterial wall retards convective movements of heparin is unknown, Pe could only be estimated to lie within upper and lower bounds. The range of Pe is less than unity, except after deendothelializing injury.
Arteries were perfused with and without transmural pressure gradients, and direct comparison of the transmural heparin transfer from each of these sets of data would experimentally confirm the relative importance of convection and diffusion. However, under physiological pressure, the arteries are significantly thinner (Table⇑) and larger in perimeter than when there is no pressure gradient, decreasing the length over which heparin must migrate and increasing the area perpendicular to transport. Thus, any hindrance to mass transfer attributable to convection is overwhelmed by these effects.
To circumvent the artifact generated by these morphological changes, a nondimensional parameter (ψ), which evaluated how much of the observed mass transfer was due to diffusion alone, was defined. ψ equals the right side of Equation 2 normalized by the left side: This value represents the mass transfer nondimensionalized by the diffusive driving potential and diffusive resistances. Note that this characterization only incorporates diffusive terms; therefore, if diffusion is the only driving force, ψ equals 1. Conversely, if convection is the only driving force, then ψ equals 0, because in these perfusions the concentration gradient of heparin was directed against the hydraulic flux. Certainly, ψ should equal 1 for the experiments performed with ΔH=0. The coefficient of Rend (bend) is 0 after a denuding injury and 1 with intact native arteries. The ψ parameter was computed for native and deendothelialized arteries, with and without a physiological transmural pressure gradient (Fig 6⇓). ψ was approximately unity; however, when there was a pressure gradient and deendothelialization, the value dropped by 20%.
The structure of the blood vessel wall would lead one to believe that there are substantial barriers to entry and transport of macromolecules. Previous data designed to examine endothelial permeability to low-density lipoprotein, albumin, and horseradish peroxidase support this notion.19 20 21 22 23 24 However, physiological and therapeutic compounds vary widely in size, conformation, and charge. Therefore, local transport properties of one compound may not be applicable to another. Indeed, the blood vessel might exclude large pathogenic molecules and allow entry of smaller, endogenous regulators, such as heparin. Although the above studies have been valuable in understanding the infiltration of lipoproteins in the course of atherogenesis, careful analysis of the transport of specific vasotherapeutic compounds is warranted when considering local therapy for vascular disease.
There has been tremendous excitement over the potential of local vascular drug delivery systems that augment mechanical intervention and limit the vascular response to injury.2 3 6 25 26 These systems can be categorized into endovascular and perivascular modes of administration. Yet there is no rigorous means by which to evaluate one modality with respect to the other. We have attempted to approach the choice between these two modes in an analytic and quantitative manner. An in vitro perfusion apparatus was used to control the environment inside the lumen and around the artery, to assess the balance between diffusive and convective mechanisms of transmural transport, and to measure the diffusive resistance of each arterial layer. We found that heparin rapidly traverses the arterial wall, diffusion exclusively controls the transmural distribution of heparin under normal conditions, convective forces rise to one quarter the magnitude of diffusive forces under extreme conditions of endothelial disruption, the diffusive barrier to heparin posed by the endothelium is minor, and the barrier to heparin transport posed by the adventitia depends on its thickness. These findings strongly suggest that drug can be administered in an equivalent manner to target tissues from either the perivascular or endovascular aspect and that the structure of the blood vessel wall does not limit their distribution from either direction.
Heparin Rapidly Traverses the Arterial Wall
Prior investigations have shown a biological effect after perivascular delivery of vasoactive agents in a number of animal models.5 6 27 However, there has been no direct evidence until now that drug traversed the adventitia and media or that drug reached the most luminal smooth muscle cells. It has been hypothesized that drugs administered in this fashion exert their effects through adventitial receptors or activate yet other compounds that diffuse to medial cells. We now implicitly demonstrate that heparin rapidly distributes across the arterial wall, invalidating the need for alternative messenger systems to explain the efficacy of the perivascular release of heparin. Steady state transmural heparin transfer was consistently observed in less than 15 minutes after administration. Thus, the time delay between release device deployment and drug extension to the intima is relatively short and should not be a factor in choosing perivascular or endovascular modes of delivery. This analysis can be extended to larger and diseased vessels as well. An order-of-magnitude analysis suggests that the time required for heparin to diffuse across the arterial wall increases with the square of its thickness. Thus, this time should increase in anatomically larger or hyperplastic arteries. For example, if intimal hyperplasia doubles the arterial wall thickness, the transmural transit time may be on the order of an hour. Even this time delay will allow distribution to occur well before the initiation of the vascular response to injury, and transport barriers imposed by the blood vessel wall should not limit the utility of locally administered pharmacological agents.
Role of Diffusion and Convection
Diffusion is an omnidirectional process resulting from random molecular movements; thus, the magnitude of diffusive forces should be independent of the aspect of delivery (Fig 1⇑). In contrast, convective forces are always aligned with the physiological hydrostatic pressure gradient across the water-permeable arterial wall and are directed from the intima toward the adventitia. Thus, at first glance, it would appear that endovascular delivery is always superior to perivascular delivery, because convective and diffusive forces appear to augment the former, whereas in the later, drug must diffuse in the face of an oncoming convective current. Yet, it is the balance between the diffusive and convective forces that will determine the appropriateness of this interpretation. If diffusive forces are much larger than convective forces, then endovascular delivery is no better than perivascular delivery. In addition, the balance between these forces will vary for each drug considered and for the state of disruption of the arterial architecture in vascular disease.
The balance between diffusive and convective forces in transmural transport is conveyed by Pe. For native uninjured arteries, the range of Pe was usually <1 (Fig 5⇑), implying that convective effects are limited by the hydraulic resistance of the arterial media and endothelium. Thus, convective forces do not enhance the distribution of heparin after endovascular delivery, nor do they limit the distribution after perivascular release. Convection can play a more significant role in thicker arteries or when the endothelial barrier to convective flux is removed. In the former, Pe will increase because the diffusive resistance increases more so than the hydraulic resistance. In the latter, Pe will increase to its theoretical maximum, irrespective of medial thickness, as the endothelial monolayer can account for a large fraction of the hydraulic resistance.28 Hence, under conditions of severe endothelial injury or dysfunction, the transmural convective currents may reach significance where they enhance heparin distribution after endovascular delivery.
It is possible to verify these theoretical considerations empirically by comparing transmural transport with and without adverse convective forces. The ψ parameter represents the measured mass transfer nondimensionalized by the diffusive driving potential and diffusive resistances, and it evaluates how much of the observed mass transfer arises from diffusion alone. If diffusion is the only driving force, ψ equals 1; if convection is the only driving force, then ψ equals 0. The data show that ψ is ≈1 with native intact arteries (Fig 6⇑), regardless of whether there is a transmural pressure gradient or not (ΔH=0 or 100 cm). After a balloon denuding injury, the introduction of a physiological transmural pressure gradient reduced ψ from 1 to 0.8. Under these circumstances, convective forces can reduce the transmural transport of heparin after perivascular delivery; thus, endovascular delivery may lead to slightly enhanced distribution of drug.
Although diffusive and convective forces determine drug transport, localization of drug is also effected by binding.15 17 Heparin avidly binds to smooth muscle cells and extracellular structures.29 30 Thus, binding can transiently slow the rate of heparin accumulation in the perfusate. In these experiments, the extravascular concentration of heparin was very large (2.5 mg/mL), so that the binding sites would quickly saturate. In addition, the mass transfer rates were observed to be steady after the first 15 minutes (R2>.97), suggesting that the transmural transport was independent of binding effects.
Barrier Function of the Adventitia and Endothelium
The resistances to heparin transport imposed by both the adventitia and the endothelium can potentially control the amount of drug deposited in the media after local administration. The adventitial resistance increases linearly with thickness (Fig 4⇑) and will vary with the extent of surgical manipulation. When the adventitia is thin, perivascular delivery may lead to more rapid deposition of drug than endovascular release, whereas at larger adventitial thicknesses, endovascular delivery may be relatively more effective. The primary resistance to transmural transport of macromolecules such as albumin, horseradish peroxidase, or low-density lipoprotein, however, is in the endothelium.19 20 21 23 The ratio of endothelial to medial diffusive resistance varies for different compounds and arteries. This value was ≈10 for albumin or low-density lipoprotein19 21 but only ≈0.5 for heparin in the rat abdominal aorta used here and ≈0.1 in arteries as thick as the rabbit thoracic aortas used in the albumin studies. In addition to the fourfold difference in size of heparin (12 to 15 kD) and albumin (60 kD), enhanced transendothelial heparin transport may arise from the flexibility and solubility of the linear highly charged compound.31 32 Phenomena such as reptation may allow heparin to pass through far smaller pores than other compounds of similar molecular weights. This distinction in transport properties illustrates the need for in-depth analysis for each compound and the danger of extrapolating from the results of studies with one molecule to another. As an example, a local delivery system for albumin would almost by necessity require perivascular delivery, whereas both endovascular and perivascular methods are viable for heparin.
The quantitative methods we have used to examine transmural drug transport may add to our understanding of fundamental structure-function relations within the blood vessel wall and drug–vascular tissue interactions and provide a rational framework for the design of local vascular drug delivery systems. We have shown that the structure of the blood vessel wall should not limit the distribution of heparin from either aspect of the artery. Yet our analysis also reveals the potential idiosyncratic behavior of individual compounds and highlights the need for individualized analysis of this type when dissimilar compounds are to be considered.
Selected Abbreviations and Acronyms
|ψ||=||measured mass transfer nondimensionalized by the diffusive driving potential and diffusive resistance|
|ΔH||=||hydrostatic head of perfusate in artery|
|ΔP||=||transmural hydrostatic pressure gradient|
|fCmed||=||convective retardation coefficient of heparin in arterial media|
|fDmed||=||diffusive retardation coefficient of heparin in arterial media|
|Al||=||cross-sectional area of arterial lumen|
|Ao||=||total open area of all the pores in the membrane|
|bend||=||coefficient of endothelial resistance (0, absent; 1, present)|
|cev||=||concentration of heparin in extravascular bath|
|ch||=||concentration of heparin in the sink chamber in diffusion cell|
|ch*||=||concentration of heparin in the source chamber in diffusion cell|
|cp||=||concentration of heparin in perfusate, endovascular compartment|
|D||=||diffusivity of heparin in aqueous solutions|
|Dadv||=||apparent diffusivity of heparin in adventitia|
|Dmed||=||apparent diffusivity of heparin in media|
|EEL||=||external elastic lamina|
|IEL||=||internal elastic lamina|
|j||=||mass transfer rate of heparin|
|L||=||length of artery|
|ladv||=||average thickness of adventitia|
|lmed||=||average thickness of media|
|lmem||=||thickness of membrane in diffusion cell|
|P||=||average perimeter of artery, length of IEL|
|Radv||=||diffusive resistance of adventitia to heparin|
|Rbl||=||boundary layer resistance to heparin transfer in lumen flow|
|Rend||=||diffusive resistance of endothelium to heparin|
|Rmed||=||diffusive resistance of media to heparin|
|U||=||transmural hydraulic velocity|
|Umed||=||heparin drift velocity in media|
|vh||=||volume of sink chamber in diffusion cell|
Boundary Layer Resistance to Heparin Transport
The boundary layer resistance results from solute that enters the lumen from points upstream and hinders the entry of solute from the wall at downstream locations (Fig 3⇑). The choice of a correlation for boundary layer resistance is determined by the flow regimes encountered in the perfusion experiments, with respect to fluid momentum and mass transport. In all of the perfusion experiments, although fluid flow in the lumen was fully developed and laminar, the artery was not long enough to consider the mass transfer fully developed.
The Sherwood number (Shd) is a nondimensional form of the resistance to mass transfer of the boundary layer33 34 : An appropriate correlation for the Shd for fully developed fluid flow and non–fully developed mass transfer is as follows33 34 : where the Reynolds number (Red) is Ūld/ν, Ūl is the average fluid velocity flowing in the lumen and equals the average volume flow rate of perfusate divided by Al, and ν is the kinematic viscosity. The hydraulic diameter (d) helps describe the flow regime through noncircular ducts:
Transmural Hydraulic Flux
The transmural hydraulic flux (U) can be determined by modeling the media and endothelium as two conductors in series as follows: where μ is the dynamic viscosity and Kmed is the specific hydraulic conductivity of the media and has been measured to be 2×10−14 cm2.35 The intrinsic hydraulic conductivity of the intact endothelium (K″end) is 8.2×10−11 cm2·s·g−1.28 Recall that bend is 0 for arteries after a denuding injury and 1 for intact native arteries.
This study was supported in part by grants from the National Institutes of Health (GM/HL-49039), the Burroughs-Wellcome Fund in Experimental Therapeutics, and the Whitaker Foundation in Biomedical Engineering. We would like to thank Prof Roger Kamm for his analytical advice and Dr Fred Bowman for the use of his perfusion apparatus.
This manuscript was sent to Leslie A. Leinwand, Consulting Editor, for review by expert referees, editorial decision, and final disposition.
- Received May 4, 1995.
- Accepted August 21, 1995.
- © 1995 American Heart Association, Inc.
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