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Department of Medical Biophysics, University of Western Ontario, London, Ontario, Canada N6A 5C1
Submitted 10 April 2003 ; accepted in final form 14 July 2003
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ABSTRACT |
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 TOP ABSTRACT METHODSRESULTS AND DISCUSSIONDISCLOSURESREFERENCES |
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avascular; vascular; oxygen diffusivity; oxygen consumption rate; heart failure
The design of replacement valves relies on a clear understanding of the characteristics of the native valve that influence its function and durability. One of the main characteristics that influences the ability of the tissue to repair structural damage, and hence determines the durability of the native valve, is tissue oxygenation. How much O2 does the valve tissue require and where does this oxygen come from?
Weind et al. (15) reported that most aortic valves have an intrinsic microcirculation located predominately in the thicker basal regions of the cusps. They speculated that this microvascular source of O2 was necessary for tissue repair. In a separate study, Weind et al. (16) found that a microvascular bed was present in regions of the cusp where tissue thickness exceeded 0.5 mm, further supporting the possibility that diffusion of O2 from the luminal blood is unable to completely satisfy the O2 demands of the tissue. Weind et al. (14) measured the O2 diffusivity (DO2) and the O2 consumption rate (VO2) of porcine aortic valve cusps and predicted, based on a simple one-dimensional diffusion model, that an intrinsic microcirculation was necessary to prevent tissue hypoxia.
Current biological replacement valves do not have an intrinsic microcirculation. Whether or not the native valves require an intrinsic microcirculation to provide additional O2 to the thicker regions of the cusps is an important consideration in the design of future, more durable, replacement valves.
This study aimed to improve understanding of the pathways for delivering O2 to aortic valve cusps. To test the hypothesis that the vascular bed found in the thicker basal region of an aortic cusp is needed to satisfy the O2 demands of the tissue, a two-dimensional (2D) finite difference model of O2 diffusion through a user-specified cusp was developed. This model used experimentally derived data for cusp thickness as well as for DO2 and VO2. A unique feature of this model is the prediction of the location and density of the microcirculation needed to prevent regions of cusp tissue from becoming hypoxic.
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METHODS |
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TOPABSTRACTMETHODSRESULTS AND DISCUSSIONDISCLOSURESREFERENCES |
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Clearly, O2 diffuses through an aortic valve in three dimensions. However, it was assumed for simplicity that the PO2 profiles of adjacent cross sections are similar. On the basis of this assumption, the flux of O2 between successive cross sections was neglected.
Governing equations. From Fick's law of diffusion and the principle of mass balance, the differential equation describing unsteady-state 2D diffusion in an O2 consuming tissue is given by
 | (1) |
2P/x2 and 2P/y2 are the second-order partial derivatives of partial pressure P with respect to x and y, P is PO2, and k is the O2 solubility of the tissue. Equation 1 was solved numerically for the complex geometry of a cusp cross section using the explicit forward-time central-space finite difference scheme. The cusp cross section was represented as a collection of small square nodes of equal size, and the PO2 profile of the cross section was calculated at every time step based on the PO2 profile of the cross section at the previous time step.
Valve geometry and three-layer structure. The modeled cusp cross section was assembled from a thickness map of a real porcine cusp obtained by Weind et al. (17). Because this thickness map was obtained from a cusp lying flat on a supporting surface, it does not represent the exact shape of a cusp in vivo in either the relaxed (open) or stressed (closed) position. However, this thickness map does provide a detailed representation of the distance that O2 must diffuse from the luminal surface to the interior of the cusp.
The aortic cusp is a composite structure made up of three distinct layers: the fibrosa, the spongiosa, and the ventricularis (6). The fibrosa, found on the aortic side of the cusp, is a fibrous layer composed predominately of collagen sheets and bundles arranged in the circumferential direction. This layer contains significantly more collagen than either of the other two layers and is thought to be the primary load-bearing layer of the cusp (10). The ventricularis, found on the ventricular side of the cusp, is also a fibrous layer. It is composed of a mixture of collagen and elastin and is thinner than the fibrosa. The spongiosa, occupying a significant portion of the cusp thickness, is found between the fibrosa and the ventricularis. It is composed of glycosaminoglycans and some fibrous proteins and is thought to be nonstructural (3).
The three layers vary in thickness from the base to the free edge of the cusp. Because of a lack of data on the thickness of the individual layers, the model approximates layer thickness as a constant fraction of the total thickness of the cusp. Figure 2 displays the location and thickness of each of the three layers within the modeled cusp cross section.
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During systole, the three cusps of the aortic valve are pushed out toward the aortic wall and blood flows from the left ventricle into the aorta. During diastole, the cusps move together to form a seal to prevent regurgitation. In this formation, the top portions of the cusps are in contact with one another. This contact area is called the coaptation region and is thought to be composed almost entirely of the ventricularis layer. The presence of the coaptation region is included in the model and is characterized by the fraction of the total cusp height that it occupies, starting from the free edge. The size and location of the coaptation region within the modeled cusp cross section is shown in Fig. 2. It is assumed that only the ventricularis is present within the coaptation region.
Boundary conditions. The aortic valve is situated between the left ventricle and the aorta. Because blood returning from the lungs is pumped from the left ventricle into the aorta, it is assumed that the surrounding luminal blood is fully oxygenated. Furthermore, because it was experimentally shown that a high concentration of vessels is present at the base of the aortic valve (13), the tissue at the base of aortic valve is assumed to be fully oxygenated. These boundary conditions are implemented by surrounding the modeled cusp cross-section with well-oxygenated blood and tissue set at a constant PO2 of 100 mmHg.
The free edge of the cusp is treated separately. There are two possible boundary conditions: a nonzero O2 flux when the valve is open and a zero O2 flux when the valve is closed (the coaptation regions of the three cusps are pressed up against each other with similar PO2 levels and so no O2 flux is expected). This model assumes the worst-case scenario by approximating the cusp as if the valve were closed at all times, thereby implementing a zero-flux boundary condition at the free edge.
Necrosis is another important consideration in the model. To simulate the real-life situation where a lack of O2 results in cell death, the VO2 of nodes in the avascular case that reach a PO2 of zero are reset to zero. The necrotic cells are no longer consuming O2; however, O2 is still diffusing through the tissue.
In the vascular model, vessels are positioned on a need basis to avoid developing necrotic regions within the tissue. If the PO2 of any node within the cusp cross section drops below an arbitrary threshold of 2 mmHg, a vessel is positioned at the point of lowest PO2. The PO2 profile of the cusp cross section is recalculated after the addition of every vessel.
Model parameters. The parameters of the model and their default values are provided in Table 1. The geometric parameters of the model include the user-specified thickness map, the distance between successive thickness measurements, the node size, the fractional thickness of the individual layers, and the fractional size of the coaptation region. The default thickness map thicknessmap.txt was obtained via a high-resolution digital radiography by Weind et al. (14). This thickness map corresponds to the thickness measured along the centerline of a porcine cusp from the free edge to the base. The successive thickness measurements are separated by a distance of 0.005 cm. A default node size of 0.005 cm x 0.005 cm is assumed based on reasonable computation time and desired model resolution.
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Because steady state can never be fully achieved with the use of numerical solving techniques, a termination condition of 
0.0001 mmHg difference between all previous and current time step PO2 values is implemented. When this termination condition is satisfied, the cusp cross section is said to have reached pseudo steady state.
Layer thicknesses for the fibrosa, spongiosa, and ventricularis of 30%, 50%, and 20% of the total cusp thickness, respectively, are assumed (10). On the basis of Weind et al.'s (16) experimental observations, 20% of the total cusp height starting from the free edge is attributed to the coaptation region.
The cusp cross section is initially set at a uniform PO2 of 50 mmHg. Because the majority of vessel diameters reported by Weind et al. (15) are <20 µm, with the occasional arteriole or venule having a diameter of up to 100 µm, all vessels in the model are set at a constant PO2 of 50 mmHg. This PO2 level corresponds to a plausible PO2 value for a capillary bed.
The diffusivities of the individual layers are assigned according to Weind et al.'s (14) experimental measurements of DO2 in porcine cusps. The cusp is also assumed to have a uniform VO2, assigned according to Weind et al.'s experimental measurements of VO2 in porcine cusps. Furthermore, the cusp is assumed to have a uniform O2 solubility equal to 90% that of plasma at 37°C (4). The external blood is assumed to have diffusivity and solubility values equal to those of plasma at 37°C (9, 4, 16).
A graphical user interface was developed to facilitate customization of the model to a specific cusp. All parameters listed in Table 1 can be modified before running a simulation, including the thickness map of the cusp under consideration.
Simulation of experimental DO2 and VO2 measurements. Simulations of the experimental measurement of cusp DO2 and VO2, as performed by Weind et al. (14), were carried out to test the model and to explore the impact of variations in cusp thickness on DO2 and VO2 measurements. Four cases were simulated: 1) a rectangular cusp with uniform DO2 to validate the model itself, 2) a notched rectangular cusp to explore the impact of thickness variability on the resulting DO2 and VO2 measurements, and cusps with Weind et al.'s experimental thickness data but with either 3) uniform DO2 values, or 4) three layers with the in vivo model parameters to determine whether the spongiosa layer influences the impact of tissue thickness variability on DO2 measurements. Because the dissected cusps in Weind et al.'s experiments were not perfused, the cusps in these model simulations were avascular with a zero-flux boundary condition along the bottom ventricularis surface, which was in contact with glass. The Clark O2 sensor output in Weind et al.'s experiments was simulated by averaging the PO2 over its reported effective area of 300 µm centered on a sensor node positioned at the glass surface. The initial condition for both the DO2 and VO2 simulations was the steady-state O2 levels in the modeled cusp with the other surfaces not in contact with the glass exposed to a fixed gas PO2. For the DO2 simulation, the sensor node PO2 output was stored for the transient response to a step change in gas PO2. For the VO2 simulation, the sensor output was recorded for the transient response to imposing a zero flux condition at all exposed surfaces. The DO2 and VO2 values for these simulations were obtained by fitting trend lines to the sensor data and performing the calculations outlined by Weind et al. (14) using
 | (2) |
 | (3) |
is the time constant of the transient and 
P/t is the fall of PO2 over time. Weind et al. determined L from the mean thickness of the cusp over the effective sample range of the O2 sensor. Software. All required programming, including the development of a graphical user interface, was performed with the use of Matlab version 6.0 (The MathWorks; Natick, MA).
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RESULTS AND DISCUSSION |
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TOPABSTRACTMETHODSRESULTS AND DISCUSSIONDISCLOSURESREFERENCES |
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The cross section of a native cusp, however, is irregular in geometry with drastic variations in thickness. Because the transient changes in PO2 associated with the DO2 simulations depend on L2 (Eq. 2), the thicker region of tissue near the sensor should slow the transient response in our 2D simulation leading to an underestimation of the predicted DO2 values. VO2 measurements are independent of the tissue thickness (Eq. 3) and should be minimally influenced by thickness variations in our simulations. The model of a rectangular cusp with a notch of variable width centered on the sensor node confirmed that DO2 measurements are very sensitive to thickness variations in the neighborhood of the sensor, whereas VO2 measurements are not (Fig. 3).
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The final two DO2 and VO2 simulations used the native valve thickness map with the simulated PO2 sensor placed at five different locations (thin, thick, and transition from thin to thick tissue). Simulated DO2 measurements for the cusp with a uniform diffusivity varied from –16 to +20% difference from the true parameter value, whereas the three-layered cusp with the spongiosa generated a greater scatter, from –38 to +21% difference. Because the same thickness profile, effective DO2 values, and sensor positions were used in both simulations, these results demonstrated that the distinct layers of a composite cusp have an effect on the DO2 measurements. It is speculated that the spongiosa, having a higher DO2 and occupying a greater portion of the total cusp thickness, enabled O2 to exchange more rapidly than the other two layers and hence exacerbated the impact of thickness variations.
The three-layer model returned measured DO2 values between 6.59 x 10–6 and 1.29 x 10–5 cm2/s, depending on sensor position. Weind et al. (16) reported experimental DO2 measurements for porcine cusps that ranged from 7.7 x 10–6 to 1.27 x 10–5 cm2/s. On the basis of the similarities between the simulation and experimental results, it is possible that the DO2 measurements obtained by Weind et al. did not properly account for the thickness variations in native cusps. Weind et al. used the mean cusp thickness in the assumed effective area of the sensor. However, because the time course for the transient response to a step change in O2 depends on L2, the thickness should be computed from the root-mean-squared equation
 | (4) |
Equation 4 was applied to the notched rectangular cusp to obtain a new mean cusp thickness and, subsequently, to calculate a new value for DO2. The averaging region was also increased from 300 to 600 µm, centered on the O2 sensor. New DO2 values were obtained within 5% of the true parameter value for all notch sizes considered. Thus, with the use of a more appropriate measure of the mean cusp thickness, it was possible to correct for the large scatter in the DO2 measurements. Further studies are required to determine how large an averaging region is necessary for accurate DO2 measurements. As a first approximation, an averaging region with a diameter at least as great as the largest thickness in the cusp cross section is proposed.
Although individual DO2 measurements made by Weind et al. may be in error, the mean DO2 reported by Weind et al. should be a good estimate of the true DO2 for the aortic valve. This mean DO2 was used for the following simulation of the native cusp in vivo.
Cusp oxygenation. The 2D finite difference model developed in this study was used to speculate on the need for an intrinsic microcirculation within native aortic valve cusps. A realistic representation of the centerline cross section of a porcine cusp was created based on a thickness map provided by Weind et al. (14). All model parameters were set to their default values outlined in Table 1. The model results are shown in Fig. 4 and Fig. 5.
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Figure 4 displays the modeled cusp cross section in the avascular case (Fig. 4A) and in the vascular case (Fig. 4B). Notice the presence of the three layers and the coaptation region in both cases. The small black markings in the vascular cusp cross section correspond to vessels systematically positioned on a need basis to avoid the development of necrotic regions. A total of 21 vessels were positioned within the cusp: 16 vessels in the spongiosa of the basal region and 5 vessels outside the basal region.
In their experimental study, Weind et al. (16) observed that most aortic cusps have an intrinsic microcirculation located predominately in the spongiosa of the basal region. They also found a significant correlation between tissue thickness and vessel density (16). The results from the model presented in this study support these observations. A high vessel density was found in the thicker basal region of the vascular cusp cross section, whereas a low vessel density was found outside the basal region. Except for a few vessels in the coaptation region, all vessels were positioned within the spongiosa. Specifically, Weind et al. (15) determined mean vessel densities of 3.98 ± 5.21 and 0.29 ± 0.59 vessels/mm2 within and outside the basal regions of the cusps, respectively. The model developed in this study obtained vessel densities that fall within these experimental ranges: 4.78 and 0.75 vessels/mm2 within and outside the basal region of the cusp, respectively.
Figure 5 displays the predicted PO2 profile of the cusp cross section at pseudo steady state in the avascular case (Fig. 5A) and in vascular case (Fig. 5B). Notice the large necrotic regions that developed in the avascular case. The onset of necrosis was prevented in the vascular case by the systematic positioning of small vessels on a need basis. Clearly, diffusion of O2 from the luminal blood alone was unable to completely satisfy the O2 demands of the tissue. These model results support the hypothesis that the vascular bed found in the thicker basal region of a native aortic cusp is needed to satisfy the O2 demands of the tissue.
In conclusion, the 2D finite difference model developed in this study was able to predict vessel locations and densities that agree with the experimentally observed vessel locations and obtained vessel densities (16) using thickness, VO2, and DO2 values from Weind et al. (14). The results from this model support the hypothesis that a microcirculation is needed to satisfy the O2 demands of the thicker regions of the native aortic valve cusps.
Simulation of experimental VO2 and DO2 measurements revealed that variations in cusp thickness greatly influenced the DO2 measurements (–38 to +21% difference). With the use of the root-mean-square average of the cusp thickness over a larger averaging region (diameter 600 µm), the error in the DO2 measurements was reduced to <5% difference from the true value. On the basis of these model results, experimental measurements of DO2 should take into account the variations in cusp thickness using the root-mean-square average.
Weind et al. (14) used a simple mean thickness in their DO2 measurements; thus it is likely that much of the variation in the reported DO2 values for porcine cusps was due to cusp thickness variations. Although individual DO2 measurements may be in error (maximum error estimated to be less than ±40%), the mean DO2 reported by Weind et al. should be a good estimate of the true DO2 for the aortic valve. Therefore, the conclusion derived from this study still holds: a microcirculation is needed to satisfy the O2 demands of the thicker regions of the native aortic valve cusps.
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DISCLOSURES |
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TOPABSTRACTMETHODSRESULTS AND DISCUSSIONDISCLOSURESREFERENCES |
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ACKNOWLEDGMENTS |
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FOOTNOTES |
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The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
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REFERENCES |
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TOPABSTRACTMETHODSRESULTS AND DISCUSSIONDISCLOSURESREFERENCES |
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This article has been cited by other articles:
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  Z. Zeng, Y. Yin, K.-M. Jan, and D. S. Rumschitzki Macromolecular transport in heart valves. II. Theoretical models Am J Physiol Heart Circ Physiol, June 1, 2007; 292(6): H2671 - H2686. [Abstract] [Full Text] [PDF]
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