Mathematical Modeling of Flow-Generated Forces in an In Vitro System of Cardiac Valve Development
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- Biechler, S.V., Potts, J.D., Yost, M.J. et al. Ann Biomed Eng (2010) 38: 109. doi:10.1007/s10439-009-9824-9
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Heart valve defects are the most common cardiac defects. Therefore, defining the mechanisms of cardiac valve development is critical to our understanding and treatment of these disorders. At early stages of embryonic cardiac development, the heart begins as a simple tube that then becomes constricted into separate atrial and ventricular regions by the formation of small, mound-like structures, called atrioventricular (AV) cushions. As valve development continues, these mounds fuse and then elongate into valve leaflets. A longstanding hypothesis proposes that blood flow-generated shear stress and pressure are critical in shaping the cushions into leaflets. Here we show results from a two-dimensional mathematical model that simulates the forces created by blood flow present in a developing chick heart and in our in vitro, tubular model system. The model was then used to predict flow patterns and the resulting forces in the in vitro system. The model indicated that forces associated with shear stress and pressure have comparable orders of magnitude and collectively produce a rotational profile around the cushion in the direction of flow and leaflet growth. Further, it was concluded that the replication of these forces on a cushion implanted in our tubular in vitro system is possible. Overall, the two-dimensional, mathematical model provides insight into the forces that occur during early cardiac valve elongation.
KeywordsHeart developmentShear stressAtrioventricular valve
Cardiac valve development is dependent upon many different factors. Both genetic and mechanical factors contribute to the proper formation of cardiac valves.3 While genetic factors have received the majority of investigators’ attention, progress has been made in delineating the effect of mechanical forces. These mechanical forces are pressure and fluid shear stress, which are both generated by blood flowing through the heart.24
In an artificial model of fluid flow in the cardiovascular system, Rodbard generated cushion and valve-like structures by altering fluid flow in silicone gel-lined tubes.17 This suggests that fluid flow alone is capable of shaping malleable substances into valve-like structures. More recent investigations have provided evidence that fluid flow is critical in shaping the developing cardiovascular system.2,9,11,16,21,22 Experiments in which hemodynamic load within the heart was altered determined that cardiac growth and development were affected.20 Generally, in these studies, a component of the circulatory system was restricted, resulting in an increase or decrease in hemodynamic pressure in the heart depending on the location of the restriction. These investigations provided data that strongly indicate normal cardiovascular development is dependent on fluid flow.
The objective of the present work is to use a simple mathematical model to simulate flow patterns and estimate pressure and shear stresses in developing cardiac valve tissue at flows consistent with published in vivo estimates. The final goal is to use the model to estimate flow parameters predicted for the aforementioned in vitro tubular culturing system in order to obtain similar shear stress values at the midpoint surface of a single, implanted cushion. Because the tubular culturing system operates under steady state flow, the model developed for in vivo estimates is representative of pseudo steady state flow and represents two different snapshots in time during the in vivo cardiac cycle of a chick embryo. These studies provide the foundation for defining the flow conditions necessary for cardiac valves to develop.
The mathematical model developed here simulates flow patterns and estimates flow-induced forces values for incompressible, Newtonian flow through two infinite slits containing idealized AV cushions. It has already been shown that for low Reynolds numbers, a two-dimensional model can accurately predict flow characteristics such as velocity and pressure.4,6 The simplified symmetrical flow through two infinite slits is used to represent flow along the centerline of a three-dimensional tube. The AV cushion is idealized as a mound extending infinitely in the z-direction.
Results and Discussion
The first objective of the study was to simulate flow in a chick heart containing two transverse cushions. At Hamburger and Hamilton10 stage 25 of chick development the AV cushions are fully developed although not yet fused (Fig. 1); therefore, this is the stage in development that was represented by the simulation shown here. The simulation for flow in a stage 25 developing chick heart will be referred to as the in vivo system. Prior experimental quantification of flow velocities have already been made for stage 25 of a developing chick heart.3 Flow parameters used were for a stage 25 chick heart during the open (diastole) and closed (systole) states of cardiac contraction. These two stages were treated mathematically as pseudo steady state snapshots in time. An estimated maximum entrance velocity of 0.2 m s−1 was published for the maximum rapid filling point in time of diastole and cushion thickness was estimated to be 250 μm.3 The slit diameter was estimated to be 1 mm and the total tube length was estimated to be 4 mm. Assuming a blood density of 1060 kg/m3 and a viscosity of 0.003 Pa·s,12 the Reynolds number for fully developed flow in the entrance region of the slit was 71, which is considered laminar. The geometry of the slit was identified in the simulation along with the fluid properties.
Figure 4c shows the force associated with pressure per unit length profile along the normalized arc length of the outer surface. The pressure profile was generated by calculating the magnitude of the normally acting force per unit length along the arc length. Pressure initially has a linear trend along the arc length but peaks to 552 dyne cm−2 upon contact with the cushion. Pressure force is normal to the cushion and creates a direct pushing force on the entrance side of the cushion. On the opposite side of the cushion, pressure drops drastically to −235 dyne cm−2 causing a normal force which acts in the opposite direction as flow. Pressure then rises to yield an outlet pressure of 0 dyne cm−2. Because pressure in the figure is gauged relative to the outlet pressure, the negative pressure on the far side of the cushion indicates a pressure smaller than the outlet pressure. Flow into a spherical obstruction will always yield a peak in pressure upon contact followed by a drop in pressure in the far side. As shown in Fig. 4, the pressure increase and decrease are of the same order of magnitude as the shear stress values indicating that both values are important to the morphogenesis of the cushion. The combination of both shear force and pressure force is expected to cause eddies in the flow patterns. Additionally, the profile of net force along the cushion surface indicates a potential rotation of cushion in the direction of flow. Toward the bottom of the inflow side of the cushion, the net force occurs upwards and away from the cushion in the direction of flow. However; toward the bottom of the outflow side of the cushion the net force pushes into the cushion. This overall profile could account for the rotation of a cushion in the direction of flow to form a leaflet.
The next objective of this study was to define flow conditions that would be necessary in the simulated in vitro tubular system8 that would result in similar shear values to the estimates described above for the in vivo stages of diastole and systole in the developing chick heart. The enriched media solution used in the tubular culturing system has a density of 1000 kg/m3 and a viscosity of 0.005 Pa·s. These characteristics were used in modeling flow in the in vitro simulation. The geometry of the in vitro tubular culturing system is composed of a 2 mm in diameter and 8 mm in length tube with only one cushion, approximately 500 μm in thickness, implanted on the inner lumen. During development, this cushion becomes elongated and forms a valve leaflet. Therefore, diastolic and systolic flow was simulated in the in vitro tubular culturing system for a cushion geometry as well as a leaflet geometry (mound shape and fin shape). The numerical, two-dimensional model was used to estimate entrance velocities that would be required in the in vitro tubular system to obtain similar shear stresses to those recovered in vivo for stage 25 chick embryos.
In the present study, a two-dimensional, mathematical model was developed to simulate flow through the developing heart. The effectiveness of the model’s ability to describe flow-generated forces was assessed. The mathematical model was found to be a capable means to estimate key forces in the heart such as pressure and shear stress. A maximum shear stress of 277 dyne cm−2 was estimated to occur in the direction of flow and tangent to developing valve cushions during the peak of diastole, while a maximum shear stress of 110 dyne cm−2 was estimated to occur during the peak of systole. Pressure was of the same order of magnitude as shear stress and was found to act in the direction of flow on the entrance side of the heart, but, interestingly, in the opposite direction of flow on the downstream side of the cushion. This net force profile on the cushion surface could be related to the leafing of the cushion in the direction of flow and eventual cushion morphogenesis could be driven by a response to these mechanical forces. Further, vortices were shown to appear on the downstream side of the cushions tissue for Reynolds numbers which were larger than approximately 4. The in vivo simulations showed large vortex formation during the diastolic snapshot and minimal formation during the systolic snapshot. These results are comparable to previous work which included modeling of flow patterns in the embryonic heart.13,19 In each study, the transition to downstream vortex formation was shown to occur at Reynolds numbers ranging from 1 to 10. The mathematical modeling in this work will be used in future experiments to predict the flow needed in the in vitro tubular system to generate similar flow-induced forces as those experienced in vivo and ultimately optimally shape implanted cushions into valve leaflets.
The authors would like to acknowledge The University of South Carolina Magellan Scholars Program for providing funding of the research project. Further gratitude is extended toward The National Institute of Health, Department of Health and Human Services for providing funding under grant number R01HL086856. Finally, the authors thank Dr. Francis Gadala-Maria and Dr. Arash Kheradvar for their helpful comments.