Author’s Accepted Manuscript Aortic Valve Dynamics using a Fluid Structure Interaction Model-The Physiology of opening and closing Govinda Balan Kalyana Sundaram, Komarakshi R. Balakrishnan, Ramarathnam Krishna Kumar www.elsevier.com/locate/jbiomech
PII: DOI: Reference:
S0021-9290(15)00289-4 http://dx.doi.org/10.1016/j.jbiomech.2015.05.012 BM7175
To appear in: Journal of Biomechanics Received date: 17 June 2014 Revised date: 28 March 2015 Accepted date: 14 May 2015 Cite this article as: Govinda Balan Kalyana Sundaram, Komarakshi R. Balakrishnan and Ramarathnam Krishna Kumar, Aortic Valve Dynamics using a Fluid Structure Interaction Model-The Physiology of opening and closing, Journal of Biomechanics, http://dx.doi.org/10.1016/j.jbiomech.2015.05.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Aortic Valve Dynamics using a Fluid Structure Interaction Model The Physiology of opening and closing
Govinda Balan Kalyana Sundaram1, M.Tech; Komarakshi R. Balakrishnan2, M.Ch and Ramarathnam Krishna Kumar3, PhD; 1
Department of Applied Mechanics, Indian Institute of Technology Madras (IITM), Chennai,
India; 2
3
Director, Cardiovascular Division, Malar Fortis Hospitals, Chennai, India; Department of Engineering Design, Indian Institute of Technology Madras (IITM),
Chennai, India;
Keywords: Aortic root; Aortic valve, replacement; Computer Applications
Correspondence to Prof. Ramarathnam Krishna Kumar, Department of Engineering Design, Indian
Institute
of
Technology
-
Madras,
Chennai
-
600036,
India.
E-mail
[email protected]
Abstract Comparative study among aortic valves requires the use of an unbiased and relevant boundary condition. Pressure and Flow boundary conditions used in literature are not 1
sufficient for an unbiased analysis. We need a different boundary condition to analyze the valves in an unbiased, relevant environment. The proposed boundary condition is a combination of the pressure and flow boundary condition methods, which is chosen considering the demerits of the pressure and flow boundary conditions. In order to study the valve in its natural environment and to give a comparative analysis between different boundary conditions, a fluid-structure interaction analysis is made using the pressure and the proposed boundary conditions for a normal aortic valve. Commercial software LS-DYNA is used in all our analysis. The proposed boundary condition ensures a full opening of the valve with reduced valve regurgitation. It is found that for a very marginal raise in the ventricular pressure caused by pumping a fixed stroke volume, the cardiac output is considerably raised. The mechanics of the valve is similar between these two boundary conditions, however we observe that the importance of the root to raise the cardiac output may be overstated, considering the importance of the fully open nodule of arantius. Our proposed boundary condition delivers all the insights offered by the pressure and flow boundary conditions, along with providing an unbiased framework for the analysis of different valves and hence, more suitable for comparative analysis.
Introduction Engineers and researchers are interested in developing prosthetics as close as possible to the natural aortic valve (Zilla et al., 2004), hence comparison studies between the normal valves and prosthetics as well as that of valve corrective surgeries have become predominant (Yoganathan et al., 2004; Ranga et al., 2006). We realized that in order to capture the intricacies in the valve operation and compare the same during the design stage, a fluid structure interaction (FSI) model of the native valve is essential (Croft & Mofrad, 2010; De 2
Hart et al., 2003b). Several studies (Yoganathan et al, 2004; Ranga et al., 2006; De Hart et al., 2003a; De Hart et al., 2003b; Lanzarone et al., 2007; Maleki, 2010; Weinberg & Mofrad, 2007) carried out on the aortic valve enable us to have a clear understanding of the fluid mechanics in the aortic valves. Other studies (Maleki, 2010; Sripathi et al., 2004) as well bring out the differences in the mechanics of stented and stentless valves. However, a persistent issue with the valve comparison studies is in their boundary condition (BC) used for analyzing the valve. FSI simulation on the aortic valves require either a pressure BC (De Hart et al., 2003b; Maleki, 2010; Weinberg & Mofrad, 2007) or a flow BC (De Hart et al., 2003a; De Tullio et al., 2011). A time varying pressure applied at the aortic and ventricular sections of the aortic valve together form the pressure BC. The flow BC is effected by explicitly specifying the time varying flow velocity at the ventricular (or aortic) section of the valve along with a timevarying standard reference pressure applied in the same ventricular section or the aortic section of the valve. These BCs are limited and do not offer a good standard for the comparison of the prosthetic valves with the natural valve. The heart pumps a fixed stroke volume for every cardiac cycle, based on the need and this is independent of the condition of the valve. Using only a pressure BC on the valve does not ensure a fixed stroke volume of the ventricle, when different valves are compared using the same BC. In other words, the flow resistance offered by these valves control the stroke volume whenever the pressure BC is used (Yoganathan et al, 2004; Maleki, 2010; Sripathi et al., 2004). In some cases, the pressure BC will not ensure the full opening of the valve and this leads to a bias in the study. If only the flow BC is specified, it over-constrains the analysis by explicitly specifying the blood regurgitation into the left-ventricle (refer Figure 7 in (De Hart et al., 2003a), and Figure 5(a) in (De Tullio et al., 2011)). In addition, the flow BC does not ensure the full closure of the valve after ventricular ejection until a proper quantity of valve regurgitation is specified by 3
the BC. Primary criteria used for the clinical evaluation of a prosthetic aortic valve are: (i) Maximum Effective Valve Orifice Area (EVOA) obtained for a given valve size; (ii) Pressure gradients needed to obtain the EVOA, which reflects the work done by the ventricle; (iii) Time taken to attain peak EVOA, the native valve takes lesser time than the stented prosthesis (Sripathi et al., 2004); (iv) Duration for which the valve is fully open (Subhani et al., 2013); (v) Regurgitation fraction and closure mechanics; and (vi) Valve's ability to increase the EVOA on demand, for example during exercise (Sripathi et al., 2004). Care must be taken that the proposed BC allows one to estimate the above said criteria, without any bias. Material and Methods Model The objective of this paper is to bring out the important differences between using a pressure BC and the new proposed BC. Although the pressure BC is analyzed, these results are not presented fully as they can be referred from other literature as well (De Hart et al., 2003b; Maleki, 2010; Weinberg & Mofrad, 2007). Using the methods laid by Thubrikar (Thubrikar, 1990), we constructed a generalized model of the aortic valve similar to the earlier models (Subhani et al., 2013; Sripathi et al., 2004; Gnyaneshwar et al., 2002) (Figure 1A) for understanding the fundamental differences in using the different BCs. A total of 3,510 4-noded shell elements form the structural model of the aortic valve, while the fluid domain is modeled using 60,000 8-node Eulerian solid elements (Figure 1A,B). Closed configuration of the leaflets (Figure 1B) is chosen as the stress-free state of the leaflets and is the configuration from which FSI analysis is carried out. In order to achieve this configuration, the valve leaflets are closed by applying a direct pressure of 85mmHg 4
(pure structural loading) on the aorta and the leaflets, and a pressure of 7mmHg on the ventricle side of leaflets and root (for the valve shown in Figure 1A). The resulting deformed mesh (Figure 1B) is extracted and used as the stress free initial configuration for the FSI model. In all our analysis, the aortic valve was constrained by a zero-displacement BC along a single layer of shell elements at the top and bottom of the valve structure (shown in Figure 1B). Materials The aortic root and the sinus were modeled using isotropic properties (Sripathi et al., 2004), and the leaflets using orthotropic properties (Grande et al., 1998). The thickness variation along the leaflets is from (Grande et al., 1998). The blood model is a Newtonian fluid with density 1050 kg/m3, and a viscosity of 4 x 10-3 Pa-s. The proposed BC for unbiased comparison of the valves The proposed BC for the clinically relevant comparative analysis between is a combination of flow BC during ventricular ejection and the pressure BC during valve closure and diastole (Figure 2B). Note that the ventricular ejection profile is uniform throughout the cross-section of the ventricle (Bronzino & Peterson, 2006). This BC ensures that the actual stroke volume of the ventricle ejects through the valve, regardless of the valve pathology and resistance. The clinical evaluation of the valve done using this BC adequately reflects the physiology. The EVOA widely used to evaluate the clinical performance of prosthetic valves (Garcia & Kadem, 2006) is an indicative measure of the increased workload by the left ventricle (LV), and is directly controlled by the flow rate rather than the pressure differential across the leaflets. Using a flow BC during systole, the pressure gradients (derived output) across the leaflets indicate a measure of the work-load\effort exerted by the LV, which gives a comparison framework. On the other hand, by using a pressure BC during systole, the 5
workload/effort applied by the left ventricle is explicitly defined by the input BCs and does not ensure an appropriate stroke volume. In other words, depending on the valve resistance the pressure differential produces a varying flow; which is not physiologically consistent in terms of demand. Another parameter used to evaluate the valve performance is the regurgitation fraction. The flow BC, if applied during the diastole (specifically the valve regurgitation phase) will specify the regurgitation volume explicitly; and will not be useful in evaluating the valves against regurgitation. Regurgitation should be a natural byproduct of the BCs and cannot be a BC, this justifies the use of a pressure BC during diastole. One can view the resistance of the aortic valve as an afterload. Under this condition, the end systolic pressure increases while the cardiac output decreases. This gives rise to a preload as there is an increase in end diastolic volume. Frank-Starling law assures an increase in contraction and cardiac output increases (Bronzino & Peterson, 2006). Hence, the afterload-preload cycle ensures the maintenance of the stroke volume. At the limit of contractility, the pressure difference required to pump the blood is not achieved and hence the valve does not open fully. The boundary condition used in this paper mimics the first case, until the limit of the contractility. The latter case, beyond the limits of contractility, requires a more involved modeling of the ventricle. Just after ventricular ejection (lasts only 0.225s), the flow BC on the ventricular side of the aortic valve is changed to a pressure BC (Figure 2B). This pressure BC during valve closure, gives a natural regurgitation. Pressure BC has its usefulness in understanding the effect of gradients on valve behavior. But, for a realistic simulation it has to be combined with the pressure – volume loop of the LV. In a sense, the cardiac output as a boundary condition enables comparison between the opening/closing mechanism of different valves. The derived pressure gradient helps to understand the 6
workload on the heart. In other words, the BC used is unbiased for comparing different valves than that used by other researchers (Ranga et al., 2006; De Hart et al., 2003a; De Hart et al., 2003b; Maleki, 2010; Weinberg & Mofrad, 2007; De Tullio et al., 2011). In this study, both the BCs are used for a comparison. The normal aortic valve is evaluated with the pressure BC (Figure 2A) and our proposed new BC (Figure 2B), the mechanics of the valve in these two BCs are compared to bring out the differences in the valve mechanics of the proposed BC. We used a timeline plot (adjusted for one cycle) to bring out the differences between the two BCs (Figure 3). This timeline plot presents the aortic/ventricular pressure, radial root displacement, Nodule of Arantius (NoA) radial displacement, axial velocity of the aortic flow measured just above the aortic valve along the central axis of the valve, and the input flow velocity (only for the proposed BC). The proposed BC is applied by specifying a uniform flow velocity (Lanzarone et al., 2007) corresponding to the cardiac output at the inlet (the ventricular domain), while at the outlet (the aortic domain) the corresponding aortic pressure (Gnyaneshwar et al., 2002) is applied (Figure 2B) until ventricular ejection. After ventricular ejection, the proposed BC at the ventricular section of the valve alternates to a pressure BC (as shown in Figure 2B) until the end of diastole, where the next cycle begins. We have not included a comparative analysis between the flow BC (De Hart et al., 2003a; De Tullio et al., 2011) and the proposed BC of the normal aortic valve, because such a BC needs to specify the unknown valve regurgitation. The simulation was done using LS-DYNA (LSTC, Livermore, CA, USA) for 6 cardiac cycles for a total time of 5 seconds, and the results of the 5th cycle were used for observations. During this analysis, penalty approach provided the suitable fluid-structure coupling between the valve and blood. An acoustic speed of 150m/s rather than 1500m/s saves the required computational time, while obtaining reliable results. This is justified, as 7
peak flow velocity in the valve is less than 5% of the acoustic speed of blood (Maleki, 2010) (Figure 3).
Results Though the same valve is considered in both cases, the valve mechanics observed using the pressure BC is significantly different from that observed using the proposed BC. This can be seen by observing the timeline of these two valve simulations (Figure 3A, B). Firstly, the timeline of the valve driven by the pressure BC does not have a plot of the ventricular velocity time curve, as the ventricular flow profile is not a constant across the section, and does not give any information on the ventricular flow-rate of the valve. Supplying a constant cardiac output of 70 ml per beat has raised the peak aortic velocity to 2.35 m/s compared to that obtained using the pressure BC, in which the peak aortic velocity is only 1.50 m/s. This flow control has opened the valve to a greater extent than the pressure BC, as seen in the NoA displacement in Figure 3A and 3B. The pressure BC does not open the valve fully, which is observed by the reduced displacement of the NoA (Figure 3A, B). This can be verified by comparing the opening and closing of the valves by the pressure and the proposed BC (Figure 4, 5). These are a direct consequence of the using the pressure BC (as shown in the Figure 2A), which offers a variable ventricular ejection depending on the valve's flow resistance. Interestingly, the root displacement of the pressure driven valve is higher than the valve driven by the proposed BC (Figure 3A, 3B), which indicates that once the valve leaflets are fully open, the root plays very minimal role in regulating the axial flow in the valve and has minimal influence over the ventricular pressure and flow rate. The contraction and relaxation of the heart changes the perimeter of the aortic valve at the base (Bronzino & Peterson, 2006), which is largest at end diastole and decreases during 8
systole. As seen from the results, the peak aortic flow velocity occurs during the initial stages of valve opening. The aortic flow velocity for the proposed BC forms a skewed profile as shown in Figure 3B, with 2 distinct peaks. On close observation, we find that this velocity profile varies with the axial distance from the valve as shown in Figure 6; and after a certain distance, we can observe only a single peak aortic flow velocity exists in the aortic region. Although the ventricular pressure waveforms between the valves are different, the peak ventricular pressure observed using the proposed BC is only 5mmHg higher than the pressure BC (Figure 3A, B). Ventricular pressure has a direct relation to the effort of the ventricle to pump blood from heart, and hence forms a better standard for measuring the valve efficiency, rather than the pressure gradient. Figure 3B also indicates that the flow resistance of the valve varies with respect to the opening stages of the valve (closed, partiallyopen, and fully-open configurations), and naturally the fully opened valve can pump more blood for the same pressure difference compared to a partially-open valve. Valve Opening The NoA opens even with a small increase in the ventricular pressure (Figure 3B). This opening is due to the initial small ventricular output that increases the ventricular pressure and dilates the root, which pulls the leaflets apart (Thubrikar, 1990; Gnyaneshwar et al., 2002). Interestingly there is no significant aortic flow developed at this stage, but a flow in the ventricular region is seen (Figure 7A), which fills up the volume created by the expanding root. As this ventricular output rises, the ventricular pressure also builds up until the NoA splits open and a strong axial flow develops in the aorta. By the time, the ventricular pressure crosses the aortic pressure the NoA displaces by 1.9mm compared to a root displacement of 0.8mm, and the axial flow develops at about 0.07m/s. The maximum NoA displacement observed is about 9.9mm and occurs at 0.060s (3.760s) after the start of systole, 9
and the maximum root displacement observed is about 1.2mm and occurs at 0.130s (3.830s) after the start of systole. In the operational resting position (defined as the fluid loaded closed resting position of the valve), there is a small difference in the leaflet curvature between each other (Figure 7A). As the blood flows out, the leaflets push the blood stagnant in the gap between the leaflets and the sinus, which creates a 'washout' (De Hart et al., 2003b; De Hart et al., 2003b) (Figure 7B, C). As the amount of opening increases, the marginal difference in the resting leaflets' curvature results in different washout for different leaflets. In one case there is a uniform washout behind the leaflets, while in the other case, there is a stagnant flow near the apex of the leaflet (Figure 7B). This causes a small delay in the total opening time of the valve, nevertheless, as the flow rate increases this stagnant flow also washed out (Figure 7C). This leads to an asymmetry during the valve opening; which exists only for a few milliseconds, and are difficult to capture in-vivo. When the orifice is small, the ventricular flow rate dictates the aortic flow velocity to be higher as observed during the early stages of valve opening (Figure 3B). In addition, no correlation between the ventricular flow and peak aortic flow velocity exists, and the peak aortic flow velocity occurs 0.030s before the peak ventricular flow. The complex interaction between the orifice area and the ventricular flow rate dictates the pressure gradients between the ventricle and aorta, if we use the proposed BC instead of pressure BC. The observations in later case may even be different (Yoganathan et al, 2004) (Figure 3A). The orifice area and ventricular ejection dictate the fall in the peak aortic velocity. After the valve opens fully, only a minimal change in the orifice area occurs until the rapid valve closure and the aortic flow velocity is almost directly proportional to the flow rate. This occurs after 0.065s from the start of systole at 3.765s (Figure 3B). 10
Our studies on dry models (pure structural analysis on the aortic valve) (Sripathi et al., 2004; Gnyaneshwar et al., 2002) observed that for a valve without stent, the root dilates even before the opening of leaflets, and it is substantial by the time the leaflet opens. This observation remains unchanged (Figure 3B). Valve Closure Phase The significant observation on valve closure is the inward movement of NoA as seen in Figure 3B, closing starts after the peak flow. As the peak aortic velocity is reached and the valve opens fully, faint reverse flow arises in the gap between the leaflet and sinus (Figure 8A) and just ahead of the leaflets a small vortex is observed at 0.105s, after the start of the systole (at 3.805s). The flow decelerates and a downward flow develops in the sinus region, which creates a vortex (Figure 8B). These vortices can be observed even when the peak aortic velocity is at 2.61m/s and by this time, the leaflets have already started to move towards the center. Unlike other observations (Yoganathan et al, 2004; Bellhouse & Bellhouse, 1968), Figure 8B clearly shows the vortex to be marginally above the leaflet and yet the valve closes rapidly with small regurgitation. Implying, the valve has a tendency to close because of reduction in flow; which gradually decreases the pressure gradient across the valve. Of course, as the valve starts to move in, the flow behind the valve has a significant role in pushing it to closure (Figure 9C). The reverse flow picks up in the valve and 0.225s after the start of systole (corresponding to 3.925s). The root also prepares for the valve closure (Figure 9A, B). As in the case of valve opening, where the compliant root helps in a smooth opening of the valve leaflets (Sripathi et al., 2004; Gnyaneshwar et al., 2002); a compliant root also helps in the quick closure of the valve leaflets with reduced regurgitation in the valve, by pushing the NoA inwards towards the center as the ventricular pressure reduces (Figure 3A, B). The axial 11
regurgitation velocity into the ventricle, indicated by the negative magnitude in the axial velocity curve, is due to the race between the closure speed, dictated by the inertia of the leaflet and reversal of velocity due to the pressure drop in the ventricle at the end of systole. The velocity is captured at the center of the valve orifice and shows a sudden negative velocity (towards the ventricle, signifying regurgitation), for a time less than 0.020s (Figure 3B). This is because in the race for reversal of motion, fluid at the center is faster, than the closure of the valve. Hence, the regurgitation velocity at the point of closure is also a function of the end systolic valve orifice area (Figure 9). If the valve is opened fully, it takes that much time to close, resulting in regurgitation during this time. If the valve does not open fully, the regurgitation is not high as the closure takes place in a shorter time (Figure 3A,B). The opening of the valve in pressure BC is smaller than in the proposed BC and has an effect on regurgitation. The valve closure consumes 0.050s, which is the same time taken for the full valve opening, as observed from the NoA displacements in Figure 3A.
Discussions Major limitation of the pressure boundary condition is the difference in valvular opening based on the structural/inertial properties. As explained in the previous section, this has an effect on regurgitation flow and ventricular output. In case of flow BC used in the literature, where the regurgitation is a part of the input, the analysis would provide a good comparison framework for the valve opening mechanics, but not for the valve closure. The regurgitation is a function of the valve properties and specifying this a priori is flawed. Flow is clearly the driving factor for efficient cardiac function based on its variation under different conditions. For example, during exercise, the flow requirement can be as high 12
as 8 L/min as compared to only 5L/min during normal activity. A normal heart develops the corresponding pressure to deliver the required flow rate. If for example, the valve is stenotic, the LV develops higher pressure and hence higher gradient. The proposed BC mimics closely the heart function. During our survey, we found two studies that consider the integration of the aortic valve with the LV and that of the whole heart model (Carmody et al., 2006; Le & Sotiropoulos, 2013). However, complete modeling of the LV in all its complexities is not a settled issue, in addition that approach with a complete ventricular modeling may only give a better inlet velocity profile. In our opinion, it will not have an effect on the outcome of the current study; as the use of LV model is primarily to develop the required flow BC for the valve operations. Consideration of the entire arterial tree for simulating the after-load on the aorta is another novel BC, which is used in the analysis of the aortic valve; however, this is computationally intensive and generally avoided. FSI studies using impedance BCs were attempted for the ventricle and aorta (some examples are (Watanabe et al., 2004; Brown, et al., 2012), even lesser studies are available for the FSI of the aortic valve. A simple Windkessel model used by Griffith (Griffith, 2012) simulates the impedance offered by the aorta. Griffith's work is commendable; but note that by using the LV pressure as the only BC (apart from the aortic pressure, which is controlled by the impedance BC), a proper cardiac output is not enforced. This means, the end systolic leaflet opening could be smaller than the actual opening of the leaflets, which could affect the valve closure mechanics as well. The phenomenon of closing is captured realistically by the current method, since the relevant end systolic valve orifice is obtained using a flow-driven BC. Regurgitation should be a derived quantity and cannot be an input. This is dictated by the mechanics of closing, which is governed by the ventricular pressure and the open valve configuration. The proposed 13
BC achieves this effect. Though the ventricular pressure in diastole is 5mm Hg, the time to reach this pressure after systolic high is a variable and can be determined for a valve only when the complete heart model along with the valve model is solved. Nevertheless, our model captures the valve opening and closing mechanics clearly. The type of valve does not affect the method proposed. The BCs should be flow controlled during systole and pressure controlled during diastole, regardless of whether the valves are mechanical or bio-prosthetic. Our approach also permits the incorporation of pressure and flow BCs obtained from 0-D, 1D models of the human circulatory system. One can obtain the big-picture variations of pressure and flow using the 0-D and 1-D models, and use this data as the BC of the FE model to study the valve in detail in a smaller environment.
Acknowledgments We acknowledge the facilities offered by Raghupati Singhania Center of Excellence (RPSCOE), HASETRI, Chennai.
Conflict of Interest: None
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Garcia, D., & Kadem, L. (2006). What do you mean by aortic valve area: geometric orifice area, effective orifice area, or gorlin area? Journal of Heart Valve Disease , 601. 15
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Rajani, R., Khattar, R., Chiribiri, A., Victor, K., & Chambers, J. (2014). Multimodality Imaging of Heart Valve Disease. Arq Bras Cardiol , 103, 251--263. Ranga, A., Bouchot, O., Mongrain, R., Ugolini, P., & Cartier, R. (2006). Computational simulations of the aortic valve validated by imaging data: evaluation of valve-sparing techniques. Interactive cardiovascular and thoracic surgery , 5, 373-378. Sripathi, V. C., Kumar, R. K., & Balakrishnan, K. R. (2004). Further insights into normal aortic valve function: role of a compliant aortic root on leaflet opening and valve orifice area. The Annals of thoracic surgery , 77, 844-851. Subhani, M., Kumar, R., & Balakrishnan, K. (2013). Normal aortic valves stay open much longer in systole than porcine substitutes. Asian cardiovascular and thoracic annals , 21, 275-280. Thubrikar, M. (1990). The aortic valve. CRC press Boca Raton, FL. Watanabe, H., Sugiura, S., Kafuku, H., & Hisada, T. (2004). Multiphysics simulation of left ventricular filling dynamics using fluid-structure interaction finite element method. Biophysical journal , 87, 2074-2085. Weinberg, E. J., & Mofrad, M. R. (2007). Transient, three-dimensional, multiscale simulations of the human aortic valve. Cardiovascular Engineering , 7, 140-155. Yoganathan, A. P., He, Z., & Casey, J. S. (2004). Fluid mechanics of heart valves. Annual review of biomedical engineering , 6, 331-362.
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Figure 1
Finite element model of the aortic valve;
(A) One-third model of the aortic valve with leaflets open as the stress free state of the leaflets, the sinus is made transparent to show the leaflets within, (B) Aortic valve closed by the application of dry pressure, this closed leaflet configuration considered as the stress-free state that is immersed in a cylindrical fluid domain.
Figure 2B
Boundary conditions applied at the inlet and outlet of the aortic valve;
(A) Pressure BC: The BC over the systole and diastole are the same shown here in unit of mmHg, (B) Proposed BC: The systole and diastole phase shown shaded in the first cycle of the valve. The ventricular output given in terms of velocity as it has a uniform profile corresponding to a stroke volume of 70 ml. The discontinuity in the ventricular output and the ventricular pressure curve indicate the on/off of the corresponding BC.
Figure 3
Time history of the root displacement, NoA displacement, blood pressure, and
flow velocities during 1 cycle of valve opening and closure; for the (A) Normal aortic valve (pressure BC), (B) Normal aortic valve (proposed BC)
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Figure 4
Opening and closing of the aortic valve leaflets using the pressure BC along
with the von-Mises leaflet stress in pascal unit.
Figure 5
Opening and closing of the aortic valve leaflets using the proposed BC along
with the von-Mises leaflet stress in pascal unit.
Figure 6
Variation in the aortic flow profile as the distance from the NoA increases (wrt
the closed position) along the axial direction. Figure 7
Opening of the normal aortic valve along with the blood velocity (m/s) map;
During (A) Closed resting position of the leaflets showing barely visible asymmetries in leaflet structure, (B) Intermediate opening stage of the valve showing asymmetrical stagnation of blood flow between the leaflet apex and the sinus forming a vortex, (C) Before fully opening the valve showing the washout of the stagnant blood near the leaflet apex.
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Figure 8
Normal aortic valve vortexes and the blood velocity (m/s) map in the sinus
region for the proposed BC; (A) Faint reverse flow developed in the sinus region near the leaflet apex during peak aortic velocity is shown in the box. (B) Development of this faint reverse flow and its gradual evolution to the deceleration vortex near the leaflet apex.
Figure 9
Stages of valve closure in a normal aortic valve;
(A) Early stage of valve closure just before reversal of aortic flow velocities, (B) Reversal of flow in the valve due to a high reverse pressure gradients, (C) Vortexes developed in the sinus cavity because of the reversal of flow in the valve.
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Figure 7C
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PII: DOI: Reference:
S0021-9290(15)00289-4 http://dx.doi.org/10.1016/j.jbiomech.2015.05.012 BM7175
To appear in: Journal of Biomechanics Received date: 17 June 2014 Revised date: 28 March 2015 Accepted date: 14 May 2015 Cite this article as: Govinda Balan Kalyana Sundaram, Komarakshi R. Balakrishnan and Ramarathnam Krishna Kumar, Aortic Valve Dynamics using a Fluid Structure Interaction Model-The Physiology of opening and closing, Journal of Biomechanics, http://dx.doi.org/10.1016/j.jbiomech.2015.05.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Aortic Valve Dynamics using a Fluid Structure Interaction Model The Physiology of opening and closing
Govinda Balan Kalyana Sundaram1, M.Tech; Komarakshi R. Balakrishnan2, M.Ch and Ramarathnam Krishna Kumar3, PhD; 1
Department of Applied Mechanics, Indian Institute of Technology Madras (IITM), Chennai,
India; 2
3
Director, Cardiovascular Division, Malar Fortis Hospitals, Chennai, India; Department of Engineering Design, Indian Institute of Technology Madras (IITM),
Chennai, India;
Keywords: Aortic root; Aortic valve, replacement; Computer Applications
Correspondence to Prof. Ramarathnam Krishna Kumar, Department of Engineering Design, Indian
Institute
of
Technology
-
Madras,
Chennai
-
600036,
India.
[email protected]
Abstract Comparative study among aortic valves requires the use of an unbiased and relevant boundary condition. Pressure and Flow boundary conditions used in literature are not 1
sufficient for an unbiased analysis. We need a different boundary condition to analyze the valves in an unbiased, relevant environment. The proposed boundary condition is a combination of the pressure and flow boundary condition methods, which is chosen considering the demerits of the pressure and flow boundary conditions. In order to study the valve in its natural environment and to give a comparative analysis between different boundary conditions, a fluid-structure interaction analysis is made using the pressure and the proposed boundary conditions for a normal aortic valve. Commercial software LS-DYNA is used in all our analysis. The proposed boundary condition ensures a full opening of the valve with reduced valve regurgitation. It is found that for a very marginal raise in the ventricular pressure caused by pumping a fixed stroke volume, the cardiac output is considerably raised. The mechanics of the valve is similar between these two boundary conditions, however we observe that the importance of the root to raise the cardiac output may be overstated, considering the importance of the fully open nodule of arantius. Our proposed boundary condition delivers all the insights offered by the pressure and flow boundary conditions, along with providing an unbiased framework for the analysis of different valves and hence, more suitable for comparative analysis.
Introduction Engineers and researchers are interested in developing prosthetics as close as possible to the natural aortic valve (Zilla et al., 2004), hence comparison studies between the normal valves and prosthetics as well as that of valve corrective surgeries have become predominant (Yoganathan et al., 2004; Ranga et al., 2006). We realized that in order to capture the intricacies in the valve operation and compare the same during the design stage, a fluid structure interaction (FSI) model of the native valve is essential (Croft & Mofrad, 2010; De 2
Hart et al., 2003b). Several studies (Yoganathan et al, 2004; Ranga et al., 2006; De Hart et al., 2003a; De Hart et al., 2003b; Lanzarone et al., 2007; Maleki, 2010; Weinberg & Mofrad, 2007) carried out on the aortic valve enable us to have a clear understanding of the fluid mechanics in the aortic valves. Other studies (Maleki, 2010; Sripathi et al., 2004) as well bring out the differences in the mechanics of stented and stentless valves. However, a persistent issue with the valve comparison studies is in their boundary condition (BC) used for analyzing the valve. FSI simulation on the aortic valves require either a pressure BC (De Hart et al., 2003b; Maleki, 2010; Weinberg & Mofrad, 2007) or a flow BC (De Hart et al., 2003a; De Tullio et al., 2011). A time varying pressure applied at the aortic and ventricular sections of the aortic valve together form the pressure BC. The flow BC is effected by explicitly specifying the time varying flow velocity at the ventricular (or aortic) section of the valve along with a timevarying standard reference pressure applied in the same ventricular section or the aortic section of the valve. These BCs are limited and do not offer a good standard for the comparison of the prosthetic valves with the natural valve. The heart pumps a fixed stroke volume for every cardiac cycle, based on the need and this is independent of the condition of the valve. Using only a pressure BC on the valve does not ensure a fixed stroke volume of the ventricle, when different valves are compared using the same BC. In other words, the flow resistance offered by these valves control the stroke volume whenever the pressure BC is used (Yoganathan et al, 2004; Maleki, 2010; Sripathi et al., 2004). In some cases, the pressure BC will not ensure the full opening of the valve and this leads to a bias in the study. If only the flow BC is specified, it over-constrains the analysis by explicitly specifying the blood regurgitation into the left-ventricle (refer Figure 7 in (De Hart et al., 2003a), and Figure 5(a) in (De Tullio et al., 2011)). In addition, the flow BC does not ensure the full closure of the valve after ventricular ejection until a proper quantity of valve regurgitation is specified by 3
the BC. Primary criteria used for the clinical evaluation of a prosthetic aortic valve are: (i) Maximum Effective Valve Orifice Area (EVOA) obtained for a given valve size; (ii) Pressure gradients needed to obtain the EVOA, which reflects the work done by the ventricle; (iii) Time taken to attain peak EVOA, the native valve takes lesser time than the stented prosthesis (Sripathi et al., 2004); (iv) Duration for which the valve is fully open (Subhani et al., 2013); (v) Regurgitation fraction and closure mechanics; and (vi) Valve's ability to increase the EVOA on demand, for example during exercise (Sripathi et al., 2004). Care must be taken that the proposed BC allows one to estimate the above said criteria, without any bias. Material and Methods Model The objective of this paper is to bring out the important differences between using a pressure BC and the new proposed BC. Although the pressure BC is analyzed, these results are not presented fully as they can be referred from other literature as well (De Hart et al., 2003b; Maleki, 2010; Weinberg & Mofrad, 2007). Using the methods laid by Thubrikar (Thubrikar, 1990), we constructed a generalized model of the aortic valve similar to the earlier models (Subhani et al., 2013; Sripathi et al., 2004; Gnyaneshwar et al., 2002) (Figure 1A) for understanding the fundamental differences in using the different BCs. A total of 3,510 4-noded shell elements form the structural model of the aortic valve, while the fluid domain is modeled using 60,000 8-node Eulerian solid elements (Figure 1A,B). Closed configuration of the leaflets (Figure 1B) is chosen as the stress-free state of the leaflets and is the configuration from which FSI analysis is carried out. In order to achieve this configuration, the valve leaflets are closed by applying a direct pressure of 85mmHg 4
(pure structural loading) on the aorta and the leaflets, and a pressure of 7mmHg on the ventricle side of leaflets and root (for the valve shown in Figure 1A). The resulting deformed mesh (Figure 1B) is extracted and used as the stress free initial configuration for the FSI model. In all our analysis, the aortic valve was constrained by a zero-displacement BC along a single layer of shell elements at the top and bottom of the valve structure (shown in Figure 1B). Materials The aortic root and the sinus were modeled using isotropic properties (Sripathi et al., 2004), and the leaflets using orthotropic properties (Grande et al., 1998). The thickness variation along the leaflets is from (Grande et al., 1998). The blood model is a Newtonian fluid with density 1050 kg/m3, and a viscosity of 4 x 10-3 Pa-s. The proposed BC for unbiased comparison of the valves The proposed BC for the clinically relevant comparative analysis between is a combination of flow BC during ventricular ejection and the pressure BC during valve closure and diastole (Figure 2B). Note that the ventricular ejection profile is uniform throughout the cross-section of the ventricle (Bronzino & Peterson, 2006). This BC ensures that the actual stroke volume of the ventricle ejects through the valve, regardless of the valve pathology and resistance. The clinical evaluation of the valve done using this BC adequately reflects the physiology. The EVOA widely used to evaluate the clinical performance of prosthetic valves (Garcia & Kadem, 2006) is an indicative measure of the increased workload by the left ventricle (LV), and is directly controlled by the flow rate rather than the pressure differential across the leaflets. Using a flow BC during systole, the pressure gradients (derived output) across the leaflets indicate a measure of the work-load\effort exerted by the LV, which gives a comparison framework. On the other hand, by using a pressure BC during systole, the 5
workload/effort applied by the left ventricle is explicitly defined by the input BCs and does not ensure an appropriate stroke volume. In other words, depending on the valve resistance the pressure differential produces a varying flow; which is not physiologically consistent in terms of demand. Another parameter used to evaluate the valve performance is the regurgitation fraction. The flow BC, if applied during the diastole (specifically the valve regurgitation phase) will specify the regurgitation volume explicitly; and will not be useful in evaluating the valves against regurgitation. Regurgitation should be a natural byproduct of the BCs and cannot be a BC, this justifies the use of a pressure BC during diastole. One can view the resistance of the aortic valve as an afterload. Under this condition, the end systolic pressure increases while the cardiac output decreases. This gives rise to a preload as there is an increase in end diastolic volume. Frank-Starling law assures an increase in contraction and cardiac output increases (Bronzino & Peterson, 2006). Hence, the afterload-preload cycle ensures the maintenance of the stroke volume. At the limit of contractility, the pressure difference required to pump the blood is not achieved and hence the valve does not open fully. The boundary condition used in this paper mimics the first case, until the limit of the contractility. The latter case, beyond the limits of contractility, requires a more involved modeling of the ventricle. Just after ventricular ejection (lasts only 0.225s), the flow BC on the ventricular side of the aortic valve is changed to a pressure BC (Figure 2B). This pressure BC during valve closure, gives a natural regurgitation. Pressure BC has its usefulness in understanding the effect of gradients on valve behavior. But, for a realistic simulation it has to be combined with the pressure – volume loop of the LV. In a sense, the cardiac output as a boundary condition enables comparison between the opening/closing mechanism of different valves. The derived pressure gradient helps to understand the 6
workload on the heart. In other words, the BC used is unbiased for comparing different valves than that used by other researchers (Ranga et al., 2006; De Hart et al., 2003a; De Hart et al., 2003b; Maleki, 2010; Weinberg & Mofrad, 2007; De Tullio et al., 2011). In this study, both the BCs are used for a comparison. The normal aortic valve is evaluated with the pressure BC (Figure 2A) and our proposed new BC (Figure 2B), the mechanics of the valve in these two BCs are compared to bring out the differences in the valve mechanics of the proposed BC. We used a timeline plot (adjusted for one cycle) to bring out the differences between the two BCs (Figure 3). This timeline plot presents the aortic/ventricular pressure, radial root displacement, Nodule of Arantius (NoA) radial displacement, axial velocity of the aortic flow measured just above the aortic valve along the central axis of the valve, and the input flow velocity (only for the proposed BC). The proposed BC is applied by specifying a uniform flow velocity (Lanzarone et al., 2007) corresponding to the cardiac output at the inlet (the ventricular domain), while at the outlet (the aortic domain) the corresponding aortic pressure (Gnyaneshwar et al., 2002) is applied (Figure 2B) until ventricular ejection. After ventricular ejection, the proposed BC at the ventricular section of the valve alternates to a pressure BC (as shown in Figure 2B) until the end of diastole, where the next cycle begins. We have not included a comparative analysis between the flow BC (De Hart et al., 2003a; De Tullio et al., 2011) and the proposed BC of the normal aortic valve, because such a BC needs to specify the unknown valve regurgitation. The simulation was done using LS-DYNA (LSTC, Livermore, CA, USA) for 6 cardiac cycles for a total time of 5 seconds, and the results of the 5th cycle were used for observations. During this analysis, penalty approach provided the suitable fluid-structure coupling between the valve and blood. An acoustic speed of 150m/s rather than 1500m/s saves the required computational time, while obtaining reliable results. This is justified, as 7
peak flow velocity in the valve is less than 5% of the acoustic speed of blood (Maleki, 2010) (Figure 3).
Results Though the same valve is considered in both cases, the valve mechanics observed using the pressure BC is significantly different from that observed using the proposed BC. This can be seen by observing the timeline of these two valve simulations (Figure 3A, B). Firstly, the timeline of the valve driven by the pressure BC does not have a plot of the ventricular velocity time curve, as the ventricular flow profile is not a constant across the section, and does not give any information on the ventricular flow-rate of the valve. Supplying a constant cardiac output of 70 ml per beat has raised the peak aortic velocity to 2.35 m/s compared to that obtained using the pressure BC, in which the peak aortic velocity is only 1.50 m/s. This flow control has opened the valve to a greater extent than the pressure BC, as seen in the NoA displacement in Figure 3A and 3B. The pressure BC does not open the valve fully, which is observed by the reduced displacement of the NoA (Figure 3A, B). This can be verified by comparing the opening and closing of the valves by the pressure and the proposed BC (Figure 4, 5). These are a direct consequence of the using the pressure BC (as shown in the Figure 2A), which offers a variable ventricular ejection depending on the valve's flow resistance. Interestingly, the root displacement of the pressure driven valve is higher than the valve driven by the proposed BC (Figure 3A, 3B), which indicates that once the valve leaflets are fully open, the root plays very minimal role in regulating the axial flow in the valve and has minimal influence over the ventricular pressure and flow rate. The contraction and relaxation of the heart changes the perimeter of the aortic valve at the base (Bronzino & Peterson, 2006), which is largest at end diastole and decreases during 8
systole. As seen from the results, the peak aortic flow velocity occurs during the initial stages of valve opening. The aortic flow velocity for the proposed BC forms a skewed profile as shown in Figure 3B, with 2 distinct peaks. On close observation, we find that this velocity profile varies with the axial distance from the valve as shown in Figure 6; and after a certain distance, we can observe only a single peak aortic flow velocity exists in the aortic region. Although the ventricular pressure waveforms between the valves are different, the peak ventricular pressure observed using the proposed BC is only 5mmHg higher than the pressure BC (Figure 3A, B). Ventricular pressure has a direct relation to the effort of the ventricle to pump blood from heart, and hence forms a better standard for measuring the valve efficiency, rather than the pressure gradient. Figure 3B also indicates that the flow resistance of the valve varies with respect to the opening stages of the valve (closed, partiallyopen, and fully-open configurations), and naturally the fully opened valve can pump more blood for the same pressure difference compared to a partially-open valve. Valve Opening The NoA opens even with a small increase in the ventricular pressure (Figure 3B). This opening is due to the initial small ventricular output that increases the ventricular pressure and dilates the root, which pulls the leaflets apart (Thubrikar, 1990; Gnyaneshwar et al., 2002). Interestingly there is no significant aortic flow developed at this stage, but a flow in the ventricular region is seen (Figure 7A), which fills up the volume created by the expanding root. As this ventricular output rises, the ventricular pressure also builds up until the NoA splits open and a strong axial flow develops in the aorta. By the time, the ventricular pressure crosses the aortic pressure the NoA displaces by 1.9mm compared to a root displacement of 0.8mm, and the axial flow develops at about 0.07m/s. The maximum NoA displacement observed is about 9.9mm and occurs at 0.060s (3.760s) after the start of systole, 9
and the maximum root displacement observed is about 1.2mm and occurs at 0.130s (3.830s) after the start of systole. In the operational resting position (defined as the fluid loaded closed resting position of the valve), there is a small difference in the leaflet curvature between each other (Figure 7A). As the blood flows out, the leaflets push the blood stagnant in the gap between the leaflets and the sinus, which creates a 'washout' (De Hart et al., 2003b; De Hart et al., 2003b) (Figure 7B, C). As the amount of opening increases, the marginal difference in the resting leaflets' curvature results in different washout for different leaflets. In one case there is a uniform washout behind the leaflets, while in the other case, there is a stagnant flow near the apex of the leaflet (Figure 7B). This causes a small delay in the total opening time of the valve, nevertheless, as the flow rate increases this stagnant flow also washed out (Figure 7C). This leads to an asymmetry during the valve opening; which exists only for a few milliseconds, and are difficult to capture in-vivo. When the orifice is small, the ventricular flow rate dictates the aortic flow velocity to be higher as observed during the early stages of valve opening (Figure 3B). In addition, no correlation between the ventricular flow and peak aortic flow velocity exists, and the peak aortic flow velocity occurs 0.030s before the peak ventricular flow. The complex interaction between the orifice area and the ventricular flow rate dictates the pressure gradients between the ventricle and aorta, if we use the proposed BC instead of pressure BC. The observations in later case may even be different (Yoganathan et al, 2004) (Figure 3A). The orifice area and ventricular ejection dictate the fall in the peak aortic velocity. After the valve opens fully, only a minimal change in the orifice area occurs until the rapid valve closure and the aortic flow velocity is almost directly proportional to the flow rate. This occurs after 0.065s from the start of systole at 3.765s (Figure 3B). 10
Our studies on dry models (pure structural analysis on the aortic valve) (Sripathi et al., 2004; Gnyaneshwar et al., 2002) observed that for a valve without stent, the root dilates even before the opening of leaflets, and it is substantial by the time the leaflet opens. This observation remains unchanged (Figure 3B). Valve Closure Phase The significant observation on valve closure is the inward movement of NoA as seen in Figure 3B, closing starts after the peak flow. As the peak aortic velocity is reached and the valve opens fully, faint reverse flow arises in the gap between the leaflet and sinus (Figure 8A) and just ahead of the leaflets a small vortex is observed at 0.105s, after the start of the systole (at 3.805s). The flow decelerates and a downward flow develops in the sinus region, which creates a vortex (Figure 8B). These vortices can be observed even when the peak aortic velocity is at 2.61m/s and by this time, the leaflets have already started to move towards the center. Unlike other observations (Yoganathan et al, 2004; Bellhouse & Bellhouse, 1968), Figure 8B clearly shows the vortex to be marginally above the leaflet and yet the valve closes rapidly with small regurgitation. Implying, the valve has a tendency to close because of reduction in flow; which gradually decreases the pressure gradient across the valve. Of course, as the valve starts to move in, the flow behind the valve has a significant role in pushing it to closure (Figure 9C). The reverse flow picks up in the valve and 0.225s after the start of systole (corresponding to 3.925s). The root also prepares for the valve closure (Figure 9A, B). As in the case of valve opening, where the compliant root helps in a smooth opening of the valve leaflets (Sripathi et al., 2004; Gnyaneshwar et al., 2002); a compliant root also helps in the quick closure of the valve leaflets with reduced regurgitation in the valve, by pushing the NoA inwards towards the center as the ventricular pressure reduces (Figure 3A, B). The axial 11
regurgitation velocity into the ventricle, indicated by the negative magnitude in the axial velocity curve, is due to the race between the closure speed, dictated by the inertia of the leaflet and reversal of velocity due to the pressure drop in the ventricle at the end of systole. The velocity is captured at the center of the valve orifice and shows a sudden negative velocity (towards the ventricle, signifying regurgitation), for a time less than 0.020s (Figure 3B). This is because in the race for reversal of motion, fluid at the center is faster, than the closure of the valve. Hence, the regurgitation velocity at the point of closure is also a function of the end systolic valve orifice area (Figure 9). If the valve is opened fully, it takes that much time to close, resulting in regurgitation during this time. If the valve does not open fully, the regurgitation is not high as the closure takes place in a shorter time (Figure 3A,B). The opening of the valve in pressure BC is smaller than in the proposed BC and has an effect on regurgitation. The valve closure consumes 0.050s, which is the same time taken for the full valve opening, as observed from the NoA displacements in Figure 3A.
Discussions Major limitation of the pressure boundary condition is the difference in valvular opening based on the structural/inertial properties. As explained in the previous section, this has an effect on regurgitation flow and ventricular output. In case of flow BC used in the literature, where the regurgitation is a part of the input, the analysis would provide a good comparison framework for the valve opening mechanics, but not for the valve closure. The regurgitation is a function of the valve properties and specifying this a priori is flawed. Flow is clearly the driving factor for efficient cardiac function based on its variation under different conditions. For example, during exercise, the flow requirement can be as high 12
as 8 L/min as compared to only 5L/min during normal activity. A normal heart develops the corresponding pressure to deliver the required flow rate. If for example, the valve is stenotic, the LV develops higher pressure and hence higher gradient. The proposed BC mimics closely the heart function. During our survey, we found two studies that consider the integration of the aortic valve with the LV and that of the whole heart model (Carmody et al., 2006; Le & Sotiropoulos, 2013). However, complete modeling of the LV in all its complexities is not a settled issue, in addition that approach with a complete ventricular modeling may only give a better inlet velocity profile. In our opinion, it will not have an effect on the outcome of the current study; as the use of LV model is primarily to develop the required flow BC for the valve operations. Consideration of the entire arterial tree for simulating the after-load on the aorta is another novel BC, which is used in the analysis of the aortic valve; however, this is computationally intensive and generally avoided. FSI studies using impedance BCs were attempted for the ventricle and aorta (some examples are (Watanabe et al., 2004; Brown, et al., 2012), even lesser studies are available for the FSI of the aortic valve. A simple Windkessel model used by Griffith (Griffith, 2012) simulates the impedance offered by the aorta. Griffith's work is commendable; but note that by using the LV pressure as the only BC (apart from the aortic pressure, which is controlled by the impedance BC), a proper cardiac output is not enforced. This means, the end systolic leaflet opening could be smaller than the actual opening of the leaflets, which could affect the valve closure mechanics as well. The phenomenon of closing is captured realistically by the current method, since the relevant end systolic valve orifice is obtained using a flow-driven BC. Regurgitation should be a derived quantity and cannot be an input. This is dictated by the mechanics of closing, which is governed by the ventricular pressure and the open valve configuration. The proposed 13
BC achieves this effect. Though the ventricular pressure in diastole is 5mm Hg, the time to reach this pressure after systolic high is a variable and can be determined for a valve only when the complete heart model along with the valve model is solved. Nevertheless, our model captures the valve opening and closing mechanics clearly. The type of valve does not affect the method proposed. The BCs should be flow controlled during systole and pressure controlled during diastole, regardless of whether the valves are mechanical or bio-prosthetic. Our approach also permits the incorporation of pressure and flow BCs obtained from 0-D, 1D models of the human circulatory system. One can obtain the big-picture variations of pressure and flow using the 0-D and 1-D models, and use this data as the BC of the FE model to study the valve in detail in a smaller environment.
Acknowledgments We acknowledge the facilities offered by Raghupati Singhania Center of Excellence (RPSCOE), HASETRI, Chennai.
Conflict of Interest: None
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Rajani, R., Khattar, R., Chiribiri, A., Victor, K., & Chambers, J. (2014). Multimodality Imaging of Heart Valve Disease. Arq Bras Cardiol , 103, 251--263. Ranga, A., Bouchot, O., Mongrain, R., Ugolini, P., & Cartier, R. (2006). Computational simulations of the aortic valve validated by imaging data: evaluation of valve-sparing techniques. Interactive cardiovascular and thoracic surgery , 5, 373-378. Sripathi, V. C., Kumar, R. K., & Balakrishnan, K. R. (2004). Further insights into normal aortic valve function: role of a compliant aortic root on leaflet opening and valve orifice area. The Annals of thoracic surgery , 77, 844-851. Subhani, M., Kumar, R., & Balakrishnan, K. (2013). Normal aortic valves stay open much longer in systole than porcine substitutes. Asian cardiovascular and thoracic annals , 21, 275-280. Thubrikar, M. (1990). The aortic valve. CRC press Boca Raton, FL. Watanabe, H., Sugiura, S., Kafuku, H., & Hisada, T. (2004). Multiphysics simulation of left ventricular filling dynamics using fluid-structure interaction finite element method. Biophysical journal , 87, 2074-2085. Weinberg, E. J., & Mofrad, M. R. (2007). Transient, three-dimensional, multiscale simulations of the human aortic valve. Cardiovascular Engineering , 7, 140-155. Yoganathan, A. P., He, Z., & Casey, J. S. (2004). Fluid mechanics of heart valves. Annual review of biomedical engineering , 6, 331-362.
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Figure 1
Finite element model of the aortic valve;
(A) One-third model of the aortic valve with leaflets open as the stress free state of the leaflets, the sinus is made transparent to show the leaflets within, (B) Aortic valve closed by the application of dry pressure, this closed leaflet configuration considered as the stress-free state that is immersed in a cylindrical fluid domain.
Figure 2B
Boundary conditions applied at the inlet and outlet of the aortic valve;
(A) Pressure BC: The BC over the systole and diastole are the same shown here in unit of mmHg, (B) Proposed BC: The systole and diastole phase shown shaded in the first cycle of the valve. The ventricular output given in terms of velocity as it has a uniform profile corresponding to a stroke volume of 70 ml. The discontinuity in the ventricular output and the ventricular pressure curve indicate the on/off of the corresponding BC.
Figure 3
Time history of the root displacement, NoA displacement, blood pressure, and
flow velocities during 1 cycle of valve opening and closure; for the (A) Normal aortic valve (pressure BC), (B) Normal aortic valve (proposed BC)
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Figure 4
Opening and closing of the aortic valve leaflets using the pressure BC along
with the von-Mises leaflet stress in pascal unit.
Figure 5
Opening and closing of the aortic valve leaflets using the proposed BC along
with the von-Mises leaflet stress in pascal unit.
Figure 6
Variation in the aortic flow profile as the distance from the NoA increases (wrt
the closed position) along the axial direction. Figure 7
Opening of the normal aortic valve along with the blood velocity (m/s) map;
During (A) Closed resting position of the leaflets showing barely visible asymmetries in leaflet structure, (B) Intermediate opening stage of the valve showing asymmetrical stagnation of blood flow between the leaflet apex and the sinus forming a vortex, (C) Before fully opening the valve showing the washout of the stagnant blood near the leaflet apex.
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Figure 8
Normal aortic valve vortexes and the blood velocity (m/s) map in the sinus
region for the proposed BC; (A) Faint reverse flow developed in the sinus region near the leaflet apex during peak aortic velocity is shown in the box. (B) Development of this faint reverse flow and its gradual evolution to the deceleration vortex near the leaflet apex.
Figure 9
Stages of valve closure in a normal aortic valve;
(A) Early stage of valve closure just before reversal of aortic flow velocities, (B) Reversal of flow in the valve due to a high reverse pressure gradients, (C) Vortexes developed in the sinus cavity because of the reversal of flow in the valve.
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