Determination of pressure data in aortic valves

Transcription

1 Determination of pressure data in aortic valves Helena Švihlová a, Jaroslav Hron a, Josef Málek a, K.R.Rajagopal b and Keshava Rajagopal c a, Mathematical Institute, Charles University, Czech Republic b, Texas A&M University, College Station TX, United States c, Memorial Hermann Texas Medical Center, Houston TX, United States MSCS Magdeburg, October 24, 2018

2 The presentation is based on the following study: H. Švihlová, J. Hron, J. Málek, K. R. Rajagopal, K. Rajagopal (2016): Determination of pressure data from velocity data with a view toward its application in cardiovascular mechanics. Part 1. Theoretical considerations. In: International Journal of Engineering Science 105, H. Švihlová, J. Hron, J. Málek, K. R. Rajagopal, K. Rajagopal (2017): Determination of pressure data from velocity data with a view towards its application in cardiovascular mechanics. Part 2: A study of aortic valve stenosis. In: International Journal of Engineering Science 114,1 15. The work was supported by KONTAKT II (LH14054) financed by MŠMT ČR. 2/37

3 Motivation Current medical methods in stenosis evaluation and pressure determination Pressure determination - continuum mechanics approach Pressure difference and energy dissipation dependence on the stenosis severity 3/37

4 Motivation Aortic valve 4/37

5 Motivation Aortic valve A. A. Basri et al.: Computational fluid dynamics study of the aortic valve opening on hemodynamics characteristics. In: 2014 IEEE Conference on Biomedical Engineering and Sciences (IECBES) (2014): F. Sturla et al.: Impact of different aortic valve calcification patterns on the outcome of transcatheter aortic valve implantation: A finite element study. In: Journal of Biomechanics (2016): S. C. Shadden, M. Astorino and J-F. Gerbeau: Computational analysis of an aortic valve jet with Lagrangian coherent structures. In: Chaos: An Interdisciplinary Journal of Nonlinear Science 20.1 (2010): /37

6 Current methods in stenosis evaluation Motivation Current medical methods in stenosis evaluation and pressure determination Pressure determination - continuum mechanics approach Pressure difference and energy dissipation dependence on the stenosis severity 6/37

7 Current methods in stenosis evaluation Current methods in stenosis evaluation anatomic stenosis severity = physiologically important stenosis ( 1 area stenotic area healthy valve area/effective orifice area additional heart work/energy dissipation ) 100% trans-stenosis pressure difference 7/37

8 Current methods in stenosis evaluation A cardiac cycle Figure: Blood pressure in the left ventricle during a cardiac cycle. (from Wikipedia, modified.) Phase 1, diastolic filling Phase 2, isovolumic contraction Phase 3, systolic ejection Phase 4, isovolumic relaxation 8/37

9 Current methods in stenosis evaluation Pressure difference Plaque Artery Wall v 1, p 1 v 2, p 2 Plaque 1 2 ρ v h 1 ρ g = 1 2 ρ v h 2 ρ g (h 1 h 2 )ρ g = 1 2 ρ v 2 2 (h 1 h 2 ) = Cv 2 2 H. Baumgartner et al.: Echocardiographic assessment of valve stenosis: EAE/ASE recommendations for clinical practice. In: European Journal of Echocardiography, 10(1) (2009): /37

10 Pressure determination Motivation Current medical methods in stenosis evaluation and pressure determination Pressure determination - continuum mechanics approach Pressure difference and energy dissipation dependence on the stenosis severity 10/37

11 Pressure determination Pressure Poisson equation ( p = ρ ( v) v v ) + div(2µ D) =: f t q ppe = div f in Ω q ppe = n f on Ω n q ppe = p on Γ out 11/37

12 Pressure determination Stokes equation ( p = ρ ( v) v v ) + div(2µ D) =: f t a + q ste = f in Ω, div a = 0 in Ω, a = 0 on Ω, q ste = p on Γ out. M. E. Cayco and R. A. Nicolaides: Finite Element Technique for Optimal Pressure Recovery from Stream Function Formulation of Viscous Flows. In: Mathematics of Computation, (1986): /37

13 Pressure determination Input data/4d MRI time resolved phase-contrast magnetic resonance imaging 4D-PCMR (4D Flow MRI) limitations in both spatial and temporal resolution of the signals A. Bakhshinejad et al.: Merging Computational Fluid Dynamics and 4D Flow MRI Using Proper Orthogonal Decomposition and Ridge Regression. In: Journal of Biomechanics 58 (2017): V. C. Rispoli et al.: Computational fluid dynamics simulations of blood flow regularized by 3D phase contrast MRI. In: BioMedical Engineering OnLine 14.1 (2015). fixing pressure (catheterization) 13/37

14 Pressure determination Reference flow incompressible unstationary Navier-Stokes equation, no-slip wall, velocity prescribed on the inlet with parabolic profile, pressure and the backflow stabilization/penalization prescribed on the outlet 14/37

15 Pressure determination Comparison on the same mesh q ppe p ref L 2 p ref L 2 q ste p ref L 2 p ref L 2 symmetric 6.40e e-14 non-symmetric 3.50e e-14 Table: Relative errors for fine data. L0 mesh for symmetric case L0 mesh for non-symmetric case pressure [mmhg] q PPE q STE p REF z coordinate [mm] pressure [mmhg] q PPE q STE p REF z coordinate [mm] 15/37

16 Pressure determination Comparison on the coarser meshes L0 L1 L2 L3 L4 L0 L1 L2 L3 L4 L4 mesh for symmetric case L4 mesh for non-symmetric case pressure [mmhg] pressure [mmhg] q PPE q STE p REF z coordinate [mm] q PPE q STE p REF z coordinate [mm] 16/37

17 Pressure determination Comparison on the coarser meshes symmetric case non-symmetric case err ppe err ste err ppe err ste err ppe err ste err ppe err ste L0N0 6.40e e e e e e11 L1N0 1.49e e e e L2N0 2.24e e e e L3N0 6.52e e e e L4N0 9.37e e e e Table: Relative errors for PPE and STE methods. err ppe = qppe p ref L 2 p ref, err ste = q ste p ref L 2 L 2 p ref L 2 17/37

18 Pressure determination Random error in the data two sources of error in velocity field: interpolation error due to the limited amount of points with known velocity velocity vectors are measured with the error/noise to simulate the latter one, we add a random number ε [ 0.05, 0.05], ε [ 0.1, 0.1] respectively, to each point where we know the velocity v exact (x) v meas (x) = (1 ± ε(x))v exact (x) ε(x) [0, 0.05] for maximal 5% error ε(x) [0, 0.1] for maximal 10% error 18/37

19 Pressure determination Random error in the data pressure [mmhg] L4 mesh for symmetric case q PPE q STE p REF z coordinate [mm] L4 mesh for non-symmetric case pressure [mmhg] q PPE q STE p REF z coordinate [mm] 19/37

20 Pressure determination Random error in the data Symmetric case Non-symmetric case relative error 10 2 relative error PPE STE PPE STE /aver node length 1/aver node length Convergence curves of relative errors for coarse data with 5% noise. Symmetric case Non-symmetric case relative error 10 2 relative error PPE STE PPE STE /aver node length 1/aver node length Convergence curves of relative errors for coarse data with 10% noise. 20/37

21 Pressure determination Comparison of two methods PPE poisson equation additional derivative of the data vector f (v) underestimate the pressure difference fixing p at the outlet limiting accuracy with noisy velocity fields STE larger linear problem less regularity requirements on the data (velocity field) better approximation q ste p L 2 fixing p at the outlet limiting accuracy with noisy velocity fields 21/37

22 Energy dissipation Motivation Current medical methods in stenosis evaluation and pressure determination Pressure determination - continuum mechanics approach Pressure difference and energy dissipation dependence on the stenosis severity 22/37

23 Energy dissipation Energy dissipation clinicalgate.com C. W. Akins, B. Travis, A. P. Yoganathan: Energy loss for evaluating heart valve performance. (2008) In: The Journal of Thoracic and Cardiovascular Surgery 136.4, /37

24 Energy dissipation Geometry J. H. Spühler et al.: 3D Fluid-Structure Interaction Simulation of Aortic Valves Using a Unified Continuum ALE FEM Model. In: Frontiers in Physiology 9 (2018). The parts of the geometry (from below): left ventricular outflow tract, ventriculo-aortic junction, aortic root (the first third should be stenotic), sinotubular junction, ascending aorta. 24/37

25 Energy dissipation Model ρ ( v t + ( v) v ) div (2µ D(v)) + p = 0 div v = 0 T = pi + 2µ D v = 0 v = v in Tn 1 ρ (v n) v = P (t)n 2 in Ω, in Ω, in Ω, on Γ wall, on Γ in, on Γ out. 25/37

26 Energy dissipation Boundary conditions Inflow velocity magnitude V (t) and outlet pressure P (t) as functions of time. 26/37

27 Energy dissipation Velocity fields NO STENOSIS 50% STENOSIS 27/37

28 Energy dissipation Pressure and energy dissipation on severity p ds Γ p = z area(γ z) pressure [mmhg] z coordinate [mm] E dis = Γ z T : D ds area(γ z) energy dissipation [Pa/s] severity 50% severity 60% severity 70% severity 80% z coordinate [mm] 28/37

29 Energy dissipation Pressure difference estimation pressure [mmhg] z coordinate [mm] NS SB sev p 1 p 2 p 1 p 2 [ mmhg] [ mmhg] 50% % % % Table: Maximal pressure difference computed through the Navier-Stokes eq. and simplified Bernoulli eq. 29/37

30 Energy dissipation Conclusion the current methods for stenosis evaluation (whether this is physiologically important or not) are based on pressure difference computed through the simplified Bernoulli equation (pressure difference proportional to v 2 ) continuum mechanics models are available to determine the pressure from the velocity field presented method, leading to the Stokes equation, allow us to compute the pressure under lower regularity requirements on the given velocity and it was shown that it provides more accurate pressure approximation the pressure and energy dissipation were computed in the geometries with narrowing up to 80% - knowing the velocity field at the inlet (the left ventricular outflow) and pressure at the outlet (the ascending aorta) 30/37

31 Energy dissipation Ongoing projects - pressure determination in patient-specific geometries relative error relative error PPE STE /average node distance PPE STE /average node distance 31/37

32 Energy dissipation Ongoing projects - wall boundary condition NOSLIP v n = 0 v τ = 0 NAVIER SLIP K=0.5 v n = 0 Tn τ = 1 K v τ PERFECT-SLIP v n = 0 Tn τ = 0 Figure: Velocity distribution on a slice of the valvular geometry without severity in time of maximal velocity (t = 0.15 s): all plug-in profiles. 32/37

33 Energy dissipation Ongoing projects - wall boundary condition NOSLIP 30% severity SLIP 30% severity Figure: Velocity distribution on a slice of the valvular geometry with 30% severity in time of maximal velocity (t = 0.15 s). 33/37

34 Energy dissipation Ongoing projects - wall boundary condition NOSLIP 50% severity SLIP 50% severity Figure: Velocity distribution on a slice of the valvular geometry with 50% severity in time of maximal velocity (t = 0.15 s). 34/37

35 Energy dissipation Ongoing projects - wall boundary condition 1 SEP SEP v ds dt Γ ds Γ 1 SEP SEP Γ ds Γ ρ v v 2 ds dt 0.8 velocity [m/s] SLIP NOSLIP 600 kinetic energy [Pa] SLIP NOSLIP z coordinate [mm] 50% STENOSIS, time-averaged values (over SEP) z coordinate [mm] 35/37

36 Energy dissipation Ongoing projects - wall boundary condition SEP SEP p ds dt Γ ds Γ pressure [mmhg] SLIP NOSLIP µ D:D ds dt SEP SEP Γ ds Γ energy dissipation [Pa/s] 10 4 SLIP NOSLIP z coordinate [mm] z coordinate [mm] 50% STENOSIS, time-averaged values (over SEP) 36/37

37 Energy dissipation Thank you for your attention. 37/37