Energy loss and turbulent formations reveal the pressure loss in coarctation flows: A novel 4D Flow MRI-Based quantification method using a finite element approach
Julio Sotelo1,2,3, Jesús Urbina1,4, Cristian Montalba1, Israel Valverde5,6, Cristian Tejos1,2,7, Pablo Irarrazaval1,2,7, Marcelo Andia1,4,7, Daniel E Hurtado3,7, and Sergio Uribe1,4,7

1Biomedical Imaging Center, Pontificia Universidad Catolica de Chile, Santiago, Chile, 2Electrical Engineering Department, Pontificia Universidad Catolica de Chile, Santiago, Chile, 3Structural and Geotechnical Engineering Departement, Pontificia Universidad Catolica de Chile, Santiago, Chile, 4Radiology Department, School of Medicine, Pontificia Universidad Catolica de Chile, Santiago, Chile, 5Pediatric Cardiology Unit, Hospital Virgen del Rocio, Seville, Spain, 6Cardiovascular Pathology Unit, Institute of Biomedicine of Seville (IBIS), Hospital Virgen del Rocio, Seville, Spain, 7Biological and Medical Engineering Institute, Schools of Engineering, Medicine and Biological Sciences, Pontificia Universidad Catolica de Chile, Santiago, Chile

Synopsis

Aortic coarctation (CoA) cause an irreversible pressure loss post-CoA given by the energy dissipation, increasing the ventricular workload. Turbulent flows through CoA generate an irreversible damage in the surrounding tissue for mechanical stresses. We implement a finite elements method to obtain 3D maps of energy loss, kinetic energy, vorticity and helicity from 4D flow data. We performed an in-vitro study that related the pressure gradient, pulse wave velocity and elastic modulus with the energy loss and vorticity and helicity parameters. Concluding that our method may allow assessing the severity of the CoA and the identification of the regions affected.

Purpose:

Invasive diagnostic catheterizations are normally required to evaluate the pressure gradient in patient with Coarctation of the Aorta (CoA), allowing the assessment of vascular function and revealing the severity of the disease1. The CoA cause an irreversible pressure loss post-CoA given by the energy dissipation, increasing the ventricular workload required to maintain a certain arterial pressure. Additionally, turbulent flows posterior to the CoA can generate an irreversible damage in the surrounding tissue under mechanical stresses fluctuation2. Some method based on MRI to calculate the energy loss and turbulence3,4 have been proposed. These methods uses finite differences, however, this technique cannot handle the smooth and complex boundaries of vessels in the cardiovascular system, and therefore induces important errors when the geometry is simplified5. In this work we proposed a method based on finite elements to obtain 3D quantitative maps of energy loss, kinetic energy, vorticity and helicity from 4D-flow data sets. We performed an in-vitro study that related the pressure gradient (PG), pulse wave velocity (PWV) and elastic modulus with the energy loss, vorticity and helicity parameters.

Methods:

We developed a finite element based computational framework to obtain the velocities gradient from 4D-flow in three orthogonal direction generating continuous 3D maps of energy loss, kinetic energy and turbulence parameters. The algorithm is based in a similar least-squares projection method previously published6. Using a MR Phillips Achieva 1.5T, the 4D-flow data was acquired in a realistic aortic phantom in normal conditions and with three different CoA 13, 11, 9 mm, one volunteer and one patient with repair CoA (Figure 1). Additionally, for the phantom with and without CoA we acquire 2D-flow (2D PC-MRI) in two position, the ascending and descending aorta in order to calculate the PWV between these two points (see figure 1A-B). The phantom model T-S-N-005 (Elastrat Sàrl, Geneva, Switzerland) is used with a pulsatile MR-compatible flow pump (Simutec, London, Ontario, Canada), and a catheterization unit to measure invasively pressures between the ascending and descending aorta in order to obtain the elastic modulus between these two points (see figure 1). A summary of the MR acquisition parameters is shown in table I. The process for creating the tetrahedral mesh from the 4D-flow images consist in: first, a generation of an angiography image, then a segmentation of the lumen of the aorta applying an intensity threshold separating the vessel of interest, and finally creating a tetrahedral mesh using the library iso2mesh in Matlab. The next step is to transfer the velocity field over each node of the tetrahedral mesh using a cubic interpolation. The finite element analysis was developed using the Python 
software, and for the visualization and analysis of the data we used the open source software Paraview.

Results and Discussion:

Result of PG, PWV and EM for the phatom data are shown in table II. We can see a reduction of the PWV if we increase the severity of the coarctation in the phantom, this is probably generated by the loss of kinetic energy of the flow that pass through the coarctation, affecting the pressure gradient between the ascending and descending aorta. Additionally, the elastic modulus decreased in the ascending aorta for small coarctations. In figure 2 we show the 3D result for vorticity, helicity density, energy loss and kinetic energy for the phantom data with and without CoA. There was an increment of turbulent flow (Vorticity and helicity density) for small CoA, that generated higher loss of energy in this location, affecting directly the pulse propagation along the vessel. The peak values of vorticity, helicity density, energy loss and kinetic energy for all cardiac phases were: 500/1500/1200/875 s-1, ±60/±420/±150/±100 m/s2, 20/200/70/46 μW and 34/240/118/60 μJ for the phantom without CoA and with CoA of 9 mm, 11 mm and 13 mm respectively. To show the applicability of our method we apply our algorithm in one patient with a repair CoA and one volunteer (Figure 3). The peak values obtained in the patient and volunteer data were: 1000/800 s-1, ±190/±70 m/s2, 120/74 μW and 120/110 μJ for vorticity, helicity density, energy loss and kinetic energy respectively.

Conclusion:

We have proposed a
 novel method that allowed us to obtain 3D maps of energy loss, kinetic energy and turbulence parameters derived from
 4D-flow data using finite elements, which has 
been validate in data obtained from an aortic phantom. From the validation process we can conclude that the 3D maps of energy loss together with the vorticity and helicity may allow assessing the severity of the CoA and the identification of the regions affected.

Acknowledgements

Anillo ACT 1416. JS thanks CONICYT and Ministry of Education of Chile, with his higher education program, for scholarship for doctoral studies.

References

1.- Tanous D, Benson LN, Horlick EM, et al. Coarctation of the aorta: evaluation and management. Curr Opin Cardiol. 2009;24(6):509-515.

2.- Malek A, Alper S, Izumo S. Hemodynamic shear stress and its role in atherosclerosis. JAMA 1999:282:2035

3.- Barker AL, van Ooij P, Bandi K, et al. Viscous energy loss in the presence of abnormal aortic flow. Magn Reson Med 2014, 72:620-628.

4.- Binter C, Gülan U, Holzner M, et al. On the accuracy of viscous and turbulent loss quantification in stenotic aortic flow using phase-contrast MRI. Magn Reson Med 8:2015.

5.- Zienkiewicz OC, Taylor RL, Zhu JZ. The finite element method: Its basis and fundamentals, sixth ed. London: Butterworth-Heinemann. 2005.

6.- Sotelo J, Urbina J, Valverde I, et al. 3D Quantification of Vorticity and Helicity from 4D Flow Data Using Finite Element Interpolations. Proc 23rd Annual Meeting ISMRM, 2015.

Figures

Figure 1. A) 2D flow in the ascending and descending aorta as well as locations were invasive pressure measurement were acquired. B) Method to calculate the Elastic Modulus from 4D flow magnitude data. C) We show the segmentation for one volunteer and one patient with a repair CoA.

Table I. 2D and 3D PC-MRI parameters for in-vitro and in-vivo data.

Table II. Result of pressure gradient, pulse wave velocity and elastic modulus for in vitro data.

Figure 2. 3D maps of velocity, vorticity, helicity density, energy loss and kinetic energy for in-vitro data. Additionally, we show the peak values of cumulative energy loss (EL) in the entire lumen of the phantom data. For visualization purposes, we adjust the color bar range.

Figure 3. 3D maps of velocity, vorticity, helicity density, energy loss and kinetic energy for in-vivo data. Additionally, we show the peak energy loss (EL) values for the aorta of the volunteer and patient. For visualization purposes, we adjust the color bar range.



Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)
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