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# Turbulent Kinetic Energy

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# Turbulent Kinetic Energy

4D flow MRI is a method to non-invasively acquire the velocity of blood flow velocity and opens new possibilities for the hemodynamic analysis. The 4D Flow Measures model supports the analysis of a velocity field acquired by the 4D flow MRI technique and allows to generate parametric maps of vorticity, helicity and energy loss. The Turbulent Kinetic Energy (TKE) model supports the analysis of the velocity field acquired by the 4D flow MRI technique and allows generating both static and multiframe TKE maps.

Turbulent Kinetic Energy

While blood flow in the human vessels is mostly considered to be laminar, it can become turbulent and ineffective in the presence of some diseases. That is because turbulent kinetic energy, the energy stored in the turbulent part of the flow, is mainly being dissipated into heat and has to be considered as a loss of energy. Such a condition occurs for example in aortic blood flow in the presence of HOCM disease (Hyperthropic Obstructive Cardiomyopathy) or artery stenosis. Hence, TKE is regarded as an important parameter to asses.

According to Reynolds' decomposition of the velocity vector field flow can be presented as a sum of two terms: the mean velocity field and the fluctuating velocity field u: The average kinetic energy per unit can be then described as the sum of the mean kinetic energy (MKE) and the turbulent kinetic energy: where ρ denotes fluid density and i the direction.

The TKE value given a specified fluid density can be calculated employing the fact that it corresponds with the intravoxel velocity standard deviation (IVSD) parameter as follows: where σi represents the directional IVSD value.

The IVSD value is calculated from the relationship between signal values of magnitude images from the MRI 4D flow acquisition with different first moment gradients, which is in general described by the equation below: where is an acquisition parameter related to the first gradient moment through the velocity encoding (VENC), and are the values of the signal for magnitude images corresponding to different first gradient moments (different VENC parameter values).

However, if one of the input magnitude images is the reference image (zero first gradient moment,VENC=0) and taking directions into account, the equation above can be simplified to: This finally leads to the following equation relating TKE to the known fluid density, acquisition parameters and the values of the magnitude images where ρ denotes fluid density, VENC velocity encoding, S the magnitude of the zero VENC reference image, Si the magnitude of the non-zero VENC in direction i.

Acquisition and Data Requirements

The images generated by 4D flow MRI sequences are magnitude images and phase images. For the usage of the Turbulent Kinetic Energy model, only the magnitude images only are required. The input data have to be in form of the magnitude image from a scan with zero first gradient moment (VENC=0, the reference image) and three magnitude images (one for each direction) from scans with non-zero first gradient moments (VENC≠0). This is the standard output of the non-symmetric four-point acquisition method.

The magnitude input images can be exported from the PGEM tool during 4D flow data processing, which supports Siemens and Philips 4D flow MRI file formats.

 Image Data MR data acquired using a 4D flow non-symmetric four-point acquisition

Model Preprocessing

When arriving at model preprocessing, the following basic information has to be specified VENC Value of VENC (velocity encoding) for the magnitude images in the X,Y,Z directions used for the non-zero VENC acquisitions. Defined in [cm/s]. Blood density Density of the blood in [kg/m3]. Default value is 1060.

Model Configuration

This model only supports TKE as parametric map. TKE Turbulent kinetic energy of the velocity field, as described above.

References

1.Dyverfeldt P., Sigfridsson A., Kvitting J.P., Ebbers T., "Quantification of Intravoxel Velocity Standard Deviation and Turbulence by Generalizing Phase-Contrast MRI", Magnetic Resonance in Medicine, 2006 October, 56(4):850-8.

2.Dyverfeldt P., Kvitting J.P., Sigifridsson A., Engvall J., Bolger A.F., Ebbers T., „Assessment of Fluctuating Velocities in Disturbed Cardiovascular Flow: In Vivio Feasibility of Generalized Phase-Contrast MRI”, Journal of Magnetic Resonance Imaging, 2008 September, 28(3):655-63

3.Dyverfeldt P., Hope M.D., Tseng E.E., Saloner D., „Magnetic Resonance Measurement of Turbulent Kinetic Energy for the Estimation of Irreversible Pressure Loss in Aortic Stenosis”, JACC Cardiovascular Imaging, 2013 Januray, 6(1): 64-71

4.Binter C., Gotschy A., Sündermann S.H., Frank M., Tanner F.C., Lüscher T.F., Manka R., Kozerke S., „Turbulent Kinetic Energy Assessed by Multipoint 4-Dimensional Flow Magnetic Resonance Imaging Provides Additional Information Relative to Echocardiography for the Determination of Aortic Stenosis Severity”, Circulation: Cardiovascular Imaging, June 2017, 10(6)