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.
Vorticity
Vorticity is the curl of the vector field for three dimensional flow, as defined by the equation below. One can imagine vorticity as the spinning move of tiny part of the fluid, while moving along with the stream.
where 
is the vorticity vector, and 
is the velocity vector.
The parametric map calculated by the 4D Flow Measures model represents the magnitude of vorticity defined as:
Helicity
Helicity is an invariant of the Euler equation of fluid flow if certain fluid restrictions are fulfilled: the fluid has to be inviscid, flow has to be incompressible and body forces acting on the fluid have to be conservative. Helicity is a pseudo-scalar value, as it changes sign with the change of coordinate system orientation and is calculated as a dot product. It can be interpreted as a measure of "knotedness" of the vortex in the fluid. It is described by the equation
where V is volume of the fluid described by the vector fields.
The parametric map calculated by the 4D Flow Measures model represents the absolute value of the helicity 
.
Energy Loss
Energy loss represents the amount of the energy lost by the fluid due to its viscosity. This energy dissipation has been described by a function, which can be related to the fluid's viscosity and volume, integrated over time. This parameter is assumed to have the most significance for turbulent flows, where the velocity field is changing rapidly developing many vortices, which in turn leads to the higher energy dissipation. The mathematical definitions are:
Energy loss equation: 
where Pl is the power loss and t represents time.
Power loss equation: 
where μ is fluid viscosity, Φ the energy dissipation function, and Vv the voxel volume (in the image data).
Dissipation function: 
where u,v,w are the velocity values in the x,y,z directions of a Cartesian coordinate system, and 
is the divergence of the velocity vector field.
Please note that the last term of the dissipation function is a non-zero component only for the compressible flows. For parametric map calculation it is assumed that the flow is incompressible so that the divergence term disappears from the equation.
Acquisition and Data Requirements
The images generated by 4D flow MRI sequences are magnitude images (providing anatomical information) and phase images (providing velocity information), which are available in different file formats and encoding systems, depending on the manufacturer. For the 4D Flow Measures model only the phase images are required. PXMOD supports the file format used by Siemens (enclosing flow information in one file, which contains velocity information in three directions, prior the VENC scaling). Alternatively, the input could also be a file exported from the PGEM tool during the 4D flow data processing (these data are already VENC-scaled with units [cm/s]). Since PGEM supports the native Philips data format (PAR-REC), it is also possible to work with Philips 4D flow data.
Image Data |
MR data acquired with a 4D flow sequence |
Model Preprocessing
When arriving at model preprocessing, the following basic information has to be specified
Scale by VENC |
Flag whether the input data needs to be scaled by the VENC parameter (velocity encoding). VENC, defined in [cm/s], is a parameter of the acquisition and should be part of the information contained in the image data. In general, one can assume that if data were not saved from the PGEM tool during the 4D flow data processing, it should be scaled. |
Blood viscosity |
Viscosity of blood in milli pascals per second. Default value is 3.0. |
Map Configuration
The following three parameters can be selected.
Vorticity (peak) |
Vorticity parameter of the velocity field, as described above. Maps are calculated for all frames and the frame with the highest values is chosen for presentation. |
Helicity (peak) |
Helicity parameter of the velocity field, as described above. Maps are calculated for all frames and the frame with the highest values is chosen for presentation. |
Energy loss |
Energy loss parameter, as described above. Map is calculated as a sum of power loss in each time point, divided by the study duration. |
References
1.Cook P. A., Bai Y., Nedjati-Gilani S., Seunarine K. K., Hall M. G., Parker G. J., Alexander D. C., "Camino: Open-Source Diffusion-MRI Reconstruction and Processing", 14th Scientific Meeting of the International Society for Magnetic Resonance in Medicine, Seattle, WA, USA, p. 2759, May 2006.
2.Markl M., Kilner P.J., Ebbers T., „Comprehensive 4D velocity mapping of the heart and great vessels by cardiovascular magnetic resonance”, Journal of Cardiovascular Magnetic Resonance 2011, 13:7
3.Barker A.J., van Ooij P., Bandi K., Garcia J., Albaghdadi M., McCarthy P., Bonow R.O., Carr J., Collins J., Malaisrie C., Markl. M, „Viscous Energy Loss in the Presence of Abnormal Aortic Flow”, Magnetic Resonance in Medicine, 2014 September ; 72(3): 620–628. doi:10.1002/mrm.24962.
4.Bird R.B., Stewart W.E., Lighfoot E.N., „Transport Phenomena”, second edition, John Wiley & Sons, 2007