Diffusion Tensor

• Dxx Diffusion Coefficient : It corresponds to the Dxx coefficient of the tensor matrix:
• Dxy Diffusion Coefficient : It corresponds to the Dxy coefficient of the tensor matrix:
• Dxz Diffusion Coefficient : It corresponds to the Dxz coefficient of the tensor matrix:
• Dyy Diffusion Coefficient : It corresponds to the Dyy coefficient of the tensor matrix:
• Dyz Diffusion Coefficient It corresponds to the Dyz coefficient of the tensor matrix:
• Dzz Diffusion Coefficient: It corresponds to the Dzz coefficient of the tensor matrix:

Mathematically, the eigen value is the factor by which a linear transformation multiplies one of its eigenvectors. In an appropriate spatial reference frame, the diffusion tensor is diagonal (contains only three nonzero elements). These elements in diffusion tensor imaging are called the eigenvalues. The vectors characterizing the reference frame are the eigenvectors.

• Maximum Eigenvalue : The diagonalization of the tensor matrix leads to the eigen system : { ( D1, e1 ), ( D2, e2 ), ( D3, e3 ) } where D1, D2, D3 are the eigenvalues with D1 < D2 < D3, and e1, e2, e3 are the normalized eigenvectors. The maximum eigenvalue corresponds to D3.
• Middle Eigenvalue : The diagonalization of the tensor matrix leads to the eigen system : { ( D1, e1 ), ( D2, e2 ), ( D3, e3 ) } where D1, D2, D3 are the eigenvalues with D1 < D2 < D3, and e1, e2, e3 are the normalized eigenvectors. The middle eigenvalue corresponds to D2.
• Minimum Eigenvalue : The diagonalization of the tensor matrixleads to the eigen system : { ( D1, e1 ), ( D2, e2 ), ( D3, e3 ) }where D1, D2, D3 are the eigenvalues with D1 < D2 < D3,and e1, e2, e3 are the normalized eigenvectors. The minimum eigenvalue corresponds to D1.

• First Invariant : It corresponds to the trace ( Dxx + Dyy + Dzz ) of the tensor matrix:

  Dxx + DYY + DZZ

• Second Invariant : It corresponds to the surface area of the diffusion ellipsoid of the tensor matrix:

  DXX*DYY + DXX*DZZ + DYY*DZZ – DXY * DXY – DXZ * DXZ – DYZ*DYZ

• Third Invariant : it corresponds to the volume of the diffusion ellipsoid of the tensor matrix:

  DXX * (DYY * DZZ – DYZ * DYZ) – DXY * ( DXY * DZZ – DXZ*DYZ) + DXZ * (DXY * DYZ – DXZ * DYY)

• Fourth Invariant : It corresponds to the square of the magnitude of the tensor matrix:

  DXX * DXX + DYY * DYY + DZZ * DZZ + 2 * (DXY * DXY + DXZ * DXZ + DYZ * DYZ)

Diffusion measures the process of movement of a water molecule from an area of high concentration to an area of lower concentration. The functions defined under this category are :

• Average Diffusion Coefficient: measures the mean diffusivity of water molecules inside tissues. It corresponds to:

  Average Diffusion Coefficient (ADC)= First Invariant /3

• Surface Diffusion Coefficient : measures the mean diffusivity of water molecules inside tissues. It corresponds to:

  Surface Diffusion Coefficient (SDC) = sqrt (Second Invariant /3)

• Volume Diffusion Coefficient : measures the mean diffusivity of water molecules inside tissues. It corresponds to:

  Volume Diffusion Coefficient (VDC) = cbrt (Third Invariant)

• Magnitude Diffusion Coefficient : measures the mean diffusivity of water molecules inside tissues. It corresponds to:

  Magnitude Diffusion Coefficient (MDC) = sqrt( Fouth Invariant / 3)

• anisotropy index: This index measures the anisotropy of the diffusion of water molecules. It corresponds to:

  ( (1e9 * MDC ) *( 1e9 * MDC) – (1e9 * VDC ) * (1e9 * VDC ) ) /2

• fractional anisotropy: This index measures the anisotropy of the diffusion of water molecules. It corresponds to:

  sqrt ( 1.5 * ( 1.0 – ( ADC * ADC) / ( MDC * MDC) ) )

• relative anisotropy: This index measures the anisotropy of the diffusion of water molecules. It corresponds to:

  sqrt ( ( MDC * MDC ) / (ADC * ADC) -1.0 )

• surface/average anisotropy: This index measures the anisotropy of the diffusion of water molecules. It corresponds to:

  sqrt ( ( SDC / ADC -1 ) * (SDC /ADC -1.0) )

• Volume / Average anisotropy: This index measures the anisotropy of the diffusion of water molecules. It corresponds to:

  sqrt ( ( VDC / ADC -1 ) * (VDC /ADC -1.0) )

• Volume ratio anisotropy: This index measures the anisotropy of the diffusion of water molecules. It corresponds to:

  1 – ( VDC / ADC ) * ( VDC / ADC ) * ( VDC / ADC ) )

• Volume / Surface anisotropy : This index measures the anisotropy of the diffusion of water molecules. It corresponds to:

  Sqrt ( ( VDC / SDC - 1 ) * ( VDC / SDC -1 ) )

Attenuation functions measure the property of weakening density of water molecules. The functions plugged in READY View under this category is:

• Exponential attenuation: This corresponds to the "mean" attenuation of the signal:

  exp ( - bValue * ADC )

• Isotropic image : It corresponds to the geometrical "mean" attenuation of the signal weighted by the T2 image:

  Isotropic_value = 1

for each orientation

Isotropic_value = isotropic_value * pow ( T2_Signal * DW_Signal , 1.0 / ( double )_nOrientation )

Each pixel having a MR signal value lower than the threshold, without any diffusion gradient, is not processed.

• T2-weighted trace : It corresponds to the "mean" attenuation of the signal weighted by the T2 image :

  Average T2Value * exp ( - Bvalue * ADC)

• Each pixel having a MR signal value lower than the threshold, without any diffusion gradient, is not processed.