Analytical Expressions for Noise Propagation in Diffusion Tensor Imaging

Dzz = (-J1-J2-J3-J4+J5+J6)/(4b), Dxx = (-J1-J2+J3+J4-J5-J6)/(4b), Dyy = (J1+J2-J3-J4-J5-J6)/(4b), with J1 = ln(S1/S0), J2 = ln(S2/S0), J3 = ln(S3/S0), J4 = ln(S4/S0), J5 = ln(S5/S0), J6 = ln(S6/S0), and b = γ T2 2 +T2 2 T3 2 +T1 2 T3 2 ), e1y = (T2T3)/sqrt(T1 2 T2 2 +T2 2 T3 2 +T1 2 T3 2 ), e1z = (T1T3)/sqrt(T1 2 T2 2 +T2 2 T3 2 +T1 2 T3 2 ), with T1 = DxyDyz-(Dyy-λ1)Dxz, T2 = DxzDyz- (Dzz-λ1)Dxy, T3 = DxzDxy-(Dxx-λ1)Dyz. Applying error propagation theory, the errors on the principal eigenvector can be determined. To illustrate typical errors occurring in DTI data we consider a prolate tensor in the corpus callosum of a volunteer dataset. The signal intensity values for S0 to S6 were determined to be 200, 138, 95, 14, 79, 42, and 23, respectively. These values produce the following eigenvalues; λ1 = 0.0017mm 2 /s, λ2 = 0.0003mm 2 /s, and λ3 = 0.0001mm 2 /s with b = 1000s/mm 2 . The eigenvalues give FA = 0.871 and RA = 0.716. The SNR was defined in the range of 20 - 120 for the non-diffusion- weighted signal S0. The noise level was thus within the range 1.7 to 10. We use the same noise level for all of S0 - S6 to determine the eigenvalue and eigenvector errors. As we can see, for the maximum noise level of 10, the SNR for the image with the greatest signal attenuation S3 is reduced to 1.4. Results Figure 1 shows the eigenvalue and principal eigenvector errors as a function of the SNR of S0. The errors for the eigenvalues and eigenvectors decrease as SNR increases. With SNR of 80, the relative error on λ1 is 10%. However for λ2 the error is 60% and for λ3 the relative error is 100%. The large relative errors for λ2 and λ3 are due to their small values rendering their values imprecise. The principal eigenvector angular error at SNR of 80 is 6°. However, this error increases to 23° when SNR falls to 20. Figure 2 shows the FA and RA error as a function of SNR of S0. The relative error for FA is clearly smaller than that of RA at any given SNR. This is shown in the righthand diagram of Figure 2 with SNR(FA)/SNR(RA) = 2.02, consistent with previous theoretical analysis 6 .