Michael Smith^{1}, Ishani DasGupta^{1}, and Elise Fear^{1}

The Bland-Altman plot is a commonly-used graphical method to compare two measurement techniques and look for systematic biases or outliers. We have identified that the Bland-Altman approach of plotting the differences between the two techniques against their average will introduce false biases in an MRI context. These biases are introduced by the magnitude operation necessary to display or analyze MRI images. We demonstrate a modified Bland-Altman approach that corrects these biases.

Standard Bland-Altman plots obtained with magnitude data are shown for images that are identical except for SNRs of 16 and 20 in Fig. 1, and with the addition of mis-centred k-space data in Fig. 3. Both the plots from the centred and mis-centred k-space data sets show standard deviations changing with image intensity, unexpected for essentially identical images differing in only their SNR. The distortion in the plots at low intensity amplitude can be explained with an extension of ideas from McGibney and Smith [6]. Specifically, Gaussian white noise (GWN) superimposed upon a large intensity ROI will remain with GWN characteristics after the magnitude operation. This is in contrast with GWN on low intensity ROI, where noise rectification will occur during the magnitude operation. This causes a Rician noise distribution which has no negative value, leading to a non-zero mean and an effective standard deviation which both change with ROI intensity level.

Figs. 4
and 5 show the results from the proposed modified Bland-Altman plot that has
been adjusted for the MRI context. Although shifting the center of k-space drastically changes the
appearance of the complex valued plots between Figs. 4 and 5, neither of the
modified plots show distortions in the standard deviation of the noise, which
correctly remains constant with changes in the average intensity.

[1] M. Lustig, D. Donoho and J. M. Pauly, “Sparse MRI: The Application of Compressed Sensing for Rapid MR Imaging”, Magnetic Resonance in Medicine, 58, 1182 – 95, 2007.

[2] S. Ravishankar and Y. Bresler, "MR image reconstruction from highly undersampled k-space data by dictionary learning," IEEE transactions on medical imaging, 30(5), 1028 – 1041, 2011.

[3] Z. P. Liang et al., “Constrained Reconstruction Methods in MR Imaging”, Reviews of Magnetic Resonance in Medicine, 4, 67 – 185, 1992.

[4] P. Adibpour and M. Smith, "Total Variation Assisted Fourier Shift Manipulation to Remove Gibbs' Artifacts in Compressive Sensing Techniques", IEEE Nuclear Science Symposium / Medical Imaging Conference, Strasbourg, France, 2016.

[5] J. M. Bland. D. G. Altman, “Statistical methods for assessing agreement between two methods of clinical measurement”, Lancet, 307 - 310, 1986.

[6] G. McGibney and M. Smith, "An unbiased signal-to-noise ratio measure for magnetic resonance images", Medical Physics, 20 (4), 1077 - 1078, 1993.

Fig. 1: Standard Bland-Altman plot reveals reduced standard
deviations at low intensities suggesting biases between the reconstruction
approaches used to reconstruct images. These biases are false as the images were reconstructed from identical *k*-spaces except for $$$SNR = 20$$$ and $$$SNR = 16$$$.

Fig. 2: High resolution IDFT reconstruction from a GE phantom’s 512 x
512 *k*-space

Fig. 3: Standard Bland-Altman plot for two images that are identical
except for $$$SNR = 20$$$ and $$$SNR = 16$$$ with the k-space data miss-centered by 4 samples in both $$$kx$$$ and $$$ky$$$ directions. This apparently indicates the same biases between the two reconstruction
techniques at low intensities as shown in Fig. 1.

Fig. 4: The proposed modified Bland-Altman plot for two images that are identical except for $$$SNR = 20$$$ and $$$SNR = 16$$$. The real and imaginary components of the modified plots are offset for easier interpretation.

The standard deviations of the differences between
the two images are no longer distorted by the magnitude operator; remaining
constant with intensity level. Our modified plot has revealed a previously
undetected intensity bias at low, essentially background, intensities. Further investigation showed this bias to be real, and associated with image distortions when the original low-noise *k*-space was unintentionally overwritten by its magnitude image's *k*-space.

Fig. 5: Modified Bland-Altman plot for two images that are identical except for $$$SNR = 20$$$ and $$$SNR = 16$$$, with equivalent miss-centring of k-space data by 4 samples in both $$$kx$$$ and $$$ky$$$ directions. The real and imaginary components of the modified plots are offset for easier interpretation.

As expected, the miss-centring
of k-space data has introduced significant imaginary components into
the complex-valued MRI image. However, the standard deviation of the
differences correctly remains constant at all intensity levels . The plot still shows the intensity biases unintentionally introduced by over-writing the original low-noise* k*-space by its magnitude image's *k*-space.