A momentum-based diffeomorphic demons framework for deformable MR-CT image registration

Abstract

Neuro-navigated procedures require a high degree of geometric accuracy but are subject to geometric error from complex deformation in the deep brain-e.g. regions about the ventricles due to egress of cerebrospinal fluid (CSF) upon neuroendoscopic approach or placement of a ventricular shunt. We report a multi-modality, diffeomorphic, deformable registration method using momentum-based acceleration of the Demons algorithm to solve the transformation relating preoperative MRI and intraoperative CT as a basis for high-precision guidance. The registration method (pMI-Demons) extends the mono-modality, diffeomorphic form of the Demons algorithm to multi-modality registration using pointwise mutual information (pMI) as a similarity metric. The method incorporates a preprocessing step to nonlinearly stretch CT image values and incorporates a momentum-based approach to accelerate convergence. Registration performance was evaluated in phantom and patient images: first, the sensitivity of performance to algorithm parameter selection (including update and displacement field smoothing, histogram stretch, and the momentum term) was analyzed in a phantom study over a range of simulated deformations; and second, the algorithm was applied to registration of MR and CT images for four patients undergoing minimally invasive neurosurgery. Performance was compared to two previously reported methods (free-form deformation using mutual information (MI-FFD) and symmetric normalization using mutual information (MI-SyN)) in terms of target registration error (TRE), Jacobian determinant (J), and runtime. The phantom study identified optimal or nominal settings of algorithm parameters for translation to clinical studies. In the phantom study, the pMI-Demons method achieved comparable registration accuracy to the reference methods and strongly reduced outliers in TRE (p [Formula: see text] 0.001 in Kolmogorov-Smirnov test). Similarly, in the clinical study: median TRE = 1.54 mm (0.83-1.66 mm interquartile range, IQR) for pMI-Demons compared to 1.40 mm (1.02-1.67 mm IQR) for MI-FFD and 1.64 mm (0.90-1.92 mm IQR) for MI-SyN. The pMI-Demons and MI-SyN methods yielded diffeomorphic transformations (J > 0) that preserved topology, whereas MI-FFD yielded unrealistic (J < 0) deformations subject to tissue folding and tearing. Momentum-based acceleration gave a ~35% speedup of the pMI-Demons method, providing registration runtime of 10.5 min (reduced to 2.2 min on GPU), compared to 15.5 min for MI-FFD and 34.7 min for MI-SyN. The pMI-Demons method achieved registration accuracy comparable to MI-FFD and MI-SyN, maintained diffeomorphic transformation similar to MI-SyN, and accelerated runtime in a manner that facilitates translation to image-guided neurosurgery.

Figure 1.

Image pre-processing. (a) Example axial CT image prior to histogram stretch (120 kV, 380 mAs, 0.44 mm isotropic voxels). (b) The same CT image following histogram stretch with p = 0.05. (c) Normalized T1-weighted MR image (repetition time 420 ms, echo time 6.5 ms) following NMI rigid registration. Histograms below each image illustrate the piece-wise linear stretch applied to CT, with image intensities between −120 and+120 HU stretched according to Equation (1).

Figure 2.

Intracranial deformation in phantom. (a) Undeformed CT image of the head phantom, showing an example displacement field about two attractive force locations (k = 80 in Eq. (21)). (b)-(d) CT images showing the degree of deformation for small (k = 25), medium (k = 80), and large (k = 135) deformations. Target point locations are shown before (red) and after (blue) deformation.

Figure 3.

Selection of algorithm parameters. (a) Sensitivity of registration performance to the histogram stretch parameter p. TRE improved with reduction in p (larger range of soft-tissue intensity stretch) down to p = 0.1 (marks by black arrows). (b) Sensitivity of registration performance for the pMI-Demons algorithm to the Gaussian smoothing kernels ${\sigma }_{update}$ and ${\sigma }_{deformation}$. The circle marks the nominal / optimal values.

Figure 4.

Selection of (a) update step length, $k_{update}$, and (b) momentum-based acceleration strength, α, in the pMI-Demons method. (a) Convergence plots for various settings of update step length. (b) Sensitivity analysis of momentum strength plotted as pMI vs. number of iterations, where arrows indicate oscillation in the pMI-Demons objective function for α = 0.9 curve.

Figure 5.

Registration performance in phantom studies. (a) TRE without histogram stretch. (b) TRE with histogram stretch. (c) Jacobian determinant, with negative values indicating non-diffeomorphic distortion.

Figure 6.

Registration results in phantom studies emulating contraction throughout the central cranial vault. Each case shows the T1-weighted MR image after registration overlaid with green Canny edges from the fixed CT image. The zoomed-in region shows subtle differences in each case, with zoomed-in Jacobian determinant $\mathcal{J}$ maps in the bottom row. (a) MI-Rigid, showing a fairly large TRE following deformation. (b) MI-FFD, showing unrealistic deformations ($\left|\mathcal{J}\right|<0$ in regions marked by the arrow). (c) MI-SyN and (d) pMI-Demons, showing realistic and diffeomorphic deformations ($\left|\mathcal{J}\right|>0$ everywhere).

Figure 7.

Registration performance evaluated as a function of the magnitude of initial deformation. The identity line demarks regions below which the registration method improved upon the initial displacement.

Figure 8.

Target registration error in clinical studies. (a) TRE without histogram stretch. (b) TRE with histogram stretch. (c) Jacobian determinant, with negative values suggesting non-diffeomorphic transformation.

Figure 9.

Registration results of CT to T1-weighted MR images in clinical studies. Each case shows the T1-weighted MR image following deformation overlaid with Canny edges from the fixed CT image. Zoomed-in views along with Jacobian determinant maps shows detailed difference between each case. (a) NMI-Rigid, showing gross misalignment up to ~11.5 mm. (b) MI-FFD, with arrows marking areas of unrealistic deformation $\left(\left|\mathcal{J}\right|<0\right)$. (c) MI-SyN with histogram stretch. (d) pMI-Demons with histogram stretch.

Figure 10.

Registration results for CT to T2-weighted MR registration in clinical studies. Each case shows the T2-weighted MR image following deformation overlaid with Canny edges from the fixed CT image. The center column shows zoomed-in views of the images with Canny edge overlay, and the column at right shows the corresponding Jacobian determinant map within the zoomed view. (a) NMI-Rigid, showing gross misalignment up to ~11.5 mm. (b) Registration result for pMI-Demons, showing better alignment of CT edges to MR image in ventricles and major sulci as well as diffeomorphic transformation (non-negative Jacobian determinant).