## Abstract

### Purpose

Magnetic resonance-guided focused ultrasound (MRgFUS) of the liver during free-breathing requires spatio-temporal prediction of the liver motion from partial motion observations. The study purpose is to evaluate the prediction accuracy for a realistic MRgFUS therapy scenario, namely for human in vivo data, tracking based on MR images routinely acquired during MRgFUS and in vivo deformations caused by the FUS probe.

### Methods

In vivo validation of the motion model was based on a 3D breath-hold image and an interleaved acquisition of two MR slices. Prediction accuracy was determined with respect to manually annotated landmarks. A statistical population liver motion model was used for predicting the liver motion for not tracked regions. This model was individualized by mapping it to end-exhale 3D breath-hold images. Spatial correspondence between tracking and model positions was established by affine 3D-to-2D image registration. For spatio-temporal prediction, MR tracking results were temporally extrapolated.

### Results

Performance was evaluated for 10 volunteers, of which 5 had a dummy FUS probe put on their abdomen. MR tracking had a mean (95 %) accuracy of 1.1 (2.4) mm. The motion of the liver on the evaluation MR slice was spatio-temporally predicted with an accuracy of 1.9 (4.4) mm for a latency of 216 ms. A simple translation model performed similarly (2.1 (4.8) mm) as the two MR slices were relatively close (mean 38 mm). Temporal prediction was important (10 % error reduction), while registration effects could only partially be assessed and showed no benefits. On average, motion magnitude, motion amplitude and breathing frequency increased by 24, 16 and 8 %, respectively, for the cases with FUS probe placement. This motion increase could be reduced by the spatio-temporal prediction.

### Conclusion

The study shows that tracking liver vessels on MR images, which are also used for MR thermometry, is a viable approach.

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## Acknowledgments

This study was funded by the EU’s 7th Framework Program (FP7/2007-2013) under Grant Agreement Nos. 270186 (FUSIMO) and 611889 (TRANS-FUSIMO).

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## Ethics declarations

### Conflict of interest

Y.Z. and G.Sat are employed by GE. A.M. acknowledges research collaborations with GE, InSightec and IBSmm, and support from the Northern Research Partnership. All other authors declare no conflict of interest.

### Research involving human participants

All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki Declaration and its later amendments or comparable ethical standards. This study does not contain patient data.

### Informed consent

Informed consent was obtained from all individual participants included in the study.

## Appendix: Population 4D motion model

### Appendix: Population 4D motion model

### Liver registration

An intensity-based non-rigid registration [8, 24] was used to quantify the liver motion captured by the 4D-MRIs. The registration parameters were iteratively adjusted such that visual inspection showed misregistrations below one pixel [33]. To cope with the sliding boundaries between the liver and the abdominal wall, the B-spline transformation parameters were optimized to maximize normalized cross-correlation only within the reference liver region.

### Inter-subject correspondences

We used the approach from [34] to define inter-subject correspondences. First a fine surface mesh was extracted from the liver segmentation. Then sagittal slices were identified, which include the most lateral liver location (slice number \(S_1\)), the inferior liver tip (\(S_2\)), the bifurcation of the inferior vena cava (IVC) (\(S_3\)) or the portal vein (\(S_4\)), with \(S_1<S_2<S_3<S_4\). Four mechanically relevant landmarks (dorsal and ventral point where the liver joins the ribcage, and most inferior dorsal and ventral point) were manually identified on slices \(S_2+1\) to \(S_3-1\). A point on the liver tip replaced the two inferior landmarks for slices \(S_1\) to \(S_2\). An additional landmark was selected on the IVC for slices \(S_3\) to \(S_4\). The landmarks belonging to one location were connected by a B-spline. A prototype of the right liver lobe (46 triangles) was then mapped to the liver by aligning its four edges with the marked delineations. Finally this coarse mesh was gradually refined by a fixed number of regular subdivisions to fit to the fine surface mesh. The resulting liver meshes consist of \(N=290\) points, where \(\mathbf {p}^n_t = [p^n_{{SI}_t} \; p^n_{{AP}_t} \; p^n_{{LR}_t}]\) denotes the position of the *n*th point at time step *t* in the superior–inferior (SI), anterior–posterior (AP) and left–right (LR) direction. The mesh position at time step \(t\in \{1,2,\ldots ,T\}\) was described by \(\mathbf {p}_t=[\mathbf {p}^1_t \; \ldots \; \mathbf {p}^N_t]^T\in \mathbb {R}^{3N \times 1}\) and its motion by \({\varDelta } \mathbf {p}_t=\mathbf {p}_t-\mathbf {p}_{\mathrm{ref}}\) where \(\mathbf {p}_{\mathrm{ref}}\) denotes the position in the reference end-exhale image. Mesh interpolation via Barycentric coordinates was used for defining correspondences for any position in the mesh.

### Creation of motion model

We used the robust exemplar model [25, 31], which has shown improved performance over a principle component analysis (PCA) model for the whole population. It is based on creating subject-specific PCA models and combining their predictions according to their closeness to the tracking results.

#### Subject-specific model

Assuming that \({\varDelta } \mathbf {p}_t, t\in {1, 2 \ldots T}\), belong to a 3*N*-dimensional Gaussian distribution \( {\varDelta } \mathbf {p}_t\sim \mathcal {N}(\mu , {\varSigma })\), the prediction task is to find the most probable vector \({\varDelta } \hat{\mathbf {p}}_t \in \mathbb {R}^{3N \times 1}\), given a subset of its elements \(\mathbf {s}_t\in \mathbb {R}^{S \times 1}\), called surrogates. Decomposing all displacements \({\varDelta } \mathbf {p}_t\) and their mean \(\mu \), and covariance matrix \({\varSigma }\) into the components relating to surrogates \(\mathbf {s}_t\) and to the rest of the points (\(\mathbf {r}_t\)), we get \({\varDelta } \mathbf {p}^T_t=\left[ \begin{array}{cc} \mathbf {s}^T_t&\; \mathbf {r}^T_t \end{array} \right] \), \(\mu ^T=\left[ \begin{array}{cc} \mu _{\mathbf {s}}^T&\; \mu _{\mathbf {r}}^T \end{array} \right] \), and \({\varSigma }=\left[ \begin{array}{cc} {\varSigma }_{\mathbf {s}\mathbf {s}} &{} {\varSigma }_{\mathbf {s}\mathbf {r}} \\ {\varSigma }_{\mathbf {r}\mathbf {s}} &{} {\varSigma }_{\mathbf {r}\mathbf {r}} \\ \end{array} \right] \). The conditional distribution \({\varDelta } \mathbf {p}_t | \mathbf {s}_t\sim \mathcal {N} \left( \mu _{{\varDelta } \mathbf {p}_t |\mathbf {s}_t} , {\varSigma }_{{\varDelta } \mathbf {p} |\mathbf {s}} \right) \) [1], with its mean \(\mu _{{\varDelta } \mathbf {p}_t |\mathbf {s}_t}=\mu + \left[ \begin{array}{c} {\varSigma }_{\mathbf {s}\mathbf {s}} \\ {\varSigma }_{\mathbf {r}\mathbf {s}} \\ \end{array} \right] {{\varSigma }_{\mathbf {s}\mathbf {s}}}^{-1}(\mathbf {s}_t - \mu _{\mathbf {s}})\) providing the most probable \({\varDelta } \hat{\mathbf {p}}_t\) given \(\mathbf {s}_t\).

PCA was employed for dimensionality reduction to the *M* eigenvectors associated with the highest eigenvalues \(\hat{\lambda }^2\) leading to \({\varSigma }\approx \hat{\mathbf {E}} \hat{{\varLambda }} \hat{\mathbf {E}}^T\) where \(\hat{\mathbf {E}}\in \mathbb {R}^{3N \times M}\) and \(\hat{{\varLambda }}\in \mathbb {R}^{M \times M}\). Finally by extracting the submatrix of eigenvectors of the surrogates \(\mathbf {s}\), i.e., \(\hat{\mathbf {E}}_{\mathbf {s}}\in \mathbb {R}^{L \times M}\), the most probable prediction is given by

#### Model regularization

Regularization of the model [4, 19] was used to compensate for tracking errors. Equation (1) provides the most probable PCA coefficients \(\mathbf {c}_t\) which minimize \(||\mathbf {Q}\mathbf {c}_t - (\mathbf {s}_{t} - \mu _\mathbf {s})||^2\) where \(\mathbf {Q}=\hat{\mathbf {E}}_sdiag(\hat{\lambda }_i)\). Using ridge regression, we want to minimize instead

where \(\eta \) scales the regularization. Equation (2) can be solved by applying singular value decomposition (SVD) to get \(\mathbf {Q}=\mathbf {U}_{\mathbf {Q}}\mathbf {D}_{\mathbf {Q}}\mathbf {V}_{\mathbf {Q}}^T\), with \(\mathbf {D}_{\mathbf {Q}}=diag(d_{\mathbf {Q},i})\) and

#### Robust exemplar model

To create an exemplar model [25, 31] for a population, a subject-specific PCA model \(M^j\) was built for each subject *j*. Then the distance \(d^j_{t}\) between surrogates \(\mathbf {s}_t\) and model \(M^j\) was calculated by \(d^j_t = (\mu _{\mathbf {S}_t}-\mu ^j_{\mathbf {s}})^T{{\varSigma }_{\mathbf {ss}}^j}^{-1}(\mu _{\mathbf {S}_t}-\mu ^j_{\mathbf {s}})\), where \(\mu _{\mathbf {S}_t}\) estimates the mean of the surrogate distribution from the last \(O\le t\) observations \(\mathbf {S}_t=[\mathbf {s}_{t-O+1}^T \ldots \mathbf {s}_t^T]\) to make \(d^j_{t}\) robust to noise. To predict \({\varDelta } \mathbf {p}_t\) for a new subject given \(\mathbf {s}_t\), predictions \({\varDelta } \hat{\mathbf {p}}_t^j\) are obtained for the *K* closest models by Eq. (3) and combined by

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Tanner, C., Zur, Y., French, K. *et al.* In vivo validation of spatio-temporal liver motion prediction from motion tracked on MR thermometry images.
*Int J CARS* **11**, 1143–1152 (2016). https://doi.org/10.1007/s11548-016-1405-4

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DOI: https://doi.org/10.1007/s11548-016-1405-4