Phase-Based Registration of Cardiac Tagged MR Images by Incorporating Anatomical Constraints

Abstract

This paper presents a novel method that combines respective benefits of the tracking-based methods and the Gabor-based non-tracking approaches for improving the motion/strain quantification from tagged MR images. The “tag number constant” concept used in Gabor-based non-tracking methods is integrated into a recent phase-based registration framework. We evaluated our method on both synthetic and real data: (1) on a synthetic data of a normal heart, we found that the constraint improved both longitudinal and circumferential strains accuracies; (2) on 15 healthy volunteers, the proposed method achieved better tracking accuracy compared to three state-of-the-art methods; (3) on one patient dataset, we show that our method is able to distinguish the infarcted segments from the normal ones.

Appendix

In Eq. 3, conducting $1^{st}$-order approximations on $\mathcal {A}_k^t$ leads to:

(6)

Instead of computing $\mathcal {A}_k^t$ maps by phase unwrapping which is highly sensitive to image artifacts, we chose to circumvent the issue by (1) computing $\nabla \mathcal {A}_k^t$ from HARP phases by the method described in [2] and (2) further computing $\mathcal {A}_k^{\tau }(\mathbf {q}_{j})-\mathcal {A}_k^{\tau }(\mathbf {p}_{j})$ ($\tau =ref$ and t) by curvilinear integration of $\nabla \mathcal {A}_k^\tau$. The path of integration is easily defined using our mesh topology. Equation 6 then becomes:

(7)

where $\beta _k^j$ is known and $\delta _k^j(\mathbf {v})$ contains the model parameters.

We first replace both $\varphi ^{(i)}(\mathbf {p}_{j})$ and $\varphi ^{(i)}(\mathbf {q}_{j})$ in $\delta _k^j(\mathbf {v})$ by $g^{(i)}_j=\frac{ \varphi ^{(i)}(\mathbf {p}_{j}) + \varphi ^{(i)}(\mathbf {q}_{j})}{2}$. This is justified by the fact that $\mathbf {p}_{j}$ and $\mathbf {q}_{j}$ are symmetric to the window center (see Fig. 2), thus $\varphi ^{(i)}(\mathbf {p}_{j})\approx \varphi ^{(i)}(\mathbf {q}_{j})$. $\delta _k^j(\mathbf {v})$ then becomes:

(8)

Then, applying the Partition-of-Unity property [7] of $g_j^{(i)}$ leads directly to [7]:

$$\begin{aligned} E_c(\mathbf {v})\le \sum _{i}{\sum _{j}{g_j^{(i)}\sum _{k}{\bigg (\beta _k^j- \frac{1}{2\pi }\mathcal {L}^{(i)}_j(\mathbf {v}^{(i)})} \bigg )^2}}=\sum _{i}{E_c^{(i)}(\mathbf {v}^{(i)})} \end{aligned}$$

(9)

Where $E_{c}^{(i)}$ is quadratic since $\mathcal {L}^{(i)}_j$ is linear in the motion parameters of $\mathbf {v}^{(i)}$.