Highly undersampled peripheral Time-of-Flight magnetic resonance angiography: optimized data acquisition and iterative image reconstruction

Abstract

Object

The aim of this study was to investigate the acceleration of peripheral Time-of-Flight magnetic resonance angiography using Compressed Sensing and parallel magnetic resonance imaging (MRI) while preserving image quality and vascular contrast.

Materials and methods

An analytical sampling pattern is proposed that combines aspects of parallel MRI and Compressed Sensing. It is used in combination with a dedicated Split Bregman algorithm. This approach is compared with current state-of-the-art patterns and reconstruction algorithms.

Results

The acquisition time was reduced from 30 to 2.5 min in a study using ten volunteer data sets, while showing improved sharpness, better contrast and higher accuracy compared to state-of-the-art techniques.

Conclusion

This study showed the benefits of the proposed dedicated analytical sampling pattern and Split Bregman algorithm for optimizing the Compressed Sensing reconstruction of highly accelerated peripheral Time-of-Flight data.

Appendix: Formulation of the split problem sub-problems

Starting from Eq. (3), the inclusion of the residual errors ${\bf b}_{x},{\bf b}_{y},{\bf b}_{w}\in {\mathbb {C}}^N$ yields the formulation in two steps, given in Eqs. (4–5).

$$\begin{aligned} (\hat{{\bf x}},\hat{{\bf d}_{x}},\hat{{\bf d}_{y}},\hat{{\bf d}_{w}})=&{\mathop {{{\mathrm{\arg \!\min }}}}\limits _{{\bf x},{\bf d}_{w},{\bf d}_{x},{\bf d}_{y}}}\frac{1}{2}\left\| {\bf E} {\bf x} - {\bf m}\right\| _{L_2}^2 \\&+ \lambda _{\mathrm{t}}\sum _{\iota =1}^{N}\sqrt{({\bf d}_{x})_{\iota }^2+({\bf d}_{y})_{\iota }^2} + \lambda _{\mathrm{w}}\left\| {\bf d}_{w}\right\| _{L_1}\\&+ \frac{\alpha \lambda _{\mathrm{t}}}{2} \left( \left\| {\bf d}_{x}- \nabla _x{\bf x} - {\bf b}_{x}^j\right\| _{L_2}^2 + \left\| {\bf d}_{y}- \nabla _y{\bf x} - {\bf b}_{y}^j\right\| _{L_2}^2\right) \\&+ \frac{\alpha \lambda _{\mathrm{w}}}{2}\left\| {\bf d}_{w}- W({\bf x}) - {\bf b}_{w}^j\right\| _{L_2}^2\end{aligned}$$

(4)

$$\begin{aligned} {\bf b}_{x}^{j+1}&={\bf b}_{x}^j + \nabla _x {\bf x}^{j+1} - {\bf d}_{x}^{j+1}, \\ {\bf b}_{y}^{j+1}&={\bf b}_{y}^j + \nabla _y {\bf x}^{j+1} - {\bf d}_{y}^{j+1}, \text { and }\\ {\bf b}_{w}^{j+1}&={\bf b}_{w}^j + W({\bf x}^{j+1}) - {\bf d}_{w}^{j+1}. \end{aligned}$$

(5)

The minimization problem for the $\hbox {L}_2$ component equals

$$\begin{aligned}&({\bf x}^{j+1})={\mathop {{{\mathrm{\arg \!\min }}}}\limits _{{\bf x}}} \,{\mathcal {L}}_{{\text {SB-L2}}}({\bf x},{\bf d}_{x}^j,{\bf d}_{y}^j,{\bf d}_{w}^j, {\bf b}_{x}^j,{\bf b}_{y}^j,{\bf b}_{w}^j) \text { with }\\&\quad {\mathcal {L}}_{{\text {SB-L2}}}({\bf x},{\bf d}_{x}^j,{\bf d}_{y}^j,{\bf d}_{w}^j, {\bf b}_{x}^j,{\bf b}_{y}^j,{\bf b}_{w}^j)= \frac{1}{2} \left\| {\bf E}{\bf x} - {\bf m}\right\| _{L_2}^2\\&\qquad + \frac{\alpha \lambda _{\mathrm{t}}}{2}\left( \left\| {\bf d}_{x}^j- \nabla _x({\bf x}) -{\bf b}_{x}^j\right\| _{L_2}^2+\left\| {\bf d}_{y}^j- \nabla _y({\bf x}) - {\bf b}_{y}^j\right\| _{L_2}^2\right) + \frac{\alpha \lambda _{\mathrm{w}}}{2} \left\| {\bf d}_{w}^j- W({\bf x}) - {\bf b}_{w}^j\right\| _{L_2}^2 \end{aligned}$$

(6)

The sub-problems for the wavelet and total variation terms are formulated as

$$\begin{aligned} ({\bf d}_{x}^{j+1},{\bf d}_{y}^{j+1})={\mathop {{{\mathrm{\arg \!\min }}}}\limits _{{\bf d}_{x},{\bf d}_{y}}}&\frac{\alpha \lambda _{\mathrm{t}}}{2} \left\| {\bf d}_{x}- \nabla _x( {\bf x}^{j+1}) - {\bf b}_{x}^j\right\| _{L_2}^2 + \frac{\alpha \lambda _{\mathrm{t}}}{2} \left\| {\bf d}_{y}- \nabla _y( {\bf x}^{j+1}) - {\bf b}_{y}^j\right\| _{L_2}^2 + \lambda _{\mathrm{t}}\sum _{\iota =1}^{N}\sqrt{({\bf d}_{x})_{\iota }^2+({\bf d}_{y})_{\iota }^2} \text { and } \end{aligned}$$

(7)

$$\begin{aligned} ({\bf d}_{w}^{j+1})={\mathop {{{\mathrm{\arg \!\min }}}}\limits _{{\bf d}_{w}}}&{\frac{\alpha \lambda _{\mathrm{w}}}{2} \left\| {\bf d}_{w}-W({\bf x}^{j+1}) - {\bf b}_{w}^j\right\| _{L_2}^2 + \lambda _{\mathrm{w}}\left\| {\bf d}_{w}\right\| _{L_1}}. \end{aligned}$$

(8)