A fast deformable registration method for 4D lung CT in hybrid framework

Abstract

Purpose

A pulmonary respiration model for deformable registration of lung CT for the surgery path planning and surgical navigation is an important, difficult, and time-consuming task. This paper presents a new fast deformable registration method for 4D lung CT in a hybrid framework incorporating point set registration with mutual information registration.

Method

The point sets of the lung surface and vessels are automatically extracted. Their displacement vectors are obtained by point set registration. The sum of squared Euclidean distance between the displacement vectors of these point sets and the displacement vectors based on the B-spline transformation model is minimized as a novel similarity measure to derive the rough transformation function. Finally, the rough transformation function is refined by using the mutual information-based registration method. To evaluate the effectiveness of the proposed method, the authors performed registrations on 20 4D lung volume cases from two different CT scanners. The proposed method was compared with the point set-based method, the mutual information-based method, and the ANTS method, which is a state-of-the-art deformable registration technique.

Results

The results show that the landmark distance errors and computation time of the proposed method decreased an average of 5 and 70 %, respectively, when compared to the mutual information-alone-based method. The proposed method results in an average of 28 % lower landmark distance error than registration method based on point sets in spite of increase in computation time. Moreover, compared with ANTS, the computation time of the proposed method is reduced by an average of 93 % in the case of comparable landmark distance errors.

Conclusion

The accuracy and speed of the proposed deformable registration method indicate that the method is suitable for use in a clinical image-guided intervention system.

Appendix

We provide here the calculation the gradient $\nabla C_{\textit{SSVED}}$ as Eq. (11):

$$\begin{aligned} \nabla C_{SSVED}&= \left[ \frac{\partial C_{SSVED}}{\partial d_1},\frac{\partial C_{SSVED} }{\partial d_2},\cdots , \right. \nonumber \\&\quad \left. \frac{\partial C_{SSVED}}{\partial d_i},\cdots , \frac{\partial C_{SSVED}}{\partial d_m}\right] ^{T} \end{aligned}$$

(11)

For a given control node displacement vector $d$, if the number of control nodes is $m,\,d=(d_1, d_2, \ldots , d_i, \ldots , d_m )$, where $d_i =(d_{ix}, d_{iy}, d_{iz} )$ is the control node displacement component along direction $x, y$, and $z$. The gradient of cost function $C_{SSVED} (U_F, U_T ;d)$ respect to $d$ is written as:

$$\begin{aligned}&\frac{\partial C_{SSVED} (U_F, U_T ;d)}{\partial d_{ix}} \nonumber \\&\quad =\frac{\partial \sum _{x_{F} \in U_{F}, x_{T} \in U_{T}} {\left[ (T_p (x_F ;d)_x -x_{Tx})^{2}+(T_p (x_F ;d)_y -x_{Ty})^{2}+(T_p (x_F ;d)_z -x_{Tz})^{2}\right] } }{\partial d_{ix}} \nonumber \\&\quad =2\sum _{x_F \in U_F, x_T \in U_T } \left[ (T_p (x_F ;d)_x -x_{Tx})\frac{\partial T_p (x_F ;d)_x }{\partial d_{ix}} \right. \nonumber \\&\qquad \qquad \qquad \qquad \qquad \quad +\left( T_p (x_F ;d)_y -x_{Ty}\right) \frac{\partial T_p (x_F ;d)_y}{\partial d_{ix}} \nonumber \\&\qquad \qquad \qquad \qquad \qquad \quad \left. +\left( T_p (x_F ;d)_z -x_{Tz}\right) \frac{\partial T_p (x_F ;d)_z }{\partial d_{ix}}\right] \end{aligned}$$

(12)

Besides, we adopted a simple calculation method for reducing the computing time of optimization. If the direction of the components in $T_{P}$ is inconsistent with the direction of the components in $d_{i}$, the corresponding derivative should be zero:

$$\begin{aligned} \frac{\partial T_{P} (x_F ;d)_{x}}{\partial d_{ix}}&= \sum _{l=0}^{3} {\sum _{m=0}^{3} {\sum _{n=0}^{3} {B_l (u)}}} B_m (v)B_n (w) \end{aligned}$$

(13)

$$\begin{aligned} \frac{\partial T_P (x_F ;d)_y}{\partial d_{ix}}&= \frac{\partial T_P (x_F ;d)_z}{\partial d_{ix}}=0 \end{aligned}$$

(14)

Likewise, the derivative of $T_P$ respect to displacement components along y and z direction $\frac{\partial C_{SSVED} (U_F, U_T ;d)}{\partial d_{iy}}$ and $\frac{\partial C_{SSVED} (U_F, U_T ;d)}{\partial d_{iz}}$ is also calculated in the same way.