An error analysis perspective for patient alignment systems

Abstract

Purpose

This paper analyses the effects of error sources which can be found in patient alignment systems. As an example, an ultrasound (US) repositioning system and its transformation chain are assessed. The findings of this concept can also be applied to any navigation system.

Methods and materials

In a first step, all error sources were identified and where applicable, corresponding target registration errors were computed. By applying error propagation calculations on these commonly used registration/calibration and tracking errors, we were able to analyse the components of the overall error. Furthermore, we defined a special situation where the whole registration chain reduces to the error caused by the tracking system. Additionally, we used a phantom to evaluate the errors arising from the image-to-image registration procedure, depending on the image metric used. We have also discussed how this analysis can be applied to other positioning systems such as Cone Beam CT–based systems or Brainlab’s ExacTrac.

Results

The estimates found by our error propagation analysis are in good agreement with the numbers found in the phantom study but significantly smaller than results from patient evaluations. We probably underestimated human influences such as the US scan head positioning by the operator and tissue deformation. Rotational errors of the tracking system can multiply these errors, depending on the relative position of tracker and probe.

Conclusions

We were able to analyse the components of the overall error of a typical patient positioning system. We consider this to be a contribution to the optimization of the positioning accuracy for computer guidance systems.

Appendix: Mathematical appendix

Purely translational error in one matrix

The undistorted vs. distorted chain from Eq. (1) looks like

$$\begin{aligned} F = M_1 \ldots M_k \ldots M_7&\end{aligned}$$

(6)

$$\begin{aligned} \widetilde{F} = M_1 \ldots \widetilde{M_k} \ldots M_7&\end{aligned}$$

(7)

therefore we have

$$\begin{aligned}&F^{-1}\widetilde{F} = \underbrace{M_7^{-1} \ldots M_{k-1}^{-1}}_{A^{-1}} M_k^{-1} \underbrace{\ldots M_1^{-1} M_1 \ldots }_I \widetilde{M_k} \underbrace{M_{k+1}\ldots M_7}_{A} \end{aligned}$$

(8)

$$\begin{aligned}&F^{-1}\widetilde{F}= A^{-1} M_k^{-1} \widetilde{M_k} A \end{aligned}$$

(9)

where is a rigid body transformation. With we have that is a purely translational error in $M_k$ we have

$$\begin{aligned} F^{-1}\widetilde{F}=A^{-1} \begin{pmatrix} I &{} R^tt \\ 0 &{} 1 \end{pmatrix} A = \begin{pmatrix} I &{} R_A^tR^tt \\ 0 &{} 1 \end{pmatrix} \end{aligned}$$

(10)

In summary, the effect of a translational error $t$ in matrix $M_k$ is a translational error of magnitude $\Vert R_A^tR^tt\Vert = \Vert t\Vert$ in the repositioning matrix.

Purely translational error in US calibration

In case of a translational error matrix appearing twice in the transformation chain, as is the case for the US calibration, we have

$$\begin{aligned} F^{-1}\widetilde{F} = A^{-1}\cdot M_4^{-1} \cdot T \cdot M_4 \cdot T^{-1} \cdot A \end{aligned}$$

(11)

where and is the $^{\mathrm{US,Linac}}T_{\mathrm{US},\mathrm{CT}}$ matrix. With we have

$$\begin{aligned} F^{-1}\widetilde{F}&= A^{-1}\cdot M_4^{-1} \cdot T \cdot M_4 \cdot T^{-1}\cdot A\nonumber \\&= \begin{pmatrix} I &{} R_A^t(R^tt-t) \\ 0 &{} 1 \end{pmatrix} \end{aligned}$$

(12)

In summary, the effect of an translational error $t$ in the US-calibration matrix is a translational error of magnitude $\Vert R_A^t(R^t-I)t\Vert = \Vert (R^t-I)t\Vert \le 2\Vert t\Vert$ in the repositioning matrix, where $R$ is the rotational part of the US-calibration.

Purely rotational error in one matrix

In Eq. (9) we set . We therefore have

$$\begin{aligned} F^{-1}\widetilde{F} = \begin{pmatrix} R_A^tR^t\widetilde{R}R_A &{} R_A^t\left( R^t\widetilde{R}-I\right) T_A \\ 0 &{} 1 \end{pmatrix} \end{aligned}$$

(13)

The overall rotation will be rather small, as the axes of the laser coordinate systems in both rooms are roughly parallel to the patient’s axes. We therefore focus on the translational part of $F^{-1}\widetilde{F}$ and get

$$\begin{aligned} \text{ error} = \left\| \left( R^t\widetilde{R}-I\right) T_A \right\| \le 2\Vert T_A\Vert . \end{aligned}$$

(14)

In the case of a rotational error in the roll angle, this reduces to

$$\begin{aligned} \text{ error}(\delta _\mathrm{roll})&= \sqrt{2(1-\cos (\delta _\mathrm{roll})) \left( y_A^2+z_A^2\right) }\nonumber \\&\approx \delta _\mathrm{roll}\cdot \sqrt{y_A^2+z_A^2} \end{aligned}$$

(15)

with . Therefore, the error is linearly dependent on $\delta _\mathrm{roll}$ for small angles.