Strain Energy Decay Predicts Elastic Registration Accuracy From Intraoperative Data Constraints

Abstract

Image-guided intervention for soft tissue organs depends on the accuracy of deformable registration methods to achieve effective results. While registration techniques based on elastic theory are prevalent, no methods yet exist that can prospectively estimate registration uncertainty to regulate sources and mitigate consequences of localization error in deforming organs. This paper introduces registration uncertainty metrics based on dispersion of strain energy from boundary constraints to predict the proportion of target registration error (TRE) remaining after nonrigid elastic registration. These uncertainty metrics depend on the spatial distribution of intraoperative constraints provided to registration with relation to patient-specific organ geometry. Predictive linear and bivariate gamma models are fit and cross-validated using an existing dataset of 6291 simulated registration examples, plus 699 novel simulated registrations withheld for independent validation. Average uncertainty and average proportion of TRE remaining after elastic registration are strongly correlated ( r = 0.78 ), with mean absolute difference in predicted TRE equivalent to 0.9 ± 0.6 mm (cross-validation) and 0.9 ± 0.5 mm (independent validation). Spatial uncertainty maps also permit localized TRE estimates accurate to an equivalent of 3.0 ± 3.1 mm (cross-validation) and 1.6 ± 1.2 mm (independent validation). Additional clinical evaluation of vascular features yields localized TRE estimates accurate to 3.4 ± 3.2 mm. This work formalizes a lower bound for the inherent uncertainty of nonrigid elastic registrations given coverage of intraoperative data constraints, and demonstrates a relation to TRE that can be predictively leveraged to inform data collection and provide a measure of registration confidence for elastic methods.

Fig. 1.

Data available for registration in hepatic image guidance. Deformable registration updates the preoperative model (parenchyma – gray; portal vein – red; hepatic vein – blue) to match intraoperative data while predicting internal displacements as accurately as possible. (a) Organ shape from intraoperative CT (green) indicates the full deformed surface of the liver. (b) In the surgical setting, points on the anterior surface of the liver (black) can be measured using tracked tools or computer vision. (c) A tracked intraoperative ultrasound plane allows localization of intrahepatic vessels and the posterior surface of the liver.

Fig. 2.

RMS Pearson correlation coefficient plotted against rate factor γ for retrospective and prospective information metrics Hr and Hp, respectively. As γ grows large, H depends only on the first distance term and as γ approaches zero, H depends only on the second energy of deformation term. The existence of prominent optima suggests that both terms contribute complementary information towards predicting registration performance. At small γ, the average correlation coefficient is considerably lower for Hp than Hr because the prospective formulation approximates energy of deformation less accurately than achievable with internal strain energy. At large γ, the difference relates to rate constant computation, where the prospective metric estimates the fundamental frequency by the lowest mode response from a series of candidate deformations, whereas the retrospective metric computes a fundamental frequency from the actual activation of deformation modes in the system. An empirical characterization of the rate factor γ affords leniency in the approximations made for the prospective metric without sacrificing substantial predictive value relative to the complete retrospective approach.

Fig. 3.

Linear regressions between constraint entropy Hp and average error capacity $\overline{E}$ with each point representing one registration to a specific configuration of intraoperative data from dataset A. (a) Correlations of Hp and $\overline{E}$ for each of the nine deformation conditions of dataset A. Axes same as (b). (b) All 6291 registrations from dataset A and total regression line (black) plotted with the 699 registrations from the separate validation dataset B (red). Legend indicates the extent of intraoperative data provided to each registration.

Fig. 4.

Empirical joint distributions of paired observations between error capacity E and uncertainty Sp. (a) Empirical distributions drawn from all registrations of each of the nine deformation conditions of dataset A, plotted on the same axes as (b); (b) The total empirical joint distribution using all targets in dataset A.

Fig. 5.

Predicted TRE from median error capacity (top) and measured TRE (bottom) after elastic registration. (a,d) Error profiles of registration to surface data pattern (black). (b,e) Error profiles after data from one additional US plane is added to registration. (c,f) Error profiles with data from three US planes provided. The distributions of predicted remaining error can guide additional data collection to areas of poor expected performance for improving registration fidelity.

Fig. 6.

Results from clinical evaluation. (a) Vascular features from three patients located at the hepatic vein (blue) and portal vein (red) were measured with tracked intraoperative ultrasound. (b) Joint distribution of retrospective uncertainty metric P(Sr, E) with overlaid clinical measurements of error capacity and computed uncertainty for each registered feature. (c) Joint distribution of prospective uncertainty metric P(Sp, E) with overlaid clinical feature measurements of error capacity and computed uncertainty. (d) Conditional distributions P(E|S, θ) for both metrics, with measured error capacity from clinical data in solid red compared to the predicted distribution medians as vertical lines.

Fig. 7.

(a) Quantile-quantile plot between joint cumulative distributions of $P\left(S_r,E\mid {\widehat{\theta }}_r\right)$ and Q(S_r, E) from post-registration metric. (b) Quantile-quantile plot between joint cumulative distributions of $P\left(S_p,E\mid {\widehat{\theta }}_p\right)$ and Q(S_p, E) from pre-registration metric.