The Active Elastic Model

Continuum mechanical models have been used to regularize ill-posed problems in many applications in medical imaging analysis such as image registration and left ventricular motion estimation. In this work, we present a significant extension to the common elastic model which we call the active elastic model. The active elastic model is designed to reduce bias in deformation estimation and to allow the imposition of proper priors on deformation estimation problems that contain information regarding both the expected magnitude and the expected variability of the deformation to be estimated. We test this model on the problem of left ventricular deformation estimation, and present ideas for its application in image registration and brain deformation during neurosurgery. Continuum mechanical models have been extensively used in medical imaging applications over the last ten years, particularly within the contexts of image registration and cardiac motion estimation. More recently, similar models have been applied to the problem of brain deformation during neurosurgery. The models used have been selected either (i) because of their mathematical properties (e.g. [3, 9]) or (ii) as an attempt to model the underlying physics of the situation (e.g. [11, 17, 21]). Such models are a specific case of the quadratic regularizers used in many computer vision applications, such as in the work of Horn[12] or in the deformable models used for segmentation (see McInerney and Terzopoulos [16] for a review). The classical elastic model is derived from the properties of elastic solids such as metals. In cases of small deformations, the linear elastic model may also be applied to model biological tissue which is more hyperelastic in nature. All linear elastic models so far used in medical imaging work are passive models. These models will produce no deformation of their own and are essentially used for smoothing and/or interpolation. Using an elastic model results in an underestimation of the deformation as the model itself biases the estimates towards zero deformation. In this paper we present work to extend these elastic models to allow for non-zero bias. We call this model the ‘active elastic model’. The active elastic model is designed to be used to solve a problem of the following form: ‘Given an input of noisy, possibly sparse, displacements find a To appear in the proceedings of IPMI 2001 2 Papademetris et al dense smooth displacement field which results in a deformation which is close to a desired/expected deformation.’ This new method allows us to construct a proper prior model on the deformation that includes both a mean (the desired magnitude of the deformation) and a covariance (derived from the desired degree of smoothness). The rest of this paper reads as follows: In section 1, we review the basic mathematics of the general energy minimization framework and we compare the use of a passive and an active elastic model for estimation purposes. In section 2, we examine the problem of bias in deformation estimation and demonstrate how the active model can be used to reduce this bias. We present some preliminary results of the application of an active model to reduce the bias in left ventricular deformation estimation in section 3.1 and we conclude by discussing potential applications of this methodology in other areas such as image registration and brain deformation during neurosurgery in section 4. 1 The Energy Minimization Framework In this section we describe a framework in which the goal is to estimate a displacement field u which is a smooth approximation of a noisy displacement field u. We will assume that u is derived from some image-based algorithm, such as the shape-based tracking algorithm[20, 17], MR tagging measurements (e.g. [11]) or optical flow estimates (e.g. [12]). We can pose this problem as an approximation problem whose solution is a least-squares fit of u to u subject to some smoothness constraints and takes the form: