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III.2.1 The Deformable Model

A deformable model was constructed to represent the four cardiac chambers and surrounding myocardium. The Doo-Sabin model was used for this purpose, a generalization of quadratic B-splines originally described by Doo and Sabin [9]. The Doo-Sabin model uses control nodes and a topology between those nodes to generate the surface. The surface can be evaluated locally around each node, splitting the surface into several patches. The Doo-Sabin algorithm gives rounded, organic-looking surfaces, which are well suited for medical shape modelling, and has been used for that purpose by Orderud and Rabben [21]

as well as Dikici [7]. In addition, the surface vertices of the model are easy to compute, as are the surface normals.

Materials and Methods

Figure III.1: The four chambers put together, without the epicardium surface.

The LV is on the upper left, RV is on the upper right, LA is lower left and RA is on the lower right. The numbered nodes and lines between them shows the topology.

The model consisted of four surface meshes meant to model the endocardium of each of the four heart chambers, and an outer layer representing the epicardium.

The latter was used in order to determine the volume conservation of the cardiac tissue. The myocardium was considered to be nearly incompressible[27], and volume concervation was applied in the algorithm, as detailed in Section III.2.3.

As each of the chambers were modelled as a closed mesh, valves and vessels were not modelled.

In order to create a model where the chambers are modelled together, we devised several sub-models with shared nodes where the chambers meet. The model had 78 nodes, where 34 were associated with the LV, 40 with the RV, 27 with the LA and 20 with the RA. 21 nodes were associated with both the LV and RV, along the septum and the base. 9 nodes were associated with the LV and LA along the LV base and 6 nodes were associated with both the RV and RA along the tricuspid valve. 10 nodes were associated with both LA and RA along the atrial septum and the base. Figure III.1 shows the four chambers together, and Figure III.2 shows them individually.

III.2.2 Model Fitting

In order to fit the model to the image, an extended Kalman algorithm was used [14] [20] [2] . The Kalman algorithm is a method for estimating values based on

(a) The left ventricle (b) The right ventricle

(c) The left atrium (d) The right atrium

Figure III.2: The 4 chambers of the model. The lines around the figures marks the control polygon.

Materials and Methods

(a) (b)

Figure III.3: Two examples of models being fitted to images. For each image both end diastole and end systole fitting is shown. End diastole is at the left of each image, and end systole at the right.

both theoretical models of how a system functions, and measurements of the system. The Kalman algorithm delivers continuous updates of the state vector as the system it is modelling changes. The extended version has advantages where the system is non-linear, using the Jacobian matrices of the system as input. The extended Kalman filter used in this article will be referred to simply as a Kalman filter in the rest of this article.

The implementation of the Kalman filters are mainly the same as used by Orderud[20], with some changes detailed below. The state vectorxthat the Kalman filter outputted consists of three types of values. There are values for individual node displacement, changing the shape of the model to fit the image.

This change is applied to the model first. Next there are variables for thickness of the walls between the models, which are applied second. Finally, the global variables handles the position, scale and rotation of the model.

The Kalman filter has two stages for each time-step in the algorithm:

prediction and updating. In our case the first stage consisted of predicting the placement of the model in the current frame based on the value of the previous frame. This was done using a linear function slightly regressing the state vector towards the initial values. The updating stage used edge detection as a measurement to create an updated estimate of the state vector.

The edge detection used as input for the update state was mainly step edges, but a strongest gradient edge detection was used in the apical region. Edge detection was done in normal direction of the surface at each evaluation point with a 2 cm capture range in each direction.

Some changes were made to Orderud’s image segmentation framework. In

order to improve accuracy, restrictions were placed on the Kalman filter output.

The apex of the model was constrained to never move above the top of the image nor below the next layer of nodes, and the individual nodes were constrained not to deviate from the initial values beyond a threshold. The threshold was set so that the shape of the model should not be degenerated or self-intersect.

These restrictions were determined based on training data.

These modifications was added to the filter itself, allowing the next frame’s prediction step to use the modified values. This means that unlike the regular Kalman filter, which normally has a step of prediction based on theoretical models of behaviour and then an update based on measurements, this algorithm adds a final step of theoretical adjustments, based on the hearts physical properties.

Figure III.3 shows two examples of the model being fitted to images.

III.2.3 Modelling Thickness

We used used a thickness parameter to determine the thickness of the walls between the chambers. Each node in the model was associated with a thickness value. After the preliminary node displacement based on node movement was done, the thickness value was used to adjust node placements. This displacement was done in the normal direction of the surface mesh.

The thickness was determined by the Kalman filter, and the divergence theorem was used to estimate the volume of the region around each node. The volume was then divided by the local patch area around the node. This gave the local node thickness. The mathematical foundation of the node thickness and how it interacts with the Kalman filter was covered in detail by Bersvendsen et al[3].

The myocardium was considered to be nearly incompressible due to its high water content[27], meaning the volume should be almost constant. This was accomplished by having the noise variable of the volume be set low in the Kalman filter. Change in thickness should mainly be caused by change in patch area caused by contraction and expansion of the chambers. Some variation of volume was allowed to capture differences between different images.

III.2.4 Validation of the Algorithm

The algorithm was validated by doing a comparison with standard software for estimating cardiac function. The evaluation was done using volume metrics, and GE’s 4D AutoLVQ, AutoRVQ and AutoLAQ features (EchoPAC SoftwareOnly v204, GE Vingmed Ultrasound, Horten Norway) with manual editing was used to establish a ground truth for the LV, RV and LA respectively. Biplane Simpson was used to estimate the RA volume, as has been done previously by for instance Wang et al.[28] and Aune et al. [1].

The algorithm was evaluated on 42 images. Across those 42 images, the LV was evaluated in 31 images, the RV in 16 images, the LA in 11 images and the RA in 14 images, for a total of 72 chambers. Which chambers were evaluated was determined by manually checking which chambers were fully visible in the

Results Table III.1: Average volumes and standard deviations

(SD) for the endiastole of the ground truth.

LV RV LA RA

Average

vol-ume 190.2 ml 106.9 ml 57.7 ml 51.1 ml

Standard

de-viation 75.5 ml 43.7 ml 27.6 ml 20.1 ml

image, and the evaluation was done before the algorithm was applied to the images. Validation of the algorithm consisted of evaluating end-systolic and end-diastolic volumes, strokes volumes and ejection fraction.

To prepare for validation, the model was manually tuned on 11 3D ultrasound images showing a total of 23 chambers.This tuning determined node configuration and layout. The training images were kept separate from the validation images.

For each image a manual translation of the model was done. The model was translated to align the base of the septum wall in the LV with the image. Aside from that the algorithm was fully automatic.