a. Inverse Consistency:
For each anatomical site the original kvCT and the synthetically deformed image from ImSimQA were used to test inverse consistency of B-Splines and diffeomorphic demons algorithms. The two images are registered separately both in the forward and inverse directions. A perfect inverse consistent algorithm in theory should give a deformation map which is a true inverse of one another. However in reality this rarely occurs because most algorithms do not produce true inverse deformation maps since deformable
registration is inherently degenerative and multiple solutions may exist for a given image matching problem.
We use the concept of compositive accumulation to quantify the inverse consistency error. The details of compositive accumulation as discussed in [129, 163] are summarized below.
The concept of compositive accumulation is used to quantify the inverse consistency error. The warping by a deformation vector field D is associated with its corresponding transformation operation Δ, such that Δ Id + D, or Δ(x) x + D (x), where Id is the identity transformation such that I d(x) = x. Mathematically given two images A and B
during DIR, the objective is to find a deformation vector field D such that warping of image B by D is close to original image A or A = B ο Δ.
If D1 and D2 are two deformation fields, a single warping by compositive addition of D1
and D2 is equivalent to successive deformation of an image by D1 and then followed by
D2.
The warping by a field D is equivalent to the composition with its corresponding
transformation ∆. One can then use the composition of function in order to replace successive warpings (i.e. by different displacement fields) with a single warping (i.e. by an equivalent displacement field). Mathematically, this compositive operation denoted as
⊕, is defined as follows
D1 ⊕ D2 = ∆1 ◦ ∆2 − Id.
By construction, the deformation operation linked to the displacement field D1 ⊕ D2 is
therefore ∆1 ο ∆2. The operation ⊕ has some interesting and useful properties. First, the
neutral is of course obtained with the null displacement field, i.e. D ⊕ 0 = 0 ⊕ D = D. It
can be shown that the associative relations (D1 ⊕ D2) ⊕ D3 = D1 ⊕ (D2 ⊕ D3) = D1 ⊕
D2 ⊕ D3 for three displacement fields D1, D2 and D3.
D1 D2 = D2 + D1 o D2, meaning that that D1 D2 is equivalent to summing
deformation field D2 with the field D1 warped by D2 [163] . A simple proof is presented
below.
D1 D2 = Δ1 ο Δ2 – Id. By construction the deformation operation linked to D1 D2 is
Δ1 ο Δ2.
Since D1 o D2 = D1 ο ∆2, and D = ∆ − Id, one can easily see that:
D2 + D1 o D2 = D2 + D1 ο ∆2
= ∆2 − Id + ∆1 ο ∆2 − Id ο ∆2 = ∆1 ο ∆2 − Id
Further the compositive addition operation is associative for three deformation fields D1, D2 and D3 meaning (D1 D2) D3 = D1 D2 D3) = D1 D2 D3
For the warp, we use a linear interpolator, i.e. we add the right field to the interpolated left field for that pixel as the resulting point x will not land exactly in the grid.
For purposes of inverse consistency, if D1 and D2 are deformation fields from forward and
inverse registration, the compositive accumulation of forward and inverse deformation fields will yield the inverse consistency error (ICE). If the deformation maps are true inverses, this composition will yield zero. The L2 norm (absolute magnitude) of the composed fields is used to quantify the magnitude of inverse consistency error.
Further, ICE between the DVF arising from DIR and the synthetic DVF generated from ImSimQA software which was used to produce clinically relevant organ deformation was evaluated. The ImSimQA can also output inverse DVF of the applied deformation. This DVF was compared with the DVF generated from the inverse registration process where the roles of source and target images were switched. A compositive accumulation of the ImSimQA DVF and the DVF from registration (B-Spline and diffeomorphic demons) was done to quantify the ICE between DVFs. If the results of DIR produced a DVF which is the exact inverse of applied synthetic DVF in ImSimQA, then this composition of DVFs will be zero. The L2 norm of the composed DVFs is computed to quantify the ICE between DVFs.
b. Determinant of jacobian of the deformation Field:
The jacobian of the deformation field gives information about the image transformation consistency[134, 164]. The jacobian is a matrix given by the first partial derivatives of the transformation with
where is kronecker delta ( and Di is the ith component
of deformation field.
We computed the determinant of the jacobian of the deformation field in this study to validate the physical behavior of deformation. A negative determinant indicates
singularities in the field and corresponds to a physically unrealistic organ deformation. A determinant greater than 1 indicates expansion at that location while a value less than 1 indicates contractions.
c. Mean Harmonic Energy of the Deformation Field:
The harmonic energy captures the non-linearity of the warp i.e. deviation from an affine transformation. The mean harmonic energy is defined as the frobenius norm of the jacobian and is inversely proportional to how smoothness of the deformation field [135]. The harmonic energy at a voxel can be defined based on the first order partial derivatives of the deformation field as follows:
HE (D) = ½
where Ʋ is the domain of the deformation field.
II. Anatomical correspondence:
In radiotherapy clinical applications the accuracy of tumor and organ at risk (OAR) structures is of paramount importance. Ultimately the changes in the shape and volume of these structures and consequently the dose received by them dictate the need for adaptive radiation therapy.
We use the Dice similarity coefficient, Hausdorff distance and average surface distance as three metrics to evaluate the accuracy of tumor and OAR for each anatomical site
before and after DIR. These metrics have been previously used to compare segmentations in radiotherapy applications and are described below [96, 165-167]. The ImSimQA DVF was used to warp the original RT structures in addition to CT images. The registration DVF from both algorithms was then applied to these RT structures. If the results of DIR were perfect then the RT structures before and after DIR would be the same. The degree of mismatch indicates the quality of DIR from an anatomical correspondence perspective. a. Dice Similarity Coefficient:
The metric computes the number of pixels that overlap between the two volumes and normalizes it by the half the sum of the number of non-zero pixels in the two volumes. The result is a value between 0 (no overlap) and 1 (perfect overlap) as shown in fig 4.9 α =
where A is the gold standard segmentation which in our case refers to segmentation in kvCT fixed image, B is the segmentation mapped from the deformably registered image. The metric is symmetric and is sensitive to both differences in scale and position. While volume overlap is a good indicator of mismatch, it is a poor indicator of shape since is not a measure of distance and hence the following metrics are also evaluated to assess the overall accuracy.
b. Hausdorff distance:
The Hausdorff distance [168] is defined as the maximum of the closest distance between two volumes where the closest distance is computed for each vertex of the two volumes. The hausdorff distance H(A,B) between 2 sets of points A = {a1, .., am} and B = {b1, .., bm} is given by
where h(A,B) = maxa€Aminb€B|| a-b||
h(A,B) is the directed hausdorff distance from A to B, which unlike the hausdorff distance is not symmetric.
h(A,B) identifies the point a ∈ A that is farthest from any point in B, and then measures the distance of A to its nearest neighbor in B. The point sets A, B in our case, are the centers of the non-zero pixels in the gold standard (original kvCT) and deformably registered segmentations. Thus, the hausdorff distance is a measure of the maximum distance between two surfaces as shown in fig 4.9 a. It obeys all four properties of metric spaces and distance functions.
– Identity: H(A,A) = 0
– Positive semi-definiteness: H(A,B) = 0 – Symmetricity: H(A,B) = H(B,A)
– Triangle inequality: H(A,C) = H(A,B) + H(A,C)
The metric is very sensitive to outliers since the most mismatched point is the sole
determining criteria of the distance. Some authors use 95% Hausdorff distance (95%HD) as the outliers are rejected in 95%HD.
Fig 4.9a Hausdorff Distance is the maximum perpendicular distance between closest
points from two contours of registered images. Black line represents an external contour from one image and gray line represents an external contour from another registered image. Small circles represent corresponding closest points between each contour. Hausdorff distance represents the distance between small circles at black arrow. b) Dice coefficient similarity (DSC) is an index of overlap of two different volumes. Solid black
line represents a volume from one image and dotted black line represents a volume from
another image after registration. DSC is a value between 0 (no overlap) and 1 (perfect overlap). (Diagram above adapted from Reference [169])
c. Average Surface Distance:
This metric mitigates the outlier problem exhibited by the Hausdorff distance. The metric is the average of the absolute distance from each surface pixel in one image to its closest point on the other image. This metric is not symmetric, although it satisfies the positive semi-definite and identity properties of distance metrics.
M (A, B) = € €
III. Image Characteristics: