Quantitative Assessment of Abdominal Aortic Aneurysm Geometry

Abstract

Recent studies have shown that the maximum transverse diameter of an abdominal aortic aneurysm (AAA) and expansion rate are not entirely reliable indicators of rupture potential. We hypothesize that aneurysm morphology and wall thickness are more predictive of rupture risk and can be the deciding factors in the clinical management of the disease. A non-invasive, image-based evaluation of AAA shape was implemented on a retrospective study of 10 ruptured and 66 unruptured aneurysms. Three-dimensional models were generated from segmented, contrast-enhanced computed tomography images. Geometric indices and regional variations in wall thickness were estimated based on novel segmentation algorithms. A model was created using a J48 decision tree algorithm and its performance was assessed using ten-fold cross validation. Feature selection was performed using the χ²-test. The model correctly classified 65 datasets and had an average prediction accuracy of 86.6% (κ = 0.37). The highest ranked features were sac length, sac height, volume, surface area, maximum diameter, bulge height, and intra-luminal thrombus volume. Given that individual AAAs have complex shapes with local changes in surface curvature and wall thickness, the assessment of AAA rupture risk should be based on the accurate quantification of aneurysmal sac shape and size.

Appendices

Appendix 1: Definition of Features (i.e., Geometric Indices) Used in the Geometry Quantification Approach

1D Size Indices

D_max::

Maximum transverse diameter for all cross sections within the AAA sac

D_neck,p::

Proximal neck diameter immediately below the renal arteries

D_neck,d::

Distal neck diameter

Since typically arterial cross sections are non-circular in shape, the definition used for calculating D _max, D _neck,p, and D _neck,d is the fluid mechanics definition for hydraulic diameter:

$$ D_{i} = {\frac{{4A_{i} }}{{P_{i} }}} $$

where A _i the cross-sectional area and P _i the perimeter of the same cross section.

H_sac is the height of AAA sac, L_sac the centerline length of AAA sac, H_neck the height of AAA neck, L_neck the centerline length of AAA neck, H_b the bulge height, and d_c the distance between the lumen centroid and the centroid of the cross section where D_max is located.

2D Shape Indices

DHr::

The diameter–height ratio, an expression of the fusiform shape of the AAA sac

$$ DHr = {\frac{{D_{\max} }}{{H_{\text{neck}} + H_{\text{sac}} }}}$$

DDr::

Diameter–diameter ratio

$$ DDr = {\frac{{D_{\max} }}{{D_{\rm neck,p} }}}$$

Hr::

Height ratio, an assessment of the relative neck height in comparison with the AAA height

$$ Hr = {\frac{{H_{\rm neck} }}{{H_{\rm neck} + H_{\rm sac} }}} $$

BL::

Bulge location, provides a measure of the relative position of the maximum transverse dimension with respect to the neck

$$ BL = {\frac{{H_{\text{b}} }}{{H_\text{neck} + H_\text{sac} }}} $$

β::

Asymmetry factor

$$ \beta = 1 - {\frac{{d_\text{c} }}{{D_{\max} }}} $$

T::

Tortuosity

$$ T = {\frac{{L_{\rm neck} + L_{\rm sac} }}{d}} $$

where d is the Euclidean distance from the centroid of the cross section where D_neck,p is located on the centroid of the cross section at the AAA distal end.

3D Size Indices

V::

Vessel volume

S::

Vessel surface area

V_ILT::

Volume of intraluminal thrombus contained within AAA sac

γ::

Ratio of AAA ILT volume

$$ \gamma = {\frac{{V_{\rm ILT} }}{V}} $$

3D Shape Indices

IPR: :

Isoperimetric ratio

$$ IPR = {\frac{S}{V^{2/3}}} $$

NFI::

Non-fusiform index

$$ NFI = \frac{\frac{S}{V^{2/3}}}{\frac{S_\text{fusiform}}{V_{\text{fusiform}}{}^{2/3}}} =\frac{IPR}{IPR_{\text{fusiform}}}$$

Second-Order Curvature-Based Indices (Calculation Described in Detail in Martufi et al. 16)

GAA :

Area averaged Gaussian curvature

MAA :

Area averaged Mean curvature

GLN :

L2 norm of the Gaussian curvature

MLN :

L2 norm of the Mean curvature

$$ GAA = {\frac{{\sum\nolimits_{\text{all\;elements}} {K_{j} S_{j} } }}{{\sum\nolimits_{\text{all\;elements}}} {S_{j} } }} $$

$$ MAA = {\frac{{\sum\nolimits_{\text{all\;elements}} {M_{j} S_{j} } }}{{\sum\nolimits_{\text{all\;elements}} {S_{j} } }}} $$

$$ GLN = {\frac{1}{4\pi }}\sqrt {\sum\limits_{\text{all\;elements}} {S_{j} } \cdot \sum\limits_{\text{all\;elements}} {\left( {K_{j}^{2} S_{j} } \right)} } $$

$$ MLN = {\frac{1}{4\pi }}\sqrt {\sum\limits_{\text{all\;elements}} {\left( {M_{j}^{2} S_{j} } \right)} } $$

Wall Thickness Indices

t_w,min::

Minimum wall thickness

t_w,max::

Maximum wall thickness

t_w,ave::

Average wall thickness

Appendix 2: Brief Description of Decision Tree Models

A decision tree is a predictive machine learning model whose structure is determined by the information gain (IG) of the feature values. The IG can be calculated as:

$$ {{\text{IG}}}\left( {{{Y}}|{{X}}{:}{{t}}} \right) = {{H}}\left( {{Y}} \right) - {{H}}\left( {{{Y}}|{{X}}{:}{{t}}} \right) $$

where

$$ {H}( {{Y}}) = {{P}}\left( {{{Y}} = {\text{Ruptured}}} \right)\ast{ \log }_{ 2} {{P}}\left( {{{Y}} = {\text{Ruptured}}} \right) + {{P}}\left( {{{Y}} = {\text{Unruptured}}} \right)\ast{ \log }_{ 2} {{P}}\left( {{{Y}} = {\text{Unruptured}}} \right) $$

$$ {{H}}\left( {{{Y}}|{{X}}{:}{{t}}} \right) = {{H}}\left( {{{Y}}|{{X}} < {{t}}} \right)*{{P}}\left( {{{X}} < {{t}}} \right) + {{H}}\left( {{{Y}}|{{X}} \ge {{t}}} \right)*{{P}}\left( {{{X}} \ge {{t}}} \right) $$

where Y the class label, which in this case is the repair status (elective vs. emergent), X the feature, and t the value of the feature. H(Y) represents the entropy prior to splitting on a certain feature and feature value and ${{H}}\left( {{{Y}}|{{X}}{:}{{t}}} \right)$ represents the conditional entropy after splitting on a feature and feature value. Entropy is a measure of the impurity of an arbitrary collection of examples, and varies between 0 and 1 where 1 indicates that there are an equal number of examples in each group and 0 indicates that all examples are in the same group. The IG is calculated as the different between H(Y) and ${{H}}\left( {{{Y}}|{{X}}{:}{{t}}} \right) .$ A high IG is desirable as it indicates that the conditional entropy, H(Y|X:t), is low, which translates as a reduction in entropy after splitting on a certain feature and feature value.

The internal nodes of a decision tree denote the different features; the branches between the nodes provide the possible values that these features can have based on the data, while the terminal nodes yield the classification variable (ruptured or unruptured). The algorithm determines the internal nodes and branches by identifying the feature that discriminates the various instances most clearly; such feature is said to have the highest IG.

For the tree displayed in Fig. 4, the feature that provided the highest IG was L _sac. Six ruptured aneurysm had L _sac > 160.57 mm and all the unruptured AAAs and the remaining four ruptured AAAs had L _sac < 160.57 mm. Since the six ruptured AAAs cannot be further divided, the tree terminates at that node. The IG is recalculated using the remaining 66 unruptured AAAs and the 4 ruptured AAAs. These are then further subdivided using the feature that had the highest IG, which in this case was S.