 Research
 Open Access
Semiautomatic measurements and description of the geometry of vascular tree based on Bézier spline curves: application to cerebral arteries
 Jarosław Żyłkowski^{1},
 Grzegorz Rosiak^{1} and
 Dominik Spinczyk^{2}Email authorView ORCID ID profile
https://doi.org/10.1186/s1293801805478
© The Author(s) 2018
 Received: 11 May 2018
 Accepted: 20 August 2018
 Published: 29 August 2018
Abstract
Background
The geometry of the vessels is easy to assess in novel 3D studies. It has significant influence on flow patterns and this way the evolution of vascular pathologies such as aneurysms and atherosclerosis. It is essential to develop robust system for vascular anatomy measurement and digital description allowing for assessment of big numbers of vessels.
Methods
A semiautomatic, robust, integrated method for vascular anatomy measurements and mathematical description are presented. Bezier splines of 6th degree and continuity of C3 was proposed and distribution of control points was dependent on local radius. Due to main interest of our institution, the system was primarily used for the assessment of the geometry of the intracranial arteries, especially the first Medial Cerebral Artery division.
Results
1359 synthetic figures were generated: 381 torus and 978 spirals. Experimental verification of the proposed methodology was conducted on 400 Middle Cerebral Artery divisions.
Conclusions
In difference to other described solution all proposed methodology steps were integrated allows analysis of variability of geometrical parameters among big number of Medial Cerebral Artery bifurcations using single application. This allows for determination of significant trends in the parameters variability with age and in contrary almost no differences between men and women.
Keywords
 Geometry of cerebral vessels
 Computeraided assessment of geometry of cerebral vessels
Background
The vessel geometry is a main feature assessed by noninvasive, threedimensional angiographic studies such as computed tomography angiography (CTA) and MRA (magnetic resonance angiography). Its influence on the development of vascular diseases such as aneurysms and atherosclerosis has been widely studied through last decades [1–7]. New possibilities for studying vascular anatomy arise due to rising number of performed angiographies. They are performed in individuals of all ages, with normal or altered anatomy or with vascular diseases such as aneurysms and atherosclerosis [8–10]. Large numbers of cases allow analyses of geometrical differences between age and sex groups and between patients with and without given diseases. Assuming that differences in average values of various groups (sex and age dependent) in the whole population represent general trends in geometry changes in individuals we could better understand this processes and their influence on the vascular pathologies evolution. This kind of strategy is being used in the cosmology for studying galaxy evolution where it is not possible to analyze evolution of particular galaxy. In medicine we still do not have CTA or MRA series covering whole lifetime of any individual. The main requirement for this type of studies if fast and robust system for vascular anatomy measurement and digital description allowing for assessment of great numbers of vessels.
Computer aided, threedimensional studies of vascular anatomy have been done by many authors [4–6]. When analyzing their methods and results we found that all presented systems were combined with separated modules, each for individual step of preprocessing, measurement and calculations. The vessel analyses were robust but also laborintensive. This feature makes the application of these system in the largenumber studies not practical. In our system we reduced the number of modules to two: one for measurements and data storage and one for the data analysis and collating.
The system of the description of the vessel anatomy applied by our team is not new and is based on works of Italian and Irish teams [4–10]. It is based on centerlines of vessels and allows for proper and robust description of different types of the vascular structures (both straight and diverging vessels).
The aim of this study is to propose semiautomatic, robust, integrated and fast system for vascular anatomy measurements and mathematical description studies with large number of subjects. The application of a spline curve, consisting of Bezier segment for centerlines approximation is a new concept. Smoothing and determination of curvature and torsion was a basic concept of central lines (CLs) transformation into mathematical functions. This step was achieved by transformation into Bezier splines of degree 6. The continuity of C3 degree was proposed and distribution of control points was dependent on local radius. The system was primarily used for the assessment of the geometry of the intracranial arteries, especially the first MCA division being an area of interest of our institutions.
The paper is organized as follows: in “Methods” section, useful convention and information are introduced. Then, the proposed methodology steps are presented in details (“Preprocessing the contrast enhanced CT examination”, “Analysis of the vessel cross section”, “Determination of the centerlines”, “Description of the centerlines”, “Calculation of the division zones of the vessel”, “Mathematical analysis of the centerlines”). “Materials and validation method” section describes the data set that was used and validation approach. “Results” section shows the outcomes in a different manner for both synthetic and clinical data. The obtained results are analyzed in reference to other works addressing the subject of vessel geometry in “Discussion” section. The last chapter is “Conclusion” section which summarizes results of the study.
Methods
Vessels topology
Angles and coplanarity index calculation

Division plane (DP)—containing points 0 of each vessel engaged in the division,

Vessel directional vectors (VDV)—vector \(B_r0{}B_r1\) and \(T_r1{}T_r0\).

Division plane normal (DPN)—vector perpendicular to DP,

Branching angle (BA)—between VDV of both branches,

Vessel angle (VA)—between VDV of the trunk and particular branch,

Coplanarity index (CoI) calculated as presented in Eq. (1)
$$CoI_{N} = 1  \left {\frac{{\angle (VDV_{N} ,DPN)}}{{\frac{\pi }{2}}}} \right$$(1)
Curvature and torsion of the curve
Preprocessing the contrast enhanced CT examination
The user sets volume of interests (VOI) in which the remaining measurements were performed. Within the VOI the resolution was raised to 0.2 mm (\(\sim\) threefold typical resolution of CTA study) in each dimension and this step utilizes tricubic value approximation method. The vessels was segmented utilizing thresholding with two low levels of 100 Hounsfield Units (HU) up to 150 HU. Values of both levels were established during initial phase of the project. The lower values were chosen for strictly arterial phase studies, which was defined as enhancement of no more than 50 HU of deep Sylvian veins occupying space around MCA trunk [18, 19].
After segmentation step, calculation of maximum value of minimal distance from model edge (MOBBM) was performed in the VOI. The results of this step values were used by centerline detection algorithms. In the last step of this part of the analysis the user sets one start point on vessel level0 and one endpoint on each vessel level1.
Analysis of the vessel cross section
The purpose of this stage is to automatically orient the measuring plane perpendicular to the long axis of the vessel and measure diameter (minimal, maximal and average) and vessel section area.

Initialization—consists of manually entering the vessel crosssection into a rectangular area.

Finding the geometric center of the vessel \(P_{sc}\).

Finding the border of a vessel—the algorithm uses a specific way of distributing values across the vessel. The central part of the vessel exhibits clearly higher values, which, when away from the center, initially drop slightly, and then, closer to the edge, go down rapidly. In case of neighboring vessels when moving between them, a rise of values is observed after an initial drop subsequent to crossing of the vessels edge. Images in Fig. 6c, d show differences in the result of the algorithm when detection of neighboring vessels was set to on and off respectively.

Smoothing of the border of the vessel \(P_0\)—uses the local average distance from the geometric center of the vessel.
Determination of the centerlines
Calculation of the division zones of the vessel
The process of calculating division zones began with finding the point of decay of the central curves forming the division. The algorithm for each central curve forming the division found the last point for which the minimum distance from the second curve was less than the set value of d (assumed to be 0.2 mm—the spatial resolution of the auxiliary volumes). The split point was calculated as the geometric mean of the two points found. Its value was entered into the partition structure as a \(T_0\).
Based on the location of the points found, the parameters of the partition zone were calculated. Split zone points were also used in further analysis of curvature and torsion (described in the following paragraph). The last steps in the division zone calculation were to determine whether it describes the actual division of the MCA or whether the aneurysm was removed—one of the component curves is the curve to the aneurysm. Of the identified zones, the one for which the split point was the furthest from the origin of the MCA trunk was chosen. This meant that it was a division zone describing the departure of the aneurysm dome from the parent vessel. The second curve of this zone was the curve of the parental aneurysm vessel. In this way, the aneurysm stem was identified. After this step, due to the scope of the analysis, zones other than those describing the actual division of the MCA were rejected.
Description of the centerlines
As in the case of vessel crosssection sections, a number (type of curve) identifying the type of vessel described by it was assigned to further identify each curve. For example, during measurements of MCA divisions, the curve of type 1 was a curve to the aneurysm sac, the curves of successive branches forming the MCA obtain consecutive umbers 2, 3 and so on. Additionally it was possible to mark up to 25 control points along the curve. These point were used during algorithms assessment phase in true CTA studies. This functionality was implemented by introducing editable control points in the view (both MPR and 3D) of the central curve.
Mathematical analysis of the centerlines
The assumption of mathematical analysis of the central curves was to determine correlations between distribution of the curvature and the torsion and evolution of vascular pathologies such as aneurysms and atherosclerosis. In order to calculate the curvature and the torsion of the center lines all curves were approximated with use of the Bezier curves. In order to partially decompose the crankshaft curve, it was decided to use the spline curves consisting of Bezier curves, connected at the site of the division of the vessel. The 6th grade curves and C3 continuity class were selected. The continuity class C3 was needed to ensure the continuity of the torsion function [20]. The 6th grade was a minimum grade allowing the creation of Bspline curves of any number of segments while maintaining this continuity class. The additional effect of the approximation with the parametric curves was the smoothing of the curves.
Materials and validation method
Models were drawn within a measuring volume using a spatial brush. The brush worked within a cube area of 1.5R sides and entered a WSP value in the SWD according to the above pattern if the calculated value was higher than the current one in the voxel. For all synthetic models the following parameters were used: C = 150, a = 10.
Experimental verification of the proposed methodology was conducted on 400 MCA trunk divisions examined with the contrast enhanced CTA. Patients were randomly assigned to this group from set of over 4500 individuals examined in the Computer Tomography Lab of the Second Department of Clinical Radiology Medical University of Warsaw between June 2006 and December 2016. For the measurements step 423 studies were assigned, but 23 (5.4\(\%\)) of them were rejected due to issues concerning study: mixed arteriovenous phase or insufficient contrast enhancement of the vessels.
Results
Validation of the algorithms
Range of parameters used for generation of synthetic figures for validation of the division zones of the vessel algorithm
Model  \(D_{min}\)  \(D_{max}\)  \(D_{step}\)  \(R1_{min}\)  \(R1_{max}\)  \(R1_{step}\)  \(R2_{min}\)  \(R2_{max}\)  \(R2_{step}\) 

Torus  1  5  0.25  5  30  1  Nd  Nd  Nd 
Spiral  1.5  4  0.25  5  30  1  5  9  1 
Patterns that overlap synthetic vessels were rejected, those in which R1 < 2D or R2 < 2D.
Vessel diameter measurements validation
Summarized results of vessel diameter measurements validation
Parameter  Torus N  Torus mean  Torus min  Torus max  Torus std dev 

\(SD_D\)  381  0.03  0.00  0.33  0.051 
\(SD_P\)  381  0.10  0.00  0.74  0.12 
\(RMS_D\)  381  0.12  0.017  1.00  0.21 
\(RMS_P\)  381  0.35  0.039  1.10  0.26 
Parameter  Spiral N  Spiral mean  Spiral min  Spiral max  Spiral std dev 

\(SD_D\)  978  0.058  0.0070  1.29  0.12 
\(SD_P\)  978  0.19  0.032  0.90  0.24 
\(RMS_D\)  978  0.082  0.013  1.43  0.13 
\(RMS_P\)  978  0.31  0.058  1.48  0.29 
The results showed in Table 2 present very good mean \(RMS_D\) and \(RMS_P\) . In case of RMSD almost one order of magnitude lower than study resolution. For the same set of data, the maximal \(RMS_D\) was more than twice bigger than this resolution. The cause of this discrepancy was found in Fig. 8.
This analysis shows very good measurement accuracy for vessels with D > 1.5 mm. Below this value the accuracy of measurement is clearly decreasing, breaking down for D = 1 mm. The result is consistent with the predictions based on the way in which the measurement algorithms and the spatial resolution of the synthetic data volume operate.
Validation of the algorithm for determining the position of the points of the centerlines
Summary of the range of parameters used for generation of synthetic models in the process of validating the accuracy of the center curves
Model  \(D_{min}\)  \(D_{max}\)  \(R1_{min}\)  \(R1_{max}\)  \(R2_{min}\)  \(R2_{max}\) 

Torus  0.6  2  5  25  Nd  Nd 
Spiral  1  2  5  30  8  100 
Calculated RMS of precision of the points of the centerline positioning, separately for helix and torus models
Group  N  \(RMS_{mean}\)  \(RMS_{median}\)  \(RMS_{min}\)  \(RMS_{max}\)  \(RMS_{SD}\) 

Helix  11  0.063  0.062  0.059  0.069  0.0034 
Torus  25  0.061  0.062  0.041  0.076  0.0090 
The obtained data showed very good accuracy of the positioning of the points of the central curves by the developed algorithms. No significant RMS L differences were found for the type and model parameters.
Validation of the vessel bifurcation zone calculation
The ranges of the values of variables of ABZ used during algorithms validation process
Parameter  Min  Max 

A1  15  80 
A2  15  80 
\(A1_{pl}\)  − 15  15 
\(A2_{pl}\)  − 15  15 
\(_wA_n\)  0  0.15 
\(D_T\)  3.5  4.5 
\(D_1\)  2.5  4.2 
\(D_2\)  2.0  3.0 
\(D_{an}\)  2  6 
The results presented as absolute difference between expected and measured value for selected variables are presented in Table 6. Average, absolute difference between real and measured angles were lower than 3.4°. Best results (below 1.9) were observed for dominant branch. When maximal absolute difference were assessed the worst results oscillated around 13° for BA and VA of no dominant branch. For dominant branch it was much better—below 7°.
Results of bifurcation zone validation process as average, median, extremes and standard deviation of absolute differences between expected and measured values
Parameter  N  Average \(\Delta\)  Median \(\Delta\)  Min \(\Delta\)  Max \(\Delta\)  SD \(\Delta\) 

\(\Delta BA\)  70  3.33  2.61  0.010  13.44  2.73 
\(\Delta BA\)  70  3.33  2.61  0.010  13.44  2.73 
\(\Delta VA_{dom}\)  70  1.83  1.47  0.030  6.58  1.43 
\(\Delta VA_{ndom}\)  70  3.27  2.98  0.093  12.93  2.49 
\(\Delta C_oI_{dom}\)  70  0.033  0.027  0.0006  0.12  0.026 
\(\Delta C_oI_{ndom}\)  70  0.039  0.035  0.0002  0.15  0.033 
\(\Delta C_oI_P\)  70  0.0085  0.0068  0.0002  0.054  0.0081 
The measurements process validation on clinical data
Number and percentage of the main three types of MCA division with divide into male and female and percentage of male and female divisions in each type of division
Division type  Number  Percent  Male  Female 

Bifurcation  340  85.0  174 (51.2\(\%\))  166 (48.8\(\%\)) 
Multiple trunks  34  6.0  13 (54.2\(\%\))  11 (45.8\(\%\)) 
Trifurcation  36  9.0  13 (36.1\(\%\))  23 (63.9\(\%\)) 
For further geometrical analysis only dichotomous divisions: BIF and MT were selected which could be properly, geometrically described by previously defined parameters. TRIFs were rejected because in our opinion they could not be simply described as two combined bifurcations and need much more, also topological parameters which exceeds the scope of the presented paper.
The Shapiro–Wilk tests showed normal distribution only for vessel dimensions. The rest of the analyzed parameters presented with distribution different than normal.
Median values and 25–75\(\%\) percentile range of analyzed geometrical parameters for bifurcations and multiple trunk vessels divisions with p values for U Mann–Whitney test comparing both groups
Variable  Median values bifurcations  Median values multiple trunks  p (U Mann–Whitney test) 

BA (degree)  82.8 (68.0–104.7)  91.5 (72.6–99.0)  \(>0.4\) 
\(VA_{dom}\) (degree)  46.0 (33.9–59.1)  48.7 (26.2–62.8)  \(>0.8\) 
\(VA_{ndom}\) (degree)  55.7 (42.1–71.3)  59.9 (51.5–80.0)  \(>0.1\) 
\(CoI_{dom}\)  0.877 (0.785–0.943)  0.886 (0.822–0.974)  \(>0.3\) 
\(CoI_{ndom}\)  0.877 (0.782–0.945)  0.806 (0.764–0.935)  \(>0.3\) 
\(CoI_T\)  0.895 (0.820–0.949)  0.883 (0.803–0.953)  \(>0.8\) 
\(Kmax_{dom}\) (rad/mm)  0.246 (0.173–0.335)  0.258 (0.159–0.449)  \(>0.5\) 
\(Kav_{dom}\) (rad/mm)  0.151 (0.115–0.205)  0.187 (0.097–0.268)  \(>0.2\) 
\(Tav_{dom}\) (rad/mm)  0.321 (0.239–0.454)  0.334 (0.213–0.432)  \(>0.5\) 
\(Kmax_{ndom}\) (rad/mm)  0.308 (0.185–0.450)  0.381 (0.298–0.640)  \(<0.05\) 
\(Kav_{ndom}\) (rad/mm)  0.177 (0.125–0.236)  0.216 (0.163–0.318)  \(<0.05\) 
\(Tav_{ndom}\) (rad/mm)  0.386 (0.281–0.525)  0.440 (0.318–0.549)  \(>0.4\) 
Average values and SD of vessels average diameters for bifurcations and multiple trunks with p values for ttest comparing both groups
Variable  Average values bifurcations  Average values multiple trunks  p (ttest) 

\(D_T\) (mm)  2.9 ± 0.55  2.8 ± 0.51  \(>0.5\) 
\(D_{dom}\) (mm)  2.4 ± 0.41  2.4 ± 0.56  \(>0.6\) 
\(D_{ndom}\) (mm)  1.8 ± 0.41  1.5 ± 0.43  < 0.001 
The significant differences between both groups concern \(Kmax_{ndom}\) and \(Kav_{ndom}\) as well as diameters of nondominant branches. Both Kmax and Kav presented higher values in MT group comparing to BIFs: 0.381 rad/mm (0.298–0.640) vs. 0.308 rad/mm (0.185–0.450) and 0.216 rad/mm (0.163–0.318) vs. 0.177 rad/mm (0.125−0.236) respectively.
DT was higher (2.87 mm ± 0.55 vs. 2.79 mm ± 0.51) and Ddom lower (2.37 mm ± 0.41 vs. 2.41 mm ± 0.56) in BIFs comparing to MTs.
\(D_{ndom}\) was higher (1.8 mm ± 0.41 vs. 1.5 mm ± 0.43) in BIFs comparing to MTs.
Spearman’s correlations coefficients of analyzed variables with age for divisions of type BIF and MT with specified p values of significance test
Parameter  BIF R  BIF p  MT R  MT p 

BA  0.1940  0.0003  0.1836  0.3905 
\(VA_{dom}\)  0.1378  0.0109  0.0209  0.9228 
\(VA_{ndom}\)  0.2092  0.0001  0.1866  0.3825 
\(CoI_{dom}\)  − 0.0285  0.6006  0.1806  0.3985 
\(CoI_{ndom}\)  − 0.1023  0.0596  0.1179  0.5832 
\(CoI_{T}\)  − 0.1604  0.0030  0.3011  0.1528 
\(Kmax_{dom}\)  0.1132  0.0369  0.4172  0.0425 
\(Kav_{dom}\)  0.1526  0.0048  0.4333  0.0344 
\(Tav_{dom}\)  − 0.1489  0.0059  0.1310  0.5419 
\(Kmax_{ndom}\)  0.0979  0.0715  − 0.0191  0.9293 
\(Kav_{ndom}\)  0.0921  0.0898  0.0379  0.8606 
\(Tav_{ndom}\)  − 0.2263  0.0000  − 0.0526  0.8070 
\(D_{T}\)  0.1713  0.0015  − 0.2001  0.3484 
\(D_{dom}\)  0.1856  0.0006  − 0.2358  0.2673 
\(D_{ndom}\)  0.1557  0.0040  − 0.2315  0.2765 
Discussion
Geometrical and mathematical description of vessel geometry was addressed in current literature for a few years. Its linkage to vessel flow was confirmed in many studies [4–7, 24–27]. Most of presented systems, contrary to ours consisted of many separated modules for preprocessing, segmentation, measurement and postprocessing of the data. The differences also include different way of vessel models representation and consequently way of centerlines determination. Now we discuss limitations of presented methods, results of algorithm validation on artificial models and finally on native, medical CTA data.
Main features of vessel segmentation and centerline detection
In the presented study, opposite to works of Italian [11–16, 24] and Irish [17] groups vessels models were defined not as 3D meshes but rather group of voxels of the same index values.
The Italian group after segmentation transformed vessel models into 3D meshes and centerlines were determined utilizing Voronoi diagrams of these meshes. In this technique centerline points in vessel cross section are point maximally distant from model surface [11–17, 28, 29]. This idea of maximal distance from model surface is similar to the assumption in the proposed algorithm. Both techniques should also be resistant to errors due to close relationship between two vessels. In this case both generate two local maxima, each in geometrical center of abutting vessels.
Methodology of centerline detection of the Irish group was not clearly described [17]. Only short information, that CLs were lead over geometrical centers of vessel cross sections, is available. This tactic could cause errors in mentioned regions of abutting vessels.
The presented strategy for segmentation is very simple but sufficient as preprocessing for created algorithm for CLs.
Mathematical analysis of the CLs
The basic concept of CLs transformation into mathematical functions was the smoothing and determination of curvature and torsion. This step was realized by transformation into Bezier splines of degree 6 and continuity of \(C^{3}\) degree [20] was proposed and distribution of control points was dependent on local radius. That was different comparing to the works of Italian and Irish groups.
Transformation into mathematical function allows to avoid indetermination of the curvature and the torsion of CL which could occur in straight vessel segments when using discrete method. The group of OFlynn used transformation into 9th degree polynomials, and Antiga’s group utilizes discrete method, as it implies from documentation of the Vascular Modelling Toolkit.
The first advantage of the first approach is that process of approximation additionally cause curve smoothing and allows for easy and continuous calculation of curvature and torsion. The first approach disadvantage is that polynomial could precisely describe only relative small segment of the vessel. Increasing degree of the polynomial increases the length of approximated CL but causes problems with locality approximation. It is also difficult to dynamically change degree of fitting to avoid over and under approximation where vessel changes it diameter.
In the second approach both problems are solved, because separation of CLs points depends on local radius and CL could have any length. There are two disadvantages: before FrénetSerret frame calculation it is necessary to smooth the curve, which causes known problems with shape preservation, setting proper level of noise filtering [30, 31]. The second, discrete algorithm for Frénet frames calculation could fail on straight portions of vessels: the frame could be indefinable or there could be fast and random fluctuations of its orientation, which causes incorrect calculation of the curvature and torsion.
The approach proposed in this study seems to solve all mentioned problems. Bezier segment degree is chosen to allow construction spline of any number with preservation continuity of third derivative [20]. Approximation within each segment is local. Degree of fitting could be altered by setting control points in distances depending on local vessel diameter.
Vessels diameter measurements
The first limitation was identified during process of measurements validations on artificial models. As presented in Table 2 and in Fig. 8 vessel diameter measurements performs very well down to about 2 * voxel size, which means that measurement of vessels less than this value is not reliable. This issue is a consequence of Nyquist–Shannon sampling theorem. Above this limit presented method reaches subvoxel accuracy of measurements.
Bifurcation zone calculations
Validation of vessel bifurcation zone presented in Table 6, showed very good accuracy of measurements of presented algorithms.
Application of the algorithms to clinical data
The most important step was analysis of clinical data performed on native CTA images of patients. We have observed that the software allows for all measurements to be performed if proper (arterial) or slightly delayed phase of the study was available. In case of mixed arteriovenous phase CLs estimation was possible but required extensive user interactions, so these cases were rejected from calculations.
The percentage representation of various MCA divisions types was similar to presented in anatomical study of Rhoton [19]. The main type of division was BIF consisting of 85\(\%\) of divisions, the second was TRIF (9\(\%\)) and the last frequent MT (6\(\%\)). In the Rhoton’s study these percentages are as follows: 78\(\%\), 12\(\%\) and 10\(\%\) respectively. In each group almost equal numbers of male and female divisions were analyzed (Table 7).
Created application allowed for analysis of 400 MCA divisions. Acquired data and metadata were stored in local database and could be used in the future. In first step, the value distribution analysis, it was showed that almost all parameters had distribution different from normal. In the second step, median and average values of analyzed parameters were calculated for BIF and MT type of division. There were found significant correlations for 9 of analyzed variables with patients age utilizing Spearman correlation coefficients. It was found that BA, VA, \(Kav_{dom}\) and D increases their values with patients age. Contrary \(CoI_{P}\) and \(Tav_{ndom}\) decreases their values. Further analyzes, especially scatter plots showed that additional widening of Kav, CoI and Tav values range with age. This observation is similar to the widening of CCA bifurcation angle with age reported by Thomas [16].
Conclusions
This article presents a concept of integrated system for measurements and analysis of large amount of vascular and anatomical data. The system implements most precise methods for vascular anatomy description based on centerlines. Use of spline curve consisting of Bezier segment for centerlines approximation is a new concept. Unlike other reported solutions all proposed methodology steps were integrated which allows the analysis of variability of geometrical parameters in a large number of MCA bifurcations using only one application This allows for determination of significant trends in the parameters variability with age and differences between MCA division types.
Splines of 6th degree and continuity of C3 degree allow unlimited length approximation of the vessels centerlines. Separation of CL spline control points was set on local vessel radius which assures proper vessel geometry approximation without risk of oversampling. Taking into account the obtained results full automation of proposed methodology allows in the future to analyze large amount of medical records and use artificial intelligence for deeper analysis and support for risk stratification for vascular lesions such as aneurysms and atherosclerosis in any vascular territory.
Declarations
Authors' contributions
Conception and design of the work: JŻ, DS, data analyis: JŻ, medical background, analysis and description, data collection: JŻ, drafting the article: JŻ, GR, DS. All authors read and approved the final manuscript.
Acknowledgements
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
The data that support the findings of this study are available from corresponding author in statistical processed representation in the numerical form.
Consent for publication
Not applicable.
Ethics approval and consent to participate
The method was evaluated based on imaging data of patients from the Second Department of Clinical Radiology after acceptance by the Ethics Committee of the Medical University of Warsaw.
Funding
This research was financial supported by the Directional Research at the Second Department of Clinical Radiology at the Medical University of Warsow, Poland.
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Authors’ Affiliations
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