# Development of a model of the coronary arterial tree for the 4D XCAT phantom.

**ABSTRACT** A detailed three-dimensional (3D) model of the coronary artery tree with cardiac motion has great potential for applications in a wide variety of medical imaging research areas. In this work, we first developed a computer-generated 3D model of the coronary arterial tree for the heart in the extended cardiac-torso (XCAT) phantom, thereby creating a realistic computer model of the human anatomy. The coronary arterial tree model was based on two datasets: (1) a gated cardiac dual-source computed tomography (CT) angiographic dataset obtained from a normal human subject and (2) statistical morphometric data of porcine hearts. The initial proximal segments of the vasculature and the anatomical details of the boundaries of the ventricles were defined by segmenting the CT data. An iterative rule-based generation method was developed and applied to extend the coronary arterial tree beyond the initial proximal segments. The algorithm was governed by three factors: (1) statistical morphometric measurements of the connectivity, lengths and diameters of the arterial segments; (2) avoidance forces from other vessel segments and the boundaries of the myocardium, and (3) optimality principles which minimize the drag force at the bifurcations of the generated tree. Using this algorithm, the 3D computational model of the largest six orders of the coronary arterial tree was generated, which spread across the myocardium of the left and right ventricles. The 3D coronary arterial tree model was then extended to 4D to simulate different cardiac phases by deforming the original 3D model according to the motion vector map of the 4D cardiac model of the XCAT phantom at the corresponding phases. As a result, a detailed and realistic 4D model of the coronary arterial tree was developed for the XCAT phantom by imposing constraints of anatomical and physiological characteristics of the coronary vasculature. This new 4D coronary artery tree model provides a unique simulation tool that can be used in the development and evaluation of instrumentation and methods for imaging normal and pathological hearts with myocardial perfusion defects.

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**ABSTRACT:**Many scientists in the field of x-ray imaging rely on the simulation of x-ray images. As the phantom models become more and more realistic, their projection requires high computational effort. Since x-ray images are based on transmission, many standard graphics acceleration algorithms cannot be applied to this task. However, if adapted properly, the simulation speed can be increased dramatically using state-of-the-art graphics hardware. A custom graphics pipeline that simulates transmission projections for tomographic reconstruction was implemented based on moving spline surface models. All steps from tessellation of the splines, projection onto the detector and drawing are implemented in OpenCL. We introduced a special append buffer for increased performance in order to store the intersections with the scene for every ray. Intersections are then sorted and resolved to materials. Lastly, an absorption model is evaluated to yield an absorption value for each projection pixel. Projection of a moving spline structure is fast and accurate. Projections of size 640 × 480 can be generated within 254 ms. Reconstructions using the projections show errors below 1 HU with a sharp reconstruction kernel. Traditional GPU-based acceleration schemes are not suitable for our reconstruction task. Even in the absence of noise, they result in errors up to 9 HU on average, although projection images appear to be correct under visual examination. Projections generated with our new method are suitable for the validation of novel CT reconstruction algorithms. For complex simulations, such as the evaluation of motion-compensated reconstruction algorithms, this kind of x-ray simulation will reduce the computation time dramatically.Physics in Medicine and Biology 09/2012; 57(19):6193-210. · 2.92 Impact Factor

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Development of a Model of the Coronary Arterial Tree for the 4-D

XCAT Phantom

George S. K. Fung1, W. Paul Segars2, Grant T. Gullberg3, and Benjamin M. W. Tsui1

1Department of Radiology, Johns Hopkins University, Baltimore, MD, USA

2Department of Radiology, Duke University, Durham, NC, USA

3E.O. Lawrence Berkeley National Laboratory, Life Science Division, Berkeley, CA, USA

Abstract

A detailed three-dimensional (3-D) model of the coronary artery tree with cardiac motion has great

potential for applications in a wide variety of medical imaging research areas. In this work, we

first developed a computer generated 3-D model of the coronary arterial tree for the heart in the

extended cardiac-torso (XCAT) phantom, thereby creating a realistic computer model of the

human anatomy. The coronary arterial tree model was based on two datasets: (1) a gated cardiac

dual-source computed tomography (CT) angiographic dataset obtained from a normal human

subject, and (2) statistical morphometric data of porcine hearts. The initial proximal segments of

the vasculature and the anatomical details of the boundaries of the ventricles were defined by

segmenting the CT data. An iterative rule-based generation method was developed and applied to

extend the coronary arterial tree beyond the initial proximal segments. The algorithm was

governed by three factors: (1) statistical morphometric measurements of the connectivity, lengths,

and diameters of the arterial segments; (2) avoidance forces from other vessel segments and the

boundaries of the myocardium; and (3) optimality principles which minimize the drag force at the

bifurcations of the generated tree. Using this algorithm, the 3-D computational model of the

largest six orders of the coronary arterial tree was generated, which spread across the outer surface

of the myocardium of the left and right ventricles. The 3-D coronary arterial tree model was then

extended to 4-D to simulate different cardiac phases by deforming the original 3-D model

according to the motion vector map of the 4-D cardiac model of the XCAT phantom at the

corresponding phases. As a result, a detailed and realistic 4-D model of the coronary arterial tree

was developed for the XCAT phantom by imposing constraints of anatomical and physiological

characteristics of the coronary vasculature. This new 4-D coronary artery tree model provides a

unique simulation tool that can be used in the development and evaluation of instrumentation and

methods for imaging normal and pathological hearts with myocardial perfusion defects.

1. Introduction

The 4-D XCAT phantom (the 3-D phantom with motion) is an extension of the NURBS-

based cardiac-torso (NCAT) phantom (Segars, 2001; Segars et al, 2003; Segars and Tsui,

2009; Segars 2010), which has been widely used in medical imaging research to evaluate

and improve 4-D cardiac imaging techniques (McGurk et al, 2009; Minarik et al, 2010;

Segars et al, 2008; Segars et al, 2009a; Tang et al, 2009). The XCAT phantom provides a

realistic and flexible computer-generated model of the human anatomy and physiology

based on the “Visible Human” anatomical imaging data from the National Library of

Medicine (Ackerman 1999). The 4-D XCAT phantom also models cardiac and respiratory

gfung2@jhmi.edu.

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Published in final edited form as:

Phys Med Biol. 2011 September 7; 56(17): 5651–5663. doi:10.1088/0031-9155/56/17/012.

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patient motions based on 4-D tagged MRI data and 4-D respiratory-gated CT data (Segars,

2001; Segars, 2010).

In clinical medical imaging, it is important to estimate a patient’s potential risk for heart

attack by imaging the coronary arteries to assess for stenoses and myocardial perfusion

defects (MPDs). To achieve that, single photon emission computed tomography (SPECT)

and positron emission tomography (PET) detect MPD by directly imaging the myocardial

perfusion distribution by utilizing appropriate radiotracers. In CT, C-arm, and magnetic

resonance (MR) angiography, stenoses in coronary arteries are detected by utilizing

appropriate contrast agents. To accurately evaluate these imaging methods, it is desirable to

have a phantom that has a realistic and detailed coronary arterial tree, and presence of

stenoses and MPDs, as they are found in representative clinical cases. At present, there is no

realistic and detailed coronary arterial tree phantom available for medical imaging research.

For instance, the current XCAT phantom is only capable of modeling the regional MPD as a

pie-shaped wedge, which is likely not clinically realistic and its location is not related to any

coronary arterial tree or stenosis.

In the 3-D version of the XCAT phantom, i.e., the 4-D XCAT phantom without the cardiac

and respiratory patient motions, high-resolution dual-source CT coronary angiographic

image data of a contrast-enhanced normal human heart was used to define the anatomical

details of the cardiac layers and the initial segments of the coronary arterial tree (Segars et

al, 2007). However, even with this state-of-the-art imaging technique, only the largest two or

three generations of the coronary arterial tree of the beating heart could be delineated.

Further enhancements are required in order to incorporate a more realistic and more detailed

model of the coronary arterial tree.

In this work, an iterative rule-based generation method (Garrity et al, 2003; Kitaoka et al,

1997) that systematically utilized anatomic, morphometric, and physiologic principles was

used to generate a more complete model of the coronary arterial tree for the 4-D XCAT

phantom. A number of computer algorithms that can be utilized to generate the coronary

arterial tree of animal hearts have been proposed (Beard and Bassingthwaighte, 2000;

Kaimovitz et al, 2005; Kaimovitz et al, 2010; Smith et al, 2000). However, they were all

subject to a number of limitations: (1) the anatomy of the myocardium was based on canine

hearts, or was defined by simplified geometrical shapes, such as a cylinder or a spheroid; (2)

the initial coronary arterial trees were based on canine hearts, or were arbitrarily defined; (3)

the morphometric data of the coronary arterial tree of a porcine heart were used directly

without appropriate scaling; and (4) complicated and computationally-intensive optimization

methods were employed to implement the tree generation algorithms. The advantages of our

proposed method include: (1) the anatomic details of the myocardial surfaces and the initial

coronary arterial tree were extracted from a set of state-of-the-art cardiac CT images of a

normal human subject; (2) the statistical morphometric data of the coronary arterial tree of

an average human subject were appropriately scaled from the original porcine heart data;

and (3) a more flexible and faster rule-based iterative generation algorithm, used in our

previous airway tree generation research (Garrity et al, 2003; Kitaoka et al, 1997), was

adopted and modified for the coronary arterial tree generation. One of the potential

applications of the detailed computer generated coronary arterial tree is for the simulation of

the realistic myocardial perfusion defects (Fung et al, 2009) for medical imaging research

based on the spatial distribution of the coronary arterial tree and the locations of the stenoses

in it.

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2. Methods

2.1. Overview

Figure 1 displays a flow chart detailing our methods used to generate the coronary arterial

tree model for the 4-D XCAT phantom. The anatomical data of the coronary arterial tree and

the heart were segmented from a set of state-of-the-art cardiac CT images of a normal

human subject, and the morphometric data of the coronary arterial tree were acquired from

physiology literature (Kassab et al, 1993; Kassab et al, 1994; Kassab and Fung, 1994;

Kassab et al, 1997). The rule-based coronary generation method was developed to extend

the coronary arteries from proximal large branches to distal small branches distributed

evenly over the myocardium. By deforming the generated 3-D coronary arterial tree

according to the motion vector map of the 4-D XCAT phantom (Segars et al, 2009b), the

coronary arterial tree model was extended to 4-D to model different phases of the beating

heart.

A. Segmentation of the Coronary Arterial Tree and Heart Wall Layers—Defining

the anatomical details of the coronary arterial tree and the heart wall was essential for

generating an anatomically realistic model. A set of high spatial and temporal resolution

dual-source cardiac-gated CT images of a normal human subject (provided by Siemens

Healthcare) was used to define the initial segments of the coronary arteries and the

geometric details of the heart wall layers. The coronary arterial tree of the human subject

was right dominant, which approximately representing 70% of the general population.

Variations of the anatomy of the coronary arterial tree of human are common. The CT

dataset consisted of 371 slices of 512×512 transaxial CT images of the entire heart taken at a

0.4mm slice thickness and a pixel size of 0.32mm. A total of 100 frames of CT data over

one heart cycle were reconstructed. The representative transaxial slices of the CT data at the

end-diastolic phase are shown in Figure 2.

The main coronary arteries, including the left anterior descending (LAD) artery, the left

circumflex (LCX) artery, and the right coronary artery (RCA), were manually segmented to

define the initial segments. The heart layers, including the subepicardial and subendocardial

layers, were also manually segmented to define the geometric boundaries of the

myocardium. The 3-D NURBS surfaces were fit to the segmented structures of the coronary

arteries and heart wall layers, as shown in Figure 3. This work was done to assist in the

creation of an improved heart model for the XCAT phantom (Segars et al, 2007).

B. Utilization of Coronary Arterial Tree Morphometric Data—To extend the

segmented tree above, we developed an iterative rule-based generation method based on

morphometric data. Statistical details of the coronary arterial tree were essential for

generating a realistic coronary arterial tree. The statistical morphometric measurements,

including the connectivity matrix, lengths, and diameters of each order of the coronary

arterial tree, venous network, and capillary network, have been thoroughly studied in a

number of excised porcine hearts (Kassab et al, 1993; Kassab et al, 1994; Kassab and Fung,

1994; Kassab et al, 1997). The morphometric data of the coronary arterial tree were used to

provide the statistical guidelines for our arterial tree generation. In the animal studies

mentioned above, the hearts were excised and prepared with the silicone elastomer-casting

method. The measurements were obtained from histological specimens by optical

sectioning, or from the vascular cast. According to the findings on porcine hearts, the

branching of the arterial tree was dominated by bifurcations, and only a very small number

of branches had trifurcation, arcading, or anastomese. In our model, the approximation of

the branching of the coronary arterial tree is based solely on the bifurcations. The arterial

segments were categorized into non-overlapping diameter ranges, from the largest arteries

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(order 11) to the smallest precapillary arterioles (order 1). The diameters, lengths, and

connectivity were defined for each vessel order of each main artery. A total of 11 orders of

vessels were distributed between the coronary capillaries and the aortic sinus for the RCA

and LAD arteries, and 10 orders of vessels were distributed for the LCX artery.

In Table 1, the morphometric measurements of the LAD branch of the coronary artery tree

are shown as an example. The connectivity, C(m,n), in the morphometric data was used to

calculate the probability of a parent segment of order m having a daughter segment of order

n. Thus, the order of one of the two daughter segments at a bifurcation could be

stochastically determined based on the connectivity probability distribution. Given the order

of the daughter segment, the corresponding diameter and length were chosen from a

Gaussian distribution, as defined in Table 1. Since the morphometric measurements in

Kassab’s paper were based on porcine hearts, the diameters and lengths of the vessel

segments were required to be linearly scaled to the normal human coronary arteries (Dodge

Jr et al, 1992) before applying them to human coronary arterial tree generation.

C. Rule-based Computer Generation of a Coronary Arterial Tree

1) Optimal Branching Angle Calculation: The relationship between the diameter of a

branch segment and the flow rate through it was first proposed in (Murray, 1926) as:

(1)

where q is the flow rate, C is a constant that depends on the organ and the fluid, d is the

diameter, and n is the diameter exponent, which is a constant. Under the assumption that

blood flow obeys Poiseuille’s Law (Fung, 1997), Murray’s Law (Murray, 1926) suggests

that for arterial networks with minimized pumping power, the volumetric flow rate in an

arterial segment is proportional to the cube of the diameter, i.e., n=3. In a recent study, it

was suggested that the flow in human coronary arteries is not entirely steady and laminar

(Changizi, 2000). Instead of a constant value n, it was argued that values of n should be

considered within the range [2,3], n∈[2,3]. In this paper, we obtained the exponent n for

each bifurcation by a uniform distributed random number generator with range from 2 to 3.

As shown in Figure 4, for a bifurcation site, a parent branch of diameter d0 has two daughter

branches of d1 and d2:

(2)

Using the above relationship, the equations expressing the diameters of the two daughter

branches in terms of the parent diameter and the flow dividing ratio r, were derived (Kitaoka

et al, 1997):

(3)

(4)

The flow dividing ratio is defined as the ratio of the flow rate q1 through the daughter branch

receiving the larger flow rate, to the flow rate q0 of the parent branch, i.e.:

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(5)

where 0.5<=r<1. The diameter of the daughter branch with the smaller flow rate is less than

that with greater flow rate, i.e., d1 ≥ d2.

As suggested previously (Hacking et al, 1996), a drag force or shear stress acting on the

lumen walls is the most likely major determinant of branch angle in vascular systems.

According to the derivation in (Zamir, 1976), the optimal branching angles for minimum

shear stress were:

(6)

(7)

By substituting eqs. (3), (4) and (5) into eqs. (6) & (7), the optimal branching angles are:

(8)

(9)

However, the two optimal branching angles were not sufficient for defining the directions of

daughter segments in 3-D space. The branching plane and the “combined branching vector,”

vd, which will be defined in subsection 2.1.C.4, are required for positioning the daughter

segments in 3-D space.

2) Self Avoidance Algorithm: To guide the coronary artery tree generation, each existing

vessel segment repels the new vascular segment by growing away from its upstream position

with a strength inversely related to the distance between the existing and the newly

generated vascular segments. The newly generated vasculature tends to grow away from

existing structures and toward more sparsely fed layers (Beard and Bassingthwaighte, 2000).

As the segments were generated from the largest to the smallest orders, the segments of

higher order or previously generated segments of equal order determined the direction of the

newly generated segment. As depicted in Figure 5, to generate a new segment at bifurcation

site xc, the self avoidance vector, vs, was defined as:

(10)

where xi is the upstream position of all previously generated segments of the same or

higher order;

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xc is the upstream position of the current generating segment;

si is the vector pointing from xi to xc, i.e., si = xc − − xi, as shown in Figure 5, sp is the

one pointing from the immediate parent position;

|si| is the absolute distance between xi and xc;

ζ is the “avoidance exponent,” governing the degree to which a new segment avoids

previously generated segments and ζ ≈ 2; and

L is the expected length of the current generating segment.

3) Boundary Avoidance Algorithm: Another important factor in guiding the coronary

artery tree generation is to avoid having arterial segments extend beyond the predefined

ventricle wall. The boundaries of the myocardium must constrain the vasculature within the

volume defined by the ventricle surfaces, which are the subepicardial and the subendocardial

surfaces. As suggested (Beard and Bassingthwaighte, 2000), the larger branches (order 9 or

larger) should be confined to the subepicardial layer while the smaller branches should be

allowed to penetrate into the myocardium.

The myocardial layer is between the subendocardial layer and the subepicardial layer. The

NURBS surfaces of the heart, segmented from CT data, were approximated with triangular

surface patches, as shown in Figure 6(a). The centroid, area, and normal vector were

calculated for each triangular patch. The outward pointing normal vectors of the

subendocardial layer are depicted in Figure 6(b), and the inward pointing normal vectors of

the subepicardial layer are depicted in Figure 6(c). The larger arterial segments (order 9 and

higher) were confined so as to grow along the subepicardial layer. For the smaller arterial

segments, they were allowed to penetrate into the myocardial layer due to the combination

effect from the self avoidance from the larger segments in the subepicardial layer and the

boundary avoidance from the subepicardial layer. To generate a new daughter segment at

bifurcation site xc, we defined the boundary avoidance vector, vb, as:

(11)

where NB is the number of boundary patches within the effective range;

Aj is the area of the boundary patch j;

nj is the (outward or inward) unit normal vector for boundary patch j;

cj is the centroid of the boundary patch j; and

L is the expected length of the current generating segment.

4) Branching Plane Computation: After calculating the self avoidance and the boundary

avoidance vectors, they were linearly combined to form the “combined branching vector,”

vd, based on empirically determined weighting constants cs and cb, defined as:

(12)

where cs + cb =1.

Since the vessel segment extended into the 3-D space, a branching plane was required to

confine the direction of the two daughter segments. The normal vector of the branching

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plane, nb, was determined by the cross product sp × vd with vd, (sp could be slightly

perturbed if sp is parallel with vd) as depicted in Figure 7 and formulated as:

(13)

Then, vd was rotated around the axis nb by θ1 or θ2 in either a clockwise or counter-

clockwise direction to form the daughter branch vectors vd1 and vd2. θ1 and θ2 were the

optimal branching angles as defined in equations (8) and (9). The directions of the daughter

branches were:

(14)

or

(15)

where Rθ1, nbis the rotation matrix which rotates the vector around the axis nb by angle θ1.

The downstream point of the daughter branches was calculated by adding the current

bifurcation position to the daughter branch vectors and scaling them by the corresponding

nominal lengths L1 and L2, which were statistically determined from the morphometric data.

They were defined as:

(16)

and

(17)

Therefore, both xd1 and xd2 were on the branching plane.

5) Generation Rules: Given the morphometry of the coronary arterial tree and fluid

dynamic constraint for bifurcations, an iterative generation algorithm, which included self

avoidance and boundary avoidance algorithms, was used to set rules to guide the growth

directions of the daughter branches at bifurcations. The rules are summarized as follows:

1.

two daughter branches are produced for each parent branch;

2.

the two daughter branches lie in the branching plane;

3.

the flow rate of the parent branch is conserved after branching;

4.

the diameters of the two daughter branches are statistically determined by the

connectivity probability and nominal diameters of the morphometric data under the

flow conservation constraint;

5.

the flow-dividing ratio and branching angles are determined by the diameters of the

two daughter branches;

6.

the branch length of a given order is determined statistically by the “scaled”

nominal length of the morphometric data;

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7.

the larger branches (order 9 or larger) are confined to the subepicardial layer and

the smaller branches are allowed to penetrate into the myocardial layer;

8.

the combined branching vector is determined by the self avoidance and boundary

avoidance algorithms; and

9.

the normal vector of the branching plane is calculated from the parent branch

vector and the combined branching vector.

3. Results and discussions

The generation algorithm successively iterated until the segments of the largest six orders of

a coronary arterial tree were constructed. The generated coronary arterial tree, including the

segments down to orders 11, 10, 9, 8, 7, and 6, with the semi-transparent subendocardial

layers of left and right ventricles at the end-diastolic phase, is shown in Figure 8(a), (b), (c),

(d), (e), and (f), respectively. The six highest order models of the coronary arterial tree that

had diameters down to 120μm, consisted of 2519, 1096, and 2736 segments for the LAD

artery, the LCX artery, and RCA, respectively.

Figure 9(a) shows a 3-D rendering of the highest six orders of the generated coronary

arterial tree and the subendocardial layers of the heart defined at end-diastole. By using the

cardiac motion vectors of the 4-D XCAT phantom defined in our previous publication

(Segars et al. 2009b), the end-diastole model was deformed to end-systole as shown in

Figure 9(b). The thickened myocardium at the end-systolic phase was clearly reflected as the

distance increased between the proximal large arteries and the subendocardial layer, which

was prominent at the lateral wall of the left ventricle in Figure 9. The longitudinal

contraction of the LV toward the apex was largest at the basal region and decreased toward

the apical region. The twisting motion could be observed when the image sequence of the

generated coronary arterial tree at different phases over a cardiac cycle was displayed.

The level of realism of the generated coronary arterial tree was qualitatively evaluated by

two physicians who specialize in cardiac imaging. Their observations were summarized as

follows: (1) the locations and the tracks of the origins, the proximal, middle, and distal

segments of the LAD, LCX, and RCA branches were similar to the typical human heart; (2)

the main sub-branches, such as the first and second diagonal, the obtuse marginal, the acute

marginal, and the posterior descending branches, were located at the typical corresponding

regions of the heart; (3) the decreasing diameter of the branches from proximal to distal sub-

segments were gradual and consistent; (4) the asymmetric bifurcation pattern of the

coronary arterial tree was well matched with the typical human heart; and (5) the proximal

large branches were located on the subepicardial layer while the smaller branches penetrated

into the myocardium. The physicians also noted a number of potential improvements: (1) the

septal branch should penetrate into the septal wall, but not extend over to the right ventricle;

(2) some of the distal LAD branches should stay on the left ventricle, and not extend over to

the right ventricle; and (3) the arterial segments should be smoothly joined together and not

include any sharp turns at the bifurcations. We plan to perform a comprehensive quantitative

evaluation study on bifurcation and tapering properties of the coronary arterial tree model,

and address the potential improvements in future studies.

4. Conclusion

An anatomically, physiologically, and morphometrically realistic 3-D model of the coronary

arterial tree with cardiac motion was generated by merging knowledge from anatomic CT

data, statistical morphometric data of the coronary arterial tree, and the cardiac motion of the

4-D XCAT phantom. This new model provides an important enhancement to the current 4-D

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XCAT phantom by enabling modeling of a more complete coronary arterial tree in

simulation studies for angiography applications. It also has the capability to model MPD for

myocardial perfusion SPECT and PET imaging when a perfusion model is applied to the

generated coronary arterial tree with a predefined stenosis location (Fung et al. 2009). The

model provides a unique tool for simulating both normal and pathological hearts, which can

be used to study and evaluate medical imaging instrumentation and image reconstruction

methods for different cardiac imaging modalities.

Acknowledgments

The authors thank Takahiro Higuchi, MD, PhD and Kenji Fukushima, MD of Johns Hopkins University for his

contributions to the evaluation of the coronary arterial tree model. This research is supported by the NIH research

grants R01 EB 168 and R01 EB 121.

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Figure 1.

Flow chart diagram of the overall detailed coronary arterial tree generation methods

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Figure 2.

Sample transaxial cardiac CT images at the (a) superior, (b) mid, (c) inferior positions of a

human heart and the coronary arteries at end-diastole

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Figure 3.

The (a) front, (b) left, and (c) top views of the segmented major coronary arteries (LAD,

LCX, RCA) and left ventricle (LV) and right ventricle (RV) layers (subendocardial and

subepicardial)

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Figure 4.

The relationship between flow rate, diameter, and branching angle of parent and daughter

vessel segments

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Figure 5.

Self avoidance algorithm

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