

ORIGINAL ARTICLE 

Year : 2022  Volume
: 8
 Issue : 1  Page : 17 

A dynamic statistical cardiac atlas with adjustable shape and motion features
Peng Zhao^{1}, Hussein Y Y. Alghalban^{2}, Yufei Zhu^{2}, Yinbao Chong^{1}, Hongkai Wang^{2}
^{1} Department of Medical Engineering, Xinqiao Hospital, Army Medical University, Dalian, Liaoning, China ^{2} School of Biomedical Engineering, Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian, Liaoning, China
Date of Submission  10Jan2022 
Date of Decision  23May2022 
Date of Acceptance  24May2022 
Date of Web Publication  29Aug2022 
Correspondence Address: Hongkai Wang School of Biomedical Engineering, Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, No. 2 Linggong Street, Ganjingzi District, Dalian, Liaoning China
Source of Support: None, Conflict of Interest: None  Check 
DOI: 10.4103/digm.digm_1_22
Background and Objectives: Digital heart atlases play important roles in computational cardiac simulation and medical image analysis. During the past decades, various heart anatomy models were developed, but they mostly focused on the ventricular part. Recently, a number of wholeheart atlases were developed but they rarely modelled the motion features. This study constructed a wholeheart atlas incorporating dynamic cardiac motion. Materials and Methods: The shape and motion features of the atlas were learnt from a training set of 57 dynamic computed tomographic angiography images including 20 cardiac phases. Intersubject variations of the heart anatomy and motion were incorporated into the atlas using the statistical shape modelling approach. Clinically relevant physiological parameters (e.g., chamber volumes, ejection fraction, and percentage of systolic phase) were correlated with the shape and motion variations using the linear regression approach. The shape and motion pattern of the atlas can be adapted by adjusting the physiological parameters. Results: Quantitative experiments were conducted to measure the anatomical accuracy of the atlas for wholeheart shape reconstruction of different subjects, a mean Dice score of 0.89–0.93 and a mean surface distance of 1.02–1.91 mm were achieved for the four heart chambers, respectively. Conclusions: This atlas provides a novel computational tool with adjustable shape and motion parameters for cardiac simulation research.
Keywords: Cardiac motion modelling, Digital human, Heart atlas, Statistical shape model
How to cite this article: Zhao P, Y. Alghalban HY, Zhu Y, Chong Y, Wang H. A dynamic statistical cardiac atlas with adjustable shape and motion features. Digit Med 2022;8:17 
Introduction   
Digital heart atlases (DHA) are the useful tools for computational simulation of cardiovascular function and diseases.^{[1]} The applications of DHAbased simulations included the studies of the biomechanical and electrophysiology cardiovascular function,^{[2],[3],[4],[5],[6]} clinical diagnostics, and treatment of heart diseases.^{[7]} In the field of medical imaging, DHAs are used for simulating the heart scan process for medical imaging device development.^{[8],[9],[10],[11]} In addition, DHAs are frequently used as anatomical priors for cardiovascular image analysis (e.g., segmentation, biotracer quantification, and disease classification),^{[12],[13]} as well as providing ventricular coordinate systems for regional structure localization^{[14]} and interheart data transfer.^{[15]}
Driven by the numerous needs of computational simulation, digital atlases of the human heart have undergone decades of development. The developmental history can be roughly divided into two stages, i.e., anatomical modelling and motion modelling. Since the 1960s, primitive geometrical (sphereshaped) heart model was used for computerized radioexposure simulation.^{[16]} Later on, the usage of exvivo and invivo human images facilitated the construction of anatomically realistic models,^{[17],[18],[19],[20]} leading to the growth of 3D computational modeling which mostly focused on the left ventricle (LV).^{[21],[22],[23],[24],[25]} Lately, biventricular^{[26]} and wholeheart atlases^{[27],[28]} were developed for more comprehensive anatomical modelling. Eventually, the establishment of large population heart image datasets such as the cardiac atlas project^{[29]} and the UK biobank^{[30]} gave rise to the construction of statistical heart shape models, revealing intersubject variations of heart anatomy and facilitating patientspecific classification of cardiovascular diseases.^{[31],[32]} Especially, the UK Biobank dataset facilitates a phenomewide association study across 26,893 participants revealing the associations of structural and functional phenotypes with sex, age and major cardiovascular risk factors, earlylife factors, mental health and cognitive function, enriching the application of heart atlases based on very large population dataset.^{[33]}
The development of shape modelling approaches also facilitated the extraction of “shape biomarkers” for morphologychanging diseases such as congenital heart disease,^{[34],[35],[36],[37]} while these works mostly focused on local structures like aorta and right ventricle (RV). In the second stage, the improvement of dynamic tomographic imaging techniques enables the acquisition of fourdimensional (4D) images with better spatial and timing resolution, consequently promoted the development of atlases incorporating cardiac motion.^{[38],[39]} The establishment of 4D cardiac models enabled quantification of biventricular motion malfunction (e.g., Takotsubo cardiomyopathy, hypokinesis, and apical ballooning^{[40],[41]}) from cardiac magnetic resonance imaging (MRI), computed tomographic (CT), ultrasound, and nuclear medicine images.^{[3],[9],[10],[11],[34],[42],[43],[44],[45],[46],[47],[48]} Recently, the interactions between ventricular shape and deformation were also modelled with manifold learning approach, endowing the inherent structural and functional correlations into the heart atlas.^{[49]}
Summarizing the development history, DHA have evolved from the preliminary simple geometry ventricular models to wholeheart scale anatomically realistic models. The incorporation of motion features endows the atlases with the ability of heart beating simulation and motion disease quantification. In the era of precision medicine, a heart atlas with adjustable shape and motion is important for the simulation of different subtypes of human hearts.^{[4],[5]} To achieve this goal, the heart atlas should incorporate the following key features:
 The atlas should cover the wholeheart region, including but not limited to the structures of ventricles, atria, myocardium, heart valves, and coronary vessels so that the applications of medical imaging simulation could be conducted based on complete heart anatomy.
 The atlas should incorporate intersubject anatomical variations. This feature is important for simulating different subjects. Ideally, the heart anatomy should be adjusted according to clinically relevant anatomical parameters (such as endsystolic volume [ESV], enddiastolic volume [EDV], myocardium thickness, myocardial fiber distribution, and LV outflow tract width) to mimic the differences of heart shape.
a The atlas should be able to model the variations of dynamic motion features such as stroke volume (SV), ejection fraction (EF), the percentage of systolic phase, etc. The modelling of heart motion will provide reference heart beating patterns for the healthy heart simulation.
Unfortunately, a heart atlas fulfilling all these requirements is still rare. To meet the demand of wholeheart anatomy and motion modelling, this study develops a dynamic heart atlas incorporating intersubject variations of the heart shape and motion learning from a training set of dynamic CT angiography (CTA) images. The atlas can adapt its anatomy and motion pattern according to userspecified physiological parameters. This study mainly focuses on the healthy heart to provide a supporting tool for cardiac anatomy and function simulation.
Materials And Methods   
To model intersubject variations, we use the statistical shape modelling (SSM) method to model the anatomical variations and further extend the SSM method to 4D space for motion modelling. We also adopt the idea of physiological parameter regression to correlate the model parameters with clinically relevant parameters such as chamber volumes, EF, and systolic phase percentage.
Training data collection and preprocessing
This study uses retrospective dynamic CTA images of 57 subjects, including 30 males and 27 females. Each CTA image included 20 frames of volumetric images covering the entire cardiac cycle from the enddiastolic (ED) point to the endsystolic (ES) point. The images were acquired using a 128slice dualsource CT of 120 kV tube voltage and 369–947 mA current, reconstructed with pixel sizes ranging from 0.26 to 0.40 mm and interslice spacing of 0.75 mm. The acquisition protocol covered the scan range from the aortic arch to the heart base, including the entire heart. We collected the images of the subjects diagnosed as asymptomatic according to their medical records, ensuring that the subjects were free of coronary artery disease, arrhythmia, rheumatic heart disease, sinus arrhythmia, hypertensive heart disease, myocardial ischemia, congenital heart disease or aortic valve regurgitation. The subjects have an age range between 26 and 78 years old, body weights between 50 and 85 kg and height between 152 and 175 cm. This study was performed under the ethical approval from the university ethics committees. No patient identification information has been used in this research or presented in this article.
As a preprocessing step, the cardiac structures including LV, RV, left atrium (LA), right atrium (RA), and myocardium were segmented from the training images in [Figure 1]a. We used an automated multiatlas segmentation (MAS) method based on the online opensource software.^{[13]} This method uses 20 atlas images (provided by the software) with human expert segmented label maps. Each atlas image was registered to the target CTA image, obtaining a nonlinear spatial transform which maps the label maps from the atlas space to the target image space. Afterwards, an atlas selection procedure was conducted to select the registered atlases with similar intensity appearances to the target image. The label maps of the selected atlases were fused into a single label map as the segmentation result. This label fusion step was performed in a voxelbyvoxel manner using the local image patch similarity as the weighting factor of the voting. Details of the MAS method are referred to.^{[13]}  Figure 1: An exemplar case of wholeheart segmentation and point correspondence calculation. (a) Automated segmentation obtained using the MAS method (shown in axial and coronal slices). Different segmented regions are displayed with different pseudo colors. (b) The registered template mesh of the point correspondence result (shown in the same slices of a). (c) Surface rendering of the registered template mesh of b.
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Using the MAS method, each frame of the dynamic CTA image was segmented as a single volumetric image. To facilitate subsequent construction of the heart atlas, the surface point correspondences between different training subjects need to be established. We used an onlinepurchased finedetailed heart anatomy model (https://www.zygotebody.com) as the shape template. The template was in the form of polygonal surface meshes, thus our atlas was represented as the surface mesh structure. The shape template was registered to the segmented meshes using a markerbased shape matching (MSM) method.^{[50]} Firstly, a human expert of cardiac anatomy was invited to define a series of anatomical landmarks (including LV and RV apex points, the centers of the aortic, bicuspid and tricuspid valves and other epicardial landmarks^{[51]}) in both the template and each CTA image, and then the automatic registration based on Bspline transform was conducted to simultaneously minimize the landmark pair distances and the chamber region overlapping ratio. We noticed that registering the entire heart template to the subject CTA image using a single global nonlinear transform cannot model the sliding motion between adjacent subcardiac structures. Therefore, we separately register each subcardiac structure (i.e., the left and RVs and atria) using the MSM method instead of registering them together. This registration procedure was used to match the template to the individual shape of each training subject in all the time phases, resulting in pointwise shape correspondence across different subjects at different time points based on the same number of surface vertices and the same vertex connection topology. Finally, the registered template meshes were further finetuned by the human expert using an interactive surface editing method^{[52]} to ensure that the registered template accurately fit the image edges of individual cardiac structures. In this way, the minor intersection of adjacent chamber surfaces caused by separate chamber registration was also corrected. The result of interactive surface finetune is shown in [Figure 1]b and [Figure 1]c.
The shape and motion of the heart are modelled in a decoupled manner. We consider the ED point as the starting phase of the cardiac cycle and only performed shape modelling of the ED phase. For motion modelling, we calculated the motion vectors from the ED phase shape to other phases and extended the SSM method to 4D to model the motion vectors. Finally, we obtained both the shape model and the motion model which can be adjusted via the model parameter.
Shape modelling
To model intersubject variations of the ED phase shape, we used the statistical shape model approach based on the classical point distribution model.^{[53],[54],[55]} Starting from the registered template meshes of the ED shape, the Procrustes analysis was performed to remove intersubject differences of translation, rotation, and scaling. Afterwards, the ED shape of each subject was represented as a 3kelement shape vector X_{i} =(x_{i}_{,1}, y_{i}_{,1}, z_{i}_{,1}, x_{i}_{,2}, y_{i}_{,2}, z_{i}_{,2}, x_{i}_{,3}, y_{i}_{,3}, z_{i}_{,3})^{T}, where i is the subject index, k is the number of mesh vertices (k = 10,081 in this study). Principal Component Analysis (PCA) was performed to the shape vectors of all training subjects, obtaining the eigenvectors {ϕ_{i}}, i = 1,…, n as shape variation modes and the eigenvalues {λ_{i}}, i = 1,…1, n as the variances of each mode, where n is the number of eigenvectors and eigenvalues obtained PCA. The shape variation modes {ϕ_{i}} were ordered according to their magnitudes of variance, i.e., λ_{1} ≥ λ_{2} ≥... λ_{n}, thus mode 1 corresponds to the largest intersubject variance, mode 2 corresponds to the second largest variance, and so on. The variance percentage ratio of the mode i was defined as The SSM is expressed as an average shape plus a linear combination of the various modes.
where S(a) is an instance of the shape model represented as a shape vector (x_{1}, y_{1}, z_{1}, x_{2}, y_{2}, z_{2,} … x_{k}, y_{k}, z_{k})^{T}. S̄ is the mean shape vector of all the training subjects. ϕ = [ϕ_{1}, …ϕ_{n}] is a 3k × n matrix of the shape modes (eigenvectors) obtained via PCA. A = [a_{1}, … , a_{n}]^{T} Is the shape parameter of the model.
Different values of a correspond to different instances of the ED shape. The anatomically plausible shape of the heart can be generated by adjusting the shape parameter a_{i} within a plausible range For individualized heart shape modelling, given vector S of a specific patient, his/her shape parameter a can be calculated by solving Equation 1, i.e.
where ϕ^{+} is the pseudo inverse of ϕ. Individualized heart shape modelling can also be achieved by automatically fitting the SSM to a patient CTA image, using the Active Shape Model (ASM) approach which automatically optimizes the shape parameter a and the model's position, orientation and scaling to fit the model surface vertices to the image edges.^{[53]}
Motion modelling
To model the heart motion, we extended the SSM method to 4D. Let be the shape vector of training subject i at a time frame j. The motion vector of the frame j is defined as which is a 60 kelement (i.e., 3k × 20) vector and . However, because there are intersubject variations of heart shape, size position and direction, we cannot directly use M_{i} for cardiac motion modelling. We need to map the motion vectors of each subject into the average heart atlas space and model heart motion in the atlas space. We adopt the method of Rao et al.^{[56]} to achieve this purpose. The mean shape mesh (i.e., S̄) of the ED phase SSM is filled into a volumetric label image L̄, in which the voxel values are the labels of different cardiac structures. The ED shape mesh of each training subject i is also filled into a volumetric label image L_{i} with the same voxel labels of L̄. L̄ is registered to each L_{i} using the volumetric image registration method based on BSpline transform and mean squared difference similarity metric.^{[56]} This registration results in a nonlinear spatial transform T_{i} which maps the mean atlas into the image space of subject i. Let J_{i} be the spatial Jacobian of transform T_{i}, then motion vector M_{i} is mapped into the atlas image space as
where v and i are the indices of mesh vertex and time frame, respectively. M_{i} (v,t) denotes an infinitesimal motion vector of vertex v at frame t, and and is the mapping result of M_{i} (v,t) in the atlas space. However, since the magnitude of the motion vectors obtained via shape subtraction (i.e., ) is much larger than infinitesimal, Rao et al. developed a stepwise integration method to approximate the mapping of large deformation vectors and proved the effectiveness of this method for heart motion mapping.^{[56]} Using Rao's integration method, we obtained the mapped motion vectors of each training subject into the mean atlas space as . PCA was performed on of all training subjects. Different from the shape model, the Procrustes analysis was not performed on the motion vectors because it is not appropriate to normalize the magnitudes and directions of the motion vectors. The motion model represented as
where M (b) is an instance of the motion model which represented as M(b) = (M^{1}(b),…, M^{20})), M̄ is the mean motion vector of all the training subjects, Ψ = [Ψ_{1}, … Ψ_{n}] is a 60k × n matrix of the motion modes obtained via PCA. b = [b_{1}, … , b_{n}]^{T} is the motion parameter of the model. Different values of b correspond to different instances of heart motion. Similar to Equation 2, the individualized motion parameter of a specific patient can be calculated by b = ψ^{+}(M  M̄).
Combining the shape and motion models, a 4D instance of the heart atlas can be characterized by the combination of shape and motion parameters, i.e.
x(a,b) = (s(a) + M^{1}(b), … , S(a) + (M^{20}(b))) (5)
Physiological parameter regression
A limitation of the classical SSM method is that the model parameters (a and b) do not have intuitive physiological meanings. To build a heart model for computation simulation, it is desirable that the model parameters correspond to clinically relevant parameters (e.g., LV volume, SV, EF, etc.). To meet this requirement, we use a linear regression method to correlate the SSM parameters with clinically relevant parameters. This method is adopted from^{[57]} for human body shape modelling but is extended to 3D heart shape modelling and 4D heart motion modelling in this study. The key idea is to introduce a regression matrix R that maps the clinically relevant parameters to the SSM parameters. Taking the shape model in [Figure 1], as an example, the mapping is represented as
R_{s} P_{s} = a (6)
Where R_{s} is a n × (l+1) regression matrix of heart shape, p_{s} = [p_{1}, … p_{l}, 1]^{T} is the set of clinically relevant shaperelated parameters such as heart chamber volume and myocardium thickness, the last item 1 of p_{s} is the constant bias of the linear equation system. a = [a_{1}, a_{2}, … a_{n}]^{T} is the shape parameter defined in Equation 1. R_{s} can be calculated based on the training set.
where A is a n × m matrix of the shape parameters of m training subjects, i.e., each column of A stores the shape parameter of one training subject calculated using Equation 2. P_{s} is a (l + 1) × m matrix of the clinically relevant parameters in which each column stores the P_{s} of one training subject. P_{s}^{+} is the pseudo inverse of P_{s}. Similarly, the regression between the motion parameters and the clinically relevant parameters can be represented as.
R_{m}P_{m} = b (8)
where R_{m} and P_{m} are the regression matrix and clinically relevant parameters of heart motion, respectively. Based on R_{s} and R_{m}, we can calculate the parameters a and b of any individual patient given his/her clinically relevant parameters P_{s} and P_{m} using Equation 6 and 8 then generate the individualized 4D heart model X (a,b) using Equation 2, 4 and 5.
In this study, we define the clinically relevant shape and motion parameters as p_{s} = [EDV_{LV}, EDV_{RV}, EDV_{LA}, EDV_{RA}, 1]^{T} and p_{m} = [EF, P_{sys}, 1], where EDV_{LV}, EDV_{RV}, EDV_{LA}, EDV_{RA} means the ED volumes of the LV, RV, LA, and RA, EF denotes the EF of LV and P_{sys} represents the percentage of LV systolic phase. Precisely, the volume of a heart chamber is calculated (n_{vox} × v_{vox}),where n_{vox} is the number of segmented voxels of the chamber in the ED phase CTA image and v_{vox} is the volume of a single voxel. EF follows the standard definition of LV EF, i.e., . Where ESV_{LV} means the ESV of LV. P_{sys} is defined as , where T_{sys} and T_{cyc} are the time duration of systolic phase and the entire cardiac cycle, respectively. We chose the chamber volumes as shape parameters since they are fundamental geometrical measurements.^{[58]} Variation of the heart motion is created by adjusting the EF and P_{sys} values. Equations 8, 4 and 5 are used to generate the heart shape of all time frames, obtaining a series of heart shapes at different time points. It is worthy to emphasize that our method is not limited to the above definitions of p_{s} and p_{m}. Other clinically relevant parameters (e.g., LV myocardium thickness and SV) can be incorporated as well. We only use the above parameters to demonstrate the feasibility of this method and leave the incorporation of other parameters towards more specific clinical applications in future studies.
Results   
To validate the usability of our atlas for cardiovascular anatomy and function simulation, we first inspect the mean shape and motion of the atlas and then conducted preliminary simulations of solid mesh modeling and pseudo medical image generation. Afterward, experiments of the model parameter adjustment were conducted to test the ability of intersubject variations modeling and the accuracy of shape modeling is also validated.
Mean shape and motion
The mean shape (X̄) of the constructed atlas is demonstrated in [Figure 2]. [Figure 2]a illustrates the surface rendering of the atlas, in which different substructures are rendered with different colors, i.e., red for LV, pink for LA, light blue for RV, dark blue for RA, gray for the myocardium, red and pink for the coronary arteries and veins, respectively. Unlike many exiting atlases which model the internal myocardium surface with a smooth spherical shape, our atlas exhibits fine details of the papillary muscles and interventricular septum which facilitates more realistic simulation of myocardium structures. We also convert the surface mesh into a solid mesh using the iso2 mesh software^{[59]} to illustrate its potential for biomechanical/electrophysiology finite element modelling. The solid mesh is rendered in [Figure 2]b, in which the LV, LA, RV, RA and the myocardium are rendered in dark blue, brown, light blue, purple and gray colors, respectively. Furthermore, to demonstrate the atlas's value for medical imaging research, the mean shape model is incorporated into our previousdeveloped human torso phantom^{[55]} and filled into a pseudo volumetric CT image. Voxels inside different anatomical structures were filled with the corresponding CT values obtained from a real CTA image. [Figure 1]c shows the axial, coronal, and sagittal sections of the pseudoCT image, which visually assemble the sections of a real CTA image. Without loss of generalizability, the atlas can also be filled with other voxel intensity values to simulate other imaging modalities like magnetic resonance (MR), ultrasonic, and nuclear medicine imaging. However, to mimic more realistic texture features of the cardiac medical images, more sophisticated simulation process (e.g. Xray projections and photon Monte Carlo simulation)^{[9]} should be conducted to generate highquality pseudo images for machine learning data augmentation, imaging apparatus design, etc.  Figure 2: The mean shape of the heart atlas. (a) Surface rendering of the mean heart shape, in which different substructures are rendered with different colours, i.e. red for left ventricle, pink for left atrium, light blue for right ventricle, dark blue for right atrium, gray for the myocardium, red and pink for the coronary arteries and veins, respectively. (b) A tetrahedral solid mesh converted from the mean shape surface mesh, the left ventricle, left atrium, right ventricle, right atrium and the myocardium are rendered in dark blue, brown, light blue, purple and gray colours, respectively. (c) A pseudocomputed tomographic image generated by filling the mean shape mesh, the axial, coronal and sagittal section slices are shown from the left to the right.
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[Figure 3] illustrates the mean motion (S̄+M̄) learned from the training set. We calculated the fourchamber volumes of all 20 frames and plotted the timevolume curves of them. Five frames of the heart cycle were evenly selected, and the corresponding heart models are displayed on top of the curves. To give a clear view of the heart chamber motion, only the ventricles, atrial, and myocardium are rendered. From [Figure 3], synchronized contraction, dilation of the ventricles, and atria visualized.  Figure 3: The mean motion of the heart atlas. The models corresponding to five selected phases are rendered on top of timevolume curves of the four heart chambers. LA: Denote left atrium, LV: Left ventricle, RA: Right atrium, RV: Right ventricle.
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Anatomical variation assessment
[Figure 4]a illustrates the shape variations corresponding to the changes of the first four parameters a_{1}, a_{2}, a_{3}, and a_{4} within the anatomically plausible range In addition to the varied shapes, the mean shape of the atlas is also displayed in the top row of the figure. It can be observed that a_{1} corresponds to the size change of the entire heart, a_{2} is related to the shape changes of all four chambers, a_{3} and a_{4} controls the variation of heart size in long axis and short axis directions, respectively. It can be seen that all the variations in [Figure 4]a correspond to simultaneous shape changes of all four chambers. [Figure 4]b demonstrates the shape variations corresponding to individual chamber volume changes. We input the increased and decreased chamber volumes and calculated the corresponding SSM shape parameter using Equation 4 and then generate the model shape using Equation 1. As [Figure 4]b illustrated, our method yielded volumerelated shape changes of an individual chamber without affecting other chambers. Moreover, it is observed that the volumerelated shape change is not a simple scaling of the chamber size. Our method learned anatomically realistic shape deformation patterns related to the volume change. This feature is desirable for realistic simulation of diseaseinduced shape abnormalities.  Figure 4: The anatomical variations of the shape model. (a) Shape changes corresponding to the variations of SSM shape parameters. (b) Single chamber shape changes resulted from the variation of clinically relevant physiological parameters. LA: Denote left atrium, LV: Left ventricle, SSM: Statistical shape modelling, RA: Right atrium, RV: Right ventricle.
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Motion variation assessment
Similar to the observation of shape variations, we adjusted the motionrelated parameters to observe the motion variations of the atlas. Variation of the heart motion is created by adjusting the EF and P_{sys} values based on the mean motion. Based on the adjusted values, Equations 6 and 3, were used to generate the heart shape of all time frames (setting a = 0), and then the chamber volumes of the generated shape were calculated. [Figure 5] shows the results of EF and P_{sys} adjustments. For clarity of the figure, we only display the heart shape of two or three frames. The decrease of EF yielded higher ESV of LV, simulating the degeneration of systolic function. Similarly, the increase in P_{sys} resulted in a shortening of the diastolic phase, which is related to heart rate change.  Figure 5: The variations of heart motion generated by adjusting the EF and P_{sys} parameters. LV: Left ventricle, EF: Ejection fraction.
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Validation of anatomy modeling accuracy
Since our objective is to construct a computational model of the heart anatomy, it is necessary to evaluate how accurate the atlas mimics realistic heart shape of different people. We used the SSM to reconstruct the ED phase shape of all the testing subjects and computed the shape reconstruction accuracy. The validation was conducted using the leaveoneout strategy, i.e., each time one of the 57 subjects were selected as the testing image and the other 56 images were used to construct the atlas model. The ground truth of this validation comes from the expert segmentation, i.e., the MAS results refined by human expert manual adjustment (as described the method section). We used the ASM method to deform and fit the ED phase shape model to the segmentation of each subject. Afterward, the shape reconstruction accuracy was measured using the Dice coefficient (reflecting the volumetric recovery ratio of the four heart chambers) and averaged surface distance (ASD) between the deformed model and the ground truth, i. e.,
where R_{M} and R_{S} represent the region of the registered model and ground truth segmentation, respectively. · means the number of voxels in the region and ∩ means the intersection of two regions. n_{M} and n_{s} are the number of points on the model surface and the ground truth segmentation surface, respectively. d_{i} is the nearest distance from the i^{th} points of the model surface to ground truth surface, and d_{j} is the nearest distance from the j ^{th} point of the ground truth surface to the model surface.
In addition to the shape reconstruction accuracy, we also measured the shape correspondence accuracy of the model construction step. The rationale behind this additional measurement is that the shape reconstruction ability is inherently affected by the shape correspondence accuracy of model construction. We also used the human expert segmentation of the ED phase images as the ground truth. The correspondence accuracy was also measured via Dice and ASD. To differentiate the correspondence accuracy and the shape reconstruction accuracy, we use Dice_{crsp} and ASD_{crsp} to denote the correspondence accuracy and use Dice_{recon} and ASD_{recon} and to denote the shape reconstruction accuracy. The measurement results are reported in [Table 1]. It can be observed that the correspondence accuracy is slightly better than the reconstruction accuracy. The correspondence accuracy achieves mean Dice coefficients over 0.90 and the mean ASD values below 1.89 mm for all four chambers. Notably, RA has a mean dice above 0.93 and a mean ASD below 0.98 mm. Although LV has the maximum mean dice coefficient (0.94) among all chambers, it's mean ASD is worse than the other chambers mainly because of the inaccurate registration of detailed trabeculated structures of the papillary muscles. For shape reconstruction, the mean dice coefficients are over 0.90 for large structures such as the left and RVs and above 0.89 for smaller structures such as the left and right atria. This result indicates that the reconstructed shape has a good volumetric overlapping with the ground truth segmentation. Regarding the surface distance metric, all four chambers have mean ASD below 1.90 mm, and the RA achieves a small ASD of 1.02 mm, meaning that the atlas is accurate for modelling cardiac structures of 1–2 mm scale. The reconstructed shape has a good volumetric overlapping with the ground truth segmentation. Regarding the surface distance metric, all four chambers have mean ASD below 1.90 mm, and the RA achieves a small ASD of 1.02 mm, meaning that the atlas is accurate for modelling cardiac structures of 1–2 mm scale.  Table 1: Correspondence accuracy and shape reconstruction accuracy of the shape model
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Comparison with existing wholeheart atlases
So far, a large body of research has been conducted to build DHA, but most of them focused on the ventricular part. To the limit of our knowledge, only a few groups have developed wholeheart atlases. As a result of this, we give a comparison between our atlas and several existing whole heartatlases, including the statistical heart model by Hoogendoorn et al.,^{[60]} the volumetric mesh wholeheart model contributed by a series of authors from the euHeart project,^{[61]} the dynamic heart model included in the 4D Extended cardiactorso (XCAT) wholebody phantom developed by Segars et al.^{[9]} and the wholeheart atlas for 4D hemodynamics modeling.^{[62]} [Table 2] compares the features of our atlas and these wholeheart atlases. For quantitative comparison, because these atlases are not publicly available, we only obtained the quantitative results (averaged ASD of the four chambers) of atlas registration to individual patient images from their papers. Although the quantitative comparison is based on different test datasets and the registration error is influenced by the test image voxel resolution, it can be observed that our atlas achieved comparable level of registration accuracy to these atlases.
In [Table 2], the term “statistical modeling” means that the atlas was constructed using SSM method as described by Equation 1 of this article. Both the Hoogendoorn's et al. atlas and the euHeart project atlas were constructed in this way to incorporate intersubject shape variations, but they did not model the motions of the heart. Moreover, they did not correlate the PCA modes with clinically relevant anatomical parameters; therefore, they do not support shape adjustment according to physiological parameters (e.g., chamber volume and myocardium thickness). The euHeart atlas provides the feature of individualized myocardium thickness measurement by fitting the atlas with individual patient image, but this feature was not incorporated as a shape controlling function. The XCAT heart phantom was constructed based on a single MR image of each gender, and it cannot model intersubject anatomical variations. The XCAT heart phantom has the feature of motion modelling and motion adjustment according to heart rate, SV, and percentage of different heart beating phases, but such adjustment was performed in an artificial manner rather than based on a statistical learning approach like ours. Finally, it is worth to mention that the XCAT heart phantom and the euHeart atlas offered the function of simulating perfusion defects, scars, and vessel plagues, while such a function is not provided by our atlas and the Hoogendoorn's et al. atlas.
Discussion   
As a computational tool for cardiac biophysical simulation, the developed atlas can be used for various healthrelevant applications, including but not limited to the following aspects. This wholeatlas atlas is useful for electrophysiology simulation of current waveform propagation through the entire heart. With the ability to adjust the motion pattern, the atlas also offers the possibility to model the dynamic mechanical reaction to electrophysiological signals. The modeling of different beating patterns also facilitates the simulation and synthesis of different heart sound styles, enabling the generation of a synthesis heart sound library to assist the development of heart sound diagnosis algorithms and apparatuses.^{[63]} Besides, by adjusting the heart shape, the atlas can be used for studying the relationship between cardiac anatomy and kinetic features such as local stresses and contractile forces. In recent years, the development of deep learning techniques dramatically enhanced the ability of computerassisted cardiac disease diagnosis. The training of neural networks requires a large amount of clinical data which is difficult to collect. This atlas may be used for the synthesis of pseudo cardiovascular training images^{[64],[65]} for artificial intelligence algorithm training. Last but not least, this atlas is built for healthy people; it can be used for cardiac function simulation of healthy subjects in sports or aerospace tasks. In the future, we will make efforts to improve the atlas for more specific applications.
Our experimental results demonstrated that the linear regression approach was effective for extracting shape and motion variation patterns related to clinically relevant parameters. We correlated the shape variations with chamber volumes and correlated the motion variations with EF and P_{sys} The shape regression was performed for the deformation vectors distributed on the entire heart. Although the chamber volume regression mainly affects the deformation of a single chamber, we notice correlated shape changes of the adjacent chambers. This is because natural shape change seldom happens to only one chamber. By performing the regression analysis on the wholeheart scale, we realistically modeled the synchronized shape changes of different chambers and prevented the chambers from deforming into each other. Similar to the shape variation regression, the motion of the heart is also regressed with clinically relevant parameters. As shown in [Figure 5], the adaption of EF and P_{sys} resulted in corresponding changes of heart beating function and the LV timevolume curve. In future, we will incorporate more physiological parameters into the atlas, including myocardium thickness, detailed phases of the cardiac cycle (e.g., isovolumetric systolic and dilation phases), the cardiac output rate, EDV, and ESV normalized by body surface area, etc. Moreover, since heart rate is correlated with P_{sys}, we only modelled the P_{sys}related motion changes in this study. For the next step, we plan to model dedicated motion pattern variation related to heart rate change. The features of ventricle compliance, myocardial elasticity and blood pressure will also be incorporated if the measurement of these features can be obtained via advanced approaches like ultrasound elastography imaging and implant blood pressure sensors. Currently, the atlas is only constructed for normal subjects. In the next step we will also add training images of several clinically popular heart diseases to increase the ability of diseasespecific simulation. The function of lesion simulation is also planned to be added, learning the idea from the euHeart atlas and the XCAT heart phantom.
The correspondence of mesh vertices has to be established before the SSM. However, the accuracy of point correspondence computation affects the accuracy of shape modelling. As reported in [Table 1], the MSM registration method reaches mean ASD values between 0.98 and 1.89 mm for the heart chambers. This result means that shape details below the scale of 2 mm, such as the trabeculated structures of the papillary muscles and the cardiac valves, cannot be perfectly modeled. As a compromise, we mapped the detailed structures like the coronary vessels, valves and papillary muscles from the purchased heart template to each training subject. Therefore, the shape model only includes the template shape of these structures but cannot accurately present their shape variations. The purpose of introducing these structures into the atlas is to facilitate the simulation applications (e.g., heart sound simulation) which require the existence of complex shape details of the papillary muscles and the valves. The motion modelling also suffers from the correspondence accuracy limitation, i.e., the obtained motion vectors could only reflect the movement of large cardiac structures. Therefore, we only regress the motion vectors with the physiological parameters reflecting large scale features (e.g. EF and P_{sys}) but could not model the movements of small structures like the valves. Recently, shape modelling without point correspondences^{[35],[37],[66],[67],[68]} shed light on overcoming the correspondence accuracy problem, but so far these methods were not used for the entire heart scale. We still need to develop more precise shape modelling method for the whole heart. Moreover, limited by the CT imaging, the atlas does not include the myocardial fibers which are useful for myocardium biomechanical and electrophysiological analysis. The twisting motion component of left ventricular contraction cannot be reflected from the CTA images either. For the next step, we plan to collect diffusion tensor imaging data and tagged MRI images to acquire the myocardial fiber details and the twisting motion components. We also plan to add myocardial strain computation function to the atlas. The myocardial strain is a volumetric measure, but the current atlas is surfacebased. To compute the myocardial strain, we will add evenly spaced internal vertices into the myocardium mesh and track their motions during the heart beating.
Conclusions   
Based on a training set of dynamic CTA images, this study constructs a dynamic digital heart atlas incorporating intersubject variabilities of heart anatomy and cardiac motion. To the extent of our knowledge, this atlas for the first time incorporates wholeheart physiologicalparametercorrelated variations of heart shape and motion. Our future study will focus on incorporating more physiological parameters (e.g., myocardial strain), adding more detailed heart structures and applying the atlas to clinical studies.
Acknowledgments
The author would like to thank Yu Wang and Dongdong Deng for their precious suggestions on biomechanical and electrophysiological modelling. We also appreciate the anonymous reviewers for their helpful remarks to increase the quality of this paper.
Financial support and sponsorship
This work was supported in part by the National Key Research and Development Program No. 2016YFC0103101 and 2016YFC0103102, the general program of National Natural Science Fund of China No. 81971693 and 61971445, the Youth Program of the National Natural Science Foundation of China No. 81401475, the Fundamental Research Funds for the Central Universities Funding of Dalian University of Technology (No. DUT20YG122) and Liaoning Key Lab of IC and BME System Funding.
Conflicts of interest
There are no conflicts of interest.
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[Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5]
[Table 1], [Table 2]
