503 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 46, NO. 3, JUNE 1999

A Realistic Spline-B ased Dynamic Heart Phantom

W. Paul Segars', David S . Lalush"2, Member, IEEE, and Benjamin M. W. T ~ u i " ~ , Member, IEEE 'Department of Biomedical Engineering and 'Department of R.adiology

The University of North Carolina at Chapel Hill, Chapel Hill, NC

Abstract We develop a realistic computerized heart phantom for use

in medical imaging research. This phantom is a hybrid of realistic patient-based phantoms and flexible geometry-based phantoms. The surfaces of heart structures are defined using non-uniform rational B-splines (NURBS), as used in 3D computer graphics. The NURBS primitives define continuous surfaces allowing the phantom to be defined at any resolution. Also, by fitting NURBS to patient data, the phantom is more realistic than those based on solid geometry. An important innovation is the extension of NURBS to the fourth dimension, time, to model heart motion. Points on the surfaces of heart structures were selected from a gated MRI study of a normal patient. Polygon surfaces were fit to the points for each time frame, and smoothed. 3D NURBS surfaces were fit to the smooth polygon surfaces and then a 4D NURBS surface was fit through these surfaces. Each of the principal 4D surfaces (atria, ventricles, inner and outer walls) contains approximately 200 control points. We conclude that 4D

An appropriate primitive that will permit the kind of modeling needed is 1 he non-uniform rational B-spline (NURBS) [8,9]. The NIJRBS is widely used in 3D computer graphics to describe three-dimensional surfaces. A NURBS surface (Fig. 1) is a bidirectional parametric representation of an object. Points on a NURBS surface are defined by two parametric variables, U and v, usually representing longitude and latitude respectively.

A 3D NURBS surface of degree p in the U direction and degree q in the v direction is defined as a piecewise ratio of B- spline polynomials by the following function [8,9].

n m

O l u l l Olvll

NURBS are an efficient and flexible way to describe the heart and other anatomical objects for a realistic phantom. The P,,] represent the control points defining the surface,

w,,] are weights that deterinine a point's influence on the shape of the surface, and N,Ju) and NJIq(v) are the nonrational B- I. INTRODUCTION

used for simulation of medical imaging data. Existing phantoms can be divided into two general classes: pixel-based and geometry-based. Pixel-based phantoms [ 1, 21 are generally based on patient data, and are thus fixed to a particular anatomy, so that study of the effects of anatomical variations is limited. Also, they are fixed to a specific resolution; generation of the phantom at other resolutions requires interpolation. Geometry-based phantoms [3,4], on the other hand, allow for anatomical variation and generation at

[3,4], but will not be as realistic as pixel-based phantoms. The 4D Dynamic MCAT phantom [3, SI was developed to

permit study of the effects of variations on cardiac SPECT images [61 and study of gated SPECT [5 , 71. Unfortunately, the ellipsoid-based heart in the current version of the Dynamic MCAT is limited in its ability to model realistic heart motion and the effects of disease states on heart motion. Thus, a phantom that will model these effects must rely on a different set of primitives to describe the surfaces of the heart and other organs.

We develop a realistic computerized heart phantom to be basis functions defined On the knot vectors

(2) ,..., q p - l , 1 , . e . , 1 w

P+l

(3) Y+'

where ' = ' + p + an'' = + 4 + '. The knot vectors

directions, U and v. Nonxero length spans in the knot vectors determine the surface segments. Each segment of the surface has its own distinct set o f basis functions; therefore, the knot vectors are instrumental in determining the basis functions.

multiple resolutions. They can be made reasonably realistic contain breakpoints Of the piecewise surface the two

Figure 1 : 3D NURBS surface.

This work was supported by grant number CA39463 from the United States National Cancer Institute. Its contents are solely the responsibility of the authors and do not necessarily represent the official views of the National Cancer Institute.

0018-9499/99$10.00 0 1999 IEEE

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Using the knot vector U in Eq. (2) as an example, the B- spline basis functions are defined as a step function for a degree p of 0 and as a linear combination of two (p-1) basis functions for a degree greater than zero as indicated by the following equations.

(4)

The definition is the same in the v direction only using the V knot vector and degree q.

Because it is a continuous surface, a NURBS permits representation of a surface at any resolution. The principal disadvantage of the NURBS, compared to 3D geometric shapes, is that a NURBS requires considerably more parameters to describe a surface. As shown in the equations above, a set of control points, knot vectors, and a set of basis functions are required to describe a NURBS surface whereas a simple mathematical relationship is all that is required for geometric surfaces. Another disadvantage of NURBS is the mathematical complexities they introduce. More complicated algorithms are required to calculate intersections, produce solids, and to handle shared surfaces. With the increase in complexity, however, comes a vast increase in flexibility to model biological shapes more realistically. Also, NURBS can be altered easily via affine and other transformations to model variations in anatomy among patients. The particular transformation needs only to be applied to the set of control points through a matrix multiplication.

We develop NURBS surfaces modeling the heart based on gated MRI scans and incorporate these into a new 4D Dynamic MCAT phantom. The result is a hybrid between the realism of pixel-based phantoms and the flexibility of geometry-based phantoms. The new NURBS heart models details not present in the previous MCAT heart, including non- uniformities of the heart wall. An important innovation is the application of the NURBS in a fourth dimension, time, to model heart motion and permit generation of the heart at any temporal resolution, as well as any spatial resolution.

Figure 3: SURFdriver surface reconstmction program. MRI slices (Fig. 2) are contoured (left) then reconstructed into smooth polygon models (right). Models of the inner heart chambers are shown with right atrium (RA), right ventricle (RV), left atrium (LA), and left ventricle (LV) indicated.

II. METHODS A gated MRI cardiac data set (Fig. 2) of a normal patient

was used as the basis for this phantom. The patient study had fifteen time frames over a complete heart cycle. At each time frame, the transaxial MR images had a pixel size of 1.56 mm x 1.56 mtn and a thickness of 1 cm.

The software program SURFdriver [lo], a surface reconstruction application, was used to display these time- frame images and choose points on the surfaces of the principal parts of the heart: right and left ventricles and atria and the outer surface of the heart (Fig. 3). Due to the coarse thickness (lcm) of the MRI transaxial image slices, special processing had to be done to select points to define the boundary between the atria and the ventricles and to define the apex and base of the heart. The boundary points and endpoints were selected using the known anatomy of the heart as a guide. These points were then reconstructed into smooth polygon models using SURFdriver’s rendering engine and smoothing function.

An interactive application was developed in OpenGL (Fig. 4) to perform similar tasks as SURFdriver with the exception that it uses established methods [8,9] for generating relatively smooth 3D NURBS surfaces that pass through the specified points. The smooth polygon models were used as input to this application, and NURBS surfaces were fit to them. The surfaces were adjusted by affine transformation and manipulation of control points so that chamber volumes fit average human heart volume curves used for the dynamic MCAT phantom.

The 3D NURBS surfaces for each of the fifteen time frames were converted to solids through a scan conversion algorithm and were combined to make fifteen phantoms over the cardiac cycle. The 3D surfaces were then used as the base

Figure 4: NURBS surface reconstruction program. Slices of the polygon models in Fig. 3 are contoured (left) then reconstructed into NURBS surfaces (right). Models of the inner heart chambers are shown with right atrium (RA), right ventricle (RV), left atrium (LA),

Figure 2: One short axis slice through fifteen frames of gated M E data. Points were chosen from all slices, and then polygon models were generated from the points.

and left ventricle (LV) indicated.

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surfaces for extending the NURBS in the time dimension in the same way that 2D contours are extended into 3D surfaces, a process called “skinning”. In 2D skinning, section curves are blended together to form a 3D surface (Fig. 5). The blend direction is either longitudinal or latitudinal. In our process of 3D skinning, the NURBS surfaces are blended in the time direction to form a 3D object changing in time.

We define skinning as follows. Let S,(u,v) be a set of c(0, ..., C) NURBS surfaces as defined in equation 1. Each surface is fit through a set of (n+l) x (m+l) data points Qk,l, k = 0 ,..., n and 1 = 0 ,..., m, as selected using the application in Figure 4. The data points Qk,, should not be confused with the control points P,,] of the surface. For the time direction a degree t is chosen. Then (m+l) x (n+l) NURBS curve interpolations are done across the data points of the surfaces yielding a time based set of data points. Each point is now defined by a time curve. A surface is constructed for a particular time by calculating the set of data points from the time curves and then performing surface interpolation through them. The final result is a set of 4D NURBS surfaces, each surface defined by a set of approximately 200 4D control _ - points.

III. RESULTS

. - I

ellipsoid-based model (left) of the exterior surface of the heart. The NURBS model contains 196 control points. Right atrium (RA), right ventricle (RV). left atrium (.LA). and left ventricle (LV) are indicated.

. - , ellipsoid-based model (left) of the exterior surface of the heart. The NURBS model contains 196 control points. Right atrium (RA), right ventricle (RV), left atrium (LA), and left ventricle (LV) are indicated.

Figures 6 and 7 display the result of using 3D NURBS to represent the surface of the heart from the gated MRI cardiac dataset compared to the current ellipsoid-based version of the MCAT phantom. From these images, it is clear that the NURBS surfaces have the potential to realistically model organ shapes, more so than models based on geometric solids.

Fig. 8 illustrates representative short-axis slices of the NURBS phantom as compared to the MCAT phantom for one time frame. Non-uniformities in the outer chamber walls are modeled by the 3D NURBS surfaces; this is in contrast to the uniform walls inherent in the MCAT simulation. The MCAT also fails to realistically model the atria and their outer chambers as well as the 3D NURBS surfaces. The NURBS phantom models the space between the atria occupied by the aorta and pulmonary artery. The two phantoms produce comparable results for the inner chambers of the ventricles, which are the main area of concern in cardiac SPECT.

Figure 9 shows the volume curves for the NURBS phantoms over the cardiac cycle. The calculated volumes illustrate a cycle very similar to that of the dynamic MCAT phantom based on average human heart volume curves. The differences in the curves can be attributed to the fact that the MCAT is an average while the NURBS model is from one single patient. The flow or area under the volume curve from baseline was calculated as being constant from the left and

right ventricles in the ”URBS phantoms. The muscle mass was also found to be C o n ~ ~ e d within 32%.

Figure 5: 2D Skinning. vertically to create a 3D surface.

Horizontal section curves are splined

Figure 8: Short-axis s1ic;es of the NURBS phantom (top) and MCAT phantom (bottom) at end-diastole. Atria are shown as the top four slices in each phantom. The ventricles are shown as the remaining eight slices. Dead space between the atria in the NURBS phantom represtnts the space occupied by the aorta and oulmonarv arterv.

506

200

150 3.

g100 - P

50

I I I I 0 0.2 0.4 0.6 0.8 1

Time

Figure 9: Ventricular volume curves of the NURBS and MCAT phantoms.

Representative points in the cardiac cycle of the result of splining the fifteen time frames in the time dimension are shown in Fig. 10. 3D renderings for the inner chambers of the atria and ventricles together are shown at the two key points, end-systole and end-diastole. Thirty-one time frames were produced from the fifteen base frames. The thirty-one frames illustrate a smoother contracting and twisting motion of the myocardium as compared to the original fifteen time frames.

IV. CONCLUSIONS We have developed a realistic computerized heart phantom

that models heart motion based on 4D B-splines. The NURBS surfaces that define the phantom can be used to generate a pixelized phantom of any given resolution. This pixelized

can be used as input to a Monte-Carlo simulation. It ossible to explicitly compute path lengths across the heart model given an arbitrary ray for Monte-Carlo n. The calculation of the intersection of a ray with a urface, however, is more complex than it would be

for a simple geometry-based phantom. Methods developed for making these models will be used t

s to the heart model using more high-resolution data. Other motions of the heart such as upward creep and motion due to breathing will also be

explored. We conclude that 4D NURBS are an efficient and flexible way to describe the heart and other anatomical objects for a realistic phantom.

V. ACKNOWLEDGMENTS The authors would like to thank Wei-Chih Hu, Ph.D. of the

Department of Biomedical Engineering, Chung Yuan University, Taiwan, ROC for providing the gated MR data set used in the study.

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Figure 10: Surface renderings of interior wall of the atria and ventricles at end-systole (left) and end-diastole (right). These surfaces were generated from 4D NURBS and represent two out of the thirty-one generated time frames.