## Three-dimensional reconstructions from low-count SPECT data using deformable models

Optics Express, Vol. 2, Issue 6, pp. 227-236 (1998)

http://dx.doi.org/10.1364/OE.2.000227

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### Abstract

We demonstrate the reconstruction of a 3D, time-varying bolus of radiotracer from first-pass data obtained by the dynamic SPECT imager, FASTSPECT, built by the University of Arizona. The object imaged is a CardioWest Total Artificial Heart. The bolus is entirely contained in one ventricle and its associated inlet and outlet tubes. The model for the radiotracer distribution is a time-varying closed surface parameterized by 162 vertices that are connected to make 960 triangles, with uniform intensity of radiotracer inside. The total curvature of the surface is minimized through the use of a weighted prior in the Bayesian framework. MAP estimates for the vertices, interior intensity and background count level are produced for diastolic and systolic frames, the only two frames analyzed. The strength of the prior is determined by finding the corner of the L-curve. The results indicate that qualitatively pleasing results are possible even with as few as 1780 counts per time frame (total after summing over all 24 detectors). Quantitative estimates of ejection fraction and wall motion should be possible if certain restrictions in the model are removed, e.g., the spatial homogeneity of the radiotracer intensity within the volume defined by the triangulated surface, and smoothness of the surface at the tube/ventricle join.

© Optical Society of America

## 1. Introduction

1. W.P. Klein, H.H. Barrett, I. W. Pang, D.D. Patton, M.M. Rogulski, J.J. Sain, and W. Smith “FASTSPECT: Electrical and mechanical design of a high resolution dynamic SPECT imager,” Conference Record of the 1995 IEEE Nucl. Sci. Symp. & Med. Imaging Conf. (IEEE, Los Alamitos, 1996), **Vol. 2**, pp. 931–933. [CrossRef]

2. K.M. Hanson, “Bayesian reconstruction based on flexible prior models,” J. Opt Soc. Amer. A **10**, pp. 997–1004 (1993). [CrossRef]

## 2. The data

### 2.1 FASTSPECT

1. W.P. Klein, H.H. Barrett, I. W. Pang, D.D. Patton, M.M. Rogulski, J.J. Sain, and W. Smith “FASTSPECT: Electrical and mechanical design of a high resolution dynamic SPECT imager,” Conference Record of the 1995 IEEE Nucl. Sci. Symp. & Med. Imaging Conf. (IEEE, Los Alamitos, 1996), **Vol. 2**, pp. 931–933. [CrossRef]

**H**, that is measured by passing a small volume element of radiotracer throughout the volume being imaged, and measuring the response of every detector pixel to that source, producing an enormous amount of information, even when compressed to take advantage of the sparsity of the matrix (150 MB of disk space after compression). The system matrix used in this article was obtained by passing a [5mm]

^{3}volume element through a 43×57×39 grid. The system matrix is noisy since only a finite number of counts are obtained for each location of the source. Given enough patience and time, though, this noise could presumably be made as low as is needed.

**H**. If information is available concerning attenuating material between the radiotracer distribution and the pinholes, it can also be incorporated into

**H**, and this was done for the

**H**used to analyze the data discussed in this article. A phantom was defined, using simple geometric shapes, that approximately characterizes the torso surrounding the artificial heart system described in the next section. The geometric shapes used to simulate the lungs were assigned an attenuation length of 0.07 cm

^{-1}and the geometric shapes used to simulate the rest of the system were assigned an attenuation length of 0.15 cm

^{-1}(the attenuation length of water at 140 keV). For each (object voxel, pinhole center) pair, the exponential attenuation e

^{-(0.15*L1+0.07*L2)}was computed, where L1=distance in cm along the line connecting the object voxel to the pinhole center which passes through the water, and L2=distance in cm along the same line which passes through the simulated lung material. The attenuation factor was multiplied by the

**H**entries for that voxel and all of the detector pixels associated with that pinhole to produce new

**H**entries.

### 2.2 The imaged object and raw data

## 3. The Bayesian estimation problem

### 3.1 The object model

**x**. All 486 values in

**x**are estimated for each time. This parametric model is converted to a non-parametric uniformly-voxellated grid,

**f**, by setting the value of each voxel in

**f**to the fraction of that voxel that is contained within the volume described by the triangulated surface. We assume that the radiotracer is homogeneously distributed throughout the volume enclosed by the triangulated surface, so that only a single parameter, I, is needed for the activity level. Thus, I

**f**is a uniformly-voxellated grid with intensity I interior to the triangulated surface defined by the vertices

**x**. Obviously,

**f**is functionally dependent on the parameters,

**f**=

**f**(

**x**).

8. M. Kass, A. Witkin, and D. Terzopolous, “Snakes: active contour models,” Int. J. Comput. Vis. , pp. 321–331 (1988). [CrossRef]

### 3.2 The measurement model

**f**, and produces a set of predicted data elements,

**g**, in this case a Poisson rate for each detector pixel. The measurement model might, in general, contain many components. For example, in an x-ray radiographic system, one would expect to have line integral transformations (parallel- or divergent-beam), convolutions, exponential point transforms, etc.

**H**, along with a single additive constant that models the background (the same background constant is used for all 24 detectors), so that

**g**=I

**Hf**+s. The background, s, must be jointly estimated from the data along with the object model parameters. Much more complicated 2D spatial field models exist within the BIE, but the very low number of background counts probably make more complex models impossible to estimate well. One extension that is worth investigating is a different background constant for each detector that varies in time in a plausible way.

**Hf**dramatically since only a few percent of the voxels in the object model are nonzero. Simply skipping over

**Hf**for values of

**f**that are zero allows us to calculate

**Hf**in about 300 msec on a DEC Alpha 500/500. The same speedup applies in the adjoint direction, wherein derivatives are propagated according to the chain rule in the direction opposite to the path that transforms object parameters into predicted data [9].

### 3.3 The probability model

**g**=I

**Hf(x)**+s) and count values equal to the raw data:

**k**. The dependence of the predicted data

**g**on the underlying parameters

**x**, I, and s, is understood. Note that because the background constant in the measurement model is additive, the predicted detector pixel rates

**g**can never be equal to or less than zero as long as the activity level and background level are greater than or equal to zero, which makes the form in (2) well-defined, and makes the derivative of ϕ(

**x**,I,s) w.r.t.

**g**(and ultimately

**x**) well-behaved.

### 3.4 The estimation problem

**x**, that produce the maximum

*a posteriori*(MAP) probability, or the minimum minus-log posterior:

10. K.M. Hanson, R.L. Bilisoly, and G.S. Cunningham, “Kinky tomographic reconstruction,” Proc. SPIE (SPIE, Bellingham, 1996), **Vol. 2710**, pp. 156–166. [CrossRef]

11. D.J.C. MacKay, “Bayesian interpolation,” Neural Comput. **4**, pp. 415–447 (1992). [CrossRef]

^{ML}(α) = arg min

_{I}[ϕ(

**x**,I,s)], and s

^{ML}(α) = arg min

_{s}[ϕ(

**x**,I,s)].

## 4. Results

## 5. Discussion

10. K.M. Hanson, R.L. Bilisoly, and G.S. Cunningham, “Kinky tomographic reconstruction,” Proc. SPIE (SPIE, Bellingham, 1996), **Vol. 2710**, pp. 156–166. [CrossRef]

## 6. Conclusion

**H**matrix is applied at each iteration.

## Acknowledgments

## References and links

1. | W.P. Klein, H.H. Barrett, I. W. Pang, D.D. Patton, M.M. Rogulski, J.J. Sain, and W. Smith “FASTSPECT: Electrical and mechanical design of a high resolution dynamic SPECT imager,” Conference Record of the 1995 IEEE Nucl. Sci. Symp. & Med. Imaging Conf. (IEEE, Los Alamitos, 1996), |

2. | K.M. Hanson, “Bayesian reconstruction based on flexible prior models,” J. Opt Soc. Amer. A |

3. | Y. Bresler, J.A. Fessler, and A. Macovski, “A Bayesian approach to reconstruction from incomplete projections of a multiple object 3D domain,” IEEE Trans. Pattern Anal. Mach. Intell. |

4. | P.C. Chiao, W.L. Rogers, N.H. Clinthorne, J.A. Fessler, and A.O. Hero, “Model-based estimation for dynamic cardiac studies using ECT,” IEEE Trans. Med. Imaging |

5. | X.L. Battle, G.S. Cunningham, and K.M. Hanson, “3D tomographic reconstruction using geometrical models,” Proc. SPIE (SPIE, Bellingham, 1996), |

6. | X.L. Battle, G.S. Cunningham, and K.M. Hanson, “Tomographic reconstruction using 3D deformable models,” to appear in Phys. Med. Biol., 1998. [CrossRef] [PubMed] |

7. | |

8. | M. Kass, A. Witkin, and D. Terzopolous, “Snakes: active contour models,” Int. J. Comput. Vis. , pp. 321–331 (1988). [CrossRef] |

9. | K.M. Hanson and G.S. Cunningham, “A computational approach to Bayesian inference,” M.M. Meyer and J.L. Rosenberger, eds., |

10. | K.M. Hanson, R.L. Bilisoly, and G.S. Cunningham, “Kinky tomographic reconstruction,” Proc. SPIE (SPIE, Bellingham, 1996), |

11. | D.J.C. MacKay, “Bayesian interpolation,” Neural Comput. |

12. | P.C. Hansen and D.P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. |

**OCIS Codes**

(100.3010) Image processing : Image reconstruction techniques

(110.6960) Imaging systems : Tomography

**ToC Category:**

Focus Issue: Tomographic image reconstruction

**History**

Original Manuscript: December 9, 1997

Published: March 16, 1998

**Citation**

Gregory Cunningham, Kenneth Hanson, and X. Battle, "Three-dimensional reconstructions from low-count SPECT data using deformable models," Opt. Express **2**, 227-236 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-2-6-227

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### References

- W.P. Klein, H.H. Barrett, I. W. Pang, D.D. Patton, M.M. Rogulski, J.J. Sain, and W. Smith OFASTSPECT: Electrical and mechanical design of a high resolution dynamic SPECT imager,O Conference Record of the 1995 IEEE Nucl. Sci. Symp. & Med. Imaging Conf. (IEEE, Los Alamitos, 1996), Vol. 2, pp. 931-933. [CrossRef]
- K.M. Hanson, OBayesian reconstruction based on flexible prior models,O J. Opt Soc. Amer. A 10, pp. 997- 1004 (1993). [CrossRef]
- Y. Bresler, J.A. Fessler, and A. Macovski, OA Bayesian approach to reconstruction from incomplete projections of a multiple object 3D domain,O IEEE Trans. Pattern Anal. Mach. Intell. 11, pp. 840-858 (1989). [CrossRef]
- P.C. Chiao, W.L. Rogers, N.H. Clinthorne, J.A. Fessler, and A.O. Hero, OModel-based estimation for dynamic cardiac studies using ECT,O IEEE Trans. Med. Imaging 13, pp. 217-226 (1994). [CrossRef] [PubMed]
- X.L. Battle, G.S. Cunningham, and K.M. Hanson, OTomographic reconstruction using 3D deformable models,O to appear in Phys. Med. Biol., 1998. [CrossRef] [PubMed]
- http://www.radiology.arizona.edu/~fastspec/detectors.html
- M. Kass, A. Witkin, and D. Terzopolous, OSnakes: active contour models,O Int. J. Comput. Vis., pp. 321- 331 (1988). [CrossRef]
- K.M. Hanson and G.S. Cunningham, OA computational approach to Bayesian inference,O M.M. Meyer and
- K.M. Hanson, R.L. Bilisoly, and G.S. Cunningham, OKinky tomographic reconstruction,O Proc. SPIE (SPIE, Bellingham, 1996), Vol. 2710, pp. 156-166. [CrossRef]
- D.J.C. MacKay, OBayesian interpolation,O Neural Comput. 4, pp. 415-447 (1992). [CrossRef]
- P.C. Hansen and D.P. OOLeary, OThe use of the L-curve in the regularization of discrete ill-posed problems,O SIAM J. Sci. Comput. 14, pp. 1487-1503 (1993). [CrossRef]
- Y. Bresler, J.A. Fessler, and A. Macovski, OA Bayesian approach to reconstruction from incomplete projections of a multiple object 3D domain,O IEEE Trans. Pattern Anal. Mach. Intell. 11, pp. 840-858 (1989).

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