Skip to main content
SpringerLink
Log in

Robust recovery of myocardial kinematics using dual \(\mathcal {H}_{\infty }\) criteria

  • Published:
Multimedia Tools and Applications Aims and scope Submit manuscript

Abstract

Accurate estimation of myocardial motion can help to better understand the pathophysiological processes of ischemic heart diseases. However, because of partial and noisy image-derived measurements on the cardiac kinematics, the performance of model-based motion estimation relies heavily on the assumption of noise distribution on the measurement data. While existing studies of model-based motion estimation have often adopted the \(\mathcal {H}_{2}\) criteria (e.g. least square error) based on fixed model constraints from mathematical or mechanical nature, we present a robust estimation framework with an adaptive biomechanical model constraint using dual \(\mathcal {H}_{\infty }\) criteria for the first time. Comparing to the minimization of average gaussian error in \(\mathcal {H}_{2}\) criteria, our \(\mathcal {H}_{\infty }\) criteria aims to minimize the maximum error without regarding the noise distribution. In this work, our dual estimation framework consists of two iterative \(\mathcal {H}_{\infty }\) filters: One filter for the kinematics estimation and another one for the elasticity estimation. At each time step, heart kinematics is estimated with sub-optimal fixed material parameters, followed by an elasticity property recovering given these sub-optimal kinematic state estimates. Such coupled estimation processes are iteratively repeated as necessary until convergence. We evaluate the performance of dual estimation framework on synthetic data, cine image sequences, and human image sequence. Our dual estimation framework shows a higher tolerance of noise than the conventional extended Kalman filter. The results obtained by both synthetic data of varying noises and magnetic resonance image sequences demonstrate the accuracy and robustness of the proposed strategy.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

References

  1. Abaqus homepage: https://www.3ds.com/products-services/simulia/products/abaqus/

  2. Abd-Elmoniem K, Stuber M, Prince J (2008) Direct three-dimensional myocardial strain tensor quantification and tracking using zHARP. Med Image Anal 12(6):778–786

    Article  Google Scholar 

  3. Axel L, Dougherty L (1989) MR imaging of motion with spatial modulation of magnetization. Radiology 171(3):841–845

    Article  Google Scholar 

  4. Bathe K, Wilson E (1976) Numerical methods in finite element analysis. Prentice-Hall, New Jersey

    MATH  Google Scholar 

  5. Benayoun S, Ayache N (1998) Dense non-rigid motion estimation in sequences of medical images using differential constraints. Int J Comput Vis 26(1):25–40

    Article  Google Scholar 

  6. Britton OJ, Bueno-Orovio A, Van Ammel K, Lu HR, Towart R, Gallacher DJ, Rodriguez B (2013) Experimentally calibrated population of models predicts and explains intersubject variability in cardiac cellular electrophysiology. Proc Nat Acad Sci 110(23):E2098–E2105

    Article  Google Scholar 

  7. Burger M, Modersitzki J, Ruthotto L (2013) A hyperelastic regularization energy for image registration. SIAM J Sci Comput 35(1):B132–B148

    Article  MathSciNet  MATH  Google Scholar 

  8. Cheng TM, Savkin AV, Celler BG, Su SW, Wang L (2008) Nonlinear modeling and control of human heart rate response during exercise with various work load intensities. IEEE Trans Biomed Eng 55(11):2499–2508

    Article  Google Scholar 

  9. Chiang P, Cai Y, Mak KH, Zheng J (2013) A B-spline approach to phase unwrapping in tagged cardiac mri for motion tracking. Magn Reson Med 69 (5):1297–1309

    Article  Google Scholar 

  10. Cochet H, Bullier E, Gerbaud E, Durieux M, Godbert Y, Lederlin M, Coste P, Barat JL, Laurent F, Montaudon M (2013) Absolute quantification of left ventricular global and regional function at nuclear MPI using ultrafast CZT SPECT: initial validation versus cardiac MR. J Nuclear Med 54(4):556–563

    Article  Google Scholar 

  11. Cox M, Driessen N, Bouten C, Baaijens F (2006) Mechanical characterization of anisotropic planar biological soft tissues using large indentation: a computational feasibility study. J Biomech Eng 128(3):428–436

    Article  Google Scholar 

  12. Declerck J, Denney T, Ozturk C, O’Dell W, McVeigh E (2000) Left ventricular motion reconstruction from planar tagged MR images: a comparison. Phys Med Biol 45(6):1611–1632

    Article  Google Scholar 

  13. Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the em algorithm. J Royal Stat Soc 39(1):1–38

    MathSciNet  MATH  Google Scholar 

  14. Denney T, Prince J (1994) 3D displacement field reconstruction from planar tagged cardiac MR images. In: Proceedings of the IEEE Workshop on Biomedical Image Analysis, pp 51–60

  15. Didinsky G, Pan Z, Basar T (1995) Parameter identification for uncertain plants using \(\mathcal {H}_{\infty }\) methods. Automatica 31(9):1227–1250

    Article  MathSciNet  MATH  Google Scholar 

  16. Gao H, Allan A, McComb C, Luo X, Berry C (2014) Left ventricular strain and its pattern estimated from cine CMR and validation with DENSE. Phys Med Biol 59(13):3637–3656

    Article  Google Scholar 

  17. Gao Z, Xiong H, Liu X, Zhang H, Ghista D, Wu W, Li S (2017) Robust estimation of carotid artery wall motion using the elasticity-based state-space approach. Med Image Anal 37:1–21

    Article  Google Scholar 

  18. Gilliam A, Epstein F, Acton S (2009) Cardiac motion recovery via active trajectory field models. IEEE Trans Inf Technol Biomed 13:226–235

    Article  Google Scholar 

  19. Glad T, Ljung L (2000) Control theory. Taylor & Francis, London

    Google Scholar 

  20. Glass L, Hunter P, McCulloch A (1991) Theory of heart. Springer-Verlag, New York

    Book  Google Scholar 

  21. Gopal S, Otaki Y, Arsanjani R, Berman D, Terzopoulos D, Slomka P (2013) Combining active appearance and deformable superquadric models for LV segmentation in cardiac MRI. In: SPIE Medical Imaging, p. 86690G. International Society for Optics and Photonics

  22. Halmos PR, Sunder VS (1978) Bounded integral operators on L 2 spaces. Springer, Berlin

    Book  MATH  Google Scholar 

  23. Hameeteman K, Klooster R, Selwaness M, Lugt A, Witteman J, Niessen W, Klein S (2013) Carotid wall volume quantification from magnetic resonance images using deformable model fitting and learning-based correction of systematic errors. Phys Med Biol 58(5):1605–1623

    Article  Google Scholar 

  24. Han L, Hipwel J, Tanner C, Taylor Z, Mertzanidou T, Cardoso J, Ourselin S, Hawkes D (2012) Development of patient-specific biomechanical models for predicting large breast deformation. Phys Med Biol 57(2):455–472

    Article  Google Scholar 

  25. Honarvar M, Sahebjavaher R, Salcudean S, Rohling R (2012) Sparsity regularization in dynamic elastography. Phys Med Biol 57(19):5909–5927

    Article  Google Scholar 

  26. Jahanzad Z, Liew Y, Bilgen M, McLaughlin R, Leong C, Chee K, Aziz Y, Ung N, Lai K, Ng S, Lim E (2015) Regional assessment of LV wall in infarcted heart using tagged MRI and cardiac modelling. Phys Med Biol 60 (10):4015–4031

    Article  Google Scholar 

  27. Janzc RF, Kubertc BR, Moriarty TF, Grimma AF (1974) Deformation of the diastolic left ventricle —II. Nonlinear geometric effects. J Biomech 7(6):509–516

    Article  Google Scholar 

  28. Kerwin W, Prince J (1999) The kriging update model and recursive space-time function estimation. IEEE Trans Signal Process 47(11):2942–2952

    Article  MathSciNet  MATH  Google Scholar 

  29. Klein G, Huesman R (2000) Elastic material model mismatch effects in deformable motion estimation. IEEE Trans Nuclear Sci 47(3):1000–1005

    Article  Google Scholar 

  30. Liu H, Hu H, Sinusas A, Shi P (2012) An H ∞ approach for elasticity properties reconstruction. Med Phys 39(1):475–481

    Article  Google Scholar 

  31. Liu H, Shi P (2004) Cardiac motion and elasticity characterization with iterative sequential H ∞ criteria. In: International Conference on Medical Image Computing and Computer Assisted Intervention (MICCAI), pp 34–42

  32. Liu H, Shi P (2009) Maximum a posteriori strategy for the simultaneous motion and material property estimation of the heart. IEEE Trans Biomed Eng 56(2):378–389

    Article  Google Scholar 

  33. Marchesseau S, Delingette H, Sermesant M, Cabrera-Lozoya R, Tobon-Gomez C, Moireau P, i Ventura RMF, Lekadir K, Hernandez A, Garreau M, Donal E, Leclercq C, Duckett SG, Rhode K, Rinaldi CA, Frangi AF, Razavi R, Chapelle D, Ayacha N (2013) Personalization of a cardiac electromechanical model using reduced order unscented Kalman filtering from regional volumes. Med Image Anal 17(7):816– 829

    Article  Google Scholar 

  34. McEachen J, Duncan J (1997) Shape-based tracking of left ventricular wall motion. IEEE Trans Med Imaging 16(3):270–283

    Article  Google Scholar 

  35. McEachen J, Nehorai A, Duncan J (2000) Multiframe temporal estimation of cardiac nonrigid motion. IEEE Trans Image Process 9(4):651–665

    Article  Google Scholar 

  36. Metaxas DN, Kakadiaris IA (2002) Elastically adaptive deformable models. IEEE Trans Pattern Anal Mach Intell 24(10):1310–1321

    Article  Google Scholar 

  37. Meyer C, Bonnemains L, Marċon F, Marie PY, Felblinger J, Vuissoz PA (2014) Longitudinal myocardial peak velocities using high temporal resolution phase-contrast and simple averaging are comparable to tissue doppler echocardiography. Magn Reson Mater Phys Biol Med 27(3):211–218

    Article  Google Scholar 

  38. Meyer F, Constable R, Sinusas A, Duncan J (1996) Tracking myocardial deformation using spatially constrained velocities. IEEE Trans Med Imaging 15(4):453–465

    Article  Google Scholar 

  39. Moireau P, Chapelle D, Le Tallec P (2008) Joint state and parameter estimation for distributed mechanical systems. Comput Methods Appl Mech Eng 197:659–677

    Article  MathSciNet  MATH  Google Scholar 

  40. Moireau P, Chapelle D, Le Tallec P (2009) Filtering for distributed mechanical systems using position measurements: perspectives in medical imaging. Inverse Probl 25(3):35,010–35,035

    Article  MathSciNet  MATH  Google Scholar 

  41. Nash MP (1998) Mechanics and material properties of the heart using an anatomically accurate mathematical model. PhD thesis, The University of Auckland, New Zealand

    Google Scholar 

  42. O’Brien RT, Kiriakidis K (2003) Convergence and performance in H ∞ -based parameter identification. IFAC Proc 36(11):127–132

    Article  Google Scholar 

  43. Ozturk C, McVeigh E (2000) Four-dimensional B-spline based motion analysis of tagged MR images: introduction and in vivo validation. Phys Med Biol 45(6):1683–702

    Article  Google Scholar 

  44. Papademetris X, Sinusas A, Dione D, Constable R, Duncan J (2002) Estimation of 3-D left ventricular deformation from medical images using biomechanical models. IEEE Trans Med Imaging 21(7):786–799

    Article  Google Scholar 

  45. Pavarino E, Neves L, Machado J, de Godoy M, Shiyou Y, Momente JC, Zafalon GF, Pinto A, Valêncio CR (2013) Free tools and strategies for the generation of 3d finite element meshes: modeling of the cardiac structures. Int J Biomed Imaging 2013:1–11

    Article  Google Scholar 

  46. Pelc N, Herfkens R, Shimakawa A, Enzmann D (1991) Phase contrast cine magnetic resonance imaging. Magn Reson Q 7(4):229–254

    Google Scholar 

  47. Perreard I, Pattison A, Doyley M, McGarry M, Barani Z, Houten E, Weaver J, Paulsen K (2010) Effects of frequency- and direction-dependent elastic materials on linearly elastic MRE image reconstructions. Phys Med Biol 55(22):6801–15

    Article  Google Scholar 

  48. Sermesant M, Delingette H, Ayache N (2006) An electromechanical model of the heart for image analysis and simulation. IEEE Trans Med Imaging 25(5):612–625

    Article  Google Scholar 

  49. Sermesant M, Moireau P, Camara O, Sainte-Marie J, Andriantsimiavona R, Cimrman R, Hill D, Chapell D, Razavi R (2005) Cardiac function estimation from MRI using a heart model and data assimilation: Advances and difficulties. International Workshop on Functional Imaging and Modeling of the Heart, pp 325–337

  50. Shen X, Deng L (1999) A dynamic system approach to speech enhancement using the \(\mathcal {H}_{\infty }\) filtering algorithm. IEEE Trans Speech Audio Process 7(4):391–399

    Article  Google Scholar 

  51. Shi P, Liu H (2003) Stochastic finite element framework for simultaneous estimation of cardiac kinematics functions and material parameters. Med Image Anal 7(4):445–464

    Article  Google Scholar 

  52. Shi P, Sinusas A, Constable RT, Duncan J (1999) Volumetric deformation analysis using mechanics-based data fusion: applications in cardiac motion recovery. Int J Comput Vis 35(1):87–107

    Article  Google Scholar 

  53. Simon D (2006) Optimal state estimation: Kalman, H-infinity, and Nonlinear Approaches. Wiley, NJ

    Book  Google Scholar 

  54. Tavakoli V, Amini AA (2013) A survey of shaped-based registration and segmentation techniques for cardiac images. Comput Vis Image Underst 117(9):966–989

    Article  Google Scholar 

  55. Taylor JR Classical mechanics, University Science Books, USA

  56. Tsap L, Goldgof D, Sarkar S (2000) Nonrigid motion analysis based on dynamic refinement of finite element models. IEEE Trans Pattern Anal Mach Intell 22 (5):526–543

    Article  Google Scholar 

  57. Vaccarella A, Momi ED, Enquobahrie A, Ferrigno G (2013) Unscented Kalman filter based sensor fusion for robust optical and electromagnetic tracking in surgical navigation. IEEE Trans Instrum Measur 62(7):2067–2081

    Article  Google Scholar 

  58. Wang D, Wang B, Zhao S, Yao H, Liu H (2017) View-based 3D object retrieval with discriminative views. Neurocomputing 252:58–66

    Article  Google Scholar 

  59. Wong A, Liu H, Sinusas A, Shi P (2003) Spatiotemporal active region model for simultaneous segmentation and motion estimation of the whole heart. In: IEEE workshop on Variational, Geometric and Level Set Methods in Computer Vision, pp 193–200

  60. Young A, Kraitchman D, Dougherty L, Axel L (1995) Tracking and finite element analysis of stripe deformation in magnetic resonance tagging. IEEE Trans Med Imaging 14(3):413–421

    Article  Google Scholar 

  61. Young A, Prince J (2013) Cardiovascular magnetic resonance: Deeper insights through bioengineering. Ann Rev Biomed Eng 15:433–461

    Article  Google Scholar 

  62. Zerhouni E, Parish D, Rogers W, Yang A, Shapiro E (1988) Human heart: tagging with MR imaging – a method for noninvasive assessment of myocardial motion. Radiology 169(1):59–63

    Article  Google Scholar 

  63. Zhang Z, Sahn DJ, Song X (2013) Right ventricular strain analysis from 3D echocardiography by using temporally diffeomorphic motion estimation. In: International Conference on Functional Imaging and Modeling of the Heart. Springer, pp 474–482

  64. Zhao S, Chen L, Yao H, Zhang Y, Sun X (2015) Strategy for dynamic 3D depth data matching towards robust action retrieval. Neurocomputing 151:533–543

    Article  Google Scholar 

  65. Zhao S, Yao H, Zhang Y, Wang Y, Liu S (2015) View-based 3D object retrieval via multi-modal graph learning. Signal Process 112:110–118

    Article  Google Scholar 

  66. Zhu Y, Pelc N (1999) A spatiotemporal model of cyclic kinematics and its application to analyzing nonrigid motion with MR velocity images. IEEE Trans Med Imaging 18(7):557–569

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by National Key Research and Development Program of China (2016YFC1300302), Natural Science Foundation of China (No. 61427807, 61525106, 61771464), and Science Technology and Innovation Committee of Shenzhen for Research Projects (SGLH20150213143207911, JCYJ20151030151431727).

Author information

Authors and Affiliations

  1. Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen, China

    Zhifan Gao & Heye Zhang

  2. Department of Imaging and Interventional Radiology, The Chinese University of Hong Kong, Hong Kong, China

    Defeng Wang

  3. Department of Optical Engineering, Zhejiang University, Hangzhou, China

    Min Guo & Huafeng Liu

  4. Northwestern Lake forest Hospital, Department of Radiation Oncology, Lake forest, IL, 60045, USA

    Ling Zhuang

  5. B. Thomas Golisano College of Computing and Information Sciences, Rochester Institute of Technology, Rochester, NY, USA

    Pengcheng Shi

Authors
  1. Zhifan Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Heye Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Defeng Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Min Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Huafeng Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Ling Zhuang
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Pengcheng Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Heye Zhang or Huafeng Liu.

Appendix

Appendix

The term \(\mathcal {H}_{\infty }\) originates from the name of the mathematical space over which the optimization takes place(more details can be found in [53]). \(\mathcal {H}\) stands for ”Hardy space”. ∞ implies that it is designed to accomplish min-max restrictions. The motivation of the \(\mathcal {H}_{\infty }\) filter is to solve problems where the uncertainties of the model and inputs are problematic. For an \(\mathcal {H}_{\infty }\) criteria, the transfer function (the measure of how good of the estimator is), from the input disturbances to the estimation error, shall be required to have a system gain that conforms to an upper bound (See (16) and (30) This criteria represents a family of solutions where the peak energy gain of the transfer function from the input disturbances to the estimate error are less than an upper bound γ.

It is observed that the smaller the γ value, the smaller the estimation error. On the other hand, the Riccati equation has to have a positive definite solution. In our practical implementation and the following deviations, matrix Q is chosen to be an identity matrix. However, depending on the confidence on the inputs, Q can be of any form and it would not change the fundamental deduction of the procedure. Thus

$$\begin{array}{@{}rcl@{}} & & [A{(\mathcal{R}(t)^{-1}+D^T V^{-1} D )}^{-1}A^T+W]^{-1} - \gamma^{-2}Q >0 \end{array} $$
(42)
$$\begin{array}{@{}rcl@{}} & \rightarrow & [A{(\mathcal{R}(t)^{-1}+D^T V^{-1} D )}^{-1}A^T+W]^{-1} > \gamma^{-2}I \end{array} $$
(43)
$$\begin{array}{@{}rcl@{}} & \rightarrow & \gamma^{2}I > A{(\mathcal{R}(t)^{-1}+D^T V^{-1} D)}^{-1}A^T+W \end{array} $$
(44)
$$\begin{array}{@{}rcl@{}} & \rightarrow & \gamma^{2} > max\{eig[A{(\mathcal{R}(t)^{-1}+D^T V^{-1} D)}^{-1}A^T+W]\} \end{array} $$
(45)
$$\begin{array}{@{}rcl@{}} & \rightarrow & \gamma = \xi max \left\{ eig[A(\mathcal{R}(t)^{-1}+ D^T V^{-1} D)^{-1}A^T+W]\right\}^{0.5} \end{array} $$
(46)

where m a x{e i g(A)} denotes the maximum eigenvalue of the matrix A, and ξ is a constant larger than 1 to ensure that γ is always greater than a certain optimal performance level. If the γ value is too close to the optimal performance level, i.e. ξ ≈ 1, it might lead to numerical errors because the matrix \(\mathcal {R}(t)\) is now close to a singular matrix.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gao, Z., Zhang, H., Wang, D. et al. Robust recovery of myocardial kinematics using dual \(\mathcal {H}_{\infty }\) criteria. Multimed Tools Appl 77, 23043–23071 (2018). https://doi.org/10.1007/s11042-017-5395-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11042-017-5395-1

Keywords

Search

Navigation