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A mathematical model for estimating the axial stress of the common carotid artery wall from ultrasound images

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Abstract

Clarifying the complex interaction between mechanical and biological processes in healthy and diseased conditions requires constitutive models for arterial walls. In this study, a mathematical model for the displacement of the carotid artery wall in the longitudinal direction is defined providing a satisfactory representation of the axial stress applied to the arterial wall. The proposed model was applied to the carotid artery wall motion estimated from ultrasound image sequences of 10 healthy adults, and the axial stress waveform exerted on the artery wall was extracted. Consecutive ultrasonic images (30 frames per second) of the common carotid artery of 10 healthy subjects (age 44 ± 4 year) were recorded and transferred to a personal computer. Longitudinal displacement and acceleration were extracted from ultrasonic image processing using a block-matching algorithm. Furthermore, images were examined using a maximum gradient algorithm and time rate changes of the internal diameter and intima-media thickness were extracted. Finally, axial stress was estimated using an appropriate constitutive equation for thin-walled tubes. Performance of the proposed model was evaluated using goodness of fit between approximated and measured longitudinal displacement statistics. Values of goodness-of-fit statistics indicated high quality of fit for all investigated subjects with the mean adjusted R-square (0.86 ± 0.08) and root mean squared error (0.08 ± 0.04 mm). According to the results of the present study, maximum and minimum axial stresses exerted on the arterial wall are 1.7 ± 0.6 and −1.5 ± 0.5 kPa, respectively. These results reveal the potential of this technique to provide a new method to assess arterial stress from ultrasound images, overcoming the limitations of the finite element and other simulation techniques.

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References

  1. Bussy C, Boutouyrie P, Lacolley P, Challande P, Laurent S (2000) Intrinsic stiffness of the carotid arterial wall material in essential hypertensives. Hypertension 35:1049–1054

    Article  CAS  PubMed  Google Scholar 

  2. Chuong CJ, Fung YC (1983) Three dimensional stress distribution in arteries. J Biomech Eng 105:268–274

    Article  CAS  PubMed  Google Scholar 

  3. Cinthio M, Ahlgren AR, Jansson T, Eriksson A, Persson HW, Lindström K (2005) Evaluation of an ultrasonic echo tracking method for measurements of arterial wall movements in two dimensions. IEEE Trans Ultrason Ferroelectr Freq Cont 52:1300–1311

    Article  Google Scholar 

  4. Cinthio M, Ahlgren AR, Bergkvist J, Jansson T, Persson HW, Lindström K (2006) Longitudinal movements and resulting shear strain of the arterial wall. Am J Physiol Heart Circ Physiol 291:394–402

    Article  Google Scholar 

  5. Cox RH (1975) Anisotropic properties of the canine carotid artery in vitro. J Biomech 8:293–300

    Article  CAS  PubMed  Google Scholar 

  6. Cunningham KS, Gotlieb AI (2005) The role of shear stress in the pathogenesis of atherosclerosis. Lab Invest 85:9–23

    Article  CAS  PubMed  Google Scholar 

  7. Dajnowiec D, Sabatini PJB, van Rossum TC, Lam JTK, Zhang M, Kapus A, Langille BL (2007) Force-induced polarized mitosis of endothelial and smooth muscle cells in arterial remodeling. Hypertension 50:255–260

    Article  CAS  PubMed  Google Scholar 

  8. Eberth JF, Cardamone L, Humphrey JD (2011) Evolving biaxial mechanical properties of mouse carotid arteries in hypertension. J Biomech 44:2532–2537

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  9. Gao F, Watanabe M, Matsuzawa T (2006) Stress analysis in a layered aortic arch model under pulsatile blood flow. Biomed Eng Online 24:5–25

    CAS  Google Scholar 

  10. Gerhard-Herman M, Gardin JM, Jaff M, Mohler E, Roman M, Naqvi TZ (2006) Guidelines for noninvasive vascular laboratory testing: a report from the American Society of echocardiography and the society for vascular medicine and biology. Vasc Med 11:183–200

    Article  PubMed  Google Scholar 

  11. Golemati S, Sassano A, Lever MJ, Bharath AA, Dhanjil S, Nicolaides AN (2003) Carotid artery wall motion estimated from B-mode ultrasound using region tracking and block matching. Ultrasound Med Biol 29:387–399

    Article  PubMed  Google Scholar 

  12. Gonzalez-Velasco EA (1996) Fourier analysis and boundary value problems. Academic Press. Inc, Chap 2: 25–43

  13. Hariton I, de Botton G, Gasser TC, Holzapfel GA (2007) Stress-driven collagen fiber remodeling in arterial walls. Biomech Model Mechanobiol 6(3):163–175

    Article  CAS  PubMed  Google Scholar 

  14. Hodisa S, Zamir M (2011) Mechanical events within the arterial wall under the forces of pulsatile flow: a review. J Mech Behav Biomed Mater 4:1595–1602

    Article  Google Scholar 

  15. Holzapfel GA, Gasser TC, Ogden RW (2004) Comparison of a multi-layer structural model for arterial walls with a fung-type model, and issues of material stability. J Biomech Eng 126:264–275

    Article  PubMed  Google Scholar 

  16. Humphrey JD, Gleason RL (2005) Effects of a sustained extension on arterial growth and remodeling: A theoretical study. J Biomech 38:1255–1261

    Article  PubMed  Google Scholar 

  17. Humphrey JD, Eberth JF, Dye WW, Gleason RL (2009) Fundamental role of axial stress in compensatory adaptations by arteries. J Biomech 42:1–8

    Article  CAS  PubMed  Google Scholar 

  18. Jackson ZS, Gotlieb AI, Langille BL (2002) Wall tissue remodeling regulates longitudinal tension in arteries. Circ Res 90:918–925

    Article  CAS  PubMed  Google Scholar 

  19. Jegelevicius D, Lukosevicius A (2002) Ultrasonic measurements of human carotid artery wall intima-media thickness. Ultragras 2:43–47

    Google Scholar 

  20. Katritsis D, Kaiktsis L, Chaniotis A, Pantos J, Efstathopoulos EP, Marmarelis V (2007) Wall shear stress: Theoretical considerations and methods of measurement. Prog Cardiovasc Dis 49:307–329

    Article  PubMed  Google Scholar 

  21. Lawrence-Brown M, Stanley BM, Sun Z, Semmens JB, Liffman K (2011) Stress and strain behaviour modelling of the carotid bifurcation. ANZ J Surg 81:810–816

    Article  PubMed  Google Scholar 

  22. Marquardt DW (1963) An algorithm for least-squares estimation 409 of nonlinear parameters. J Soc Indust Appl Math 11:431–441

    Article  Google Scholar 

  23. Masson I, Boutouyrie P, Laurent S, Humphrey JD, Zidi M (2008) Characterization of arterial wall mechanical behavior and stresses from human clinical data. J Biomech 41:2618–2627

    Article  PubMed  PubMed Central  Google Scholar 

  24. Mickael ME, Heydari A, Crouch R, Johnstone S (2010) Estimation of stress-strain relationships in vascular walls using multi-layer hyperelastic modelling approach. IEEE conf Comput Cardiol, pp 577–580

  25. Oshinski JN, Curtin JL, Loth F (2006) Mean-average wall shear stress measurements in the common carotid artery. J Cardiovasc Magn Reson 8:717–722

    Article  PubMed  Google Scholar 

  26. Patel DJ, Fry DL (1966) Longitudinal tethering of arteries in dogs. Circ Res 19:1011–1021

    Article  CAS  PubMed  Google Scholar 

  27. Persson M, Ahlgren ÅR, Eriksson A, Jansson T, Persson HW, Lindström K (2002) Non-invasive measurement of arterial longitudinal movement. IEEE Ultrason Symp, pp 1783–1786

  28. Rafati M, Mokhtari-Dizaji M, Saberi H, Soleimani E (2011) Extraction the longitudinal movement of the carotid artery wall using consecutive ultrasonic images: block matching algorithm. Iran J Med Phys 8:49–59

    Google Scholar 

  29. Samijo SK, Willigers JM, Barkhuysen R, Kitslaar PJ, Reneman RS, Brands PJ, Hoeks AP (1998) Wall shear stress in the human common carotid artery as function of age and gender. Cardiovasc Res 39:515–522

    Article  CAS  PubMed  Google Scholar 

  30. Schmidt-Trucksäss A, Grathwohl D, Schmid A, Boragk R, Upmeier C, Keul J, Huonker M (1998) Assessment of carotid wall motion and stiffness with tissue Doppler imaging. Ultrasound Med Biol 24:639–646

    Article  PubMed  Google Scholar 

  31. Sokolis DP (2010) A passive strain-energy function for elastic and muscular arteries: correlation of material parameters with histological data. Med Biol Eng Comput 48:507–518

    Article  PubMed  Google Scholar 

  32. Soleimani E, Mokhtari Dizaji M, Saberi H (2011) Carotid artery wall motion estimation from consecutive ultrasonic images: comparison between block-matching and maximum-gradient algorithms. J Teh Univ Heart Ctr 6:72–78

    Google Scholar 

  33. Stoitsis J, Golemati S, Dimopoulos AK, Nikita KS (2005) Analysis and quantification of arterial wall motion from B-mode ultrasound images comparison of block-matching and optical flow. Eng Med Biol, 27th Conf, pp 1–4

  34. Stroev PV, Hoskins PR, Easson WJ (2007) Distribution of wall shear rate throughout the arterial tree: a case study. Atherosclerosis 191:276–280

    Article  CAS  PubMed  Google Scholar 

  35. Sunagawa K, Kanai H, Tanaka M (2000) Simultaneous measurement of blood flow and arterial wall vibrations in radial and axial directions. IEEE Ultrason Symp, pp 1541–1544

  36. Taber LA (2008) Nonlinear theory of elasticity: Application in biomechanics. World Scientific Pub Co Inc, Chap 4: 145–176

  37. Tozzi P, Hayoz D, Oedman C, Mallabiabarrena I, Von Segesser LK (2001) Systolic axial artery length reduction: an overlooked phenomenon in vivo. Am J Physiol Heart Circ Physiol 280:2300–2305

    Google Scholar 

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Acknowledgments

This study was supported by Faculty of Medical Sciences, Tarbiat Modares University.

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Authors and Affiliations

  1. Department of Medical Physics, Faculty of Medical Sciences, Tarbiat Modares University, Tehran, Iran

    Effat Soleimani & Manijhe Mokhtari-Dizaji

  2. Department of Radiology, Faculty of Radiology, Imam Khomeini Hospital, Tehran University of Medical Sciences, Tehran, Iran

    Hajir Saberi & Shervin Sharif-Kashani

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  1. Effat Soleimani
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  2. Manijhe Mokhtari-Dizaji
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  4. Shervin Sharif-Kashani
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Correspondence to Manijhe Mokhtari-Dizaji or Hajir Saberi.

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Soleimani, E., Mokhtari-Dizaji, M., Saberi, H. et al. A mathematical model for estimating the axial stress of the common carotid artery wall from ultrasound images. Med Biol Eng Comput 54, 1205–1215 (2016). https://doi.org/10.1007/s11517-015-1409-1

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  • DOI: https://doi.org/10.1007/s11517-015-1409-1

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