2014, 11(3): 427-448. doi: 10.3934/mbe.2014.11.427

Model validation for a noninvasive arterial stenosis detection problem

1. 

Center for Research in Scientific Computation, Center for Quantitative Sciences in Biomedicine, North Carolina State University, Raleigh, NC 27695-8212

2. 

Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695-8212, United States

3. 

Brunel Institute of Computational Mathematics, Brunel University, Uxbridge, UB8 3PH, United Kingdom, United Kingdom, United Kingdom

4. 

Blizard Institute, Barts and the London School of Medicine and Dentistry, Queen Mary, University of London, United Kingdom, United Kingdom

5. 

Clinical Physics, Barts Health Trust, United Kingdom

Received  January 2013 Revised  May 2013 Published  January 2014

A current thrust in medical research is the development of a non-invasive method for detection, localization, and characterization of an arterial stenosis (a blockage or partial blockage in an artery). A method has been proposed to detect shear waves in the chest cavity which have been generated by disturbances in the blood flow resulting from a stenosis. In order to develop this methodology further, we use one-dimensional shear wave experimental data from novel acoustic phantoms to validate a corresponding viscoelastic mathematical model. We estimate model parameters which give a good fit (in a sense to be precisely defined) to the experimental data, and use asymptotic error theory to provide confidence intervals for parameter estimates. Finally, since a robust error model is necessary for accurate parameter estimates and confidence analysis, we include a comparison of absolute and relative models for measurement error.
Citation: H. Thomas Banks, Shuhua Hu, Zackary R. Kenz, Carola Kruse, Simon Shaw, John Whiteman, Mark P. Brewin, Stephen E. Greenwald, Malcolm J. Birch. Model validation for a noninvasive arterial stenosis detection problem. Mathematical Biosciences & Engineering, 2014, 11 (3) : 427-448. doi: 10.3934/mbe.2014.11.427
References:
[1]

M. Ainsworth, P. Monk and W. Muniz, Dispersive and dissipative properties of discontinuous Galerkin finite element methods for the second-order wave equation,, Journal of Scientific Computing, 27 (2006), 5. doi: 10.1007/s10915-005-9044-x.

[2]

M. Akay, Noninvasive Detection of Coronary Artery Disease Using Advanced Signal Processing Methods,, PhD Dissertation, (1990).

[3]

M. Akay, Y. Akay, W. Welkowitz, J. Semmlow and J. Kostis, Application of adaptive filters to noninvasive acoustical detection of coronary occlusions before and after angioplasty,, IEEE Trans. on Biomed. Eng., 39 (1992), 176. doi: 10.1109/10.121649.

[4]

M. Akay, Y. Akay, W. Welkowitz, J. Semmlow and J. Kostis, Noninvasive detection of coronary artery disease using neural networks,, Proc. IEEE Eng. in Med. & Biol. Soc., (1991), 1434. doi: 10.1109/IEMBS.1991.684531.

[5]

M. Akay, M. Bauer, J. Semmlow, W. Welkowitz and J. Kostis, Analysis of diastolic heart sounds before and after angioplasty,, Proc. IEEE Eng. in Med. & Biol. Soc., (1988), 257. doi: 10.1109/IEMBS.1988.94505.

[6]

M. Akay, W. Welkowitz, J. Semmlow and J. Kostis, Application of the ARMA method to acoustic detection of coronary artery disease,, Med. & Biol. Eng. & Comput., 29 (1991), 365. doi: 10.1007/BF02441656.

[7]

H. T. Banks, A Functional Analysis Framework for Modeling, Estimation and Control in Science and Engineering,, CRC Press, (2012). doi: 10.1201/b12209.

[8]

H. T. Banks, J. H. Barnes, A. Eberhardt, H. Tran and S. Wynne, Modeling and computation of propagating waves from coronary stenoses,, Comp. and Appl. Math., 21 (2002), 767.

[9]

H. T. Banks and K. Bihari, Modelling and estimating uncertainty in parameter estimation,, Inverse Problems, 17 (2001), 95. doi: 10.1088/0266-5611/17/1/308.

[10]

H. T. Banks and B. G. Fitzpatrick, Inverse problems for distributed systems: Statistical tests and ANOVA,, Proc. International Symp. on Math. Approaches to Envir. and Ecol. Problems, 81 (1989), 88. doi: 10.1007/978-3-642-46693-9_18.

[11]

H. T. Banks, K. Holm and D. Robbins, Standard error computations for uncertainty quantification in inverse problems: Asymptotic theory vs. Bootstrapping,, Math. and Comp. Modelling, 52 (2010), 09. doi: 10.1016/j.mcm.2010.06.026.

[12]

H. T. Banks, S. Hu and Z. R. Kenz, A brief review of elasticity and viscoelasticity for solids,, Adv. in Applied Math. and Mech., 3 (2011), 1.

[13]

H. T. Banks, S. Hu, Z. R. Kenz, C. Kruse, S. Shaw, J. R. Whiteman, M. P. Brewin, S. E. Greenwald and M. J. Birch, Material parameter estimation and hypothesis testing on a 1D viscoelastic stenosis model: Methodology,, J. Inverse and Ill-Posed Problems, 21 (2012), 12. doi: 10.1515/jip-2012-0081.

[14]

H. T. Banks, S. Hu, Z. R. Kenz, C. Kruse, S. Shaw, J. R. Whiteman, M. P. Brewin, S. E. Greenwald and M. J. Birch, Model validation for a noninvasive arterial stenosis detection problem,, CRSC-TR12-22, (2012), 12.

[15]

H. T. Banks, Z. R. Kenz and W. C. Thompson, A review of selected techniques in inverse problem nonparametric probability distribution estimation,, J. of Inverse and Ill-Posed Problems, 20 (2012), 429. doi: 10.1515/jip-2012-0037.

[16]

H. T. Banks, Z. R. Kenz and W. C. Thompson, An extension of RSS-based model comparison tests for weighted least squares,, CRSC-TR12-18, 79 (2012), 12.

[17]

H. T. Banks and N. Luke, Modeling of propagating shear waves in biotissue employing an internal variable approach to dissipation,, Communication in Computational Physics, 3 (2008), 603.

[18]

H. T. Banks, N. Medhin and G. Pinter, Multiscale considerations in modeling of nonlinear elastomers,, Inter. J. for Comp. Methods in Eng. Science and Mechanics, 8 (2007), 53. doi: 10.1080/15502280601149346.

[19]

H. T. Banks, N. Medhin and G. Pinter, Nonlinear reptation in molecular based hysteresis models for polymers,, Quarterly of Applied Math., 62 (2004), 767.

[20]

H. T. Banks and G. A. Pinter, A probabilistic multiscale approach to hysteresis in shear wave propagation in biotissue,, Multiscale Modeling and Simulation, 3 (2005), 395. doi: 10.1137/040603693.

[21]

H. T. Banks and J. R. Samuels, Jr, Detection of cardiac occlusions using viscoelastic wave propagation,, Advances in Appl. Math. and Mech., 1 (2009), 1.

[22]

H. T. Banks and H. T. Tran, Mathematical and Experimental Modeling of Physical and Biological Processes,, CRC Press, (2009).

[23]

J. D. De Basabe, M. K. Sen and M. F. Wheeler, The interior penalty discontinuous Galerkin method for elastic wave propagation: grid dispersion,, Geophys J. Int., 175 (2008), 83.

[24]

A. O. Borisyuk, Noise field in the human chest due to turbulent flow in a larger blood vessel,, Flow, 61 (1999), 269. doi: 10.1016/S0889-9746(03)00056-2.

[25]

A. O. Borisyuk, Experimental study of voise produced by steady flow through a simulated vascular stenosis,, J. of Sound and Vibration, 256 (2002), 475.

[26]

A. O. Borisyuk, Model study of noise field in the human chest due to turbulent flow in a larger blood vessel,, J. of Fluids and Structures, 17 (2003), 1095. doi: 10.1016/S0889-9746(03)00056-2.

[27]

M. P. Brewin, M. J. Birch and S. E. Greenwald, et al., Characterization of the uniaxial elastic properties of an agar-based tissue mimicking material,, in preparation., ().

[28]

S. Catheline, J.-L. Gennisson, G. Delon, M. Fink, R. Sinkus, S. Abouelkaram and J. Culioli, Measurement of viscoelastic properties of homogeneous soft solid using transient elastography: an inverse problem approach,, J. Acoust. Soc. Am, 116 (2004), 3734. doi: 10.1121/1.1815075.

[29]

S. Catheline, L. Sandrin, J.-L. Gennisson, M. Tanter and M. Fink, Ultrasound-based noninvasive shear elasticity probe for soft tissues,, IEEE Ultrasonics Symposium, 2 (2000), 1799. doi: 10.1109/ULTSYM.2000.921672.

[30]

S. Chen, M. Fatemi and J. Greenleaf, Quantifying elasticity and viscosity from measurement of shear wave speed dispersion,, J. Acoust. Soc. Am., 115 (2004), 2781. doi: 10.1121/1.1739480.

[31]

S. Chen, M. Urban, C. Pislaru, R. Kinnick, Y. Zheng, A. Yao and J. Greenleaf, Shearwave dispersion ultrasound vibrometry (SDUV) for measuring tissue elasticity and viscosity,, IEEE Trans. on Ultrason., 56 (2009), 55.

[32]

T. Cheng, Diastolic murmur caused by coronary artery stenosis,, Ann. Int. Med, 72 (1970). doi: 10.7326/0003-4819-72-4-543.

[33]

T. Deffieux, G. Montaldo and M. Tanter, Shear wave spectroscopy for in vivo quantification of human soft tissues visco-elasticity,, IEEE Trans. on Medical Imag., 28 (2009), 313. doi: 10.1109/TMI.2008.925077.

[34]

B. El-Asir, L. Khadra, A. Al-Abbasi and M. Mohammed, Time-frequency analysis of heart sounds,, Proc. IEE TENCON, (1996), 553. doi: 10.1109/TENCON.1996.608401.

[35]

Y. C. Fung, Biomechanics: Mechanical Properties of Living Tissues,, Springer-Verlag, (1993). doi: 10.1115/1.3138285.

[36]

A. Góral-Wójcicka, W. Borgieł, Z. Małota and Z. Nawrat, On the acoustic phenomena produced by turbulence in the flowing blood,, Polish J. Med. Phys. & Eng., 8 (2002), 29.

[37]

A. Karpiouk, S. Alglyamov, Y. Illinskii, E. Zabolotskaya and S. Emelianov, Assessment of shear modulus of tissue using ultrasound radiation force acting on a spherical acoustic inhomogeneity,, IEEE Trans. on Ultrason., 56 (2009), 2380. doi: 10.1109/TUFFC.2009.1326.

[38]

C. Kruse, S. Shaw and J. R. Whiteman, High order space-time finite element schemes for acoustic and viscodynamic wave equations with temporal decoupling,, in preparation., ().

[39]

T. S. Lee, W. Liao and H. T. Low, Numerical simulation of turbulent flow through series stenoses,, Inter. J. for Numer. Methods in Fluids, 42 (2003), 717. doi: 10.1002/fld.550.

[40]

S. Levinson, M. Shinagawa and T. Sato, Sonoelastic determination of human skeletal muscle elasticity,, J. Biomechanics, 28 (1995), 1145. doi: 10.1016/0021-9290(94)00173-2.

[41]

S. Lundin, R. Metcalf and C. Hartley, Effects of severity and eccentricity of carotid stenosis on pulsatile blood flow,, Proc. Joint EMBS/BMES, (2003), 1311. doi: 10.1109/IEMBS.2002.1106403.

[42]

N. Luke, Modeling Shear Wave Propagation in Biotissue: An Internal Variable Approach to Dissipation,, PhD Dissertation, (2006).

[43]

S. E. Nissen, Application of intravascular ultrasound to characterize coronary artery disease and assess the progression or regression of atherosclerosis,, Am. J. Cardiol., 89 (2002). doi: 10.1016/S0002-9149(02)02217-8.

[44]

S. E. Nissen and P. Yock, Intravascular ultrasound: Novel pathophysiological insights and current clinical applications,, Circulation, 103 (2001), 604. doi: 10.1161/01.CIR.103.4.604.

[45]

N. Owsley and A. Hull, Beamformed nearfield imaging of a simulated coronary artery containing a stenosis,, IEEE Trans. Med. Imaging, 17 (1998), 900. doi: 10.1109/42.746623.

[46]

N. Owsley, A. J. Hull, M. H. Ahmed and J. Kassal, A proof of concept experiment for the detection of occluded coronary arteries using array sensor technology,, Engr. in Medicine and Biol. Society, 1 (1995), 145. doi: 10.1109/IEMBS.1995.575042.

[47]

V. Padmanabhan and J. Semmlow, A dedicated system for acoustic detection of coronary artery disease,, Proc. Eng. in Med. & Biol. Soc., (1992), 457. doi: 10.1109/IEMBS.1992.595658.

[48]

T. Pedley, Mathematical modelling of arterial fluid dynamics,, J. of Eng. Math., 47 (2003), 419. doi: 10.1023/B:ENGI.0000007978.33352.59.

[49]

C. Prado, S. Ramos, J. Elias, Jr., and M. Rossi, Turbulent blood flow plays an essential localizing role in the development of atherosclerotic lesions in experimentally induced hypercholesterolaemia in rats,, Int. J. Exp. Path., 89 (2008), 72. doi: 10.1111/j.1365-2613.2007.00564.x.

[50]

J. R. Samuels, Jr., Inverse Problems and Post Analysis Techniques for a Stenosis-Driven Acoustic Wave Propagation Model,, PhD Dissertation, (2008).

[51]

L. Sandrin, S. Catheline, M. Tanter, X. Hennequin and M. Fink, Time-resolved pulsed elastography with ultrafast ultrasonic imaging,, Ultrasonic Imaging, 21 (1999), 259.

[52]

J. Sangster and C. Oakley, Diastolic murmur of coronary artery stenosis,, Brit. Heart J., 35 (1973). doi: 10.1136/hrt.35.8.840.

[53]

J. Semmlow and K. Rahalkar, Acoustic detection of coronary artery disease,, Annu. Rev. Biomed. Eng, 9 (2007), 449. doi: 10.1146/annurev.bioeng.9.060906.151840.

[54]

J. Semmlow, W. Welkowitz, J. Kostis and J. Mackenzie, Coronary artery disease-correlates between diastolic auditory characteristics and coronary artery stenoses,, IEEE Trans. on Biomed. Eng., 30 (1983), 136. doi: 10.1109/TBME.1983.325211.

[55]

C. Taylor, T. Hughes and C. Zarins, Finite element modeling of three-dimensional pulsatile flow in the abdominal aorta: Relevance to atherosclerosis,, Annals of Biomed. Eng., 26 (1998), 975. doi: 10.1114/1.140.

[56]

J. Verburg and E. van Vollenhoven, Phonocardiography: Physical and technical aspects and clinical uses,, in Noninvasive Physiological Measurements (ed. P Rolfe), (1979), 213.

[57]

H. Vermarien and E. van Vollenhoven, The recording of heart vibrations: A problem of vibration measurement on soft tissue,, Med. & Biol. Eng. & Comput., 22 (1984), 168. doi: 10.1007/BF02446821.

[58]

A. Voss, A. Mix and T. Hübner, Diagnosing aortic valve stenosis by parameter extraction of heart sound signals,, Annals of Biomed. Eng., 33 (2005), 1167. doi: 10.1007/s10439-005-5347-x.

[59]

J.-Z. Wang, B. Tie, W. Welkowitz, J. Semmlow and J. Kostis, Modeling sound generation in stenosed coronary arteries,, IEEE Trans. on Biomed. Eng., 37 (1990), 1087. doi: 10.1109/10.61034.

[60]

J. Wloka, Partial Differential Equations,, Cambridge University Press, (1987).

[61]

M. Zia, B. Griffel, V. Fridman, C. Saponieri and J. Semmlow, Noise detection in heart sound recordings,, Conf. Proc. IEEE Eng. Med. Biol. Soc., (2011), 5880. doi: 10.1109/IEMBS.2011.6091454.

show all references

References:
[1]

M. Ainsworth, P. Monk and W. Muniz, Dispersive and dissipative properties of discontinuous Galerkin finite element methods for the second-order wave equation,, Journal of Scientific Computing, 27 (2006), 5. doi: 10.1007/s10915-005-9044-x.

[2]

M. Akay, Noninvasive Detection of Coronary Artery Disease Using Advanced Signal Processing Methods,, PhD Dissertation, (1990).

[3]

M. Akay, Y. Akay, W. Welkowitz, J. Semmlow and J. Kostis, Application of adaptive filters to noninvasive acoustical detection of coronary occlusions before and after angioplasty,, IEEE Trans. on Biomed. Eng., 39 (1992), 176. doi: 10.1109/10.121649.

[4]

M. Akay, Y. Akay, W. Welkowitz, J. Semmlow and J. Kostis, Noninvasive detection of coronary artery disease using neural networks,, Proc. IEEE Eng. in Med. & Biol. Soc., (1991), 1434. doi: 10.1109/IEMBS.1991.684531.

[5]

M. Akay, M. Bauer, J. Semmlow, W. Welkowitz and J. Kostis, Analysis of diastolic heart sounds before and after angioplasty,, Proc. IEEE Eng. in Med. & Biol. Soc., (1988), 257. doi: 10.1109/IEMBS.1988.94505.

[6]

M. Akay, W. Welkowitz, J. Semmlow and J. Kostis, Application of the ARMA method to acoustic detection of coronary artery disease,, Med. & Biol. Eng. & Comput., 29 (1991), 365. doi: 10.1007/BF02441656.

[7]

H. T. Banks, A Functional Analysis Framework for Modeling, Estimation and Control in Science and Engineering,, CRC Press, (2012). doi: 10.1201/b12209.

[8]

H. T. Banks, J. H. Barnes, A. Eberhardt, H. Tran and S. Wynne, Modeling and computation of propagating waves from coronary stenoses,, Comp. and Appl. Math., 21 (2002), 767.

[9]

H. T. Banks and K. Bihari, Modelling and estimating uncertainty in parameter estimation,, Inverse Problems, 17 (2001), 95. doi: 10.1088/0266-5611/17/1/308.

[10]

H. T. Banks and B. G. Fitzpatrick, Inverse problems for distributed systems: Statistical tests and ANOVA,, Proc. International Symp. on Math. Approaches to Envir. and Ecol. Problems, 81 (1989), 88. doi: 10.1007/978-3-642-46693-9_18.

[11]

H. T. Banks, K. Holm and D. Robbins, Standard error computations for uncertainty quantification in inverse problems: Asymptotic theory vs. Bootstrapping,, Math. and Comp. Modelling, 52 (2010), 09. doi: 10.1016/j.mcm.2010.06.026.

[12]

H. T. Banks, S. Hu and Z. R. Kenz, A brief review of elasticity and viscoelasticity for solids,, Adv. in Applied Math. and Mech., 3 (2011), 1.

[13]

H. T. Banks, S. Hu, Z. R. Kenz, C. Kruse, S. Shaw, J. R. Whiteman, M. P. Brewin, S. E. Greenwald and M. J. Birch, Material parameter estimation and hypothesis testing on a 1D viscoelastic stenosis model: Methodology,, J. Inverse and Ill-Posed Problems, 21 (2012), 12. doi: 10.1515/jip-2012-0081.

[14]

H. T. Banks, S. Hu, Z. R. Kenz, C. Kruse, S. Shaw, J. R. Whiteman, M. P. Brewin, S. E. Greenwald and M. J. Birch, Model validation for a noninvasive arterial stenosis detection problem,, CRSC-TR12-22, (2012), 12.

[15]

H. T. Banks, Z. R. Kenz and W. C. Thompson, A review of selected techniques in inverse problem nonparametric probability distribution estimation,, J. of Inverse and Ill-Posed Problems, 20 (2012), 429. doi: 10.1515/jip-2012-0037.

[16]

H. T. Banks, Z. R. Kenz and W. C. Thompson, An extension of RSS-based model comparison tests for weighted least squares,, CRSC-TR12-18, 79 (2012), 12.

[17]

H. T. Banks and N. Luke, Modeling of propagating shear waves in biotissue employing an internal variable approach to dissipation,, Communication in Computational Physics, 3 (2008), 603.

[18]

H. T. Banks, N. Medhin and G. Pinter, Multiscale considerations in modeling of nonlinear elastomers,, Inter. J. for Comp. Methods in Eng. Science and Mechanics, 8 (2007), 53. doi: 10.1080/15502280601149346.

[19]

H. T. Banks, N. Medhin and G. Pinter, Nonlinear reptation in molecular based hysteresis models for polymers,, Quarterly of Applied Math., 62 (2004), 767.

[20]

H. T. Banks and G. A. Pinter, A probabilistic multiscale approach to hysteresis in shear wave propagation in biotissue,, Multiscale Modeling and Simulation, 3 (2005), 395. doi: 10.1137/040603693.

[21]

H. T. Banks and J. R. Samuels, Jr, Detection of cardiac occlusions using viscoelastic wave propagation,, Advances in Appl. Math. and Mech., 1 (2009), 1.

[22]

H. T. Banks and H. T. Tran, Mathematical and Experimental Modeling of Physical and Biological Processes,, CRC Press, (2009).

[23]

J. D. De Basabe, M. K. Sen and M. F. Wheeler, The interior penalty discontinuous Galerkin method for elastic wave propagation: grid dispersion,, Geophys J. Int., 175 (2008), 83.

[24]

A. O. Borisyuk, Noise field in the human chest due to turbulent flow in a larger blood vessel,, Flow, 61 (1999), 269. doi: 10.1016/S0889-9746(03)00056-2.

[25]

A. O. Borisyuk, Experimental study of voise produced by steady flow through a simulated vascular stenosis,, J. of Sound and Vibration, 256 (2002), 475.

[26]

A. O. Borisyuk, Model study of noise field in the human chest due to turbulent flow in a larger blood vessel,, J. of Fluids and Structures, 17 (2003), 1095. doi: 10.1016/S0889-9746(03)00056-2.

[27]

M. P. Brewin, M. J. Birch and S. E. Greenwald, et al., Characterization of the uniaxial elastic properties of an agar-based tissue mimicking material,, in preparation., ().

[28]

S. Catheline, J.-L. Gennisson, G. Delon, M. Fink, R. Sinkus, S. Abouelkaram and J. Culioli, Measurement of viscoelastic properties of homogeneous soft solid using transient elastography: an inverse problem approach,, J. Acoust. Soc. Am, 116 (2004), 3734. doi: 10.1121/1.1815075.

[29]

S. Catheline, L. Sandrin, J.-L. Gennisson, M. Tanter and M. Fink, Ultrasound-based noninvasive shear elasticity probe for soft tissues,, IEEE Ultrasonics Symposium, 2 (2000), 1799. doi: 10.1109/ULTSYM.2000.921672.

[30]

S. Chen, M. Fatemi and J. Greenleaf, Quantifying elasticity and viscosity from measurement of shear wave speed dispersion,, J. Acoust. Soc. Am., 115 (2004), 2781. doi: 10.1121/1.1739480.

[31]

S. Chen, M. Urban, C. Pislaru, R. Kinnick, Y. Zheng, A. Yao and J. Greenleaf, Shearwave dispersion ultrasound vibrometry (SDUV) for measuring tissue elasticity and viscosity,, IEEE Trans. on Ultrason., 56 (2009), 55.

[32]

T. Cheng, Diastolic murmur caused by coronary artery stenosis,, Ann. Int. Med, 72 (1970). doi: 10.7326/0003-4819-72-4-543.

[33]

T. Deffieux, G. Montaldo and M. Tanter, Shear wave spectroscopy for in vivo quantification of human soft tissues visco-elasticity,, IEEE Trans. on Medical Imag., 28 (2009), 313. doi: 10.1109/TMI.2008.925077.

[34]

B. El-Asir, L. Khadra, A. Al-Abbasi and M. Mohammed, Time-frequency analysis of heart sounds,, Proc. IEE TENCON, (1996), 553. doi: 10.1109/TENCON.1996.608401.

[35]

Y. C. Fung, Biomechanics: Mechanical Properties of Living Tissues,, Springer-Verlag, (1993). doi: 10.1115/1.3138285.

[36]

A. Góral-Wójcicka, W. Borgieł, Z. Małota and Z. Nawrat, On the acoustic phenomena produced by turbulence in the flowing blood,, Polish J. Med. Phys. & Eng., 8 (2002), 29.

[37]

A. Karpiouk, S. Alglyamov, Y. Illinskii, E. Zabolotskaya and S. Emelianov, Assessment of shear modulus of tissue using ultrasound radiation force acting on a spherical acoustic inhomogeneity,, IEEE Trans. on Ultrason., 56 (2009), 2380. doi: 10.1109/TUFFC.2009.1326.

[38]

C. Kruse, S. Shaw and J. R. Whiteman, High order space-time finite element schemes for acoustic and viscodynamic wave equations with temporal decoupling,, in preparation., ().

[39]

T. S. Lee, W. Liao and H. T. Low, Numerical simulation of turbulent flow through series stenoses,, Inter. J. for Numer. Methods in Fluids, 42 (2003), 717. doi: 10.1002/fld.550.

[40]

S. Levinson, M. Shinagawa and T. Sato, Sonoelastic determination of human skeletal muscle elasticity,, J. Biomechanics, 28 (1995), 1145. doi: 10.1016/0021-9290(94)00173-2.

[41]

S. Lundin, R. Metcalf and C. Hartley, Effects of severity and eccentricity of carotid stenosis on pulsatile blood flow,, Proc. Joint EMBS/BMES, (2003), 1311. doi: 10.1109/IEMBS.2002.1106403.

[42]

N. Luke, Modeling Shear Wave Propagation in Biotissue: An Internal Variable Approach to Dissipation,, PhD Dissertation, (2006).

[43]

S. E. Nissen, Application of intravascular ultrasound to characterize coronary artery disease and assess the progression or regression of atherosclerosis,, Am. J. Cardiol., 89 (2002). doi: 10.1016/S0002-9149(02)02217-8.

[44]

S. E. Nissen and P. Yock, Intravascular ultrasound: Novel pathophysiological insights and current clinical applications,, Circulation, 103 (2001), 604. doi: 10.1161/01.CIR.103.4.604.

[45]

N. Owsley and A. Hull, Beamformed nearfield imaging of a simulated coronary artery containing a stenosis,, IEEE Trans. Med. Imaging, 17 (1998), 900. doi: 10.1109/42.746623.

[46]

N. Owsley, A. J. Hull, M. H. Ahmed and J. Kassal, A proof of concept experiment for the detection of occluded coronary arteries using array sensor technology,, Engr. in Medicine and Biol. Society, 1 (1995), 145. doi: 10.1109/IEMBS.1995.575042.

[47]

V. Padmanabhan and J. Semmlow, A dedicated system for acoustic detection of coronary artery disease,, Proc. Eng. in Med. & Biol. Soc., (1992), 457. doi: 10.1109/IEMBS.1992.595658.

[48]

T. Pedley, Mathematical modelling of arterial fluid dynamics,, J. of Eng. Math., 47 (2003), 419. doi: 10.1023/B:ENGI.0000007978.33352.59.

[49]

C. Prado, S. Ramos, J. Elias, Jr., and M. Rossi, Turbulent blood flow plays an essential localizing role in the development of atherosclerotic lesions in experimentally induced hypercholesterolaemia in rats,, Int. J. Exp. Path., 89 (2008), 72. doi: 10.1111/j.1365-2613.2007.00564.x.

[50]

J. R. Samuels, Jr., Inverse Problems and Post Analysis Techniques for a Stenosis-Driven Acoustic Wave Propagation Model,, PhD Dissertation, (2008).

[51]

L. Sandrin, S. Catheline, M. Tanter, X. Hennequin and M. Fink, Time-resolved pulsed elastography with ultrafast ultrasonic imaging,, Ultrasonic Imaging, 21 (1999), 259.

[52]

J. Sangster and C. Oakley, Diastolic murmur of coronary artery stenosis,, Brit. Heart J., 35 (1973). doi: 10.1136/hrt.35.8.840.

[53]

J. Semmlow and K. Rahalkar, Acoustic detection of coronary artery disease,, Annu. Rev. Biomed. Eng, 9 (2007), 449. doi: 10.1146/annurev.bioeng.9.060906.151840.

[54]

J. Semmlow, W. Welkowitz, J. Kostis and J. Mackenzie, Coronary artery disease-correlates between diastolic auditory characteristics and coronary artery stenoses,, IEEE Trans. on Biomed. Eng., 30 (1983), 136. doi: 10.1109/TBME.1983.325211.

[55]

C. Taylor, T. Hughes and C. Zarins, Finite element modeling of three-dimensional pulsatile flow in the abdominal aorta: Relevance to atherosclerosis,, Annals of Biomed. Eng., 26 (1998), 975. doi: 10.1114/1.140.

[56]

J. Verburg and E. van Vollenhoven, Phonocardiography: Physical and technical aspects and clinical uses,, in Noninvasive Physiological Measurements (ed. P Rolfe), (1979), 213.

[57]

H. Vermarien and E. van Vollenhoven, The recording of heart vibrations: A problem of vibration measurement on soft tissue,, Med. & Biol. Eng. & Comput., 22 (1984), 168. doi: 10.1007/BF02446821.

[58]

A. Voss, A. Mix and T. Hübner, Diagnosing aortic valve stenosis by parameter extraction of heart sound signals,, Annals of Biomed. Eng., 33 (2005), 1167. doi: 10.1007/s10439-005-5347-x.

[59]

J.-Z. Wang, B. Tie, W. Welkowitz, J. Semmlow and J. Kostis, Modeling sound generation in stenosed coronary arteries,, IEEE Trans. on Biomed. Eng., 37 (1990), 1087. doi: 10.1109/10.61034.

[60]

J. Wloka, Partial Differential Equations,, Cambridge University Press, (1987).

[61]

M. Zia, B. Griffel, V. Fridman, C. Saponieri and J. Semmlow, Noise detection in heart sound recordings,, Conf. Proc. IEEE Eng. Med. Biol. Soc., (2011), 5880. doi: 10.1109/IEMBS.2011.6091454.

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