Time-of-Flight Approximation for Medical Ultrasonic Imaging

  • Steven R. Broadstone
  • R. Martin Arthur
Part of the Acoustical Imaging book series (ACIM, volume 16)


Solutions of the acoustic wave equation linearized using the Born assumptions require determination of the time of flight from the transmitting transducer to the field point and back to the receiver. The time of flight within a plane in a medium that has a constant phase velocity is found by taking the square root of the sum of squares of scaled range and azimuth distances. The paraxial approximation significantly reduces calculation time and provides good estimates near the propagation axis of the transducer and at long range. We matched the time-of-flight profile with a second-degree, two-dimensional polynomial containing 9 coefficients that was more accurate than the paraxial approximation. Coefficients were found using the method of moments. In its forward difference form the polynomial can produce a new time-of-flight approximation in a single-addition time. If the division in the paraxial approximation is replaced by a scale factor, the result can also be expressed as a forward difference and evaluated in a single-addition time. Assuming a 1500 m/s background velocity, the maximum errors over a 3 x 3 cm region on the transducer axis fell below 1/3 the wavelength of a 3.5 MHz signal for ranges greater than 3 and 22 cm, for the first-difference form of the moments and paraxial approximations, respectively. At angles up to 30 degrees off axis the moments error fell below 1/3 wavelength beyond a range of 6 cm. At an angle of 15 degrees, the usual limit for the paraxial approximation, its maximum error was more than 10 wavelengths at distances up to 30 cm.


Arithmetic Operation Moment Approximation Backscatter Signal Forward Difference Paraxial Approximation 
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  1. 1.
    F. Calogero and A. Degasperis, “Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolution Equations,” Vol. I, North-Holland Publishing Company, New York (1982).MATHGoogle Scholar
  2. 2.
    J. F. Claerbout, “Imaging the Earth’s Interior,” Blackwell Scientific Publishing Company, London (1985).Google Scholar
  3. 3.
    P. D. Corl, G. S. Kino, C. S. DeSilets, and P. M. Grant, A digital synthetic focus acoustic imaging system, in: “Acoustic Imaging,” 8:39–53, A. F. Metherell, ed., Plenum Press, New York (1980).Google Scholar
  4. 4.
    P. M. Morse and K. U. Ingard, “Theoretical Acoustics,” McGraw-Hill, St. Louis (1968).Google Scholar
  5. 5.
    J. A. Fawcett, Inversion of N-Dimensional Spherical Averages, SIAM J. of Appl. Math., 45: 336–341 (1985).MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    A. Ozbek and B. C. Levy, Inversion of Parabolic and Paraboloidal Projections, IEEE Trans, on Acoust., Speech and Sig. Proc., in press (1987).Google Scholar
  7. 7.
    M. Kaveh, M. Soumekh, and R. K. Mueller, A Comparison of Born and Rytov Approximations in Acoustic Tomography, in: “Acoustic Imaging,” 11:325–335, J. P. Powers ed., Plenum Press, New York (1982).Google Scholar
  8. 8.
    P. Laiily, Migration methods: Partial but efficient solutions to the seismic inverse problem, in: “Inverse Problems of Acoustic and Elastic Waves,” pp. 182–214, SIAM, Philadelphia (1984).Google Scholar
  9. 9.
    D. de Vries, and A. J. Berkhout, A note on the effect of velocity errors in computerized acoustic focusing techniques, J. Acoust. Soc. Am., 74:353–356 (1983).ADSCrossRefGoogle Scholar
  10. 10.
    S. R. Broadstone and R. M. Arthur, An approach to real-time reflection tomography using the complete dataset, in: “IEEE 1986 Ultrasonics Symposium Proceedings,” 2:829–831, ed. B. R. McAvoy, IEEE No. 86CH2375-4, New York (1986).Google Scholar

Copyright information

© Plenum Press, New York 1988

Authors and Affiliations

  • Steven R. Broadstone
    • 1
  • R. Martin Arthur
    • 1
  1. 1.Biomedical Computer Laboratory and Department of Electrical EngineeringWashington UniversitySt. LouisUSA

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