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    MathWorks Inc matlab stable toolbox
    Phase velocity and attenuation in breast and liver obtained from the dispersion relation in Eq. (15) for the Treeby–Cox <t>wave</t> <t>equation</t> ( ∗ ), the approximations to the dispersion relation for the Treeby–Cox wave equation given in Eqs. (17) and (18) (•), and the attenuation and phase velocity for the <t>power</t> <t>law</t> wave equation given in Eqs. (1) and(2) (○), respectively.
    Matlab Stable Toolbox, supplied by MathWorks Inc, used in various techniques. Bioz Stars score: 90/100, based on 1 PubMed citations. ZERO BIAS - scores, article reviews, protocol conditions and more
    https://www.bioz.com/result/matlab stable toolbox/product/MathWorks Inc
    Average 90 stars, based on 1 article reviews
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    Article Title: Time-domain analysis of power law attenuation in space-fractional wave equations

    Journal: The Journal of the Acoustical Society of America

    doi: 10.1121/1.5047670

    Phase velocity and attenuation in breast and liver obtained from the dispersion relation in Eq. (15) for the Treeby–Cox wave equation ( ∗ ), the approximations to the dispersion relation for the Treeby–Cox wave equation given in Eqs. (17) and (18) (•), and the attenuation and phase velocity for the power law wave equation given in Eqs. (1) and(2) (○), respectively.
    Figure Legend Snippet: Phase velocity and attenuation in breast and liver obtained from the dispersion relation in Eq. (15) for the Treeby–Cox wave equation ( ∗ ), the approximations to the dispersion relation for the Treeby–Cox wave equation given in Eqs. (17) and (18) (•), and the attenuation and phase velocity for the power law wave equation given in Eqs. (1) and(2) (○), respectively.

    Techniques Used: Dispersion

    Time-domain Green's functions scaled by 4πr calculated for breast with y = 1.5, α0 = 0.086 Np/cm/MHz1.5, and c0 = 1450 m/s at (a) r = 1 nm, (b) r = 10 nm, (c) r = 100 nm, (d) r = 1 μm, (e) r = 100 μm, (f) r = 1 mm, (g) r = 1 cm, and (h) r = 10 cm with the power law (solid line), Chen–Holm (dashed line), and Treeby–Cox (dot-dashed line) wave equations. No noncausal behavior is observed at any distance in the time-domain Green's functions for the Chen–Holm and Treeby–Cox wave equations evaluated for breast, and beyond about r = 100 μm, the time-domain Green's functions for the power law wave equation and the Treeby–Cox wave equation are nearly indistinguishable.
    Figure Legend Snippet: Time-domain Green's functions scaled by 4πr calculated for breast with y = 1.5, α0 = 0.086 Np/cm/MHz1.5, and c0 = 1450 m/s at (a) r = 1 nm, (b) r = 10 nm, (c) r = 100 nm, (d) r = 1 μm, (e) r = 100 μm, (f) r = 1 mm, (g) r = 1 cm, and (h) r = 10 cm with the power law (solid line), Chen–Holm (dashed line), and Treeby–Cox (dot-dashed line) wave equations. No noncausal behavior is observed at any distance in the time-domain Green's functions for the Chen–Holm and Treeby–Cox wave equations evaluated for breast, and beyond about r = 100 μm, the time-domain Green's functions for the power law wave equation and the Treeby–Cox wave equation are nearly indistinguishable.

    Techniques Used:

    Time-domain Green's functions scaled by 4πr calculated for liver with y = 1.139, α0 = 0.0459 Np/cm/MHz1.139, and c0 = 1540 m/s at (a) r = 100 zm, (b) r = 1 am, (c) r = 10 am, (d) r = 100 am, (e) r = 100 μm, (f) r = 1 mm, (g) r = 1 cm, and (h) r = 10 cm with the power law (solid line), Chen–Holm (dashed line), and Treeby–Cox (dot-dashed line) wave equations. No noncausal behavior is observed at any distance in the time-domain Green's functions for the Chen–Holm and Treeby–Cox wave equations evaluated for liver, and beyond about r = 100 μm, the time-domain Green's functions for the power law wave equation and the Treeby–Cox wave equation are nearly indistinguishable.
    Figure Legend Snippet: Time-domain Green's functions scaled by 4πr calculated for liver with y = 1.139, α0 = 0.0459 Np/cm/MHz1.139, and c0 = 1540 m/s at (a) r = 100 zm, (b) r = 1 am, (c) r = 10 am, (d) r = 100 am, (e) r = 100 μm, (f) r = 1 mm, (g) r = 1 cm, and (h) r = 10 cm with the power law (solid line), Chen–Holm (dashed line), and Treeby–Cox (dot-dashed line) wave equations. No noncausal behavior is observed at any distance in the time-domain Green's functions for the Chen–Holm and Treeby–Cox wave equations evaluated for liver, and beyond about r = 100 μm, the time-domain Green's functions for the power law wave equation and the Treeby–Cox wave equation are nearly indistinguishable.

    Techniques Used:

    The (a), (c) amplitudes and (b), (d) FWHM values of the time-domain Green's functions calculated for the power law, Chen–Holm, and Treeby–Cox wave equations in (a), (b) breast and (c), (d) liver. The amplitudes of all three time-domain Green's functions decrease as the distance increases while the FWHM values of all three time-domain Green's functions increase as the distance increases. The amplitudes of the time-domain Green's functions for all three of these fractional wave equations are very similar at each distance, and the FWHM values are all approximately the same at longer distances, although there is a small difference in the FWHM values at shorter distances that diminishes with increasing distance.
    Figure Legend Snippet: The (a), (c) amplitudes and (b), (d) FWHM values of the time-domain Green's functions calculated for the power law, Chen–Holm, and Treeby–Cox wave equations in (a), (b) breast and (c), (d) liver. The amplitudes of all three time-domain Green's functions decrease as the distance increases while the FWHM values of all three time-domain Green's functions increase as the distance increases. The amplitudes of the time-domain Green's functions for all three of these fractional wave equations are very similar at each distance, and the FWHM values are all approximately the same at longer distances, although there is a small difference in the FWHM values at shorter distances that diminishes with increasing distance.

    Techniques Used:

    Simulated three-cycle Hanning-weighted pulses with center frequencies (a) f0 = 7.5 MHz and (b) f0 = 29 MHz convolved with time-domain Green's functions for the power law, Chen–Holm, and Treeby–Cox wave equations multiplied by 4πr evaluated in breast at r = 1 cm.
    Figure Legend Snippet: Simulated three-cycle Hanning-weighted pulses with center frequencies (a) f0 = 7.5 MHz and (b) f0 = 29 MHz convolved with time-domain Green's functions for the power law, Chen–Holm, and Treeby–Cox wave equations multiplied by 4πr evaluated in breast at r = 1 cm.

    Techniques Used:

    Simulated three-cycle Hanning-weighted pulses with center frequencies (a) f0 = 7.5 MHz and (b) f0 = 29 MHz convolved with time-domain Green's functions for the power law, Chen–Holm, and Treeby–Cox wave equations multiplied by 4πr evaluated in liver at r = 1 cm.
    Figure Legend Snippet: Simulated three-cycle Hanning-weighted pulses with center frequencies (a) f0 = 7.5 MHz and (b) f0 = 29 MHz convolved with time-domain Green's functions for the power law, Chen–Holm, and Treeby–Cox wave equations multiplied by 4πr evaluated in liver at r = 1 cm.

    Techniques Used:



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    Phase velocity and attenuation in breast and liver obtained from the dispersion relation in Eq. (15) for the Treeby–Cox <t>wave</t> <t>equation</t> ( ∗ ), the approximations to the dispersion relation for the Treeby–Cox wave equation given in Eqs. (17) and (18) (•), and the attenuation and phase velocity for the <t>power</t> <t>law</t> wave equation given in Eqs. (1) and(2) (○), respectively.
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    Phase velocity and attenuation in breast and liver obtained from the dispersion relation in Eq. (15) for the Treeby–Cox wave equation ( ∗ ), the approximations to the dispersion relation for the Treeby–Cox wave equation given in Eqs. (17) and (18) (•), and the attenuation and phase velocity for the power law wave equation given in Eqs. (1) and(2) (○), respectively.

    Journal: The Journal of the Acoustical Society of America

    Article Title: Time-domain analysis of power law attenuation in space-fractional wave equations

    doi: 10.1121/1.5047670

    Figure Lengend Snippet: Phase velocity and attenuation in breast and liver obtained from the dispersion relation in Eq. (15) for the Treeby–Cox wave equation ( ∗ ), the approximations to the dispersion relation for the Treeby–Cox wave equation given in Eqs. (17) and (18) (•), and the attenuation and phase velocity for the power law wave equation given in Eqs. (1) and(2) (○), respectively.

    Article Snippet: For the power law wave equation, the time-domain Green's function is calculated in Matlab with the STABLE toolbox, 19 which numerically evaluates the expression for the stable pdf in Eq. (21) .

    Techniques: Dispersion

    Time-domain Green's functions scaled by 4πr calculated for breast with y = 1.5, α0 = 0.086 Np/cm/MHz1.5, and c0 = 1450 m/s at (a) r = 1 nm, (b) r = 10 nm, (c) r = 100 nm, (d) r = 1 μm, (e) r = 100 μm, (f) r = 1 mm, (g) r = 1 cm, and (h) r = 10 cm with the power law (solid line), Chen–Holm (dashed line), and Treeby–Cox (dot-dashed line) wave equations. No noncausal behavior is observed at any distance in the time-domain Green's functions for the Chen–Holm and Treeby–Cox wave equations evaluated for breast, and beyond about r = 100 μm, the time-domain Green's functions for the power law wave equation and the Treeby–Cox wave equation are nearly indistinguishable.

    Journal: The Journal of the Acoustical Society of America

    Article Title: Time-domain analysis of power law attenuation in space-fractional wave equations

    doi: 10.1121/1.5047670

    Figure Lengend Snippet: Time-domain Green's functions scaled by 4πr calculated for breast with y = 1.5, α0 = 0.086 Np/cm/MHz1.5, and c0 = 1450 m/s at (a) r = 1 nm, (b) r = 10 nm, (c) r = 100 nm, (d) r = 1 μm, (e) r = 100 μm, (f) r = 1 mm, (g) r = 1 cm, and (h) r = 10 cm with the power law (solid line), Chen–Holm (dashed line), and Treeby–Cox (dot-dashed line) wave equations. No noncausal behavior is observed at any distance in the time-domain Green's functions for the Chen–Holm and Treeby–Cox wave equations evaluated for breast, and beyond about r = 100 μm, the time-domain Green's functions for the power law wave equation and the Treeby–Cox wave equation are nearly indistinguishable.

    Article Snippet: For the power law wave equation, the time-domain Green's function is calculated in Matlab with the STABLE toolbox, 19 which numerically evaluates the expression for the stable pdf in Eq. (21) .

    Techniques:

    Time-domain Green's functions scaled by 4πr calculated for liver with y = 1.139, α0 = 0.0459 Np/cm/MHz1.139, and c0 = 1540 m/s at (a) r = 100 zm, (b) r = 1 am, (c) r = 10 am, (d) r = 100 am, (e) r = 100 μm, (f) r = 1 mm, (g) r = 1 cm, and (h) r = 10 cm with the power law (solid line), Chen–Holm (dashed line), and Treeby–Cox (dot-dashed line) wave equations. No noncausal behavior is observed at any distance in the time-domain Green's functions for the Chen–Holm and Treeby–Cox wave equations evaluated for liver, and beyond about r = 100 μm, the time-domain Green's functions for the power law wave equation and the Treeby–Cox wave equation are nearly indistinguishable.

    Journal: The Journal of the Acoustical Society of America

    Article Title: Time-domain analysis of power law attenuation in space-fractional wave equations

    doi: 10.1121/1.5047670

    Figure Lengend Snippet: Time-domain Green's functions scaled by 4πr calculated for liver with y = 1.139, α0 = 0.0459 Np/cm/MHz1.139, and c0 = 1540 m/s at (a) r = 100 zm, (b) r = 1 am, (c) r = 10 am, (d) r = 100 am, (e) r = 100 μm, (f) r = 1 mm, (g) r = 1 cm, and (h) r = 10 cm with the power law (solid line), Chen–Holm (dashed line), and Treeby–Cox (dot-dashed line) wave equations. No noncausal behavior is observed at any distance in the time-domain Green's functions for the Chen–Holm and Treeby–Cox wave equations evaluated for liver, and beyond about r = 100 μm, the time-domain Green's functions for the power law wave equation and the Treeby–Cox wave equation are nearly indistinguishable.

    Article Snippet: For the power law wave equation, the time-domain Green's function is calculated in Matlab with the STABLE toolbox, 19 which numerically evaluates the expression for the stable pdf in Eq. (21) .

    Techniques:

    The (a), (c) amplitudes and (b), (d) FWHM values of the time-domain Green's functions calculated for the power law, Chen–Holm, and Treeby–Cox wave equations in (a), (b) breast and (c), (d) liver. The amplitudes of all three time-domain Green's functions decrease as the distance increases while the FWHM values of all three time-domain Green's functions increase as the distance increases. The amplitudes of the time-domain Green's functions for all three of these fractional wave equations are very similar at each distance, and the FWHM values are all approximately the same at longer distances, although there is a small difference in the FWHM values at shorter distances that diminishes with increasing distance.

    Journal: The Journal of the Acoustical Society of America

    Article Title: Time-domain analysis of power law attenuation in space-fractional wave equations

    doi: 10.1121/1.5047670

    Figure Lengend Snippet: The (a), (c) amplitudes and (b), (d) FWHM values of the time-domain Green's functions calculated for the power law, Chen–Holm, and Treeby–Cox wave equations in (a), (b) breast and (c), (d) liver. The amplitudes of all three time-domain Green's functions decrease as the distance increases while the FWHM values of all three time-domain Green's functions increase as the distance increases. The amplitudes of the time-domain Green's functions for all three of these fractional wave equations are very similar at each distance, and the FWHM values are all approximately the same at longer distances, although there is a small difference in the FWHM values at shorter distances that diminishes with increasing distance.

    Article Snippet: For the power law wave equation, the time-domain Green's function is calculated in Matlab with the STABLE toolbox, 19 which numerically evaluates the expression for the stable pdf in Eq. (21) .

    Techniques:

    Simulated three-cycle Hanning-weighted pulses with center frequencies (a) f0 = 7.5 MHz and (b) f0 = 29 MHz convolved with time-domain Green's functions for the power law, Chen–Holm, and Treeby–Cox wave equations multiplied by 4πr evaluated in breast at r = 1 cm.

    Journal: The Journal of the Acoustical Society of America

    Article Title: Time-domain analysis of power law attenuation in space-fractional wave equations

    doi: 10.1121/1.5047670

    Figure Lengend Snippet: Simulated three-cycle Hanning-weighted pulses with center frequencies (a) f0 = 7.5 MHz and (b) f0 = 29 MHz convolved with time-domain Green's functions for the power law, Chen–Holm, and Treeby–Cox wave equations multiplied by 4πr evaluated in breast at r = 1 cm.

    Article Snippet: For the power law wave equation, the time-domain Green's function is calculated in Matlab with the STABLE toolbox, 19 which numerically evaluates the expression for the stable pdf in Eq. (21) .

    Techniques:

    Simulated three-cycle Hanning-weighted pulses with center frequencies (a) f0 = 7.5 MHz and (b) f0 = 29 MHz convolved with time-domain Green's functions for the power law, Chen–Holm, and Treeby–Cox wave equations multiplied by 4πr evaluated in liver at r = 1 cm.

    Journal: The Journal of the Acoustical Society of America

    Article Title: Time-domain analysis of power law attenuation in space-fractional wave equations

    doi: 10.1121/1.5047670

    Figure Lengend Snippet: Simulated three-cycle Hanning-weighted pulses with center frequencies (a) f0 = 7.5 MHz and (b) f0 = 29 MHz convolved with time-domain Green's functions for the power law, Chen–Holm, and Treeby–Cox wave equations multiplied by 4πr evaluated in liver at r = 1 cm.

    Article Snippet: For the power law wave equation, the time-domain Green's function is calculated in Matlab with the STABLE toolbox, 19 which numerically evaluates the expression for the stable pdf in Eq. (21) .

    Techniques: