Advanced Imaging 1024 Jan. 7, 2009 Ultrasound Lectures History and tour Wave equation Diffraction theory Rayleigh-Sommerfeld Impulse response Beams Lecture 1:Fundamental acoustics DG: Jan 7 Absorption Reflection Scatter Speed of sound Image formation: - signal modeling - signal processing - statistics Lecture 2:Interactions of ultrasound with tissue and image formation DG: Jan 14

The Doppler Effect Scattering from Blood CW, Pulsed, Colour Doppler Lecture 3: Doppler Ultrasound I DG: Jan 21 Velocity Estimators Hemodynamics Clinical Applications Lecture 4: Doppler US II DG: Jan 28 Lecture 5: Special Topics Mystery guest: Feb 4 or 11

ULTRASOUND LECTURE 1 Physics of Ultrasound Waves: The Simple View

ULTRASOUND LECTURE 1 Physics of Ultrasound: Longitudinal and Shear Waves

ULTRASOUND LECTURE 1 Physics of Ultrasound Waves: Surface waves

ULTRASOUND LECTURE 1 Physics of Ultrasound Waves 1)The wave equation y z u t u x u+Δ u Particle Displacement = Particle Velocity = Particle Acceleration = u x

Equation of Motion dV dxdydz x p    Net force = ma p = pressure = P – P 0 density or (1)

Definition of Strain S = strain Also  TermsNonlinear S C S B SAp... !3 32  bulk modulus Taking the derivative wrt time of (2) = (4) (2) (3)

Substituting for from Eq (1) (in one dimension) Substituting from or ; wave velocity 3 dimensions (3) (5)

For the one dimensional case solutions are of the form ; (6) for the forward propagating wave. A closer look at the equation of state and non-linear propagation Assume adiabatic conditions (no heat transfer) P = P 0 P                  Condensation = S Gamma=ratio of specific heats

P = P, Expand as a power series P = P  3 0 ' 0 ' 0 ' ! SCPSPSP B A .. B/A = Depends solely on thermodynamic factors

MaterialB/A Water 5 Soft Tissues 7.5 Fatty Tissues 11 Champagne (Bubbly liquid)

Additional phase term small for small x and increasingly significant as = shock distance (8) In terms of particle velocity, v, Fubini developed a non linear solution given by: Nonlinear Wave Equation (7)

Mach # (9) At high frequencies the plane wave shock distance can be small. So for example in water: Shock distance = 43 mm Shock Distance, l

Where (11) Thus the explicit solution is given by (12) We can now expand (Eq. 8) in a Fourier series (10)

Hamilton and Blackstock Nonlinear Acoustics 1998 Aging of an Ultrasound Wave

Hamilton and Blackstock Nonlinear Acoustics 1998 Relative Amplitude Harmonic Amplitude vs Distance (narrow band, plane wave)

Focused Circular Piston 2.25 MHz, f/4.2, Aperture = 3.8 cm, focus = 16 cm Hamilton and Blackstock Nonlinear Acoustics 1998

Propagation Through the Focus

Nonlinear Propagation: Consequences _______________________ Generation of shock fronts Generation of harmonics Transfer of energy out of fundamental

RADIATION OF ULTRASOUND FROM AN APERTURE We want to consider how the ultrasound propagates in the field of the transducer. This problem is similar to that of light (laser) in which the energy is coherent but has the added complexity of a short pulse duration i.e. a broad bandwidth. Start by considering CW diffraction theory based on the linear equation in 1 dimension

The acoustic pressure field of the harmonic radiator can be written as: Where is a complex phasor function satisfying the Helmholtz Equation (13) (14)

To solve this equation we make use of Green’s functions (15) s = surface area= normal derivative = pressure at the aperture

Rayleigh Sommerfeld Theory Assume a planar radiating surface in an infinite “soft” baffel Aperture Field Point Conjugate field point n use as the Greens function = 0 in the aperture

(16) Equation 15 can now be written:

Example Consider the distribution of pressure along the axis of a plane circular source: radius = a

From Equation (16) in r,, z coordinates The integrand here is an exact differential so that

The pressure amplitude is given by the magnitude of this expression (17)

To look at the form of (17) find an approximation for:

Maxima m = 1, 3, 5,...

Minima ; m = 0, 2, 4, 6

Eq. 17 M:

THE NEAR FIELD (off axis)

Fresnel approximation (Binomial Expansion) (19) From (16) we have

Note that the r’ in the denominator is slowly varying and is therefore ~ equal to z Grouping terms we have  )( ),( zK yx z jk jkz e zj e yxP  

Where (20) We can eliminate the quadratic term by “focusing” the transducer Thus the diffraction limit of the beam is given by:

Consider a plane circular focused radiator in cylindrical coordinates radius a Circular Aperture

Where FWH M (21)

Square Aperture

Need to consider the wideband case. Returning to Eq. 16 we have:

This is a tedious integration over 3 variables even after significant approximations have been made There must be a better way!

Impulse Response Approach to Field Computations Begin by considering the equation of motion for an elemental fluid volume i.e. Eq. 1 (22) Now let us represent the particle velocity as the gradient of a scalar function. We can write

Where is defined as the velocity potential we are assuming here that the particle velocity is irrotational i.e. ~ no turbulence ~ no shear waves ~ no viscosity Rewrite as (22) (23)

The better way: Impulse response method

Impulse Response where (24) Thus (25)

Useful because is short! convolution easy Also is an analytic function No approximations! Can be used in calculations You will show that for the CW situation (26)

IMPULSE RESPONSE THEORY EXAMPLE Consider a plane circular radiator is the shortest path to the transducer near edge of radiating surface far edge of radiating surface

Also let So that

(27) * * a very powerful formula while the wavefront lies between and Thus we have: (next page) ie

Planar Circular Aperture

otherwise small large Consider the on axis case:

Recall Equ 26 so that the pressure wave form is given by small large

Off axis case

5 mm radius disk, z = 80 mm

Spherically focused aperture - relevant to real imaging devices

Spherically focused aperture impulse response

Spherically focused aperture impulse responses

Spherically focused aperture pressure distribution a.f/2 b.f/2.4 c.f/3 Frequency = 3.75 MHz