non-linear acoustics and contrast agents

what are the assumptions in linear acoustic (5)

system is linear:

1. doubling source amplitude doubles pressure everywhere

2. total acoustic pressure is the sum of individual waves

3. two pulses can cross without interacting

4. a propagating plane wave stays the same shape unless absorption occurs

5. sound speed depends only on the medium

what are some non-linear effects (6)

1. radiation pressure

2. streaming

3. cavitation

4. harmonic generation

5. shock formation

6. wave steepening

when does non-linear effects occur

1. high frequencies → non-linear effects are noticeable even at low acoustic pressures

2. large amplitude waves

   1. material non-linearity: fluids become harder to compress the higher the pressure

   2. convective non-linearity: change in wave speed due to particle movement

explain convective non-linearity

it is the change in waves speed (C) due to particle movement

1. in areas of high pressure in the wave (compression) particles have increased velocity → sound wave travels faster $C_{+}=C_{o}+|u|$ → peak catches up with trough → gradient steepens

2. in areas of low pressure in the wave (rarefaction): particles have decreased velocity → wave travels slower $C_{-}=C_{o}-|u|$ → trough lags behind peaks → gradient shallows (decreases)

results in a sawtooth pattern which can lead to a shock (straight line)

convective non-linearity in the wave equation

affects the conservation of mass equation $\frac{d\rho}{dt}=-\rho_{o}\nabla\cdot u-2\rho\nabla\cdot u$

explain material non-linearities

when a fluid is compressed sufficiently → stiffness increases → sound speed increases at higher pressures → peaks travel faster than troughs

affects constitutive equation: $p=c_o²(\rho+\frac{1}{2}\frac{B}{A}\frac{\rho²}{\rho_0})$ where $\frac{B}{A}$ is the non-linearity parameter

see notes for derivation

why is material and convective nonlinearities the only nonlinearities that are kept

• because their effect is cumulative as the wave travel → cumulative non-linearities

• all other non-linear effect do not carry forward with the wave and only affect local waves → local-nonlinearities

what is the non-linearity coefficient?

$\beta$ = $1+\frac{1}{2}\frac{B}{A}$ it is a measure of how a medium distorts an ultrasound wave as it propagates

what is the effect size of convective and material nonlinearities

the effect of material non-linearities is larges than convective ones

how does non-linearity affect frequency

• Single frequency input, : As the wave steepens, harmonics are generated

  • Energy from the lowest fundamental frequency gets moved up into higher frequencies

  • Frequencies appear at $2f_o,3f_0…$

• dual frequency input: $f_1~and~f_2$ simultaneously

  • harmonics from each frequency is generated

  • sum and differences of the frequencies also appear

what is a shock

when the gradient is steep/large enough for the pressure change to occur on an intermolecular scale

what is the shock formation distance

$$x=\frac{c_{o}^2}{\omega\beta u_{o}}=\frac{\rho_{o}c_{o}^3}{2\pi f\beta p_{o}}$$

why is shock formation not likely?

as the wave energy is pushed to higher frequencies absorption also increases with frequencies → energy never reaches threshold for shock formation

where is non-linear effects most likely to occur

at the focus because they require high energy acoustic pressure

what is tissue harmonic imaging

in normal imaging you send and listen for the fundamental frequency

in harmonic imaging you listen for the second harmonic

what are the advantages of harmonic imaging

1. harmonics are not generated near the probe → less noise at tissue surface

2. harmonic beams are thinner → better lateral resolution

does doubling frequency have the same effect as the first harmonic

no

1. increased grating lobe

2. increased clutter and noise at the surface of the tissue

how is the 1st harmonic isolated

1. A narrowband transmit pulse is used → transmit spectrum does not overlap with that of the second harmonic

2. 2 pulses are sent with inverted phase:

• non-linear effect causes part of the signal to move to higher frequency

• add the two signals together → fundamental parts sum to 0 → leaves non-linear component

• no change in pulse length → no change in axial resolution

what are the drawbacks of harmonic imaging

1. use of narrowband pulse → longer pulse length → worse axial resolution

2. need to send 2 pulses → reduced frame rate

instead of 2 inverted pulse what is another way of isolating the harmonic

amplitude modulation:

• send a pulse with twice the amplitude of the 1st one

• linear parts are subtracted forming an image from $B_2-2B_1$

what are the disadvantages of amplitude modulation

1. halves frame rate

2. amplitude energy needs to be limited for safety

what are contrast agents in US

microbubbles → small enough to pass through the lungs and long lived enough to pass to the organs → micron sized

what is the pressure inside the bubble

pressure of the gas inside the bubble is greater than the liquid outside → due to surface tension

$$P_{Gas}=P_{liquid}+P_{laplace}=P_{liquid}+\frac{2\delta}{R}$$

where:

• delta: surface tension

• R: bubble radius

how do you prolong bubble life

high internal gas pressure causes diffusion out of the bubble → dissolve in under 1s

1. lower surface tension → reduced laplace pressure → reduce diffusion

• ex: coating surface with amphiphilic molecules that act as a surfactant (ex: mixing patient blood with microbubble) → lowers surface tension + creates a barrier to gas diffusion

2. make the bubble larger

3. use large molecular weight gas → makes diffusion more difficult

bubble behaviour with frequency

• <0.1MPa: linear behaviour and linear cavitation

• >0.1MPa: high amplitude pulsation → slight non-linear effect

  • pulsation is not symmetrical as when the bubble is compressed the gas acts as a spring → produces large amplitude waves

  • generates harmonics

how does coating affect non-linear effect

• phospholipid coating → non-linear behaviour at low acoustic pressures

• polymer coating → non-linear behaviour at higher acoustic pressures

when are subharmonics generated

at very large frequencies:

• bubble shape may distort instead of bouncing

• surface oscillations

• bubble may miss some driving cycles

leads to the generation of subharmonics → fractions of the fundamental frequency

what is loss of correlation imaging

colour doppler mode is used to produce high pressure waves popping the contrast agent

• released gas produces strong reflections

• doppler interprets this as large random velocity fluctuations

• used for perfusion studies → areas without bubbles → not perfused

very sensitive but low frame rate

dynamic contrast enhancement ultrasound

1. Method 1: Bolus injection

   1. A bolus of USCA is injected

   2. The Time-Intensity Curve shows wash-in/wash-out time

2. Method 2: Flash replenishment

   1. USCA in image plane destroyed by a high amplitude pulse.

   2. Time-Intensity Curve shows the replenishment time