Biomedical Ultrasound (MPHY0018)
Acoustics Basics
• wavenumber k (rad/m) and wavelength λ (m) are related by k = 2π/λ
• temporal frequency, ω (rad/s) or f (Hz), and time period of a wave T (s) are related by f = ω/2π = 1/T
• wavelength, λ, frequency, f and wave speed c0 are related by c0 = f λ
• p is the acoustic pressure (Pa), u is the acoustic particle velocity (vector) (m/s), ρ is the acoustic density (kg/m3 )
• total field is given by the sum of ambient and acoustic parts, e.g., pT = p0 + p
• instantaneous acoustic intensity is a vector defined as Iinst (x, t) = p(x, t)u(x, t)
RT
• time-averaged acoustic intensity is Iav (x) = (1/T ) 0 pudt where T is the acoustic period
p
• sound pressure level Lp = 20log10 pref
dB, sound power (or intensity) level LP = 10log10 PPref dB
Deriving the Wave Equation
• material derivative D/Dt = ∂/∂t + (u · ∇)
• linearised equation of state p = c20 ρ
• linearised mass equation ∂ρ/∂t = −ρ0 ∇ · u, or ∂ρ/∂t = −ρ0 ∂u/∂x in 1D
• linearised momentum equation ∂u/∂t = −(1/ρ0 )∇p, or ∂u/∂t = −(1/ρ0 )∂p/∂x in 1D
• wave equation (∂ 2 /∂t2 − c20 ∇2 )p = 0, or (1/c20 )∂ 2 p/∂t2 = ∂ 2 p/∂x2 in 1D
• (linear) sound speed c20 = (∂p/∂ρ|s,0 = B/ρ0 where B is the bulk modulus
Solutions to the 1D Wave Equation (plane waves)
• plane wave solutions g(x ± c0 t) where ± indicates propagation towards x = ∓∞
• harmonic plane wave p = Re Aei(kx±ωt) = A cos(kx ± ωt)
• acoustic pressure and particle velocity are related by p = ∓ρ0 c0 u
• instantaneous acoustic intensity, Iinst (t) = p2 /ρ0 c0
• time-averaged acoustic intensity, Iav = A2 /2ρ0 c0
• first-order accurate forward difference ∂f /∂x|j ≈
fj+1 −fj
∆x
Wave Phenomena
• characteristic acoustic impedance Z = ρ0 c0
• normal incidence pressure reflection coefficient, R = (Z2 − Z1 )/(Z2 + Z1 )
• normal incident pressure transmission coefficient, T = 2Z2 /(Z2 + Z1 )
• for pressure reflection coefficients: 1 + R = T
• energy reflection coefficient: Re = |R|2
• energy transmission coefficient: Te = 1 − |R|2
• conservation of energy: Re + Te = 1
• Snell’s law: sin θi /c1 = sin θt /c2 where θi is the incidence angle and θt transmission angle
Attenuation and Absorption
• plane wave, p = Aei(kx−ωt) e−αx where α is the absorption coefficient
• intensity Iattenuated (x) = Iunattenuated e−2αx
• α in dB/cm = 8.7 × (α in Nepers/cm)
Ultrasound Transducers
• centre frequency fc = c/2L where L is the width of the piezoelectric element
• matched-load bandwidth (full-width-half-maximum) ∆f = 2c/3L
• fractional bandwidth = ∆f /fc × 100
• Quality factor Q = fc /∆f
• quarter-wave matching layer impedance ZML =
√
ZP ZT
• quarter-wave matching layer thickness LML = λML /4 = cML /4f
• matched load receive response S(f ) ∝ sinc(πf L/c)
• matched load receive response minima at f = nc/L,
n = 1, 2, . . .
Ultrasound Fields
• Here, a is the radius of a single element plane piston transducer, or, in the multiple element case, either half the
width of the array or half the elevation height of a single element.
• unfocussed beam, width of near field region W ≈ 2a
• unfocussed beam, position of last axial maximum (start of far field) zn = (4a2 − λ2 )/(4λ)
• unfocussed beam, far field directivity function D(θ) = 2J1 (ka sin θ)/ka sin θ where θ is the angle with the beam
axis, and J1 is a first-order Bessel function of the first kind
• unfocussed beam, far field beam divergence angle θd = sin−1 (0.61λ/a)
√
• unfocussed beam, acoustic pressure amplitude on beam axis |p(z)| = 2ρ0 c0 Un sin(k( a2 + z 2 − z)/2) where Un
is the amplitude of the normal particle velocity at the transducer surface
• focussed beam, width of focal zone W = 1.41λF/2a where F is the focal length
• focussed beam, length of focal zone D = 9.7λ(F/2a)2
• For an axial beam, grating lobes angles θ = sin−1 (nλ/d), n = 1, 2, . . . where d is the pitch
Diagnostic Ultrasound Imaging & Doppler
• pulse repetition frequency, PRF = c0 /2Lmax where Lmax is the imaging depth
• frame rate, FR = PRF/N where N is the number of scan lines
• axial resolution zmin = tp c0 /2 where tp is the pulse envelope width
• Doppler shift fd = 2f v cos θ/c0 where v is the flow velocity (which can be positive or negative), θ is the angle
between the flow and the transducer beam axis, and f is the transmit frequency
• pulse wave Doppler, maximum measurable Doppler shift fd,max = PRF/2
• pulse wave Doppler, maximum measurable velocity, vmax = c20 /(8f Lmax cos θ) where Lmax is the imaging depth
Nonlinear Acoustics
• convective and material nonlinearities c ≈ c0 + u + 0.5(B/A)u = c0 + βu
• coefficient of nonlinearity β ≡ 1 + 0.5(B/A) where B/A is the nonlinearity parameter
• shock distance xshock = ρ0 c30 /(2πf0 βp) where f0 is the frequency (Hz), p is the pressure amplitude.
Ultrasound Biophysics & Safety
• time-averaged acoustic intensity in 1D (in the presence of absorption) Iav (x) = (A2 /2ρ0 c0 )e−2αx
2
2
(x)
e−2αx = αPρA0 c(x)
where PA (x) = Ae−αx is the local
= ραA
• volume rate of heat deposition Q = − dIav
dx
0 c0
0
pressure amplitude
• temperature rise (no diffusion) ∆T = αPA2 t/ρ20 c0 C, where C (Jkg−1 K−1 ) is specific heat capacity, t (s) time
• thermal dose (cumulative equivalent minutes) t43 = tR43−T where T is the (constant) temperature in ◦ C, and
R = 0.5(T ≥ 43◦ C) or R = 0.25(T < 43◦ C)
• thermal index TI = P0 /Pdeg where P0 is the time averaged power, Pdeg is the power to raise tissue by 1 ◦ C
√
• mechanical index, MI = Pr / f depends on peak rarefactional pressure Pr in MPa, and f is frequency in MHz.
• volume radiation force F V = Q/c0
• surface radiation force Fabsorbing = P0 /c0 , Freflecting = 2P0 /c0
Elastography
• shear wave speed cs =
p
G/ρ0 where G is the shear modulus
• Young’s modulus in soft tissue E ≈ 3G
Metrology
• pulse duration td = 1.25 × (t90% − t10% ), where tx% is the time the pressure squared integral pi reaches x% of
its maximum.
• derated intensity I.3 = Ie−0.069fc z , where I is the intensity measured in water, fc (MHz) is the centre frequency
of the pulse used, and z (cm) is distance travelled.