Ultrasound, Doppler and dose numbers

Pulse-echo — depth = c t / 2 at the assumed 1540 m/s (65 μs → 5 cm, the book’s example); wavelength λ = c / f (5 MHz → 0.3 mm); the depth limit from 0.5 dB/cm/MHz attenuation, counted both ways, over a 60 dB dynamic range (3 MHz → 20 cm, 10 MHz → 6 cm) with a plot of echo level against depth for several probe frequencies; and the frame rate when every line waits for the deepest echo (15 cm and 128 lines → about 40 frames per second).

Reflection — R = Z₂ − Z₁)/(Z₂ + Z₁² between any two materials in the impedance table, or your own Z values, with the transmitted fraction, the amplitude ratio (negative means the echo is inverted) and the echo level in dB. The book’s three boundaries — fat/muscle, soft tissue/bone, soft tissue/air — are checked against the table.

Doppler — Δf = 2 f₀ v cos θ / c (the book’s carotid example: 3.8 MHz, 0.88 m/s, 60° → 2.2 kHz), the velocity a machine reports from a measured shift, and the effect of entering the angle wrong: the change in cos θ and in the reported velocity, plotted against the true angle.

Air ranging — the chapter 16 exercise with a 40 kHz ultrasonic distance sensor: distance from the echo pulse width at 343 m/s (or from the air temperature), the echo time to expect for a distance, how far a fixed 343 m/s is off in cold or warm air, and the wavelength.

Dose — an effective dose in mSv as days of natural background, as a range because background is 2–3 mSv a year, with the book’s dose table on one log scale.

Attenuation and distance — I = I₀ e^−μx with half- and tenth-value layers for water, cortical bone and lead at 60 and 100 keV, or your own μ; and the inverse-square law, D₂ = D₁ (d₁/d₂)².

Counting noise — a Monte Carlo of Poisson counting at means of 10 to 10⁴ (or your own list): the sample mean, standard deviation and mean/SD against √N, on a log–log plot, with a histogram of the smallest mean.

1540 m/s, 0.5 dB/cm/MHz, 60 dB, the reflection and Doppler equations, the worked examples, the dose table and 2–3 mSv a year background, 343 m/s in air: Chapter 16 and Appendix C. Inverse square: Chapter 17.

Acoustic impedances (MRayl): air 0.0004, lung 0.18, fat 1.34, water 1.48, kidney 1.63, blood 1.65, liver 1.65, muscle 1.71, bone 7.8 — Radiopaedia, “Acoustic impedance”, after Bushberg et al., The Essential Physics of Medical Imaging. The book gives no Z values.

μ/ρ: NIST X-ray mass attenuation coefficients (Hubbell and Seltzer) — water 0.2059 and 0.1707, cortical bone (ICRU-44, 1.92 g/cm³) 0.3148 and 0.1855, lead (11.35 g/cm³) 5.021 and 5.549 cm²/g at 60 and 100 keV.

Speed of sound in dry air: c ≈ 331.3 √(1 + T/273.15) m/s (ideal gas, T in °C).

Straight-line physics: no beam spreading, focusing, refraction, speckle, nonlinear propagation or frequency-dependent speed of sound; attenuation is linear in frequency.

Reflection is for a flat boundary at normal incidence. Tissue impedances vary between sources and between people.

The Doppler tab assumes one velocity along one beam; real spectra spread, and aliasing limits the highest measurable shift.

Beer–Lambert values are narrow-beam and monoenergetic; a real X-ray spectrum hardens as it passes through material, and scatter adds to a broad beam.

Effective doses are typical adult values for comparison only. They vary widely with protocol, equipment and body size, and say nothing about any one person.