Acoustic output, bioeffects and regulatory measurement: MI, TI, derating and the 510(k) output table
Every transmit sequence ends in a number a reviewer checks: a derated intensity, a mechanical index, three thermal indices. This module rebuilds where each number comes from (the bioeffect it stands for, the measurement behind it, the derating and the depth where it is evaluated), what the current FDA guidance and IEC standards actually say, and how coded excitation, plane-wave ultrafast imaging, push pulses and contrast imaging move the numbers.
You will be able to
- State the thermal and mechanical bioeffect mechanisms and the AIUM numbers that anchor them (1.5 °C, 4 °C for 5 min, MI 0.4 with gas bodies)
- Compute \(p_{r.3}\), \(\mathrm{PII}_{.3}\), \(I_{\mathrm{SPTA.3}}\), \(I_{\mathrm{SPPA.3}}\), pulse duration and duty factor from a waveform, a pulse rate and a scan geometry
- Derate correctly (pressure ÷20, intensity ÷10) and find the depth where each derated quantity peaks
- Evaluate MI and the TIS, TIB and TIC variants (scanning, non-scanning, break-point depth) for a defined mode
- Quote the FDA Track 3 limits, the ophthalmic limits and the Track 1 application-specific table exactly, with their conditions
- Plan a hydrophone and radiation-force measurement, correct for spatial averaging, and run a worst-case search
Assumes
- Hydrophone spatial averaging, pulse-echo test analysis and the derating/MI rule (Module 21, Simulation & characterization, Demo 21.6)
- Coded excitation and the fixed-MI energy argument (Module 10, Matched filtering)
- Nonlinear propagation and harmonics (Module 15, Wave physics & harmonics); plane-wave compounding (Module 23, Beams, k-space, compounding); microbubble response (Module 28, Microbubbles & contrast)
Numbers attributed to the FDA guidance (February 2023: Table 3, Sections 5.2.4–5.2.8) and the AIUM statements (2021; 2016, reapproved 2022) were checked against the current published text (October 2026). IEC clause-level details (formula constants, table layouts) are given only where a public source confirmed them; where the module relies on the ODS lineage or on a secondary source, it says so (the IEC 62359 definition of \(z_{\mathrm{MI}}\) as the depth of maximum attenuated PII is taken from published papers that cite the 2010 edition, and it matches the FDA glossary). Two items could not be confirmed from any public source and are flagged where they appear: the constant \(C_{\mathrm{TIB}}\) of the IEC 62359 \(\sqrt{P_{\alpha} I_{\mathrm{ta},\alpha}}\) form of TIB, and the surface-temperature limits of IEC 60601-2-37:2024. Always work from the edition of each standard your device declares conformity to.
22.1Two ways ultrasound acts on tissue: heat and mechanics
Heating. A wave of intensity I in a medium with intensity absorption coefficient \(\mu_a\) (twice the amplitude coefficient, in Np/m) deposits a power density \(q = \mu_a I\). Temperature follows the Pennes bioheat equation (22.1): conduction carries heat away over the beam radius, perfusion removes it over a perfusion length L of order 1 cm. Both are slow compared with any pulse, so heating responds to the time-average intensity or power, never to the peak.
- \(\rho c\)
- volumetric heat capacity of tissue (density × specific heat; here \(c\) is the specific heat, not the sound speed), J/(m³·K)
- \(T,\ t\)
- temperature (°C) and time (s)
- \(k\)
- thermal conductivity, W/(m·K)
- \(w_b,\ c_b\)
- blood perfusion rate, kg/(m³·s), and blood specific heat, J/(kg·K)
- \(T_a\)
- arterial blood temperature, °C
- \(\mu_a\)
- intensity absorption coefficient, Np/m
- \(I_{\mathrm{ta}}\)
- time-average intensity, W/m²
- \(\nabla^{2}T\)
- Laplacian of temperature (second spatial derivative), K/m²
- \(\partial T/\partial t\)
- rate of temperature rise, K/s
- \(Q\)
- power of a point heat source, W
- \(r\)
- distance from the source, m
- \(L\)
- perfusion length, m (of order 1 cm)
- \(\Delta T(r)\)
- steady-state temperature rise above baseline, K
- \(k,\ w_b c_b\)
- as in (22.1)
- \(\pi\)
- 3.14159
- \(a\)
- beam radius, mm
- \(\kappa\)
- thermal diffusivity, mm²/s
- \(\tau\)
- conduction time constant, s
- \(k,\ \rho c\)
- as in (22.1)
- \(t_i\)
- time spent at temperature \(T_i\), min
- \(T_i\)
- tissue temperature during interval \(i\), °C
- \(R\)
- dimensionless base
- \(\mathrm{CEM43}\)
- cumulative equivalent minutes at 43 °C, min
- \(\sum_i\)
- sum over time intervals i
What is safe thermally (AIUM, Statement on Biological Effects of Ultrasound in Vivo, 2021). A maximum in situ rise of 1.5 °C above 37 °C may be used clinically without reservation on thermal grounds. A fetal rise of 4 °C for 5 min or more can cause severe developmental defects; with commercial equipment this is unlikely at a fetal site if TI ≤ 2.5 and the dwell time is ≤ 4 min, and there is no time limit when TI stays below 0.7. Heating increases from B-mode to colour Doppler to M-mode to spectral Doppler and ARFI. Heating near bone is much larger than in soft tissue, so obstetric practice follows TIS to 10 weeks after the last menstrual period and TIB after.
Mechanics. Inertial cavitation needs a gas nucleus. For an optimally sized free gas nucleus in the diagnostic range, the threshold peak rarefactional pressure grows roughly as \(\sqrt{f}\) (Apfel & Holland 1991), which is why the mechanical index divides \(p_r\) by \(\sqrt{f}\). In tissue without gas bodies the AIUM finds no adverse non-thermal effects below MI 1.9, in vivo inertial cavitation thresholds at or above about 3.6 MPa for f ≥ 0.5 MHz, and lowest empirical thresholds of MI > 4 (blood-cell extravasation in mouse kidney). Gas bodies change everything: pulmonary capillary haemorrhage in some mammals has a threshold MI near 0.4, intestinal effects near 1.4; the separate AIUM contrast-agent statement (reapproved 2021) reports bioeffects with gas-body contrast agents above MI 0.4. Radiation force and streaming are the third mechanism; ARFI and SWE use them on purpose (Section 22.8).
Duty factor and the heating time constant
Approx. modelfirst-order lumped response \(d\Delta T/dt = (\Delta T_{\mathrm{ss}}\, s(t) - \Delta T)/\tau\); lower panel re-plotted from AIUM (2021) Table 1Heat integrates energy over seconds to minutes, so it scales with power and \(I_{\mathrm{ta}}\) and is indexed by TI. Cavitation responds within a single rarefaction half-cycle, so it scales with \(p_r\) and frequency and is indexed by MI. Two modes with the same MI can differ by 100× in \(I_{\mathrm{SPTA}}\), and two modes with the same \(I_{\mathrm{SPTA}}\) can differ by 10× in MI.
22.2The exposure quantities: pressures, PII, intensities, pulse duration, duty factor
All quantities come from a hydrophone waveform \(p(t)\) at a point in water, converted to intensity with the plane-wave relation \(i(t) = p^{2}(t)/(\rho c)\); the definitions follow in (22.5)–(22.10). That relation is an approximation (exact for a plane progressive wave, good in the focal region and far field, poor very close to the aperture), and every intensity in a submission inherits it.
- \(p(t)\)
- hydrophone pressure waveform at the point, MPa
- \(p_c\)
- peak compressional pressure, MPa
- \(p_r\)
- peak rarefactional pressure, MPa
- \(\max,\ \min\)
- maximum and minimum of the waveform over time
- \(\mathrm{PII}\)
- pulse-intensity integral (energy fluence per pulse), J/m² in SI; the FDA tables use J/cm² (1 J/cm² = 10⁴ J/m²)
- \(\rho c\)
- characteristic acoustic impedance of water, Rayl (1.5 MRayl)
- \(p(t)\)
- pressure waveform, as in (22.5)
- \(\int dt\)
- integral over the whole pulse
- \(t\)
- time, µs
- \(t_d\)
- pulse duration, µs
- \(t_{10},\ t_{90}\)
- times at which the cumulative \(\int p^{2}\,dt\) reaches 10 % and 90 % of its final value, µs
- \(1.25\)
- scale factor: converts the 10 % to 90 % energy interval to the full pulse duration (1/0.8)
- \(I_{\mathrm{pa}}\)
- pulse-average intensity, W/cm²
- \(I_{\mathrm{ta}}\)
- temporal-average intensity, mW/cm²
- \(\dot N\)
- pulses per second at the point, s⁻¹
- \(\mathrm{PII}\)
- pulse-intensity integral, as in (22.6)
- \(t_d\)
- pulse duration, as in (22.7)
- \(\times\)
- multiplication
- \(\mathrm{DF}\)
- duty factor at the point, dimensionless
- \(t_d,\ \dot N\)
- as in (22.7) and (22.8)
- \(I_{\mathrm{pa}},\ I_{\mathrm{ta}}\)
- as in (22.8)
- \(f_{\mathrm{frame}}\)
- frame rate, Hz
- \(w_{-6}\)
- −6 dB beam width, mm
- \(\Delta\)
- line spacing, mm
- \(0.75\)
- dimensionless factor \(\sqrt{\pi/2}/1.665\): pulses per frame summed over a Gaussian beam profile
- \(\dot N\)
- pulses per second at the point, as in (22.8)
The subscripts carry the meaning. SP (spatial peak) is the maximum over the field, TA a temporal average over the acoustic repetition period, PA an average over the pulse duration, ".3" or "α" the derated value (Section 22.3). In a non-scanning mode (PW Doppler, M-mode) every pulse passes the point; in a scanning mode (B-mode, colour) only the lines whose beams overlap the point contribute; in plane-wave and diverging-wave imaging every transmit insonifies every point, so the pulse rate at the point equals the PRF.
\(I_{\mathrm{SPTA}}\) from PII × pulses per second at the point
Approx. modeldefinitions of PII, \(t_d\) and \(I_{\mathrm{ta}}\) applied to Gaussian-envelope pulses; Gaussian line overlap for scanned modesFor a scanned focused mode, multiplying PII by the system PRF overstates \(I_{\mathrm{ta}}\) by the number of lines divided by the overlap count (128/3.8 ≈ 34 in the example). Multiplying PII by the frame rate understates it by the overlap count. Plane-wave modes are the opposite case: every transmit counts, and compounding angles do not dilute the exposure.
22.3Derating: 0.3 dB/cm/MHz, applied correctly
Fields are measured in water, which barely attenuates; tissue does. The derated ("attenuated", subscript α in IEC notation, ".3" in FDA notation) value multiplies the water value at depth z by the loss of a homogeneous model tissue with 0.3 dB cm\(^{-1}\) MHz\(^{-1}\) along the beam axis (22.11)–(22.12). The FDA guidance states the assumption in those words. The value is deliberately below typical soft tissue (about 0.5–1 dB/cm/MHz, Duck 1990) so that paths with long fluid segments, such as a full bladder or amniotic fluid, are not badly underestimated.
- \(A\)
- derating loss, dB
- \(f_{\mathrm{awf}}\)
- acoustic working frequency, MHz
- \(z\)
- depth along the beam axis from the transducer face, cm
- \(0.3\)
- derating coefficient, dB/(cm·MHz)
- \(p\)
- pressure in water, MPa
- \(I,\ \mathrm{PII},\ P\)
- intensity, pulse-intensity integral and power in water
- \(_{.3}\)
- derated value (IEC: subscript \(\alpha\))
- \(A\)
- derating loss, dB, as in (22.11)
- \(z,\ f_{\mathrm{awf}}\)
- as in (22.11)
Derating calculator: water, derated and in situ profiles
Approx. modellinear Gaussian-beam field of a circular aperture (source 1/e half-width 0.4 × aperture); the 0.3 dB/cm/MHz derating applied exactly as defined(1) Using ÷10 for pressure or ÷20 for intensity. (2) Derating the water spatial peak at its own depth instead of searching for the maximum of the derated profile. (3) Using the transducer nominal frequency instead of the measured acoustic working frequency. (4) Derating a waveform measured at high drive in water, where nonlinear propagation has already moved energy into harmonics and partly saturated the peak: the linear derating then misrepresents the in situ field, most for long low-attenuation paths. (5) Measuring z from the wrong origin: depth is measured from the transducer face (or lens surface) along the beam axis.
22.4The mechanical index
IEC 62359 defines MI from the attenuated peak rarefactional pressure at a specific depth \(z_{\mathrm{MI}}\), the depth on the beam axis where the attenuated pulse-intensity integral is largest, and the acoustic working frequency \(f_{\mathrm{awf}}\) (from the hydrophone waveform at the position of spatial-peak temporal-peak pressure, as defined in IEC 62127-1). The FDA guidance (Appendix A glossary) uses the same depth in its formula. Its one-line prose definition ("the spatial-peak value of the peak rarefactional pressure, derated by 0.3 dB/cm-MHz at each point along the beam axis") reads like a maximum over depth, but the formula that follows fixes the point: \(\mathrm{MI} = p_{r.3}(z_{\mathrm{sp}})/\sqrt{f_c}\), where \(z_{\mathrm{sp}}\) is "the point on the beam axis where the pulse intensity integral (\(\mathrm{PII}_{.3}\)) is maximum" and \(f_c\) is the centre frequency (22.13). MI is therefore not the maximum of \(p_{r.3}(z)\) over depth; it is \(p_{r.3}\) read at the depth where the derated PII peaks.
- \(p_{r,\alpha}\)
- attenuated (derated) peak rarefactional pressure, MPa
- \(z_{\mathrm{MI}}\)
- depth on the beam axis where the attenuated PII is largest, cm
- \(f_{\mathrm{awf}}\)
- acoustic working frequency, MHz
- \(C_{\mathrm{MI}}\)
- normalizing constant
- \(\mathrm{MI}\)
- mechanical index, dimensionless
- \(\sqrt{\cdot}\)
- square root of \(f_{\mathrm{awf}}\) in MHz
- \(z_{\mathrm{MI}}\)
- as in (22.13), cm
- \(\mathrm{PII}_{\alpha}(z)\)
- derated pulse-intensity integral at depth \(z\), as in (22.12)
- \(\arg\max_z\)
- the depth \(z\) at which the quantity is largest
- \(p_{r,\alpha}(z)\)
- derated peak rarefactional pressure, MPa, as in (22.13)
- \(p_r(z)\)
- peak rarefactional pressure in water at depth \(z\), MPa, as in (22.5)
- \(f,\ z\)
- acoustic working frequency (MHz) and depth (cm)
- \(0.3\)
- derating coefficient, dB/(cm·MHz)
- \(\mathrm{MI}\)
- mechanical index, as in (22.13)
- \(p_{r.3}\)
- derated peak rarefactional pressure, MPa (subscript .3 as in (22.12))
- \(f\)
- acoustic working frequency, MHz
- \(\Leftrightarrow\)
- equivalent to
- \(1.9\)
- upper MI limit (non-ophthalmic), dimensionless
MI indexes the likelihood of inertial cavitation of free gas nuclei and the scaling of gas-body effects, which is why it is displayed. It is not a threshold for shelled contrast microbubbles, whose response depends on resonance, shell and size distribution (Module 28), and it is evaluated at one depth with 0.3 dB/cm/MHz, not at the bubble with the real attenuation (Section 22.8). Nonlinear propagation combined with the focal phase shift of the harmonics makes \(p_c\) exceed \(p_r\) in a focused field (a plane-wave Fubini waveform stays symmetric), so reading \(p_c\) as the MI pressure overstates it.
22.5The thermal indices: TIS, TIB, TIC
A thermal index is a power ratio: \(\mathrm{TI} = W/W_{\mathrm{deg}}\), the relevant (attenuated, bounded) acoustic power divided by the power that raises the model tissue by 1 °C. It is not a temperature, but the AIUM states that TI predicts the maximum temperature rise within a factor of 2, with TIS usually overestimating or close, and underestimating by up to 2 in fetal examinations through low-attenuation fluid. Three tissue models define three indices.
| Index | Tissue model | \(W_{\mathrm{deg}}\) idea | Typical use |
|---|---|---|---|
| TIS (soft tissue) | Homogeneous soft tissue, 0.3 dB/cm/MHz | ≈ \(210/f_{\mathrm{awf}}\) mW (\(C_{\mathrm{TIS}}\) = 210 mW·MHz); heating ∝ absorbed power ∝ \(f\cdot W\) | Abdominal, vascular, first-trimester obstetric |
| TIB (bone at depth) | Soft tissue path, bone at the depth where heating is greatest | ≈ 40 mW/cm × beam diameter at the bone (all power absorbed in a thin bone layer) | Obstetric after 10 weeks LMP; pulsed Doppler near bone |
| TIC (bone at surface) | Bone immediately under the transducer | 40 mW/cm × \(D_{\mathrm{eq}}\) of the aperture (\(C_{\mathrm{TIC}}\) = 40 mW/cm) | Transcranial, paediatric head |
All three are power ratios \(W/W_{\mathrm{deg}}\) built on the standards' simplified models, not temperatures; the AIUM quotes agreement with measured maximum rise within about a factor of 2. Soft tissue: heating peaks near the surface for scanned modes and beyond \(z_{\mathrm{bp}} = 1.5\,D_{\mathrm{eq}}\) for large unscanned apertures (22.16)–(22.20). Bone at focus: power absorbed in a thin bone layer over the −6 dB beam diameter at \(z_b\). Bone at surface: the same over the aperture. Not to scale.
- \(P_1\)
- power through a 1 cm wide window of the scan, mW
- \(C_{\mathrm{TIS}}\)
- 210 mW·MHz
- \(f_{\mathrm{awf}}\)
- acoustic working frequency, MHz
- \(\mathrm{TIS}\)
- soft-tissue thermal index (scanning), dimensionless
- \(P\)
- total acoustic power, mW
- \(A_{\mathrm{aprt}}\)
- radiating aperture area, cm²
- \(\mathrm{TIS}\)
- soft-tissue thermal index (non-scanning, small aperture), dimensionless
- \(C_{\mathrm{TIS}},\ f_{\mathrm{awf}}\)
- as in (22.16)
- \(P_{\alpha}(z)\)
- attenuated power at depth \(z\), mW
- \(I_{\mathrm{ta},\alpha}(z)\)
- attenuated temporal-average intensity, mW/cm²
- \(z_{\mathrm{bp}}\)
- break-point depth, cm
- \(D_{\mathrm{eq}}\)
- equivalent aperture diameter, cm
- \(\mathrm{TIS}\)
- soft-tissue thermal index (non-scanning, large aperture), dimensionless
- \(\max,\ \min\)
- largest over depths beyond \(z_{\mathrm{bp}}\); smaller of the two bracketed quantities (mW)
- \(C_{\mathrm{TIS}},\ f_{\mathrm{awf}}\)
- as in (22.16)
- \(A_{\mathrm{aprt}}\)
- radiating aperture area, cm², as in (22.17)
- \(1\ \mathrm{cm^{2}}\)
- reference area that converts intensity (mW/cm²) to power (mW)
- \(d_{-6}(z)\)
- −6 dB beam diameter at depth \(z\), cm
- \(\mathrm{TIB}\)
- bone thermal index, dimensionless
- \(\mathrm{TIB}_{\mathrm{scan}},\ \mathrm{TIS}_{\mathrm{scan}}\)
- scanning-mode bone and soft-tissue thermal indices
- \(P_{\alpha}(z)\)
- attenuated power, mW, as in (22.18)
- \(40\ \mathrm{mW/cm}\)
- bone normalizing constant \(C_{\mathrm{TIC}}\) of (22.20)
- \(P\)
- total acoustic power, mW
- \(C_{\mathrm{TIC}}\)
- 40 mW/cm
- \(D_{\mathrm{eq}}\)
- equivalent aperture diameter, cm
- \(\mathrm{TIC}\)
- cranial-bone thermal index, dimensionless
- \(A_{\mathrm{aprt}}\)
- radiating aperture area, cm², as in (22.17)
- \(\pi\)
- 3.14159
MI and TI calculator for a defined mode
Approx. modellinear separable Gaussian-beam field; \(\mathrm{MI} = p_{r.3}/\sqrt{f}\) at \(z_{\mathrm{MI}}\); \(\mathrm{TIS} = P f/210\) (scanned: \(P_1\); unscanned \(A > 1\) cm²: \(\min[P_{\alpha}, I_{\mathrm{ta},\alpha}\cdot 1\ \mathrm{cm^{2}}]\) beyond \(z_{\mathrm{bp}}\)); \(\mathrm{TIB} \approx P_{\alpha}/(40\ \mathrm{mW/cm}\times d_{-6})\); \(\mathrm{TIC} = P/(40\,D_{\mathrm{eq}})\). The standards' simplified formulas, not the full IEC 62359 procedure22.6The regulatory frame: FDA guidance, IEC standards, ALARA
FDA. Marketing Clearance of Diagnostic Ultrasound Systems and Transducers, issued February 21, 2023 (originally issued June 27, 2019; the 2019 version superseded the September 9, 2008 guidance; docket FDA-2017-D-5372). It defines two tracks. Track 1: devices that do not conform to the Output Display Standard in IEC 60601-2-37; output is compared with application-specific preamendments levels. Track 3: devices that conform to the Output Display Standard in IEC 60601-2-37; global maximum limits apply and MI/TI are displayed so the user can apply ALARA.
| FDA guidance (2023) | \(I_{\mathrm{SPTA.3}}\) (mW/cm²) | \(I_{\mathrm{SPPA.3}}\) (W/cm²) | MI | Condition |
|---|---|---|---|---|
| Track 3, global maximum (all applications except ophthalmic) | ≤ 720 | ≤ 190 | ≤ 1.9 | \(I_{\mathrm{SPTA.3}} \le 720\) and either \(\mathrm{MI} \le 1.9\) or \(I_{\mathrm{SPPA.3}} \le 190\) |
| Track 3, ophthalmic | ≤ 50 | n/a | ≤ 0.23 | \(\mathrm{TI} = \max(\mathrm{TIS}_{\mathrm{as}}, \mathrm{TIC}) \le 1\) |
| Track 1: peripheral vessel | 720 | 190 | 1.9 | Preamendments levels (Table 3), derated; \(I_{\mathrm{SPPA.3}}\) or MI |
| Track 1: cardiac | 430 | 190 | 1.9 | |
| Track 1: fetal imaging & other | 94 | 190 | 1.9 | |
| Track 1: ophthalmic | 17 | 28 | 0.23 |
Other statements in the guidance that matter at the bench: an explanation is expected for any TI above 6.0; if no system/transducer combination can exceed a TI of 1.0 or an MI of 1.0 in any mode, the global maximum values are reported instead of the full per-mode tables; the Table 3 levels are themselves estimated to carry ±30 % (intensities) and ±15 % (MI) uncertainty, so a manufacturer whose measurement uncertainty is within those margins need not subtract it, and one whose uncertainty exceeds them should reduce the levels by the excess; measurements should use a spot-poled membrane or capsule hydrophone unless another type is shown equivalent or better; the protocol must assure that global maximum output conditions are identified, especially in autoscanning and combined modes; labelling should state the ALARA precaution; for transcranial (cephalic) use with \(I_{\mathrm{SPTA.3}}\) above 94 mW/cm², an estimate of the maximum temperature rise and on-screen precautions (scanning through the eye, burr holes, fontanels, foramen magnum) are recommended; fetal heart-rate monitors (CW) are held to 20 mW/cm² \(I_{\mathrm{SATA}}\) at the transducer face.
IEC documents, current editions. IEC 60601-2-37:2024 (Ed. 3.0, July 2024; replaces Ed. 2 of 2007 and Amendment 1:2015; clause 201.11 on excessive temperatures fully revised): particular safety requirements and the output display. IEC 62359:2010 + AMD1:2017 (Ed. 2.1): test methods for TI and MI; a third edition that adds "and acoustic intensities" to the title was circulated as a draft for comment in 2023 (check whether it has published before citing it). IEC 62127-1:2022 (Ed. 2.0): hydrophone measurement of medical fields; it removed the 40 MHz upper limit, made sensitivity complex-valued, and amended the spatial-averaging correction. IEC 62127-2 (hydrophone calibration) and IEC 62127-3 (hydrophone properties). IEC 61161:2013 (Ed. 3.0): power measurement with radiation force balances.
History in one paragraph. The Track 1 levels descend from devices marketed before the 1976 Medical Device Amendments. The AIUM/NEMA Output Display Standard (UD 3) introduced the real-time display of MI and TI so that users, not fixed application limits, manage exposure; its 1998 (Revision 1) and 2004 (Revision 2) versions were later superseded by IEC 60601-2-37, and the FDA notes that the AIUM has withdrawn its equivalent standard. The physics of the indices (\(W_{\mathrm{deg}}\) models, break-point depth, MI from Apfel & Holland) is unchanged from the ODS.
| Acoustic-output section of a submission | What the reviewer looks for |
|---|---|
| Track and applications | Track 3 conformity to IEC 60601-2-37 (or Track 1 with application-specific levels); intended uses matching the indications table |
| Global maxima per transducer/mode | MI, TIS, TIB, TIC, \(I_{\mathrm{SPTA.3}}\), \(I_{\mathrm{SPPA.3}}\) (or MI) against Track 3; TI > 6 explained |
| Reporting tables | Per IEC 60601-2-37 example: indices with the parameters behind them (\(p_{r.\alpha}\), P, \(P_1\), \(z_{\mathrm{MI}}\), \(z_s\), \(z_b\), \(f_{\mathrm{awf}}\), prr, srr, beam dimensions, aperture) for each combination whose displayed MI or TI exceeds 1.0 |
| Measurement protocol | Hydrophone type and calibration, spatial-averaging and bandwidth corrections, power method, the global-maximum search, Type A and Type B uncertainties |
| Display accuracy | Displayed MI/TI versus measured values across settings |
| Labelling | ALARA statement, output tables in the user documentation, application-specific precautions |
As Low As Reasonably Achievable means the user, guided by the displayed indices, lowers output and dwell time to the minimum that answers the clinical question. The AIUM dwell-time tables (reapproved 2022) turn the displayed TI into a time budget: obstetric TI ≤ 0.7 no limit, 1.0–1.5 under 30 min, 2.5–3.0 under 1 min, above 3.0 not recommended; non-obstetric TI ≤ 1.5 no limit, 4–5 under 1 min, 5–6 under 15 s, above 6 not recommended; reduce dwell by 33 % for ARFI and pulsed Doppler when bone is near the focus.
22.7Measurement: hydrophones, spatial averaging, radiation force, the worst-case search
The tank. Degassed (and for high pressures deionized) water, absorbing walls, a 3-axis positioner with angular alignment, and a hydrophone with its preamplifier and calibration file. Alignment and the search for the beam axis and spatial peak come first, then axial scans, then lateral and elevation scans at the depths that matter (\(z_{\mathrm{MI}}\), \(z_s\), \(z_b\)).
Hydrophones. Spot-poled membrane hydrophones (with capsule hydrophones, the FDA's stated default for pulsed waveforms) have the most uniform frequency response; needle hydrophones are small and robust but their response is less flat and their effective radius can differ from the geometric one; fibre-optic hydrophones survive high pressures and reach small sizes. The calibration is the complex end-of-cable sensitivity \(M_L(f)\) (IEC 62127-2), so p(t) is obtained by deconvolution. Finite size averages the field over the sensitive area (spatial averaging, below) and finite bandwidth drops and phase-shifts harmonics (Module 21).
Typical tank about 60 × 40 × 40 cm of degassed water with absorbing lining; lateral steps of 0.05–0.1 mm through a focus and 0.5–1 mm axially; \(f_{\mathrm{awf}}\) from the waveform at the position of the \(p_r\) peak. The hydrophone voltage is deconvolved with the complex end-of-cable sensitivity \(M_L(f)\) and corrected for spatial averaging before any derating. Power comes from a radiation force balance (absorbing target, \(F = P/c\) (22.23): 100 mW gives 66.7 µN) per IEC 61161. Not to scale.
- \(w\)
- Gaussian focal spot parameter: amplitude \(e^{-r^{2}/w^{2}}\), −6 dB width \(W_{-6} = 1.665\,w\), mm
- \(b\)
- hydrophone radius, mm
- \(\langle p\rangle\)
- pressure averaged over the hydrophone face, MPa
- \(p(0)\)
- true on-axis peak pressure, MPa
- \(r\)
- radial distance from the beam axis, mm
- \(e\)
- exponential function
- \(w,\ b\)
- Gaussian spot parameter and hydrophone radius, mm
- \(\text{intensity error}\)
- fractional intensity (pressure squared) reading relative to the true peak; below 1 means under-reading
- \(w_{\mathrm{meas}}\)
- measured (broadened) spot parameter, mm
- \(w,\ b\)
- as in (22.21), mm
- \(\text{peak}\)
- peak pressure reading corrected for spatial averaging, MPa
- \(F\)
- radiation force on an absorbing target, N
- \(P\)
- acoustic power, W
- \(c\)
- speed of sound in water, m/s
Hydrophone spatial averaging versus element size and beam width
Approx. modelelliptical Gaussian focal spot; numerical disc average over the hydrophone face; Gaussian-kernel correctionThe worst-case (global maximum) search. Each index has its own maximizing setting. MI usually peaks at the highest transmit voltage, the shortest focus that still lies beyond the near-field region, and the lowest frequency of a multi-frequency probe; TIS in scanned modes at the highest frame rate and widest power; TIB in Doppler at a moderate focus with a narrow beam and the highest PRF; TIC at the largest total power and smallest aperture. A defensible protocol enumerates the parameters (voltage, focal depth, aperture, frequency, pulse length, PRF, line density, sector width, combined modes, zoom), uses the system's own output model to rank candidates, and measures the top candidates for every index. Combined modes (B + colour + PW) sum contributions and are where FDA asks most often for evidence that the global maximum was found.
Surface temperature. Transducer self-heating is limited separately from TI, and the FDA guidance points to IEC 60601-2-37 for it. For IEC 60601-2-37:2007 + A1:2015 the applied-part limits are usually quoted as 43 °C when coupled to a tissue-mimicking test object and 50 °C in still air (from secondary sources; the 2007 clause text was not available for this module). Flag: the foreword of IEC 60601-2-37:2024 (Ed. 3.0) states that clause 201.11 on excessive temperatures "has been fully revised", and the revised limits and test conditions could not be confirmed from any public source (October 2026). Do not carry 43 °C / 50 °C into a test plan against the 2024 edition without reading its clause 201.11.
Each index has its own maximizing setting, so the search runs per index and per transducer/mode, with combined modes summed. The output model only ranks candidates; the reported numbers are measured. The FDA guidance asks that the protocol assure global maximum conditions are found, especially in autoscanning and combined modes, and expects reporting tables for every combination whose global maximum MI or TI exceeds 1.0.
22.8New sequences, new numbers: coding, ultrafast, push pulses, contrast and ULM
Coded excitation (Module 10). At a fixed peak pressure, MI is unchanged while PII, \(I_{\mathrm{SPTA}}\) and power scale with the energy ratio (the compression gain). A TB-50 chirp at MI 1.9 and 200 pulses/s at a point goes from 27 to about 2000 mW/cm², so a coded mode is usually limited by \(I_{\mathrm{SPTA.3}}\) or TI, not MI.
Plane-wave and diverging-wave ultrafast imaging. The unfocused transmit has a low peak pressure (low MI) but every transmit reaches every point, so \(I_{\mathrm{ta}} = \mathrm{PII}\times\mathrm{PRF}\). At 5 kHz this is 40–50 times the pulse rate a focused B-mode delivers to a point, and long acquisitions (Doppler ensembles of hundreds of frames, ULM acquisitions of seconds to minutes) turn that into TI and dwell-time questions.
Push pulses (ARFI, SWE). A push is a focused burst of hundreds of microseconds (hundreds to a few thousand cycles), often near the MI limit. Its PII is hundreds of times that of an imaging pulse, so a few pushes per second dominate \(I_{\mathrm{SPTA.3}}\) and focal heating, and the tracking ensemble adds on top. The AIUM places ARFI with spectral Doppler at the top of the heating ranking and asks for 33 % shorter dwell times when bone is near the focus. Report the push and the tracking sequence as one combined mode, and evaluate heating at the push focus and at the transducer surface.
Contrast imaging and ULM. Low-MI contrast imaging is designed to keep bubbles oscillating without destroying them; high-MI flashes destroy them for replenishment studies. The pressure at the bubble is the in situ pressure at its depth, with the real attenuation of the path, not the derated pressure at \(z_{\mathrm{MI}}\): through a fluid path it is higher than the derated value, through liver it is lower. The AIUM contrast statement (reapproved 2021) reports bioeffects above MI 0.4 with gas-body agents and recommends the minimal MI and examination time; BMUS (2009) names MI 0.7 as the cavitation-risk threshold with contrast. ULM adds long plane-wave acquisitions at kHz rates: low MI per pulse and an \(I_{\mathrm{SPTA}}\) set by the PRF (a time average, independent of duration), while heating, and with it the TI dwell-time budget, accumulates over the whole acquisition.
| Sequence | MI | \(I_{\mathrm{SPTA.3}}\) | TI | What binds first |
|---|---|---|---|---|
| Focused B-mode | High (focused, short pulses) | Low (few pulses per point) | Low TIS (power spread over the scan) | MI |
| Coded B-mode | Same as B-mode at equal peak | × compression gain | × compression gain | \(I_{\mathrm{SPTA.3}}\) or TI |
| PW Doppler | Moderate | High (every pulse at the gate) | TIB highest near bone | TIB, \(I_{\mathrm{SPTA.3}}\) |
| Plane-wave ultrafast | Low (unfocused) | \(\propto\) PRF | Grows with PRF and duration | TI and dwell time |
| ARFI/SWE push | Near limit | Pushes dominate | Focal and surface heating | TI, surface temperature, \(I_{\mathrm{SPTA.3}}\) |
| Low-MI contrast / ULM | Very low (bubble survival) | Low per pulse, × PRF | Low | Bubble destruction (physics), not regulation |
22.9Visual summary
Fourteen pictures that carry the module. Each card is a question to answer in five seconds, the answer with its number, and a figure computed in the page; the badge says whether it is a definition or published table plotted as is, a simulation or an approximate model (water \(\rho c\) = 1.5 MRayl, \(c\) = 1500 m/s unless stated).
22.10How experts think about it
An expert keeps two ledgers. The mechanical ledger has one entry per pulse: the derated peak rarefactional pressure at the depth the standard specifies, divided by \(\sqrt{f}\) (22.13). The thermal ledger integrates: PII times the number of pulses that actually pass the point, power through the right window, divided by the power that heats the model tissue by a degree. Every number in a 510(k) table is one of these, derated with the right exponent at the right depth, measured with a hydrophone whose size and bandwidth have been accounted for, and taken at the global maximum over settings.
Conduction and perfusion time constants are seconds to minutes (\(a^{2}/4\kappa \approx\) 1.9 s for a 1 mm beam, (22.3)), while a pulse lasts a microsecond, so tissue temperature sees only the average absorbed power. Bubble growth in a rarefaction half-cycle happens within that microsecond, so only the peak negative pressure and the time available (\(1/f\)) matter. A B-mode line and a PW Doppler gate at the same \(p_r\) differ in \(I_{\mathrm{ta}}\) by the pulse-rate ratio: 113 versus 5000 pulses/s is 44×.
The derating is a loss in dB, the same number of dB for pressure and intensity; only the conversion to a linear factor differs. \(A(z) = 0.3\,f z\) (22.11): at 7.5 MHz and 4 cm it is 9.0 dB, pressure × 0.355 and intensity × 0.126. Mixing the exponents misstates MI by the square root (or the square) of the intended factor, a 40 % error at 4.5 dB.
Multiplying a focal gain curve by a falling exponential moves its maximum toward the probe; the shift grows with f, with depth and with a weak focal gain. For the Gaussian-beam model of a 12 mm circular aperture focused at 8 cm, the water maximum is at 6.5 cm (focal shift), the derated maximum at 5.1 cm, and the derated pressure there is 0.49 of the water peak. A test plan that only scans around the water focus can miss \(z_{\mathrm{MI}}\).
Invert the definition (22.13) with \(C_{\mathrm{MI}} = 1\ \mathrm{MPa\cdot MHz^{-1/2}}\). The water pressure needed at 4 cm is larger by \(10^{0.3\cdot f\cdot 4/20}\): 4.98 MPa at 3 MHz, 8.5 MPa at 5 MHz and 14.7 MPa at 7.5 MHz. High-frequency probes rarely reach MI 1.9 at depth; they reach it in the first centimetres where derating is small.
For a Gaussian pulse \(I_{\mathrm{pa}} = 0.782\,p^{2}/(2\rho c)\). At MI 1.9 that is 188 W/cm² at 2 MHz, 282 at 3 MHz and 471 at 5 MHz: above 2 MHz most short-pulse modes at the MI limit exceed 190 W/cm² and comply through the MI branch. A low-frequency long-burst mode can do the reverse. Both conditions sit under the \(I_{\mathrm{SPTA.3}}\) limit, which always applies.
A point is hit by every line whose beam covers it, weighted by the beam profile. For a Gaussian profile \(e^{-2x^{2}/w^{2}}\) the sum over lines spaced \(\Delta\) is \(\sqrt{\pi/2}\,w/\Delta\), and \(w = w_{-6}/1.665\), so \(n = 0.753\,w_{-6}/\Delta\) (22.10). At 30 frames/s the point sees 113 pulses/s, not the 3840/s the system fires; the factor of 34 is the most common error in hand calculations.
The definition is energy-based, so ringing tails count only through their energy. For a Gaussian envelope the cumulative \(p^{2}\) is an error function with standard deviation \(\sigma_t/\sqrt{2}\); its 10–90 % width is \(1.812\,\sigma_t\), times 1.25 gives \(2.266\,\sigma_t\). For a rectangular burst of length \(T\), the factor 1.25 makes \(t_d = T\) exactly (22.7), which is the reason for the 1.25.
\(\mathrm{DF} = t_d\) × (pulses per second at the point) (22.9). A 3-cycle, 3 MHz pulse lasts 0.96 µs: at 5 kHz DF = 0.48 %, so 104 W/cm² pulse-average becomes 502 mW/cm². In scanned B-mode at 113 pulses/s the DF is \(1.1\times 10^{-4}\). Plane-wave ultrafast reaches PW-Doppler duty factors with short pulses, and push sequences reach them with a few long bursts.
\(W_{\mathrm{deg}}\) comes from steady-state models with fixed absorption, conduction and perfusion. Integrating the steady Pennes solution for a collimated Gaussian beam carrying \(W_{\mathrm{deg}} = 210/f\) mW (\(k\) = 0.5 W/m/K, \(L\) = 1 cm) gives a peak rise of 0.5–1.6 °C depending on beam radius (0.5–4 mm), and only 20–30 % lower at 7.5 MHz than at 1 MHz, because the absorbed power per unit length \(\propto f\cdot W\) is the same 210 mW·MHz at every frequency. The index is useful for ranking and ALARA; it is not a thermometer.
Absorbed power per unit length is \(\mu_a P\) with \(\mu_a \propto f\), so the power for a given rise scales as \(1/f\). In a scanned mode the heat source is the beam swept across the scan, and the maximum rise is near the surface where the swept power per unit width is highest, hence the bounded power through a 1 cm window: a 4 cm scan radiating 80 mW at 3.5 MHz has \(P_1\) = 20 mW and TIS = 0.33 (22.16).
Bone absorbs almost all incident power in a thin layer, so the rise depends on power per unit beam diameter, not on frequency. For a 1 cm² aperture \(D_{\mathrm{eq}}\) = 1.128 cm and TIC reaches 1 at 45 mW (22.20); a transcranial probe radiating 200 mW through 2 cm² has TIC = 3.1. TIB uses the same idea at the depth where a narrow beam meets bone, which is why Doppler TIB often exceeds TIS by 3–5×.
Near a large aperture the power is spread over more than 1 cm², so the relevant heat source is the intensity times 1 cm²; deeper, the beam has converged and the whole attenuated power counts. The smaller of the two, maximized over \(z \ge z_{\mathrm{bp}}\), approximates the hottest point (22.18). For a 19.2 × 10 mm aperture \(z_{\mathrm{bp}}\) = 2.35 cm; in the PW preset the maximum is at \(z_{\mathrm{bp}}\) and TIS = 0.34 while TIB = 1.5.
The hydrophone reports the average pressure over its face; squaring the average underestimates the peak intensity by the square of the pressure loss. For a round Gaussian focus \(\langle p\rangle/p(0) = (w^{2}/b^{2})\left(1 - e^{-b^{2}/w^{2}}\right)\) (22.21). A 0.4 mm hydrophone on a 7.5 MHz, F/3 focus (\(W_{-6} \approx 1.0\,\lambda F_{\#} \approx\) 0.6 mm) is at \(d/W_{-6}\) = 0.67: −1.3 dB in pressure and 26 % low in \(I_{\mathrm{SPTA}}\) for a round beam, −0.7 dB and 15 % for a 3:1 elongated one.
A plane wave carries momentum flux \(P/c\) (22.23); an absorbing target takes all of it, a perfect reflector normal to the beam twice, and a 45° conical reflector \(P/c\) along the axis (\(2\cos^{2}45^{\circ} = 1\)). At \(c\) = 1500 m/s, 10 mW is 6.7 µN or 0.68 mg, so diagnostic power needs a microgram balance, draught shielding and temperature control. Focusing and steering reduce the axial component and need the corrections in IEC 61161.
A focused B-mode point sees about frame rate × \(0.75\,w_{-6}/\Delta\) pulses per second; a plane-wave sequence at PRF 5 kHz delivers 5000. Compounding angles do not dilute this: each angle insonifies the field. The low unfocused peak pressure keeps MI small, but \(I_{\mathrm{SPTA}}\), power and TI grow linearly with PRF and with acquisition length, which is what an ultrafast or ULM mode must document.
\(\mathrm{PII} = p^{2}/(2\rho c)\times\int \mathrm{env}^{2}\,dt\) grows with burst length. A 1200-cycle push at 3 MHz (400 µs at −6 dB) carries 600 times the PII of a 2-cycle pulse at the same pressure. At \(p_{r.3}\) = 3 MPa (MI 1.73) one push is 0.090 J/cm²; two per second give 181 mW/cm² from the pushes alone. Push repetition, not push amplitude, is usually the lever that keeps an SWE mode inside 720 mW/cm² and within TI limits.
Fetal bone ossifies through the second trimester and absorbs strongly, so the bone-at-focus index becomes the relevant one (AIUM, reapproved 2022). The dwell tables encode the AIUM \(\Delta T\)–time data (fetal 4 °C: 2 min; 2 °C: 30 min; 1.5 °C: 1 h) with margin, using the displayed TI as a conservative proxy for \(\Delta T\).
Pre-existing gas removes the nucleation barrier, so effects appear at far lower pressures than in gas-free tissue, where in vivo inertial cavitation needs about 3.6 MPa or more. The AIUM (2021) puts the pulmonary capillary haemorrhage threshold in laboratory mammals near 0.4 and intestinal effects near 1.4; its contrast-agent statement (reapproved 2021) reports contrast-agent bioeffects (PVCs, microvascular leakage) above 0.4. BMUS (2009) uses 0.3 for neonatal lung and intestine and 0.7 for contrast cavitation risk.
22.11Cheat sheet
| Concept | Formula or number | Use |
|---|---|---|
| Plane-wave intensity | \(i(t) = p^{2}(t)/(\rho c)\) | Every intensity in the output table |
| Pulse-intensity integral | \(\mathrm{PII} = \int p^{2}/(\rho c)\,dt\) | \(z_{\mathrm{MI}}\), \(I_{\mathrm{ta}}\), \(I_{\mathrm{pa}}\) |
| Pulse duration | \(t_d = 1.25\,(t_{90} - t_{10})\) of cumulative \(\int p^{2}\) | \(I_{\mathrm{pa}} = \mathrm{PII}/t_d\); Gaussian \(t_d = 2.27\,\sigma_t\) |
| Temporal average | \(I_{\mathrm{ta}} = \mathrm{PII}\) × pulses/s at the point \(= I_{\mathrm{pa}}\times\mathrm{DF}\) | Scanned: × frame × \(0.75\,w_{-6}/\Delta\); plane wave: × PRF |
| Derating | \(A = 0.3\,f z\) dB; \(p\times 10^{-A/20}\), \(I\times 10^{-A/10}\) | All ".3" / "α" quantities |
| Mechanical index | \(\mathrm{MI} = p_{r,\alpha}(z_{\mathrm{MI}})/\sqrt{f_{\mathrm{awf}}}\) (MPa, MHz) | Displayed; Track 3 ≤ 1.9 (or \(I_{\mathrm{SPPA.3}} \le 190\)) |
| Thermal index | \(\mathrm{TI} = W/W_{\mathrm{deg}}\) | Displayed; explain TI > 6 |
| TIS | \(P_{(1)} f/210\) (mW, MHz); large non-scanned: \(\min[P_{\alpha}, I_{\mathrm{ta},\alpha}\cdot 1\ \mathrm{cm^{2}}]\) beyond \(z_{\mathrm{bp}} = 1.5\,D_{\mathrm{eq}}\) | Soft tissue |
| TIB, TIC | \(\mathrm{TIB} \approx P_{\alpha}/(40\,d)\) at \(z_b\); \(\mathrm{TIC} = P/(40\,D_{\mathrm{eq}})\) (mW, cm) | Fetal bone after 10 weeks; transcranial |
| Track 3 global maxima | \(I_{\mathrm{SPTA.3}} \le\) 720 mW/cm²; \(\mathrm{MI} \le 1.9\) or \(I_{\mathrm{SPPA.3}} \le\) 190 W/cm² | Ophthalmic: \(\mathrm{TI} \le 1\), \(I_{\mathrm{SPTA.3}} \le 50\), \(\mathrm{MI} \le 0.23\) |
| Track 1 \(I_{\mathrm{SPTA.3}}\) | 720 / 430 / 94 / 17 mW/cm² (peripheral vessel / cardiac / fetal & other / ophthalmic) | Non-ODS devices |
| AIUM thermal anchors | 1.5 °C without reservation; fetal 4 °C ≥ 5 min harmful | Dwell-time tables, ALARA |
| Thermal dose | \(\mathrm{CEM43} = \sum t\,R^{43-T}\), \(R\) = 0.25 / 0.5 | Comparing heating histories |
| Hydrophone averaging | \(\langle p\rangle/p_0 = (w^{2}/b^{2})\left(1 - e^{-b^{2}/w^{2}}\right)\) | \(d \le W_{-6}/4\) for ≈ 4 % in \(I\) |
| Radiation force | \(F = P/c\) (absorber) | 100 mW → 66.7 µN (6.8 mg) |
22.12Interview and qualifying-exam questions
I1Define MI and explain every factor in it. · screen
\(\mathrm{MI} = p_{r,\alpha}(z_{\mathrm{MI}})/(C_{\mathrm{MI}}\sqrt{f_{\mathrm{awf}}})\) with \(C_{\mathrm{MI}} = 1\ \mathrm{MPa\cdot MHz^{-1/2}}\) (22.13): the peak rarefactional pressure derated by 0.3 dB/cm/MHz, taken (IEC 62359 \(z_{\mathrm{MI}}\), FDA \(z_{\mathrm{sp}}\)) at the depth where the derated pulse-intensity integral peaks, divided by the root of the acoustic working frequency. Rarefaction because bubbles grow under tension; \(\sqrt{f}\) because the inertial-cavitation threshold of an optimally sized free nucleus rises roughly as \(\sqrt{f}\) (Apfel & Holland 1991). MI 1.9 at 3 MHz is 3.29 MPa derated.
Follow-up: does MI predict contrast-bubble destruction? Only loosely: shelled bubbles respond by resonance and shell properties, and the pressure at the bubble uses the real path attenuation.
I2Why is the derating 0.3 dB/cm/MHz when soft tissue is closer to 0.5–0.7? · screen
It is a deliberately low average so that exposure is not underestimated for paths with long fluid segments (full bladder, amniotic fluid) where attenuation is far below tissue values. The FDA states it as an assumed average attenuation along the beam axis. Pressure is derated by \(10^{-0.3fz/20}\) and intensity by \(10^{-0.3fz/10}\) (22.12): 4.5 dB at 3 MHz and 5 cm, × 0.596 and × 0.355.
Follow-up: when does derating still underestimate? Very low-loss paths and nonlinear water measurements at high drive, where energy already went into harmonics before derating.
I3State the Track 3 limits exactly, including the ophthalmic case. · regulatory
FDA guidance (February 2023): derated global maximum \(I_{\mathrm{SPTA.3}} \le\) 720 mW/cm², and either \(\mathrm{MI} \le 1.9\) or derated \(I_{\mathrm{SPPA.3}} \le\) 190 W/cm². Ophthalmic: \(\mathrm{TI} = \max(\mathrm{TIS}_{\mathrm{as}}, \mathrm{TIC}) \le 1\), \(I_{\mathrm{SPTA.3}} \le\) 50 mW/cm², \(\mathrm{MI} \le 0.23\). Any TI above 6.0 should be explained. Track 3 applies to devices conforming to the Output Display Standard in IEC 60601-2-37.
Follow-up: a short-pulse 3.5 MHz mode at MI 1.85 has \(I_{\mathrm{SPPA.3}}\) = 300 W/cm². Compliant? Yes, through the MI branch, provided \(I_{\mathrm{SPTA.3}} \le 720\).
I4Track 1 or Track 3: what is the difference and when would you choose Track 1? · regulatory
Track 1 is for devices that do not conform to the IEC 60601-2-37 output display; their derated output is held to application-specific preamendments levels (\(I_{\mathrm{SPTA.3}}\) 720 peripheral vessel, 430 cardiac, 94 fetal imaging and other, 17 mW/cm² ophthalmic; \(I_{\mathrm{SPPA.3}}\) 190, or 28 W/cm² ophthalmic, or MI 1.9 / 0.23). Track 3 replaces these with global limits plus an on-screen MI/TI display. Track 1 suits simple, fixed-output devices (some Doppler-only or single-purpose devices) where a display adds cost but the low application limit is easy to meet.
Follow-up: a fetal-imaging device on Track 1 at 200 mW/cm²? Not acceptable: the fetal level is 94.
I5When must MI and TI be displayed, and what is reported if they never exceed 1? · regulatory
The display requirement comes from IEC 60601-2-37 (the Output Display Standard), which, as public secondary sources summarize it (the IEC text is not public), ties the obligation to whether the system can exceed an index value of 1.0. The FDA guidance follows the same threshold: if no system/transducer combination can exceed a TI of 1.0 or an MI of 1.0 in any operating mode, the global maximum values are submitted instead of per-mode reporting tables; tables are provided for combinations whose global maximum displayed MI or TI exceeds 1.0. Display accuracy against measurement is part of the submission.
Follow-up: which TI is displayed? The one appropriate to the application (TIS, TIB or TIC), with the user able to select it.
I6You add an SWE mode to a cleared system. Plan the acoustic-output work. · onsite
Treat push plus tracking as one combined mode. Measure the push waveform with a membrane hydrophone at the push focus and along the axis to find \(z_{\mathrm{MI}}\); derate; compute PII per push (a 400 µs push at 3 MPa is about 0.09 J/cm²), \(I_{\mathrm{SPTA.3}}\) from push and tracking rates, \(I_{\mathrm{SPPA.3}}\), MI, and the TIs at the push and tracking powers. Search the global maximum over push voltage, duration, focal depth, push rate and region-of-interest size. Measure surface temperature under the worst sequence. Expect TI and \(I_{\mathrm{SPTA.3}}\), not MI, to set the push repetition limit; check AIUM guidance on reduced dwell for ARFI near bone for labelling.
Follow-up: why can a compliant \(I_{\mathrm{SPTA.3}}\) still be a heating concern? The average hides heating within a push and at the face; the TI and surface tests catch it.
I7Estimate \(I_{\mathrm{SPTA.3}}\) for a 5 MHz plane-wave mode at 5 kHz PRF with \(p_{r.3}\) = 0.8 MPa and 2-cycle pulses. · onsite
\(\sigma_t = 2/(2.3548\cdot 5\ \mathrm{MHz}) = 0.170\ \mathrm{\mu s}\), \(\int \mathrm{env}^{2} = 0.301\ \mathrm{\mu s}\). \(\mathrm{PII} = \frac{p^{2}}{2\rho c}\cdot 0.301\ \mathrm{\mu s}\) \(= \frac{(0.8\ \mathrm{MPa})^{2}}{3\ \mathrm{MRayl}}\times 0.301\ \mathrm{\mu s}\) \(= 0.0642\ \mathrm{J/m^{2}}\) = 6.4 µJ/cm². Every transmit reaches the point: \(I_{\mathrm{SPTA.3}}\) = 6.4 µJ/cm² × 5000/s = 32 mW/cm², \(\mathrm{MI} = 0.8/\sqrt{5} = 0.36\). A focused B-mode at the same pressure and 113 pulses/s would be 0.7 mW/cm².
Follow-up: a 60 s ULM acquisition at the same settings: \(I_{\mathrm{SPTA}}\) is the same, but TI and dwell time now govern; the AIUM tables apply to the displayed TI.
I8How do you choose a hydrophone for a 15 MHz linear array, and what corrections do you apply? · onsite
At 15 MHz and F/2 the focal \(W_{-6}\) is about 0.2 mm, so the effective diameter should be at most 50–100 µm (a spot-poled membrane, or a fibre-optic or small needle hydrophone with known effective radius), and the calibrated bandwidth should extend to several harmonics (≥ 60 MHz) because the focal waveform is nonlinear. Deconvolve with the complex sensitivity, apply the spatial-averaging correction of IEC 62127-1:2022, and include both in the uncertainty budget. A 200 µm hydrophone at \(W_{-6}\) = 0.2 mm would under-read p by 2.8 dB and intensity by about 48 %.
Follow-up: why do harmonics make it worse? Their beams are narrower, so averaging removes more of them and distorts \(p_c\) and \(p_r\) differently.
I9TIS, TIB or TIC: which applies when, and why does Doppler TIB exceed TIS? · screen
TIS for soft tissue (abdomen, vessels, early pregnancy), TIB when bone may lie at the beam's focus (obstetrics after 10 weeks LMP, Doppler near bone), TIC when bone is under the transducer (transcranial). Bone absorbs nearly all incident power in a thin layer, so its \(W_{\mathrm{deg}}\) scales with beam diameter (≈ 40 mW/cm × \(d\)), whereas soft tissue \(W_{\mathrm{deg}} = 210/f\) mW. A narrow Doppler focus therefore gives a small bone \(W_{\mathrm{deg}}\); in the PW preset TIB = 1.5 against TIS = 0.34.
Follow-up: why is TIB in scanned modes set equal to TIS? Sweeping the beam spreads the bone heating, and the surface soft-tissue case dominates.
I10What does the ±30 % / ±15 % statement in the FDA guidance mean for your uncertainty budget? · regulatory
The Track 1 preamendments levels are themselves estimated to be uncertain by ±30 % for intensities and ±15 % for MI. A manufacturer whose expanded measurement uncertainty is within those margins does not have to subtract it from the limits; if it exceeds them, the levels are reduced by the excess (an intensity uncertainty of 40 % reduces the level by 10 %). The protocol must describe Type A and Type B uncertainties: hydrophone calibration, spatial averaging, bandwidth, positioning, temperature and the power measurement.
Follow-up: which term usually dominates at high frequency? Hydrophone calibration and spatial averaging.
I11For CEUS and ULM, what pressure matters at the bubble, and why is the displayed MI a poor guide? · research
The bubble responds to the in situ pressure at its depth, with the real attenuation and the local beam, and to its resonance relative to \(f\); the displayed MI is the 0.3-derated pressure at one depth divided by \(\sqrt{f}\), a scaling derived for free gas nuclei. Through a fluid path (\(\alpha \approx\) 0.05) at 3.5 MHz and 8 cm the in situ pressure exceeds the derated value by 7.0 dB; through liver-like tissue (\(\alpha \approx\) 0.7) it is 11.2 dB lower. Researchers calibrate the in situ pressure (hydrophone plus a tissue model) and report it with the frequency and pulse length.
Follow-up: what MI threshold do safety statements give for contrast? The AIUM reports bioeffects above MI 0.4 with gas-body agents and recommends minimal MI and time; BMUS uses 0.7 as the cavitation-risk threshold.
22.13MATLAB lab
Base MATLAB only (no toolboxes); each script runs as-is in MATLAB R2019b+ and GNU Octave 8. Paste into the editor and press Run.
m22_derating_mi.mDerating table, focused-piston field in water vs derated, \(z_{\mathrm{MI}}\) and MI, allowed water pressure vs frequency · change f0, a, d, p_water · runtime ≈ 2 s
%% m22_derating_mi.m - Derating at 0.3 dB/cm/MHz, the depth z_MI, and the mechanical index
% Purpose : (1) print the derating factors for pressure (20 log) and intensity (10 log);
% (2) compute the on-axis CW field of a focused circular piston in water by a
% numerical Rayleigh ring sum, derate it, find the depth of the derated maximum
% (z_MI) and the MI; (3) show how MI and z_MI move with frequency for the same
% water peak pressure, and the water pressure that MI = 1.9 allows.
% Change : f0 (MHz), a (mm), d (mm), p_water (MPa).
% Expect : 3 MHz at 5 cm: 4.50 dB, p x 0.596, I x 0.355. For a = 6 mm, d = 70 mm, 3.5 MHz,
% 4 MPa water peak: water peak at 46.0 mm (focal shift), z_MI = 41.5 mm, MI = 1.26.
% Notes : Linear CW model (one frequency), so p_r^2 is proportional to PII at every depth and
% the IEC depth (max derated PII) equals the depth of the max derated p_r.
rng(1);
c = 1500; % m/s, water
%% 1. Derating table
fs = [2 3 3.5 5 7.5 10]; % MHz
zs = [2 5 10]; % cm
fprintf('Derating 0.3 dB/cm/MHz: loss dB | pressure factor | intensity factor\n');
for f = fs
for z = zs
A = 0.3*f*z;
fprintf(' f = %4.1f MHz, z = %4.1f cm: %5.2f dB | p x %.3f | I x %.3f\n', f, z, A, 10^(-A/20), 10^(-A/10));
end
end
%% 2. Focused piston, on-axis field (ring sum of the Rayleigh integral)
f0 = 3.5; a = 6; d = 70; p_water = 4.0; % MHz, mm, mm, MPa
nr = 400; r = ((1:nr) - 0.5)/nr*a*1e-3; % m, ring radii
dr = a*1e-3/nr;
z = (2:0.25:150)*1e-3; % m, depth
onaxis = @(f) abs(sum(bsxfun(@times, exp(-1i*2*pi*f*1e6/c*(sqrt(bsxfun(@plus, z.^2, r'.^2)) - (sqrt(r'.^2 + (d*1e-3)^2) - d*1e-3)*ones(size(z)))) ./ sqrt(bsxfun(@plus, z.^2, r'.^2)), r'*dr), 1));
pw = onaxis(f0); pw = p_water*pw/max(pw); % MPa, water
zcm = z*100;
p3 = pw.*10.^(-0.3*f0*zcm/20); % derated pressure
pii3 = p3.^2; % proportional to derated PII
[~, iw] = max(pw); [~, im] = max(pii3);
MI = p3(im)/sqrt(f0);
fprintf('\nFocused piston a = %g mm, d = %g mm, f = %.1f MHz, water peak %.1f MPa\n', a, d, f0, p_water);
fprintf(' water peak at z = %.1f mm\n', z(iw)*1e3);
fprintf(' derated maximum (z_MI) at z = %.1f mm, loss there %.2f dB\n', z(im)*1e3, 0.3*f0*zcm(im));
fprintf(' p_r.3 = %.2f MPa, MI = %.2f\n', p3(im), MI);
fprintf(' (wrong) water peak derated at its own depth: MI = %.2f\n', pw(iw)*10^(-0.3*f0*zcm(iw)/20)/sqrt(f0));
%% 3. Same geometry and water peak at other frequencies
fl = 2:0.5:7; MIf = zeros(size(fl)); zMI = MIf; zW = MIf; pAllow = MIf;
for k = 1:numel(fl)
q = onaxis(fl(k)); q = p_water*q/max(q);
q3 = q.*10.^(-0.3*fl(k)*zcm/20);
[~, j] = max(q3); [~, jw] = max(q);
MIf(k) = q3(j)/sqrt(fl(k)); zMI(k) = z(j)*1e3; zW(k) = z(jw)*1e3;
pAllow(k) = p_water*1.9/MIf(k); % water peak that gives MI = 1.9 (linear scaling)
end
fprintf('\n f (MHz) | water peak z (mm) | z_MI (mm) | MI at %.1f MPa water | water peak for MI 1.9 (MPa)\n', p_water);
for k = 1:2:numel(fl)
fprintf(' %4.1f | %5.1f | %5.1f | %.2f | %.2f\n', fl(k), zW(k), zMI(k), MIf(k), pAllow(k));
end
fprintf('\nMI = 1.9 means p_r.3 = 1.9*sqrt(f): 3 MHz %.2f, 5 MHz %.2f, 7.5 MHz %.2f MPa\n', 1.9*sqrt(3), 1.9*sqrt(5), 1.9*sqrt(7.5));
%% Figures
figure('Position', [100 100 900 360]);
subplot(1,2,1);
plot(z*1e3, pw, 'k', z*1e3, p3, 'b', 'LineWidth', 1.5); hold on;
plot(z(iw)*1e3, pw(iw), 'ko', z(im)*1e3, p3(im), 'bo', 'MarkerFaceColor', 'b');
xlabel('depth z (mm)'); ylabel('p_r (MPa)'); legend('water', 'derated 0.3 dB/cm/MHz'); grid on; xlim([0 150]);
title(sprintf('a = %g mm, d = %g mm, %.1f MHz: MI = %.2f', a, d, f0, MI));
subplot(1,2,2);
plot(fl, MIf, 'b-o', 'LineWidth', 1.5); hold on; plot(fl, 1.9*ones(size(fl)), 'r--');
xlabel('frequency (MHz)'); ylabel('MI at fixed water peak'); grid on;
title(sprintf('water peak %.1f MPa, same geometry', p_water));
What to look for. The derating table (3 MHz at 5 cm: 4.50 dB, × 0.596 and × 0.355). For a 6 mm radius piston focused at 70 mm, 3.5 MHz and 4 MPa water peak: water peak at 46.0 mm (focal shift), \(z_{\mathrm{MI}}\) = 41.5 mm, \(p_{r.3}\) = 2.36 MPa, MI = 1.26; derating the water peak at its own depth gives 1.23, 2 % low here and more for weaker focusing. At the same water peak MI falls from 2.27 at 2 MHz to 0.39 at 7 MHz while \(z_{\mathrm{MI}}\) moves from 31 to 53 mm; the water peak allowed by MI 1.9 rises from 3.4 to 19.3 MPa.
m22_intensity_parameters.m\(p_c\), \(p_r\), PII, pulse duration, \(I_{\mathrm{pa}}\), \(f_{\mathrm{awf}}\) from a waveform; \(I_{\mathrm{ta}}\) and DF for scanned, PW, plane-wave and push modes · change f0, ncyc, prf, frame rate · runtime ≈ 1 s
%% m22_intensity_parameters.m - From a hydrophone waveform to PII, t_d, I_pa, I_ta and duty factor
% Purpose : compute the exposure quantities exactly as defined for an output report:
% p_c, p_r, PII = int p^2/(rho c) dt, pulse duration t_d = 1.25 (t90 - t10) of the
% cumulative p^2 integral, I_pa = PII/t_d, acoustic working frequency (zero crossings
% and -3 dB spectral mean), then I_ta and DF for scanned, PW, plane-wave and push modes.
% Change : f0, ncyc (cycles at -6 dB of the Gaussian envelope), p0, h2 (2nd-harmonic level),
% frame rate, beam width W6 and line spacing dx, prf.
% Expect : 3 MHz, 3 cycles, 2 MPa (linear): PII = 0.100 mJ/cm^2; t_d = 0.905 us from the waveform
% (0.962 us from the envelope-only closed form: the carrier ripple in p^2 shifts t10 and
% t90 for few-cycle pulses); PW at 5 kHz 502 mW/cm^2; B-mode 30 Hz 11.3 mW/cm^2.
rng(1);
rhoc = 1.5e6; % Rayl, water
f0 = 3e6; ncyc = 3; p0 = 2e6; % Hz, cycles at -6 dB, Pa
fs = 500e6; t = (-12e-6:1/fs:12e-6); % s
sig = ncyc/(2.3548*f0); % s, envelope standard deviation
env = exp(-t.^2/(2*sig^2));
%% 1. Linear pulse and a pulse with a 2nd harmonic (nonlinear-looking, p_c > p_r)
h2 = 0.35; % relative 2nd-harmonic amplitude
P = [p0*env.*cos(2*pi*f0*t); p0*env.*(cos(2*pi*f0*t) + h2*env.*cos(2*pi*2*f0*t))];
names = {'linear', 'with 2nd harmonic'};
for k = 1:2
p = P(k,:);
pc = max(p); pr = -min(p);
cum = cumsum(p.^2)/fs/rhoc; PII = cum(end); % J/m^2
t10 = t(find(cum >= 0.1*PII, 1)); t90 = t(find(cum >= 0.9*PII, 1));
td = 1.25*(t90 - t10); Ipa = PII/td;
% acoustic working frequency: zero crossings within the -12 dB envelope, and -3 dB spectral mean
sel = find(env > 0.25); ps = p(sel); ts = t(sel);
ic = find(ps(1:end-1).*ps(2:end) < 0); % sign changes
tc = ts(ic) - ps(ic).*(ts(ic+1) - ts(ic))./(ps(ic+1) - ps(ic)); % interpolated crossing times
fzc = (numel(tc) - 1)/(2*(tc(end) - tc(1)));
N = 2^nextpow2(numel(p))*4; S = abs(fft(p, N)); fr = (0:N-1)*fs/N; half = 1:N/2;
S = S(half); fr = fr(half); [Sm, im] = max(S);
il = find(S(1:im) < Sm/sqrt(2), 1, 'last') + 1; iu = im - 1 + find(S(im:end) < Sm/sqrt(2), 1);
fawf = (fr(il) + fr(iu))/2;
fprintf('%s pulse:\n', names{k});
fprintf(' p_c = %.2f MPa, p_r = %.2f MPa\n', pc/1e6, pr/1e6);
fprintf(' PII = %.4f mJ/cm^2, t_d = %.3f us, I_pa = %.1f W/cm^2\n', PII*1e3/1e4, td*1e6, Ipa/1e4);
fprintf(' f_awf: zero-crossing %.2f MHz, -3 dB spectral mean %.2f MHz\n', fzc/1e6, fawf/1e6);
if k == 1
fprintf(' closed forms: t_d = 2.2655 sigma = %.3f us, PII = p^2/(2 rho c) sigma sqrt(pi) = %.4f mJ/cm^2, I_pa = 0.782 p^2/(2 rho c) = %.1f W/cm^2\n', ...
2.2655*sig*1e6, p0^2/(2*rhoc)*sig*sqrt(pi)*1e3/1e4, 0.7824*p0^2/(2*rhoc)/1e4);
PIIlin = PII; tdlin = td; cumlin = cum; t10l = t10; t90l = t90;
end
end
% envelope closed form vs exact t_d as the pulse gets longer
fprintf('\n cycles | t_d exact / closed form\n');
for nc = [2 3 5 10 20]
sg = nc/(2.3548*f0); q = exp(-t.^2/(2*sg^2)).*cos(2*pi*f0*t); cq = cumsum(q.^2); cq = cq/cq(end);
tdq = 1.25*(t(find(cq >= 0.9, 1)) - t(find(cq >= 0.1, 1)));
fprintf(' %4d | %.3f\n', nc, tdq/(2.2655*sg));
end
%% 2. Pulses per second at the point, I_ta and duty factor
frame = 30; W6 = 1.5e-3; dx = 0.3e-3; % Hz, m, m
w = W6/1.6651; % 1/e^2 intensity half-width
lines = (-60:60)*dx; neff = sum(exp(-2*lines.^2/w^2)); % exact sum over lines
fprintf('\nScanned B-mode: overlap sum %.3f pulses/frame (approx 0.7527 W6/dx = %.3f)\n', neff, 0.7527*W6/dx);
modes = {'B-mode 30 Hz', 'colour 10 Hz x ens 10', 'PW 5 kHz', 'plane wave 5 kHz'};
rate = [frame*neff, 10*neff*10, 5000, 5000];
fprintf(' mode | pulses/s at point | I_ta (mW/cm^2) | DF\n');
for k = 1:numel(modes)
Ita = PIIlin*rate(k);
fprintf(' %-22s | %10.1f | %10.2f | %.2e\n', modes{k}, rate(k), Ita*1e3/1e4, tdlin*rate(k));
end
% push pulse: 1200 cycles at -6 dB (400 us), 3 MPa, 3 MHz, 2 per second
sp = 1200/(2.3548*3e6); PIIp = (3e6)^2/(2*rhoc)*sp*sqrt(pi);
fprintf(' push 400 us, 3 MPa, 2/s | PII = %.3f J/cm^2 | I_ta = %.0f mW/cm^2 | DF = %.2e\n', PIIp/1e4, 2*PIIp*1e3/1e4, 2*2.2655*sp);
fprintf(' push / 2-cycle PII ratio at equal pressure = %.0f\n', 1200/2);
fprintf('Track 3: I_SPTA.3 <= 720 mW/cm^2 and (MI <= 1.9 or I_SPPA.3 <= 190 W/cm^2)\n');
%% Figure
figure('Position', [100 100 900 330]);
subplot(1,2,1);
plot(t*1e6, P(1,:)/1e6, 'b', t*1e6, P(2,:)/1e6, 'm'); grid on; xlim([-1.5 1.5]);
xlabel('t (\mus)'); ylabel('p (MPa)'); legend(names); title('waveforms');
subplot(1,2,2);
plot(t*1e6, cumlin/PIIlin, 'b', 'LineWidth', 1.5); hold on;
plot([t10l t10l]*1e6, [0 1], 'k--', [t90l t90l]*1e6, [0 1], 'k--'); grid on; xlim([-1.5 1.5]);
xlabel('t (\mus)'); ylabel('cumulative \int p^2 (norm.)');
title(sprintf('t_d = 1.25 (t_{90} - t_{10}) = %.3f \\mus', tdlin*1e6));
What to look for. Linear 3-cycle pulse: \(p_c\) = 2.00, \(p_r\) = 1.85 MPa (the carrier peak is not at a rarefaction), PII = 0.1004 mJ/cm², \(t_d\) = 0.905 µs (envelope closed form 0.962), \(I_{\mathrm{pa}}\) = 111 W/cm², \(f_{\mathrm{awf}}\) 3.00 MHz by both methods. Adding a second harmonic raises \(p_c\) to 2.70 and lowers \(p_r\) to 1.35 MPa with PII almost unchanged: MI and \(I_{\mathrm{pa}}\) move differently. The overlap sum is 3.763 pulses per frame (rule \(0.7527\,W_{-6}/\Delta = 3.764\)): B-mode 11.3, colour 37.8, PW and plane wave 502 mW/cm²; one push 0.090 J/cm², two per second 181 mW/cm².
m22_hydrophone_spatial_averaging.mDisc-average error vs diameter for round and elongated Gaussian foci, closed-form check, Gaussian-kernel correction, jinc focus · change W6, aspect, d · runtime ≈ 2 s
%% m22_hydrophone_spatial_averaging.m - How a finite hydrophone under-reads a focus, and a correction
% Purpose : average a focal pressure field over a disc of diameter d (the hydrophone face) and
% report the pressure (MI) and intensity (I_SPTA, I_SPPA) errors versus d/W6, where W6 is
% the -6 dB beam width; compare with the closed form for a round Gaussian focus; apply
% the Gaussian-kernel correction that uses only the measured width; repeat for a jinc
% (uniform circular aperture) focus with the same W6.
% Change : W6 (mm), aspect (elevation/azimuth width), the list of d/W6 ratios.
% Expect : round Gaussian, d = W6/4: -0.19 dB in p, -0.38 dB (4.2 %) in I;
% d = W6/2: -0.74 dB, -1.48 dB (15.7 %); corrected within 0.02 dB at d = W6/2.
rng(1);
W6 = 1.0; % mm, azimuth -6 dB width (pressure or intensity: same point)
aspect = 3; % elevation width / azimuth width
w = W6/1.6651; % mm, 1/e amplitude half-width: p = exp(-r^2/w^2)
ratios = [0.1 0.25 0.5 0.67 0.75 1.0 1.2];
nr = 40; nt = 64; % polar quadrature over the disc
discavg = @(fun, b) sum(sum(fun(((1:nr)'-0.5)/nr*b*cos(2*pi*(0:nt-1)/nt), ((1:nr)'-0.5)/nr*b*sin(2*pi*(0:nt-1)/nt)) .* repmat(((1:nr)'-0.5), 1, nt))) / (nt*sum((1:nr)-0.5));
g_round = @(x, y) exp(-(x.^2 + y.^2)/w^2);
g_elong = @(x, y) exp(-x.^2/w^2 - y.^2/(aspect*w)^2);
% jinc focus with the same -6 dB width: p = 2 J1(u)/u, u = k r; -6 dB at u = 2.2151
u6 = fzero(@(u) 2*besselj(1,u)/u - 0.5, 2.2);
kj = u6/(W6/2);
jincf = @(x, y) 2*besselj(1, kj*sqrt(x.^2 + y.^2 + 1e-12))./(kj*sqrt(x.^2 + y.^2 + 1e-12));
fprintf('-6 dB point of 2J1(u)/u at u = %.4f\n', u6);
fprintf(' d/W6 | round: num closed (dB p) | I error %% | corrected (dB) | elongated %g:1 (dB p) | jinc (dB p)\n', aspect);
res = zeros(numel(ratios), 5);
for k = 1:numel(ratios)
b = ratios(k)*W6/2;
pr = discavg(g_round, b);
pcf = (w^2/b^2)*(1 - exp(-b^2/w^2));
% Gaussian-kernel correction: measured width^2 = w^2 + b^2/2 per axis, peak scales by w/w_meas per axis
cr = pr*(w^2 + b^2/2)/w^2;
pe = discavg(g_elong, b);
pj = discavg(jincf, b);
res(k,:) = [20*log10(pr), 20*log10(pcf), 20*log10(cr), 20*log10(pe), 20*log10(pj)];
fprintf(' %4.2f | %6.3f %6.3f | %5.1f | %6.3f | %6.3f | %6.3f\n', ...
ratios(k), res(k,1), res(k,2), 100*(1 - pr^2), res(k,3), res(k,4), res(k,5));
end
% Example: 0.4 mm hydrophone on a 0.6 mm focus (7.5 MHz, F/2)
b = 0.2; wx = 0.6/1.6651;
p1 = discavg(@(x, y) exp(-(x.^2 + y.^2)/wx^2), b);
p3 = discavg(@(x, y) exp(-x.^2/wx^2 - y.^2/(3*wx)^2), b);
fprintf('\n0.4 mm hydrophone, W6 = 0.6 mm: round %.2f dB (I %.1f %% low), 3:1 elongated %.2f dB (I %.1f %% low)\n', ...
20*log10(p1), 100*(1-p1^2), 20*log10(p3), 100*(1-p3^2));
%% Measured lateral profile for one case (d = 0.5 W6, round)
b = 0.25*W6; xs = linspace(-1.6*W6, 1.6*W6, 161); meas = zeros(size(xs));
for i = 1:numel(xs)
meas(i) = discavg(@(x, y) g_round(x + xs(i), y), b);
end
c6 = meas(81)/2; i6 = find(meas(81:end) < c6, 1) + 79;
W6m = 2*interp1(meas(i6:i6+1), xs(i6:i6+1), c6);
fprintf('d = W6/2: measured W6 = %.3f mm (true %.3f), predicted sqrt(W6^2 + 1.6651^2 b^2/2) = %.3f mm\n', W6m, W6, 1.6651*sqrt(w^2 + b^2/2));
%% Figure
figure('Position', [100 100 900 340]);
subplot(1,2,1);
plot(ratios, res(:,1), 'b-o', ratios, 2*res(:,1), 'r-o', ratios, res(:,4), 'b--', ratios, res(:,5), 'k-.', ratios, res(:,3), 'g-s', 'LineWidth', 1.3);
grid on; xlabel('hydrophone diameter / W_{-6}'); ylabel('reading error (dB)');
legend('pressure, round', 'intensity, round', 'pressure, elongated', 'pressure, jinc focus', 'corrected pressure', 'Location', 'southwest');
title('spatial-averaging error');
subplot(1,2,2);
plot(xs, g_round(xs, 0), 'k', xs, meas, 'b', 'LineWidth', 1.5); grid on;
xlabel('x (mm)'); ylabel('p (norm.)'); legend('true', 'measured, d = W_{-6}/2');
title(sprintf('W_{-6} = %.2f mm reads as %.3f mm', W6, W6m));
What to look for. The numerical disc average matches the closed form to 0.001 dB. Round Gaussian focus: −0.19, −0.74, −1.32, −2.84 dB in pressure at d/\(W_{-6}\) = 0.25, 0.5, 0.67, 1 (intensity 4.2, 15.7, 26.1, 48.0 % low). A 3:1 elongated focus loses about half as many dB; a jinc focus with the same \(W_{-6}\) slightly less than the Gaussian. The Gaussian-kernel correction leaves −0.02 dB at d = \(W_{-6}\)/2 and −0.25 dB at d = \(W_{-6}\); the measured width at d = \(W_{-6}\)/2 is 1.044 mm for a true 1.000 mm, as predicted.
m22_ti_estimates.mThe Demo 22.4 Gaussian-beam model: MI, \(I_{\mathrm{SPTA.3}}\), \(I_{\mathrm{SPPA.3}}\), TIS, TIB, TIC for the PW, B-mode and push presets; TIB vs focal depth · change preset fields · runtime ≈ 2 s
%% m22_ti_estimates.m - MI, derated intensities and TIS / TIB / TIC for a defined mode
% Purpose : the Gaussian-beam output model of Demo 22.4. A separable Gaussian source (1/e amplitude
% half-width 0.4 x aperture in each axis) focused in azimuth (Fx) and elevation (Fy)
% radiates a Gaussian-envelope pulse in water. From it: p_r.3(z), PII.3(z), I_ta.3(z),
% I_pa.3(z), P_alpha(z); MI at z_MI (max derated PII); I_SPTA.3, I_SPPA.3; TIS (scanning:
% P1 f/210; non-scanning large aperture: max over z >= z_bp of min[P_alpha, I_ta,alpha x 1 cm^2]
% f/210), TIB (model P_alpha/(40 mW/cm x d_-6), scanning TIB = TIS), TIC = P/(40 D_eq).
% Change : the preset structs (f MHz, Lx Ly mm, Fx Fy mm, p0 MPa source pressure, N cycles at -6 dB,
% mode 'scan' | 'unscan' | 'plane', rate or lines/frame/fov).
% Expect : PW preset: MI 0.60 at 4.80 cm, I_SPTA.3 428 mW/cm^2, P 39.2 mW, TIS 0.34 (z_s 2.36 cm),
% TIB 1.49, TIC 0.63. B-mode: MI 1.16, TIS = TIB 0.12, TIC 0.32. Push: MI 1.62,
% I_SPTA.3 159 mW/cm^2, I_SPPA.3 206 W/cm^2. sqrt-form TIB cross-check (C_TIB = 50):
% 1.063 x the diameter form for these Gaussian beams.
% Note : TIB here uses the diameter model (40 mW/cm, Nowicki 2020 eq. 6). IEC 62359 computes
% non-scanning TIB as sqrt(P_alpha I_ta,alpha)/C_TIB at the depth where the product is
% largest. C_TIB = 50 mW/cm is the commonly quoted ODS value but was NOT confirmed from a
% public source for this course; the script prints that form only as a cross-check.
% Use the standard's own text for submission numbers.
rhoc = 1.5e6; cw = 1500;
zs = 0.1:0.02:15; % cm
der = @(f, z, pw) 10.^(-0.3*f*z/(10*pw + 20*(1-pw))); % pw = 1: intensity (10 log), 0: pressure
den = @(ws, F, lam) (1/F)^2 + (lam/(pi*ws^2))^2; % complex beam parameter q = q0 + z
qr0 = @(ws, F, lam) (-1/F)/den(ws, F, lam); % Re q0 (converging: R = -F)
qi0 = @(ws, F, lam) (lam/(pi*ws^2))/den(ws, F, lam); % Im q0
gw = @(ws, F, lam, z) sqrt(lam*((qr0(ws, F, lam) + z).^2 + qi0(ws, F, lam)^2)/(pi*qi0(ws, F, lam))); % 1/e half-width (m)
% presets (same as the page)
P1 = struct('name','PW Doppler','f',3.0,'Lx',19.2,'Ly',10,'Fx',50,'Fy',60,'p0',0.45,'N',8,'mode','unscan','rate',6000,'lines',0,'frame',0,'fov',0);
P2 = struct('name','B-mode','f',3.5,'Lx',19.2,'Ly',10,'Fx',50,'Fy',60,'p0',0.9,'N',2,'mode','scan','rate',0,'lines',128,'frame',40,'fov',38.4);
P3 = struct('name','push 2/s','f',3.0,'Lx',19.2,'Ly',10,'Fx',40,'Fy',60,'p0',1.0,'N',1200,'mode','unscan','rate',2,'lines',0,'frame',0,'fov',0);
P4 = struct('name','plane 5 kHz','f',5,'Lx',38.4,'Ly',5,'Fx',1e6,'Fy',20,'p0',0.8,'N',2,'mode','plane','rate',5000,'lines',0,'frame',0,'fov',0);
presets = {P1, P2, P3, P4};
out = cell(1, numel(presets));
for k = 1:numel(presets)
P = presets{k};
lam = cw/(P.f*1e6); wsx = 0.4*P.Lx*1e-3; wsy = 0.4*P.Ly*1e-3;
wx = gw(wsx, P.Fx*1e-3, lam, zs*1e-2); wy = gw(wsy, P.Fy*1e-3, lam, zs*1e-2); % m
amp = sqrt(wsx*wsy./(wx.*wy));
sig = P.N/(2.3548*P.f*1e6); Ee = sig*sqrt(pi); td = 2.2655*sig;
p0 = P.p0*1e6; E = p0^2/(2*rhoc)*Ee*pi*wsx*wsy/2; % J per pulse
if strcmp(P.mode, 'scan')
pps = P.lines*P.frame; dx = P.fov*1e-3/P.lines; rate = zeros(size(zs));
for i = 1:numel(zs)
L = ceil(4*wx(i)/dx); l = -L:L; rate(i) = P.frame*sum(exp(-2*(l*dx).^2/wx(i)^2));
end
else
pps = P.rate; rate = pps*ones(size(zs));
end
Pw = E*pps*1e3; % mW
p = p0*amp; pii = p.^2/(2*rhoc)*Ee;
pr3 = p.*der(P.f, zs, 0); pii3 = pii.*der(P.f, zs, 1); ita3 = pii.*rate.*der(P.f, zs, 1);
ipa3 = pii/td.*der(P.f, zs, 1); P3a = Pw*der(P.f, zs, 1);
[~, iMI] = max(pii3); MI = pr3(iMI)/1e6/sqrt(P.f);
[ispta3, iT] = max(ita3*0.1); isppa3 = max(ipa3)/1e4; % mW/cm^2, W/cm^2
tisA = P3a*P.f/210; tisB = ita3*0.1*P.f/210; tib = P3a./(40*1.6651*sqrt(wx.*wy)*100);
Aap = P.Lx*P.Ly/100;
if ~strcmp(P.mode, 'unscan')
if strcmp(P.mode, 'scan'), fovcm = P.fov/10; else, fovcm = P.Lx/10; end
Pb = Pw*min(1, 1/fovcm); TIS = Pb*P.f/210; TIB = TIS; Deq = sqrt(4*(fovcm*P.Ly/10)/pi); TIC = Pw/(40*Deq); zsv = NaN; zbv = NaN; zbp = NaN;
else
Deq = sqrt(4*Aap/pi); zbp = 1.5*Deq; Pb = NaN;
if Aap <= 1
TIS = Pw*P.f/210; zsv = NaN;
else
m = min(tisA, tisB); m(zs < zbp) = 0; [TIS, j] = max(m); zsv = zs(j);
end
[TIB, jb] = max(tib); zbv = zs(jb); TIC = Pw/(40*Deq);
CTIB = 50; % mW/cm, UNVERIFIED (see header)
[TIBsq, jq] = max(sqrt(P3a.*ita3*0.1)/CTIB); % sqrt(mW x mW/cm^2) = mW/cm
end
fprintf('%-12s: MI %.2f at z_MI %.2f cm | I_SPTA.3 %.1f mW/cm^2 (z %.2f) | I_SPPA.3 %.1f W/cm^2 | P %.1f mW\n', ...
P.name, MI, zs(iMI), ispta3, zs(iT), isppa3, Pw);
fprintf(' TIS %.3f (z_s %.2f, z_bp %.2f) | TIB %.3f (z_b %.2f) | TIC %.3f (D_eq %.2f cm)\n', TIS, zsv, zbp, TIB, zbv, TIC, Deq);
if strcmp(P.mode, 'unscan')
fprintf(' cross-check sqrt(P_a I_ta,a)/50: TIB %.3f at %.2f cm (ratio to diameter form %.3f)\n', TIBsq, zs(jq), TIBsq/TIB);
end
ok = ispta3 <= 720 && (MI <= 1.9 || isppa3 <= 190);
if ok, fprintf(' Track 3: within\n'); else, fprintf(' Track 3: exceeded\n'); end
out{k} = struct('pr3', pr3, 'tisA', tisA, 'tisB', tisB, 'tib', tib, 'zbp', zbp);
end
%% TIB and MI versus azimuth focal depth (PW preset)
Fl = [20 30 40 50 70 100];
fprintf('\nPW preset, azimuth focus sweep:\n Fx (mm) | MI | TIS | TIB\n');
for F = Fl
P = P1; P.Fx = F;
lam = cw/(P.f*1e6); wsx = 0.4*P.Lx*1e-3; wsy = 0.4*P.Ly*1e-3;
wx = gw(wsx, P.Fx*1e-3, lam, zs*1e-2); wy = gw(wsy, P.Fy*1e-3, lam, zs*1e-2);
amp = sqrt(wsx*wsy./(wx.*wy)); sig = P.N/(2.3548*P.f*1e6); Ee = sig*sqrt(pi);
p0 = P.p0*1e6; E = p0^2/(2*rhoc)*Ee*pi*wsx*wsy/2; Pw = E*P.rate*1e3;
pii = (p0*amp).^2/(2*rhoc)*Ee; pii3 = pii.*der(P.f, zs, 1); [~, iMI] = max(pii3);
MI = p0*amp(iMI)*der(P.f, zs(iMI), 0)/1e6/sqrt(P.f);
P3a = Pw*der(P.f, zs, 1); tisA = P3a*P.f/210; tisB = pii3*P.rate*0.1*P.f/210;
Deq = sqrt(4*(P.Lx*P.Ly/100)/pi); m = min(tisA, tisB); m(zs < 1.5*Deq) = 0;
fprintf(' %4d | %.2f | %.2f | %.2f\n', F, MI, max(m), max(P3a./(40*1.6651*sqrt(wx.*wy)*100)));
end
%% Figure
o = out{1};
figure('Position', [100 100 900 330]);
subplot(1,2,1); plot(zs, o.pr3/1e6, 'b', 'LineWidth', 1.5); grid on; xlabel('z (cm)'); ylabel('p_{r.3} (MPa)'); title('PW preset: derated pressure');
subplot(1,2,2); plot(zs, o.tisA, 'Color', [0.8 0.5 0], 'LineWidth', 1.3); hold on; plot(zs, o.tisB, 'Color', [0 0.6 0.5], 'LineWidth', 1.3);
plot(zs, o.tib, 'm', 'LineWidth', 1.3); plot([o.zbp o.zbp], [0 1.1*max(o.tib)], 'k--'); grid on;
xlabel('z (cm)'); ylabel('index units'); legend('P_\alpha f/210', 'I_{ta,\alpha} 1 cm^2 f/210', 'TIB(z) model', 'z_{bp}'); title('TI integrands');
What to look for. The same numbers as Demo 22.4. PW: MI 0.60 at 4.80 cm, \(I_{\mathrm{SPTA.3}}\) 428 mW/cm², P 39.2 mW, TIS 0.344 at \(z_s\) = 2.36 cm (just beyond \(z_{\mathrm{bp}}\) = 2.35 cm), TIB 1.49 at 4.70 cm, TIC 0.63. B-mode: MI 1.16, \(I_{\mathrm{SPTA.3}}\) 14.8 mW/cm² at the surface (many lines overlap there), TIS = TIB = 0.125, TIC 0.32. Push: MI 1.62, \(I_{\mathrm{SPPA.3}}\) 206 W/cm² (compliant through the MI branch), \(I_{\mathrm{SPTA.3}}\) 159 mW/cm². Focus sweep: TIB 3.62 at 20 mm, 1.49 at 50 mm, 1.05 at 100 mm, TIS constant at 0.34. The cross-check line evaluates the IEC form \(\sqrt{P_{\alpha} I_{\mathrm{ta},\alpha}}/C_{\mathrm{TIB}}\) with the unverified \(C_{\mathrm{TIB}}\) = 50 mW/cm (Section 22.5): PW 1.586, push 0.523, both 1.063 times the diameter form.
m22_bioheat_toy.mSteady Pennes rise for \(W = 210/f\) mW vs beam radius and frequency; duty-cycle heating with a first-order model; CEM43 · change k, L, a, tau · runtime ≈ 3 s
%% m22_bioheat_toy.m - Is TIS = 1 about 1 degC? Pennes steady state, duty cycle and thermal dose
% Purpose : (1) integrate the steady-state Pennes Green's function G(r) = exp(-r/L)/(4 pi k r) over
% the heat source q = mu_a I of a collimated Gaussian beam carrying W = W_deg = 210/f mW
% in tissue attenuating 0.3 dB/cm/MHz (all absorbed), and report the peak rise versus
% beam radius and frequency; (2) a first-order duty-cycle heating model and CEM43.
% Change : k (W/m/K), L (perfusion length, m), radii a, frequencies f, tau, duty, period.
% Expect : peak rise of order 0.5-1.6 degC (within the AIUM "factor of 2" of TI = 1), weakly
% dependent on f (absorbed power per length ~ f W = 210 mW MHz for all f), larger for
% narrow beams. Duty cycle: period << tau follows the mean DF*dTss.
% Note : infinite homogeneous medium, no surface boundary or transducer self-heating.
rng(1);
k = 0.5; L = 0.01; % W/(m K), m
radii = [0.5 1 2 4]*1e-3; % m, 1/e^2 intensity radius
freqs = [1 2 3.5 5 7.5]; % MHz
dT = zeros(numel(radii), numel(freqs));
for ia = 1:numel(radii)
a = radii(ia);
dr = a/24; r = ((1:6*24) - 0.5)*dr; % m
for jf = 1:numel(freqs)
f = freqs(jf); W = 210/f*1e-3; % W
mu = 0.3*log(10)/10*f*100; % 1/m, intensity attenuation (all absorbed)
dz = min(a/4, 5e-4); zmax = min(0.15, 8/mu + 0.05); z = ((1:round(zmax/dz)) - 0.5)*dz;
[R, Z] = meshgrid(r, z);
q = mu*2*W/(pi*a^2)*exp(-2*R.^2/a^2).*exp(-mu*Z); % W/m^3
best = 0;
for z0 = 0:0.0025:0.04 % search the on-axis maximum
D = sqrt(R.^2 + (Z - z0).^2);
v = sum(sum(q.*exp(-D/L)./(4*pi*k*D).*2*pi.*R))*dr*dz;
best = max(best, v);
end
dT(ia, jf) = best;
end
end
fprintf('Steady Pennes rise (degC) for W = 210/f mW (TIS = 1 if all power counted), k = %.1f, L = %.0f mm\n', k, L*1e3);
fprintf(' radius (mm) |'); fprintf(' %4.1f MHz', freqs); fprintf('\n');
for ia = 1:numel(radii)
fprintf(' %4.1f |', radii(ia)*1e3); fprintf(' %6.2f ', dT(ia,:)); fprintf('\n');
end
fprintf(' range %.2f to %.2f degC; variation from 1 to 7.5 MHz at 1 mm: %.0f %%\n', min(dT(:)), max(dT(:)), 100*(1 - dT(2,end)/dT(2,1)));
%% Convergence check (one case, finer grid)
a = 1e-3; f = 3.5; W = 210/f*1e-3; mu = 0.3*log(10)/10*f*100;
dr = a/48; r = ((1:6*48) - 0.5)*dr; dz = a/8; z = ((1:round(0.1/dz)) - 0.5)*dz; [R, Z] = meshgrid(r, z);
q = mu*2*W/(pi*a^2)*exp(-2*R.^2/a^2).*exp(-mu*Z); best = 0;
for z0 = 0:0.0025:0.04
D = sqrt(R.^2 + (Z - z0).^2); best = max(best, sum(sum(q.*exp(-D/L)./(4*pi*k*D).*2*pi.*R))*dr*dz);
end
fprintf(' convergence (a = 1 mm, 3.5 MHz): coarse %.3f, fine %.3f degC\n', dT(2,3), best);
%% Time constants
kappa = k/(1050*3600); % m^2/s
fprintf('\nkappa = %.3f mm^2/s; tau = a^2/(4 kappa): a = 0.5 mm %.2f s, 1 mm %.2f s, 5 mm %.1f s; L^2/kappa = %.0f s\n', ...
kappa*1e6, (0.5e-3)^2/(4*kappa), (1e-3)^2/(4*kappa), (5e-3)^2/(4*kappa), L^2/kappa);
%% Duty-cycle model: dT' = (dTss*s(t) - dT)/tau
dTss = 3; tau = 20; D = 0.5; Tdur = 600; dt = 0.01;
periods = [2 60];
figure('Position', [100 100 900 330]);
subplot(1,2,2); hold on;
for ip = 1:numel(periods)
Tp = periods(ip); t = 0:dt:Tdur; s = mod(t, Tp) < D*Tp;
th = zeros(size(t)); e = exp(-dt/tau);
for n = 2:numel(t), th(n) = dTss*s(n) + (th(n-1) - dTss*s(n))*e; end
T = 37 + th; R = 0.25*(T < 43) + 0.5*(T >= 43); cem = sum(dt/60*R.^(43 - T));
fprintf('period %5.1f s: peak dT %.2f degC, mean (last half) %.2f, DF*dTss = %.2f, CEM43 = %.3g min\n', ...
Tp, max(th), mean(th(t > Tdur/2)), D*dTss, cem);
plot(t, th, 'LineWidth', 1.2);
end
xlabel('t (s)'); ylabel('\DeltaT (\circC)'); grid on; legend('period 2 s', 'period 60 s'); title(sprintf('\\tau = %g s, DF = %g, \\DeltaT_{ss} = %g', tau, D, dTss));
fprintf('CEM43 checks: 40 degC for 128 min = %.2f; 41 degC for 30 min = %.3f; 45 degC for 1 min = %.1f\n', 128*0.25^3, 30*0.25^2, 0.5^-2);
subplot(1,2,1);
plot(radii*1e3, dT, '-o', 'LineWidth', 1.3); hold on; plot([0 4.5], [1 1], 'k--', [0 4.5], [2 2], 'k:', [0 4.5], [0.5 0.5], 'k:');
xlabel('beam radius a (mm)'); ylabel('peak \DeltaT (\circC)'); grid on;
legend('1 MHz', '2 MHz', '3.5 MHz', '5 MHz', '7.5 MHz'); title('Pennes steady rise at W = 210/f mW');
What to look for. Peak rise for \(W = 210/f\) mW: 1.56 °C (0.5 mm radius, 1 MHz) down to 0.48 °C (4 mm, 7.5 MHz); 23 % variation from 1 to 7.5 MHz at 1 mm; the grid is converged (1.110 versus 1.112 °C). Time constants 0.47 s, 1.89 s and 47 s for 0.5, 1 and 5 mm radii; \(L^{2}/\kappa\) = 756 s. Duty cycle 50 % with τ = 20 s: a 2 s period stays at the 1.50 °C mean (peak 1.54), a 60 s period swings to 2.45 °C with the same mean. CEM43 checks: 2.00, 1.875, 4.0.
22.14Self-check
Q1A 5 MHz probe measures \(p_r\) = 4.0 MPa in water at 3 cm. What are \(p_{r.3}\) and the MI if this is \(z_{\mathrm{MI}}\)?
\(A = 0.3\times 5\times 3 = 4.5\) dB, factor \(10^{-4.5/20} = 0.596\): \(p_{r.3}\) = 2.38 MPa, \(\mathrm{MI} = 2.38/\sqrt{5} = 1.07\).
Q2A water \(I_{\mathrm{SPTA}}\) of 600 mW/cm² is measured at 4 cm at 3.5 MHz. Is the derated value within Track 3?
\(A = 0.3\times 3.5\times 4 = 4.2\) dB; intensity factor \(10^{-0.42} = 0.380\): \(I_{\mathrm{SPTA.3}}\) = 228 mW/cm² ≤ 720, provided 4 cm is where the derated \(I_{\mathrm{ta}}\) peaks (search the profile, not one point).
Q3A scanned linear-array mode radiates 80 mW at 3.5 MHz across a 40 mm scan. What is TIS?
\(P_1 = 80\times(10/40) = 20\) mW through a 1 cm window; \(\mathrm{TIS} = 20\times 3.5/210 = 0.33\).
Q4Why does multiplying PII by the system PRF overstate \(I_{\mathrm{SPTA}}\) for focused B-mode but not for plane-wave imaging?
In focused scanning only the lines whose beams overlap the point deliver energy there (about \(0.75\,w_{-6}/\Delta\) per frame); in plane-wave imaging every transmit insonifies every point, so pulses per second at the point equal the PRF.
Q5A transcranial probe has a 2 cm² aperture and radiates 120 mW. What is TIC, and what does the AIUM non-obstetric table allow?
\(D_{\mathrm{eq}} = \sqrt{4\times 2/\pi} = 1.60\) cm; \(\mathrm{TIC} = 120/(40\times 1.60) = 1.88\). Non-obstetric (adult transcranial included): \(1.5 < \mathrm{TI} \le 2.0\), under 120 min dwell.
Q6A 0.5 mm hydrophone measures a round Gaussian focus with \(W_{-6}\) = 1.0 mm. How low are the pressure and intensity readings?
\(d/W_{-6} = 0.5\): \(\langle p\rangle/p_0 = 0.918\), −0.74 dB in pressure and the same −0.74 dB in intensity (\(0.918^{2} = 0.843\), 15.7 % low); MI reads 8 % low.
Q7Tissue sits at 41 °C for 30 min. What is the thermal dose in CEM43?
\(R\) = 0.25 below 43 °C: \(30\times 0.25^{2} = 1.9\) CEM43.
22.15Figures from the literature
The regulatory tables and published figures behind the numbers of this module. Links only. Table, figure and equation numbers were checked against the cited documents (October 2026).
FDA Track 1 table and the Track 3 limits
Table 3 (Section 5.2.7) lists the preamendments derated levels by use: peripheral vessel 720 mW/cm², cardiac 430, fetal imaging and other 94, ophthalmic 17, with \(I_{\mathrm{SPPA.3}}\) 190 W/cm² or MI 1.9 (ophthalmic 28 W/cm² or 0.23). Section 5.2.8 gives the Track 3 global maxima (\(I_{\mathrm{SPTA.3}} \le 720\) and \(\mathrm{MI} \le 1.9\) or \(I_{\mathrm{SPPA.3}} \le 190\); ophthalmic \(\mathrm{TI} = \max(\mathrm{TIS}_{\mathrm{as}}, \mathrm{TIC}) \le 1\), 50 mW/cm², MI 0.23); Section 5.2.4 gives the ±30 % / ±15 % uncertainty rule and the membrane or capsule hydrophone recommendation. The card "Track 1 or Track 3 limit?" re-plots these values.
U.S. FDA, Marketing Clearance of Diagnostic Ultrasound Systems and Transducers, guidance issued February 21, 2023 (originally June 27, 2019). fda.gov/media/71100
AIUM: temperature rise versus exposure time without adverse effects
Table 1 pairs each rise with the longest exposure without observed adverse effects, fetal and postnatal: 9.6 °C for under 1 s (postnatal 5 s), 6 °C for 8 s (1 min), 5 °C for 30 s (4 min), 4 °C for 2 min (16 min), 3 °C for 8 min (1 h), 2 °C for 30 min (4 h), 1.5 °C for 1 h (≥ 50 h). Demo 22.1 and the card "How long may tissue be heated by \(\Delta T\)?" re-plot it. The same statement gives the 1.5 °C, 4 °C for 5 min, TI ≤ 2.5 for 4 min and TI < 0.7 anchors, and the MI thresholds 0.4 (lung), 1.4 (intestine), 1.9 and MI > 4.
AIUM, Statement on the Biological Effects of Ultrasound in Vivo, approved August 19, 2021. aium.org
AIUM dwell-time tables for the displayed TI
Table 1 (obstetric, also neonatal transcranial and spinal) and Table 2 (adult transcranial, abdominal, peripheral vascular and other) give maximum dwell times by TI range, with the 33 % reduction for ARFI and pulsed Doppler when bone is near the focus and the TIS-to-TIB switch at 10 weeks LMP. Rule card 17 re-plots both tables.
AIUM, Recommended Maximum Scanning Times for Displayed Thermal Index (TI) Values, approved October 30, 2016, reapproved September 26, 2022. aium.org
How well does TIS predict the temperature maximum?
Fig. 13 plots the computed maximum steady-state temperature rise against the unscanned-mode TIS for rectangular sources grouped by f-number, all at \(I_{\mathrm{SPTA.3}}\) = 720 mW/cm²; Fig. 14 plots the rise against the axial location of the maximum for circular sources. The spread of \(\Delta T_{\mathrm{max}}\) at a given TIS is the quantitative content of "TI is a ratio, not degrees" (rule card 9), and the axial locations show why the break-point depth exists. These are model calculations summarized in a review, not in vivo measurements.
W. D. O'Brien Jr., "Ultrasound–biophysics mechanisms," Prog. Biophys. Mol. Biol. 93(1–3), 212–255 (2007; online 2006). doi:10.1016/j.pbiomolbio.2006.07.010
The simplified TI formulas with their constants
Eqs. (4)–(7) give \(\mathrm{TIS} = P f/210\) mW·MHz (scanned, 1 cm window; stationary beam), \(\mathrm{TIB} = W(z)/(d_{-6}\times 40\ \mathrm{mW/cm})\) and \(\mathrm{TIC} = W_0/(D\times 40\ \mathrm{mW/cm})\), and Eq. (10) gives MI with \(C_{\mathrm{MI}} = 1\ \mathrm{MPa\cdot MHz^{-1/2}}\). These are the forms Demo 22.4 implements. The paper does not give the \(\sqrt{P_{\alpha} I_{\mathrm{ta},\alpha}}\) form of TIB used by IEC 62359 (see the flag in Section 22.5).
A. Nowicki, "Safety of ultrasonic examinations; thermal and mechanical indices," Med. Ultrason. 22(2), 203–210 (2020). doi:10.11152/mu-3272
22.16Go deeper
- U.S. FDA, Marketing Clearance of Diagnostic Ultrasound Systems and Transducers, guidance for industry and FDA staff, issued February 21, 2023 (originally issued June 27, 2019), docket FDA-2017-D-5372.
- IEC 60601-2-37:2024, Medical electrical equipment – Part 2-37: Particular requirements for the basic safety and essential performance of ultrasonic medical diagnostic and monitoring equipment, Ed. 3.0.
- IEC 62359:2010 + AMD1:2017, Ultrasonics – Field characterization – Test methods for the determination of thermal and mechanical indices related to medical diagnostic ultrasonic fields, Ed. 2.1.
- IEC 62127-1:2022, Ultrasonics – Hydrophones – Part 1: Measurement and characterization of medical ultrasonic fields, Ed. 2.0; with IEC 62127-2 (calibration) and IEC 62127-3 (hydrophone properties).
- IEC 61161:2013, Ultrasonics – Power measurement – Radiation force balances and performance requirements, Ed. 3.0.
- AIUM official statements: Statement on Biological Effects of Ultrasound in Vivo (2021); Recommended Maximum Scanning Times for Displayed Thermal Index (TI) Values (2016, reapproved 2022); Statement on Biological Effects in Tissues with Ultrasound Contrast Agents (2002, reapproved 2021).
- AIUM/NEMA UD 3-2004, Standard for Real-Time Display of Thermal and Mechanical Acoustic Output Indices on Diagnostic Ultrasound Equipment, Revision 2 (historical; superseded by IEC 60601-2-37).
- British Medical Ultrasound Society, Guidelines for the safe use of diagnostic ultrasound equipment (2009 revision).
- J. G. Abbott, "Rationale and derivation of MI and TI: a review," Ultrasound Med. Biol. 25, 431–441 (1999). Where the \(W_{\mathrm{deg}}\) models, the break-point depth and the \(\sqrt{f}\) of MI come from.
- W. L. Nyborg, "Biological effects of ultrasound: development of safety guidelines. Part II: general review," Ultrasound Med. Biol. 27, 301–333 (2001).
- R. E. Apfel and C. K. Holland, "Gauging the likelihood of cavitation from short-pulse, low-duty cycle diagnostic ultrasound," Ultrasound Med. Biol. 17, 179–185 (1991).
- S. A. Sapareto and W. C. Dewey, "Thermal dose determination in cancer therapy," Int. J. Radiat. Oncol. Biol. Phys. 10, 787–800 (1984).
- H. H. Pennes, "Analysis of tissue and arterial blood temperatures in the resting human forearm," J. Appl. Physiol. 1, 93–122 (1948).
- T. A. Bigelow et al., "The thermal index: its strengths, weaknesses, and proposed improvements," J. Ultrasound Med. 30, 714–734 (2011).
- A. Nowicki, "Safety of ultrasonic examinations; thermal and mechanical indices," Med. Ultrason. 22, 203–210 (2020).
- C. Kollmann et al., "Ultrasound output: thermal (TI) and mechanical (MI) indices," Ultraschall Med. 34, 422–434 (2013).
- NCRP Report No. 140, Exposure Criteria for Medical Diagnostic Ultrasound: II. Criteria Based on All Known Mechanisms (2002).
- F. A. Duck, Physical Properties of Tissue (Academic Press, 1990); T. L. Szabo, Diagnostic Ultrasound Imaging: Inside Out, 2nd ed. (Academic Press, 2014), chapters on acoustic output and bioeffects.