Appendix A — Formula Compendium

Every formula used in this book, with units, assumptions, and a worked example. Constants here are authoritative for the calculator source in the repository.

1. Physics

Wave equation

c = fλ

c = 1540 m/s in soft tissue (assumed by all scanners); f = frequency (Hz); λ = wavelength (m).

Attenuation ≈ 0.5 dB/cm/MHz in soft tissue.

Reflection coefficient

R = ((Z₂ - Z₁) / (Z₂ + Z₁))²        Z = ρc

Maximum frame rate

FR_max = c / (2 D N)

D = imaging depth (m); N = scan lines per frame.

Mechanical index

MI = P_r / √(f)

Regulatory cap 1.9. Contrast imaging requires low MI (0.1–0.3).

2. Doppler

Doppler equation

v = (Δf · c) / (2 f₀ cosθ)

Simplified Bernoulli

ΔP = 4v²

ΔP in mmHg, v in m/s. Valid only when v₁ < 1.5 m/s.

Expanded Bernoulli — mandatory when v₁ ≥ 1.5 m/s:

ΔP = 4(v₂² - v₁²)

Nyquist limit

v_max = (c · PRF) / (4 f₀ cosθ)        PRF_max = c / (2D)

3. Flow and cardiac output

CSA = π (D/2)² = 0.785 D²

SV = CSA × VTI

CO = (SV × HR) / 1000

CI = CO / BSA        SVI = SV / BSA

BSA = 0.007184 × height(cm)^0.725 × weight(kg)^0.425     (Du Bois)

SVR = ((MAP - CVP) × 80) / CO     (dyn·s·cm⁻⁵)

Qp/Qs = (CSA_RVOT × VTI_RVOT) / (CSA_LVOT × VTI_LVOT)

🧮 Worked example — cardiac output

LVOT diameter 2.0 cm; LVOT VTI 18 cm; HR 90/min; BSA 1.8 m².

CSA = 0.785 × 2.0² = 3.14 cm²

SV = 3.14 × 18 = 56.5 mL

CO = (56.5 × 90)/1000 = 5.09 L/min

CI = 5.09/1.8 = 2.83 L/min/m²; SVI = 56.5/1.8 = 31.4 mL/m² → below 35, a low-flow state despite a "normal" cardiac output.

4. Left ventricular systolic function

EF = ((EDV - ESV) / EDV) × 100

FS = ((LVIDd - LVIDs) / LVIDd) × 100

dP/dt = 32 / Δt(s)      (mmHg/s; from the MR jet, 1 to 3 m/s)

LV mass = 0.8 × [1.04 × ((LVIDd + PWTd + SWTd)³ - LVIDd³)] + 0.6

RWT = (2 × PWTd) / LVIDd

Dimensions in cm, mass in g. The 32 mmHg constant is 4(3²) − 4(1²).

🧮 Worked example — LV mass

LVIDd 5.0 cm, PWTd 1.0 cm, SWTd 1.0 cm.

0.8 × [1.04 × (7.0³ − 5.0³)] + 0.6 = 0.8 × [1.04 × 218] + 0.6 = 182.0 g

RWT = (2 × 1.0)/5.0 = 0.40 (≤ 0.42, eccentric geometry)

5. Ventriculo-arterial coupling

P_es ≈ 0.9 × SBP

Ea = P_es / SV        Ees = P_es / (ESV - V₀)

Ea/Ees = ESV / SV = ESV / (EDV - ESV) = (1 / EF) - 1     (assuming V₀ ≈ 0)

Normal Ea 2–2.5 mmHg/mL; normal Ees 2–3 mmHg/mL; normal Ea/Ees ≈ 0.6–1.0. Ratio > 1.5 indicates decoupling.

🧮 Worked example — coupling

SBP 110 mmHg; EDV 140 mL; ESV 95 mL → SV 45 mL; EF 32%.

P_es = 0.9 × 110 = 99 mmHg

Ea = 99/45 = 2.2 mmHg/mL

Ees (simplified) = 99/95 = 1.04 mmHg/mL

Ea/Ees = 95/45 = 2.11 → severe decoupling. Cross-check: 1/0.32 − 1 = 2.13. ✓

6. Right heart and pulmonary circulation

PASP = 4 v²(TR) + RAP

mPAP = 4 v²(PR early) + RAP

PADP = 4 v²(PR end) + RAP

PVR (WU) ≈ (v_TR / VTI_RVOT) × 10 + 0.16

FAC = ((RVEDA - RVESA) / RVEDA) × 100

RV–PA coupling = TAPSE / PASP      (mm/mmHg)

MPI (Tei) = (IVCT + IVRT) / ET

v_TR in m/s, VTI in cm. TAPSE/PASP 0.3–0.4 mm/mmHg indicates uncoupling. RV MPI abnormal ≥ 0.55 (tissue Doppler) or ≥ 0.40 (pulsed Doppler).

🧮 Worked example — PASP

TR V_max 3.6 m/s; IVC 2.4 cm with < 50% collapse → RAP 15 mmHg.

TR gradient = 4 × 3.6² = 51.8 mmHg; PASP = 51.8 + 15 = 66.8 mmHg.

With TAPSE 1.2 cm: TAPSE/PASP = 12/66.8 = 0.18 mm/mmHg → severe RV–PA uncoupling.

7. Diastolic function

E/e′ thresholds (ASE 2025): septal ≥ 15, lateral ≥ 13, average ≥ 14 indicate elevated LAP.

LAV = (8/3π) × (A_4C × A_2C) / L        LAVi = LAV / BSA

LA stiffness = (E/e′) / LARS

LAP across an interatrial shunt = 4V² + RAP

L = the shorter of the two long-axis lengths. LAVi abnormal > 34 mL/m².

8. Valve quantification

AVA = (CSA_LVOT × VTI_LVOT) / VTI_AV

DI = VTI_LVOT / VTI_AV        (severe AS < 0.25)

MVA = 220 / PHT               PHT = 0.29 × DT

Flow_PISA = 2πr² × V_aliasing

EROA = (2πr² × V_aliasing) / V_peak(regurgitant jet)

RegVol = EROA × VTI(regurgitant jet)

RF = (RegVol / SV through the regurgitant valve) × 100

ELI = (AVA × A_a) / ((A_a - AVA) × BSA)

r in cm, velocities in cm/s, EROA in cm², RegVol in mL. Mean gradient is machine-integrated across the envelope — it is not 4 × (mean velocity)².

🧮 Worked example — AVA and DI

LVOT diameter 2.1 cm; LVOT VTI 16 cm; AV VTI 88 cm; BSA 1.7 m².

CSA = 0.785 × 2.1² = 3.46 cm²; SV = 3.46 × 16 = 55.4 mL

AVA = 55.4/88 = 0.63 cm²; AVAi = 0.37 cm²/m²

DI = 16/88 = 0.18 → severe AS, concordant with AVA.

SVI = 55.4/1.7 = 32.6 mL/m² → low-flow state; interpret gradients accordingly (Chapter 18).

🧮 Worked example — MVA by PHT

PHT 180 ms → MVA = 220/180 = 1.22 cm² (severe mitral stenosis is ≤ 1.5 cm²).

9. Volumes and geometry

Biplane method of discs (Simpson's)

V = (π/4) × Σ (aᵢ × bᵢ) × (L/n)

aᵢ, bᵢ = orthogonal disc diameters; L = LV long axis; n = number of discs (typically 20).

Teichholz — reproduced for completeness; not recommended for EF estimation:

V = (7.0 / (2.4 + LVID)) × LVID³

10. Fluid responsiveness indices

ΔV_peak = ((V_max - V_min) / ((V_max + V_min)/2)) × 100      threshold ≥ 12%

dIVC = ((D_max - D_min) / D_min) × 100                        ≥ 18%

cIVC = ((D_max - D_min) / D_max) × 100                        > 40–50%

SVC collapsibility = ((D_max - D_min) / D_max) × 100           ≥ 36%

Passive leg raise: ΔVTI ≥ 10%. End-expiratory occlusion: ΔVTI ≥ 5%. Mini fluid challenge (100 mL): ΔVTI ≥ 6%.

11. Miscellaneous

PHT = 0.29 × DT

CPO = (MAP × CO) / 451      (watts; < 0.6 W adverse in cardiogenic shock)

PAPi = (PASP - PADP) / RAP

Wall stress (Laplace): σ ∝ (P × r) / 2h

Aortic regurgitation severity by PHT: < 200 ms severe; > 500 ms mild.

Error propagation reference