Cardiovascular Biomechanics, Cardiology & Vascular Surgery
Law of Laplace Cardiac & Aneurysm Wall Stress Formula
Calculate Spherical Left-Ventricular Myocardial Afterload Wall Stress (P*r)/(2*h), Cylindrical Aortic Aneurysm Circumferential Hoop Stress (P*r)/h in mmHg and kPa, and Geometric Stress Amplifier r/(2*h).
Distinguish spherical ventricular afterload (P*r / 2h) from cylindrical aneurysm hoop stress (P*r / h)!
Two of cardiology and vascular surgery’s most vital clinical phenomena follow directly from the Young-Laplace thin-shell biomechanics law! In a roughly spherical Left Ventricle, two orthogonal curvatures share the pressure load so Myocardial Wall Stress (true ventricular afterload and oxygen-demand driver!) equals sigma_sphere = (P * r) / (2 * h)—explaining why concentric hypertrophy (thickening h in aortic stenosis) normalizes wall stress, whereas eccentric dilated cardiomyopathy (enlarging r while thinning h) catastrophically triples afterload! Conversely, in a cylindrical abdominal aortic aneurysm (AAA), longitudinal curvature vanishes along the vessel axis, DOUBLING Circumferential Hoop Stress to sigma_cylinder = (P * r) / h (1 mmHg = 0.13332 kPa) and explaining why aneurysm rupture risk accelerates past 5.5 cm diameter!
Verify at Peak Systolic Pressure P = 150 mmHg (B2), Inner_Lumen_Radius r = 3.0 cm (B3), and Wall_Thickness h = 1.0 cm (B4): Spherical Ventricular Wall Stress is (150 * 3) / (2 * 1) = 225.00 mmHg (E2); Cylindrical Hoop Stress is (150 * 3) / 1 = 450.00 mmHg (E3), equal to 450 * 0.13332 = 59.99 kPa (E4); and Spherical Geometric Amplifier r / (2*h) is 1.5000 (E5)!
| Laplace Biomechanical Metric (P=150 mmHg, r=3.0 cm, h=1.0 cm) | Formula | Verified Output |
|---|---|---|
| Spherical LV Wall Stress (mmHg) | (Pressure * Radius) / (2 * Thickness) | 225.00 mmHg (675.00 @ r=4.5, h=0.5) |
| Cylindrical Hoop Stress (mmHg) | (Pressure * Radius) / Thickness | 450.00 mmHg (1,350.00 @ Dilated) |
| Cylindrical Hoop Stress (kPa) | Cylindrical_mmHg * 0.13332 | 59.99 kPa |
| Spherical Geometric Ratio r/(2h) | Inner_Radius / (2 * Wall_Thickness) | 1.5000 |
Cylindrical Hoop Stress (kPa) & Spherical Wall Stress Formula
Validate strictly positive Transmural_Pressure P, Inner_Radius r, and Wall_Thickness h!
=LET(pr_m, B2, rd_c, B3, th_c, B4, IF(OR(pr_m <= 0, rd_c <= 0, th_c <= 0), "Invalid Hemodynamic Geometry", ROUND(((pr_m * rd_c) / th_c) * 0.13332, 2)))Dilated Cardiomyopathy / Thin-Wall Aneurysm Remodeling Check (r = 4.5 cm, h = 0.5 cm)
Test chamber dilation B3 = 4.5 cm paired with wall thinning B4 = 0.5 cm at the exact same 150 mmHg blood pressure: Spherical LV Stress E2 triples to 675.00 mmHg and Cylindrical Hoop Stress E3 triples to 1,350.00 mmHg!
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0), "Invalid Hemodynamic Geometry", ROUND((B2 * B3) / (2 * B4), 2))Try the example
Use the original values or change them to test your understanding.