Laplace law

**Formula (high-yield forms):** - Wall tension (thin-walled cylinder/vessel): T = P × r - Wall stress (σ) = (P × r) / (2 × h), where h = wall thickness. Often tested as: wall stress ∝ (pressure × radius) / wall thickness. - Sphere (e.g., alveolus): P = 2T / r. (Some texts give P = 4T/r for a soap bubble with two surfaces.)

**Pathophysiology and key applications:** 1. **Cardiac (ventricular wall stress):** Wall stress = (P × r) / (2h). Increased preload (volume) dilates the ventricle (↑ r) and increases wall stress. Increased afterload (↑ P) also raises wall stress. - Pressure overload (e.g., aortic stenosis, HTN) leads to concentric hypertrophy (parallel sarcomere addition), which increases h and normalizes wall stress. - Volume overload (e.g., aortic/mitral regurgitation) leads to eccentric hypertrophy (series sarcomere addition) with ↑ r. - Dilated cardiomyopathy: ↑ r means ↑ wall stress and ↑ myocardial O2 demand, so a failing, dilated heart is energetically inefficient. This is why preload reduction (nitrates, diuretics) helps in HF. 2. **Aneurysms:** Larger radius means higher wall tension, which creates a positive feedback loop that promotes further expansion and rupture. Abdominal aortic aneurysm repair is generally indicated at ≥5.5 cm (or rapid growth, >0.5 cm in 6 months, or symptoms). 3. **Pulmonary (alveoli):** Collapsing pressure P = 2T / r. Small alveoli have higher collapsing pressure and tend to empty into larger ones (atelectasis). **Surfactant** (DPPC, from type II pneumocytes) reduces surface tension more in smaller alveoli, stabilizing them. Deficiency leads to neonatal RDS (prematurity, low lecithin:sphingomyelin ratio <1.5-2). 4. **Vessels (capillaries):** Capillaries have a very small radius, so they withstand high pressure with very thin walls and little tension. Large vessels (aorta) need thick walls. 5. **Other hollow organs:** Bladder and GI tract (e.g., toxic megacolon, where dilation increases perforation risk at a cecal diameter of ≥12 cm and a colonic diameter of >6 cm). Gallbladder and bowel wall distension similarly. 6. **Esophageal varices / Zenker or other diverticula:** A larger radius increases wall tension and the risk of rupture or bleeding. Variceal wall tension is related to portal pressure, vessel radius, and wall thickness.

**Clinical pearls:** - Cardiac wall stress is a major determinant of myocardial O2 consumption (along with HR, contractility). - Compensatory hypertrophy normalizes wall stress by increasing h, but it can eventually lead to diastolic dysfunction and ischemia. - Remember the contrast: Poiseuille's law (resistance ∝ 1/r⁴, determines flow), whereas Laplace's law describes wall tension.

**Exam tip:** If a question asks why the dilated ventricle is at a disadvantage or why a larger aneurysm is more likely to rupture, the answer is Laplace's law. If it asks about alveolar stability, think surfactant and Laplace.