Key Formulas & Equations
Ohm's Law (Bioimpedance): Z = V/I
Bioimpedance: Z = √(R² + Xc²)
Capacitive Reactance: Xc = 1/(2πfC)
Poiseuille's Law: Q = (πr⁴ΔP)/(8ηL)
Flow Rate: Q = A·v = ΔP/R
Reynolds Number (Blood): Re = ρvD/μ
Oxygen Saturation: SaO₂ = [HbO₂]/([Hb] + [HbO₂]) × 100%
Heart Rate (from ECG): HR = 60/R-R interval (seconds)
Shear Stress (Blood Vessel): τ = 4ηQ/(πr³)
Wall Tension (Laplace): T = P·r (cylinder); T = P·r/2 (sphere)
Pressure Gradient: ΔP = 8ηLQ/(πr⁴)
Doppler Equation: Δf = 2f₀v·cos(θ)/c
Ultrasound Velocity: c = √(B/ρ)
Attenuation: I = I₀e^(-μx)
Core Concepts
Biomechanics
The study of mechanical laws relating to the movement or structure of living organisms. Includes joint forces, bone mechanics, and prosthetic design.
Biomedical Instrumentation
Design of devices to measure physiological signals: ECG, EEG, EMG, blood pressure, pulse oximetry. Involves sensors, signal conditioning, and noise reduction.
Biomaterials
Materials engineered to interact with biological systems for medical purposes. Must be biocompatible, non-toxic, and mechanically appropriate.
Medical Imaging
Technologies for visualizing the human body: X-ray, CT, MRI, ultrasound, PET. Each has specific physics principles, advantages, and limitations.
Biomedical Signal Processing
Analysis of physiological signals using digital signal processing: filtering, Fourier transforms, feature extraction, and pattern recognition.
Physiological Systems Modeling
Mathematical representation of biological systems: cardiovascular, respiratory, neural, and metabolic models for simulation and analysis.
Prosthetics and Orthotics
Design and fitting of artificial devices to replace or support missing or impaired body parts. Involves mechanics, materials, and human factors.
Cell and Tissue Engineering
Applying engineering principles to biological tissues: scaffolds, bioreactors, stem cell culture, and regenerative medicine.
Worked Examples
Example 1: Nernst Potential
Problem: Calculate equilibrium potential for K⁺ (inside = 150 mM, outside = 5 mM, T = 37°C).
Solution:
E = (RT/zF)ln([K⁺]out/[K⁺]in) = (61.54/z)log(5/150)
E = 61.54 × log(0.0333) = 61.54 × (-1.477) = -90.9 mV
Example 2: Doppler Shift
Problem: Ultrasound (5 MHz) reflects from blood moving at 0.5 m/s toward probe (c = 1540 m/s, θ = 0°). Calculate frequency shift.
Solution:
Δf = 2f₀v·cos(θ)/c = 2 × 5×10⁶ × 0.5 × 1/1540 = 3247 Hz ≈ 3.25 kHz
Example 3: Beer-Lambert Law
Problem: A solution (ε = 1000 M⁻¹cm⁻¹, l = 1 cm) has absorbance 0.5. Calculate concentration.
Solution:
A = εlc → c = A/(εl) = 0.5/(1000 × 1) = 0.0005 M = 0.5 mM
Practice Problems
Problem 1
Calculate Reynolds number for blood (ρ = 1060 kg/m³, μ = 0.0035 Pa·s) flowing at 0.3 m/s in a 5 mm artery.
Answer: Re = 1060 × 0.3 × 0.005/0.0035 = 454 (laminar flow)
Problem 2
A capillary (radius 4 μm, length 1 mm, η = 0.004 Pa·s) has pressure drop 2000 Pa. Calculate flow rate using Poiseuille's Law.
Answer: Q = π × (4×10⁻⁶)⁴ × 2000/(8 × 0.004 × 0.001) = 5.03×10⁻¹³ m³/s
Problem 3
Calculate heart rate from ECG with R-R interval of 0.8 seconds.
Answer: HR = 60/0.8 = 75 bpm
Problem 4
Calculate wall tension in an aorta (radius 1.5 cm, pressure 120 mmHg = 16,000 Pa).
Answer: T = P × r = 16000 × 0.015 = 240 N/m
Problem 5
A 2 MHz ultrasound probe detects a 500 Hz shift. Calculate blood velocity (c = 1540 m/s, θ = 60°).
Answer: v = Δf × c/(2f₀cosθ) = 500 × 1540/(2 × 2×10⁶ × 0.5) = 0.385 m/s
Problem 6
Calculate oxygen saturation if [HbO₂] = 15 g/dL and [Hb] = 3 g/dL.
Answer: SaO₂ = 15/(15+3) × 100% = 83.3%
Problem 7
A material has E = 20 GPa and experiences strain of 0.002. Calculate stress.
Answer: σ = E × ε = 20×10⁹ × 0.002 = 40 MPa