1. Mechanical oscillations and wave motion. 2. Thermodynamics of biological systems 3. PHYSICAL BASES OF STRUCTURE AND FUNCTIONING OF BIOLOGICAL MEMBRANES. TRANSPORT PHENOMENON 4. BIOELECTRIC POTENTIALS. Resting and Active Potential. 5. Biomechanics. HOOKE’S LAW; DEFORMATION TYPES 6. Biomechanics of lever system in the human body 7. HYDRODYNAMICS OF VISCOUS LIQUID Explain every topic with equations
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The library is a clinical medicine collection and doesn't include biophysics textbooks, so the explanations below are drawn from my training knowledge in medical biophysics. Here is a full, equation-rich treatment of all 7 topics.
Medical Biophysics - Complete Topic Guide
1. Mechanical Oscillations and Wave Motion
Simple Harmonic Motion (SHM)
A system undergoes SHM when the restoring force is proportional to displacement:
F = -kx
where k is the spring constant (N/m) and x is displacement from equilibrium.
Newton's second law gives the equation of motion:
m(d²x/dt²) = -kx
d²x/dt² + (k/m)x = 0
The solution is:
x(t) = A cos(ωt + φ)
where:
- A = amplitude (maximum displacement)
- ω = angular frequency (rad/s)
- φ = initial phase angle
Angular frequency: ω = √(k/m)
Period: T = 2π/ω = 2π√(m/k)
Frequency: f = 1/T = (1/2π)√(k/m)
Velocity: v(t) = -Aω sin(ωt + φ)
Acceleration: a(t) = -Aω² cos(ωt + φ) = -ω²x
Energy in SHM:
- Potential energy: E_p = (1/2)kx² = (1/2)kA²cos²(ωt + φ)
- Kinetic energy: E_k = (1/2)mv² = (1/2)mω²A²sin²(ωt + φ)
- Total mechanical energy: E = E_p + E_k = (1/2)kA² (constant)
Damped Oscillations
In real biological systems, oscillations lose energy due to friction/viscosity:
m(d²x/dt²) + b(dx/dt) + kx = 0
where b is the damping coefficient (N·s/m).
Introducing: β = b/(2m) (damping factor), ω₀ = √(k/m) (natural frequency):
d²x/dt² + 2β(dx/dt) + ω₀²x = 0
For underdamped case (β < ω₀):
x(t) = Ae^(-βt) cos(ω₁t + φ)
where the damped frequency: ω₁ = √(ω₀² - β²)
The amplitude decays exponentially. Quality factor: Q = ω₀/(2β)
Forced Oscillations and Resonance
An external periodic force F₀cos(ωt) drives the system:
m(d²x/dt²) + b(dx/dt) + kx = F₀cos(ωt)
Amplitude of steady-state oscillations:
A = (F₀/m) / √[(ω₀² - ω²)² + (2βω)²]
Resonance occurs when ω ≈ ω₀, giving maximum amplitude. This is relevant to the resonance of bones and organs exposed to mechanical vibrations or ultrasound.
Wave Motion
A transverse or longitudinal wave traveling in the +x direction:
y(x, t) = A sin(kx - ωt + φ)
where the wave number k = 2π/λ and wavelength λ = vT = v/f
The relationship: v = λf = ω/k
Wave equation:
∂²y/∂t² = v² (∂²y/∂x²)
Wave speed in a medium:
- In an elastic solid: v = √(E/ρ), where E = elastic modulus, ρ = density
- In a fluid: v = √(B/ρ), where B = bulk modulus
Sound intensity:
I = P/S (W/m²), where P = power, S = area
Sound level in decibels: L = 10 log(I/I₀), where I₀ = 10⁻¹² W/m² (threshold of hearing)
Ultrasound (f > 20 kHz) is used in medical diagnostics. The Doppler effect:
f' = f₀ · (v ± v_observer)/(v ∓ v_source)
Used in Doppler echocardiography and blood flow measurement.
2. Thermodynamics of Biological Systems
First Law of Thermodynamics
Energy conservation for a biological system:
ΔU = Q - W
- ΔU = change in internal energy
- Q = heat absorbed by the system
- W = work done by the system
For biological cells, work includes mechanical work (muscle contraction), osmotic work, and electrical work.
In differential form: dU = δQ - δW
For a process at constant pressure (most biological processes): Q_p = ΔH (enthalpy)
H = U + pV, so ΔH = ΔU + pΔV
Second Law of Thermodynamics and Entropy
ΔS ≥ Q/T (Clausius inequality)
For a reversible process: dS = δQ_rev / T
Living organisms are open systems far from equilibrium. They maintain order (low entropy internally) by exporting entropy to the environment:
ΔS_universe = ΔS_system + ΔS_surroundings ≥ 0
Gibbs Free Energy
At constant T and p, the criterion for spontaneity is the Gibbs free energy:
G = H - TS
ΔG = ΔH - TΔS
- If ΔG < 0: process is spontaneous (exergonic) - e.g., ATP hydrolysis
- If ΔG > 0: process requires energy input (endergonic)
- If ΔG = 0: system is at equilibrium
Standard free energy: ΔG° = -RT ln K_eq
where R = 8.314 J/(mol·K), T = absolute temperature, K_eq = equilibrium constant.
ATP hydrolysis: ATP + H₂O → ADP + Pᵢ, ΔG° = -30.5 kJ/mol
Coupled reactions: If ΔG_reaction1 > 0 and ΔG_reaction2 < 0, coupling is possible when:
ΔG_total = ΔG₁ + ΔG₂ < 0
Chemical Potential
For a component i in a mixture:
μᵢ = μᵢ° + RT ln(aᵢ)
where aᵢ is the activity (concentration for dilute solutions).
Electrochemical potential (for ions):
μ̃ᵢ = μᵢ° + RT ln(cᵢ) + zᵢFψ
where zᵢ = charge number, F = Faraday's constant (96,485 C/mol), ψ = electric potential.
Thermodynamics of Metabolism
The efficiency of energy conversion in biological systems:
η = W_useful / Q_total
Mitochondrial efficiency for ATP synthesis is approximately 40%. The rest is released as heat, maintaining body temperature.
Basal metabolic rate (BMR) follows the allometric scaling:
BMR ≈ k · M^(3/4) (Kleiber's law)
where M is body mass.
3. Physical Bases of Biological Membrane Structure and Transport Phenomena
Membrane Structure
Biological membranes are lipid bilayers ~7-8 nm thick. The fluid-mosaic model describes a phospholipid bilayer with embedded proteins.
Membrane capacitance per unit area:
C_m = ε₀ ε_r / d
where ε₀ = 8.85 × 10⁻¹² F/m, ε_r ≈ 2-3 (relative permittivity of lipid), d ≈ 7 nm.
Typical value: C_m ≈ 0.01 F/m² = 1 μF/cm²
Passive Transport - Fick's Laws of Diffusion
Fick's First Law (steady-state flux):
J = -D (dc/dx)
- J = molar flux (mol/m²·s)
- D = diffusion coefficient (m²/s)
- dc/dx = concentration gradient
Fick's Second Law (non-steady state):
∂c/∂t = D (∂²c/∂x²)
For diffusion across a membrane of thickness d:
J = -D (c₂ - c₁)/d = P(c₁ - c₂)
where P = D/d is the permeability coefficient (m/s).
Osmosis
The osmotic pressure across a semipermeable membrane (van't Hoff equation):
π = iCRT
where i = van't Hoff factor, C = molar concentration (mol/m³), R = 8.314 J/(mol·K), T = temperature (K).
For plasma: π ≈ 780 kPa (≈ 7.7 atm). This drives water across cell membranes.
Water flux (osmotic flow):
J_w = L_p (ΔP - Δπ)
where L_p is the hydraulic conductivity (water permeability).
Electrodiffusion - Nernst-Planck Equation
When both concentration and electrical gradients drive ion movement:
J_i = -D_i [dc_i/dx + (z_i F)/(RT) · c_i · dψ/dx]
At equilibrium (J = 0), integrating gives the Nernst equation for the equilibrium potential of ion i:
E_i = (RT)/(z_i F) · ln(c_i^out / c_i^in)
At 37°C, RT/F ≈ 26.7 mV:
E_i = (26.7 mV / z_i) · ln(c_i^out / c_i^in)
For K⁺: E_K ≈ -90 mV; for Na⁺: E_Na ≈ +60 mV; for Cl⁻: E_Cl ≈ -70 mV.
Active Transport
Active transport moves ions against their electrochemical gradient, requiring ATP.
Na⁺/K⁺-ATPase: pumps 3 Na⁺ out and 2 K⁺ in per ATP hydrolyzed. It is electrogenic (net current outward).
Work done per cycle:
W = 3(μ̃_Na^in - μ̃_Na^out) + 2(μ̃_K^out - μ̃_K^in)
Energy balance: W ≤ |ΔG_ATP| = 50-60 kJ/mol under physiological conditions.
4. Bioelectric Potentials - Resting and Action Potential
Resting Membrane Potential (RMP)
The resting membrane potential of most excitable cells is between -60 to -90 mV (inside negative).
When multiple ions contribute, the resting potential is given by the Goldman-Hodgkin-Katz (GHK) equation:
V_m = (RT/F) · ln [ (P_K[K⁺]_o + P_Na[Na⁺]_o + P_Cl[Cl⁻]_i) / (P_K[K⁺]_i + P_Na[Na⁺]_i + P_Cl[Cl⁻]_o) ]
At rest: P_K : P_Na : P_Cl ≈ 1 : 0.04 : 0.45
Dominant term is K⁺ due to high resting permeability, so V_m ≈ E_K ≈ -90 mV (slightly more positive due to Na⁺ leak).
Equivalent Circuit of the Membrane
The membrane is modeled electrically as:
- Capacitance: C_m (lipid bilayer)
- Conductances: g_K, g_Na, g_Cl (ion channels)
- Batteries: E_K, E_Na, E_Cl (Nernst potentials)
Current through each channel: I_ion = g_ion (V_m - E_ion)
Total membrane current:
I = C_m (dV_m/dt) + g_K(V_m - E_K) + g_Na(V_m - E_Na) + g_Cl(V_m - E_Cl)
Action Potential (AP)
The action potential has four phases:
Phase 1 - Depolarization: Na⁺ channels open rapidly.
Threshold potential ≈ -55 mV. Above threshold, positive feedback (Hodgkin cycle):
↑V_m → ↑g_Na → ↑I_Na (inward) → ↑V_m (further depolarization)
The overshoot reaches ≈ +30 mV (approaching E_Na ≈ +60 mV).
Phase 2 - Repolarization: Na⁺ channels inactivate; K⁺ channels (delayed rectifiers) open.
K⁺ efflux repolarizes the membrane back toward E_K.
Phase 3 - Hyperpolarization (afterpotential): K⁺ channels remain open briefly, V_m dips below resting level (≈ -95 mV).
Hodgkin-Huxley model (1952) describes conductance changes with gating variables:
g_Na = g̅_Na · m³ · h (m = activation gate, h = inactivation gate)
g_K = g̅_K · n⁴ (n = activation gate)
Each gating variable follows: dα/dt = α_α(V)(1-α) - β_α(V)·α
where α_α, β_α are voltage-dependent rate constants.
Full Hodgkin-Huxley equation:
C_m (dV/dt) = -g̅_Na m³h(V - E_Na) - g̅_K n⁴(V - E_K) - g_L(V - E_L) + I_ext
Propagation of the Action Potential
Cable equation (for nerve axon):
(λ²)(∂²V/∂x²) = τ_m(∂V/∂t) + V
Space constant: λ = √(r_m / r_a) = √(ρ_m d / (4ρ_i))
Time constant: τ_m = r_m · C_m
Conduction velocity: θ = λ/τ_m (unmyelinated fibers), proportional to √d.
For myelinated fibers (saltatory conduction): θ ∝ d, much faster (up to 120 m/s).
5. Biomechanics - Hooke's Law and Deformation Types
Stress and Strain
Normal (tensile/compressive) stress:
σ = F/A (Pa = N/m²)
Normal strain:
ε = ΔL/L₀ (dimensionless)
Shear stress:
τ = F_t/A (tangential force per area)
Shear strain:
γ = Δx/h = tan(φ)
where φ is the angle of deformation.
Hooke's Law
Within the elastic (linear) region:
σ = E · ε (Young's modulus, E)
- Cortical bone: E ≈ 17-20 GPa
- Cartilage: E ≈ 1-10 MPa
- Tendon: E ≈ 1.5 GPa
- Muscle: E ≈ 10-100 kPa
For shear: τ = G · γ (G = shear modulus)
For volumetric compression: P = -K · (ΔV/V₀) (K = bulk modulus)
Relations between elastic constants:
E = 2G(1 + ν) = 3K(1 - 2ν)
where ν = Poisson's ratio: ν = -(lateral strain)/(axial strain) = -ε_transverse/ε_axial
For most biological tissues: ν ≈ 0.4-0.5 (nearly incompressible).
Types of Deformation
1. Tensile/Compressive: Force applied along the axis.
- Axial deformation: ΔL = (F · L₀)/(E · A)
- Relevant to bone and tendon loading.
2. Shear deformation: Force parallel to the surface.
- τ = G · γ, where G = E/(2(1+ν))
3. Bending (flexural):
Normal stress in a beam: σ = M·y/I
- M = bending moment, y = distance from neutral axis, I = second moment of area.
- Maximum bending stress: σ_max = M · c / I where c = distance to outer fiber.
For a solid circular cross-section: I = πd⁴/64
4. Torsion:
Shear stress in a shaft: τ = T·r/J
- T = torque, r = radial distance, J = polar moment of inertia = πd⁴/32.
5. Volumetric compression:
ΔV/V₀ = -P/K
Stress-Strain Curve for Biological Tissues
Typical soft tissue curve has:
- Toe region (low stress, large strain): wavy collagen fibers straighten (nonlinear, J-shaped curve)
- Linear (elastic) region: Hooke's law applies
- Yield point: permanent deformation begins
- Failure point (ultimate stress σ_u)
Elastic energy stored per unit volume (strain energy density):
u = (1/2) σ ε = σ²/(2E)
Fatigue: Repeated cyclic loading at σ < σ_u can cause failure. Described by the S-N (Wöhler) curve.
6. Biomechanics of the Lever System in the Human Body
Lever Principles
A lever consists of: load (resistance), effort (force), and fulcrum (pivot).
Condition of static equilibrium (torques balance):
ΣM = 0: F_E · d_E = F_L · d_L
Mechanical advantage (MA):
MA = F_L / F_E = d_E / d_L
Classes of Levers in the Human Body
Class I lever - Fulcrum between effort and load.
- Example: Head nodding on the atlanto-occipital joint.
- Effort (neck extensors) acts posteriorly; load (weight of face/skull) acts anteriorly; fulcrum = atlantooccipital joint.
- MA can be > or < 1.
Class II lever - Load between fulcrum and effort.
- Example: Standing on tiptoe (rising onto the balls of the feet).
- Fulcrum = metatarsophalangeal joints; load = body weight at ankle; effort = calf muscles (gastrocnemius/soleus).
- MA > 1 (force advantage).
- Equation: F_E · d_E = W · d_W → F_E = W · (d_W/d_E) where d_W < d_E.
Class III lever - Effort between fulcrum and load (most common in the body).
- Example: Bicep curl (forearm flexion).
- Fulcrum = elbow joint; effort = biceps muscle insertion (~5 cm from elbow); load = weight in hand (~35 cm from elbow).
- MA < 1, meaning the muscle exerts a force LARGER than the load.
- Equation: F_biceps · 0.05 m = F_load · 0.35 m
- F_biceps = F_load × 7 (biceps must exert 7× the load)
This design sacrifices force for speed and range of motion.
Torque and Joint Reaction Forces
For the forearm held horizontal with a weight W in hand:
Torque by weight about elbow: M_W = W · L (L = distance from elbow to hand)
Torque by muscle: M_m = F_m · d (d = muscle moment arm)
Equilibrium: F_m = W · L / d
Joint reaction force (R): The net force at the joint:
R = F_m - W (approximately, for vertical forces only)
Since d << L, R >> W. For example, at the shoulder joint during arm elevation, joint reaction forces can reach 8-10 × body weight.
Moment of Inertia and Rotation
For limb rotation, the equation of motion:
I · α = ΣM_net
where I = Σ(m_i · r_i²) is the moment of inertia of the limb and α is angular acceleration.
For a uniform rod of mass m, length L (approximating a limb segment):
I = (1/3)mL² (about one end)
Limb angular velocity and linear velocity of distal end: v = ω · L
7. Hydrodynamics of Viscous Liquids
Ideal Fluid - Continuity Equation
For incompressible flow, conservation of mass gives:
A₁v₁ = A₂v₂ (continuity equation)
where A = cross-sectional area, v = flow velocity. Blood flows slower in capillaries than in the aorta because of the much larger total cross-sectional area.
Bernoulli's Equation (Ideal Fluid)
Energy conservation along a streamline:
p + (1/2)ρv² + ρgh = constant
or: p₁ + (1/2)ρv₁² + ρgh₁ = p₂ + (1/2)ρv₂² + ρgh₂
- p = static pressure, (1/2)ρv² = dynamic pressure, ρgh = hydrostatic pressure.
In blood vessels: stenosis (narrowing) increases velocity (continuity) and decreases static pressure (Bernoulli). This creates the risk of vessel collapse (venous collapse, Starling resistor effect).
Viscosity
Real (viscous) fluids resist flow. Dynamic viscosity η (Pa·s):
τ = η (dv/dy)
Newtonian fluid: shear stress τ is proportional to shear rate dv/dy.
Blood is a non-Newtonian fluid (shear-thinning) but approximated as Newtonian in large vessels.
Blood viscosity: η_blood ≈ 3-4 × 10⁻³ Pa·s; water: η_water = 1 × 10⁻³ Pa·s.
Kinematic viscosity: ν = η/ρ (m²/s)
Poiseuille's Law (Hagen-Poiseuille)
For laminar, steady flow of a viscous fluid through a straight circular tube:
Q = (π r⁴ ΔP) / (8 η L)
where:
- Q = volumetric flow rate (m³/s)
- r = tube radius (m)
- ΔP = pressure difference (Pa)
- η = dynamic viscosity (Pa·s)
- L = tube length (m)
The r⁴ dependence is critical in physiology: Halving the radius reduces flow 16-fold at the same pressure.
Hydraulic resistance:
R = ΔP/Q = 8ηL/(πr⁴)
Analogous to Ohm's law: ΔP = Q · R.
For vessels in series: R_total = R₁ + R₂ + ...
For vessels in parallel: 1/R_total = 1/R₁ + 1/R₂ + ...
Mean flow velocity:
v̄ = Q/A = r² ΔP / (8ηL)
Velocity profile (parabolic for Poiseuille flow):
v(r) = (ΔP / 4ηL)(R² - r²)
Maximum velocity at center: v_max = R² ΔP / (4ηL) = 2v̄
Reynolds Number
Dimensionless parameter predicting laminar vs. turbulent flow:
Re = ρvd/η = vd/ν
- Re < 2000: laminar flow (parabolic profile, quiet)
- 2000 < Re < 4000: transitional
- Re > 4000: turbulent flow (chaotic, energy wasted, produces sounds - bruits, heart murmurs)
In the aorta: Re ≈ 2000-4000 during peak systole (near-turbulent).
In arterioles and capillaries: Re << 1 (strongly laminar).
Turbulence and Korotkoff sounds in sphygmomanometry: when cuff pressure exceeds systolic pressure, artery collapses; when cuff pressure falls below systolic, turbulent jets form → Korotkoff sounds → blood pressure measurement.
Stokes' Law
Drag force on a sphere moving through a viscous fluid:
F_drag = 6π η r v
where r = sphere radius, v = velocity.
At terminal velocity (gravity = drag + buoyancy):
(4/3)πr³(ρ_particle - ρ_fluid)g = 6π η r v_t
v_t = 2r²(ρ_particle - ρ_fluid)g / (9η)
This underlies erythrocyte sedimentation rate (ESR): larger or aggregated red cells (rouleaux formation) sediment faster. Used clinically as a nonspecific marker of inflammation.
Vascular Compliance
Blood vessels are elastic, not rigid tubes. Compliance C describes their ability to expand:
C = ΔV/ΔP (m³/Pa)
The windkessel model treats the arterial tree as a compliant reservoir (capacitor) + peripheral resistance:
C (dP/dt) = Q_in - P/R
During diastole (Q_in ≈ 0):
P(t) = P₀ e^(-t/RC)
where RC = τ = vascular time constant (typically ~0.1-0.2 s). This explains the diastolic decay of arterial blood pressure.
Pulse wave velocity (speed of the pressure wave, not blood):
c = √(Eh/(2ρr))
where E = vessel wall elastic modulus, h = wall thickness, ρ = blood density, r = vessel radius.
Typical values: aorta ≈ 4-6 m/s; peripheral arteries ≈ 8-12 m/s. PWV increases with arterial stiffness (aging, hypertension), making it a biomarker of cardiovascular risk.
Summary Table of Key Equations
| Topic | Key Equation | Variables |
|---|---|---|
| SHM | x = A cos(ωt+φ), T = 2π√(m/k) | A=amplitude, ω=angular frequency |
| Wave propagation | v = λf, y = A sin(kx-ωt) | λ=wavelength, f=frequency |
| Thermodynamics | ΔG = ΔH - TΔS | G=Gibbs energy, H=enthalpy, S=entropy |
| Diffusion | J = -D(dc/dx) | D=diffusion coeff, c=concentration |
| Nernst potential | E = (RT/zF)ln(c_out/c_in) | z=valence, F=Faraday's constant |
| Goldman eq. | V_m = (RT/F)ln[(P_K[K]_o+...)/(P_K[K]_i+...)] | P=permeability |
| Hooke's law | σ = Eε | E=Young's modulus |
| Lever equilibrium | F_E × d_E = F_L × d_L | d=moment arm |
| Poiseuille's law | Q = πr⁴ΔP/(8ηL) | r=radius, η=viscosity |
| Reynolds number | Re = ρvd/η | ρ=density, v=velocity, d=diameter |
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