Musculoskeletal Biophysics

Derivation: Force from Cross-Bridge Cycling

Step 1 — Single cross-bridge force: Each myosin head undergoes a conformational change (the “power stroke”) upon binding actin, generating a displacement $d \approx 10$ nm and a force $f_{\text{cb}} \approx 2$ pN. The work done per stroke is:

$w_{\text{cb}} = f_{\text{cb}} \times d \approx 2 \times 10^{-12} \times 10 \times 10^{-9} = 2 \times 10^{-20} \text{ J}$

This is comparable to the free energy of ATP hydrolysis under physiological conditions ($\Delta G_{\text{ATP}} \approx 50 \text{ kJ/mol} = 8.3 \times 10^{-20}$ J per molecule), giving a thermodynamic efficiency of ~25%.

Step 2 — Total muscle force: The total isometric force is the product of the number of attached cross-bridges and the force per bridge:

$F = n \times f_{\text{cb}}$

where $n$ is the number of attached cross-bridges at any instant. The fraction attached depends on the attachment rate constant $f_+$ and detachment rate constant $g_+$:

$n = N_{\text{total}} \times \frac{f_+}{f_+ + g_+}$

For a typical half-sarcomere with ~300 myosin heads and ~50% attached: $n \approx 150$, giving $F_{\text{half-sarc}} = 150 \times 2 = 300$ pN. A whole muscle contains ~$10^{10}$ parallel half-sarcomeres, generating forces up to ~3000 N.

Derivation: Hill's Force-Velocity Equation

Step 3 — Force-velocity relationship: A.V. Hill (1938) discovered empirically that the force-velocity relationship of shortening muscle is a rectangular hyperbola:

$\boxed{(F + a)(v + b) = (F_0 + a)b}$

where $F_0$ is the isometric force (at $v = 0$), $v$ is shortening velocity, and $a$ and $b$ are constants with dimensions of force and velocity respectively.

Step 4 — Physical meaning of the constants: Rearranging:

$v = b \cdot \frac{F_0 - F}{F + a}$

At $F = 0$: the maximum shortening velocity is $v_{\max} = b \cdot F_0/a$. The ratio $a/F_0 \approx 0.25$ for most muscles, meaning $a \approx F_0/4$.

Step 5 — Cross-bridge interpretation: From Huxley's model, the constants relate to cross-bridge kinetics:

$a = \frac{f_+ \cdot n_0 \cdot k \cdot h^2}{2(f_+ + g_+)}, \quad b = \frac{f_+ \cdot h}{2}$

where $k$ is the cross-bridge stiffness, $h$ is the power stroke distance, and $n_0$ is the myosin density. As shortening velocity increases, cross-bridges have less time to attach and complete the power stroke, so force decreases. At $v_{\max}$, cross-bridges are cycling so fast that no net force is generated.

Power output: The mechanical power is $P = F \times v$, which peaks at approximately $v \approx v_{\max}/3$ and $F \approx F_0/3$:

$P_{\max} \approx \frac{F_0 \cdot v_{\max}}{4(1 + a/F_0)^2} \approx 0.1 \cdot F_0 \cdot v_{\max}$

Derivation: Length-Tension Relationship from Filament Overlap

Step 1 — Filament geometry: In each half-sarcomere:

• Thick filament (myosin): total length = 1.6 $\mu$m, bare zone = 0.2 $\mu$m, cross-bridge region = 0.7 $\mu$m per side
• Thin filament (actin): length = 1.0 $\mu$m from Z-line
• Z-line width: ~0.1 $\mu$m

Step 2 — Optimal length: Maximum overlap of the cross-bridge bearing region with actin occurs at sarcomere length:

$L_0 = 2 \times 1.0 + 0.2 = 2.2 \, \mu\text{m}$

At this length, every myosin head in the cross-bridge region can reach an actin binding site. Force is 100% of maximum.

Step 3 — Descending limb (L > L₀): As the sarcomere stretches beyond $L_0$, the overlap zone decreases linearly. The number of available cross-bridges is proportional to the overlap:

$\frac{F}{F_{\max}} = \frac{L_{\max} - L}{L_{\max} - L_0} \quad \text{for } L_0 \leq L \leq L_{\max}$

where $L_{\max} = 2 \times 1.0 + 1.6 = 3.6 \, \mu$m (no overlap at all). This is the descending limb of the length-tension curve.

Step 4 — Ascending limb (L < L₀): Below $L_0$, two effects reduce force. Between 2.0 and 2.2 $\mu$m, thin filaments enter the bare zone (no cross-bridges to interact with). Below 2.0 $\mu$m, thin filaments from opposite sides begin to overlap and interfere with each other. Below ~1.7 $\mu$m, thick filaments collide with Z-lines, causing steep force decline.

Step 5 — Series elastic element: The tendon acts as a series elastic element (SEE) with stiffness $k_{\text{SE}}$:

$F = k_{\text{SE}} \times (L_{\text{tendon}} - L_{0,\text{tendon}})$

The SEE stores elastic energy during isometric contraction and releases it during rapid shortening, contributing to muscle efficiency. In activities like running and jumping, tendon elasticity recovers 60–90% of the stored energy.

Derivation: Stress-Strain and Beam Bending in Long Bones

Step 1 — Elastic properties: Cortical bone behaves as a linear elastic material up to ~1% strain:

$\sigma = E\varepsilon$

with Young's modulus $E \approx 17$ GPa (longitudinal), ultimate tensile strength$\sigma_{\text{ult}} \approx 135$ MPa, and ultimate compressive strength $\approx 205$ MPa. The asymmetry reflects bone's composite nature: hydroxyapatite crystals resist compression while collagen fibres resist tension.

Step 2 — Beam bending theory: Long bones (femur, tibia) are loaded primarily in bending during activities like walking and running. For a beam loaded in bending, the stress at distance $c$ from the neutral axis is:

$\sigma = \frac{Mc}{I}$

where $M$ is the bending moment and $I$ is the second moment of area. The maximum stress occurs at the outer surface ($c = c_{\max}$):

$\sigma_{\max} = \frac{M c_{\max}}{I}$

Step 3 — Hollow tube advantage: Long bones are approximately hollow cylinders with outer radius $R_o$ and inner radius $R_i$. The second moment of area is:

$I = \frac{\pi}{4}(R_o^4 - R_i^4)$

For the same cross-sectional area (same weight), a hollow tube has a much larger $I$ than a solid rod, dramatically reducing bending stress. With $R_i/R_o \approx 0.5$ (typical for the femoral shaft):

$\frac{I_{\text{hollow}}}{I_{\text{solid, same area}}} = \frac{R_o^2 + R_i^2}{R_o^2 - R_i^2} \cdot \frac{R_o^2}{R_o^2 + R_i^2} \approx 1.67$

Step 4 — Safety factor: The safety factor against fracture is:

$SF = \frac{\sigma_{\text{ult}}}{\sigma_{\max}} \approx 2 \text{ to } 5$

During normal walking, peak femoral stress is ~30–50 MPa, giving SF $\approx 4$. During vigorous activity (jumping, sprinting), SF can drop to ~2. This explains why stress fractures occur with repetitive loading at moderate stress levels (fatigue failure).

Wolff's Law and Bone Remodelling

Wolff's law (1892): Bone tissue remodels along the principal stress trajectories. The internal architecture of trabecular bone in the proximal femur beautifully illustrates this: trabeculae align with the principal compressive and tensile stress directions.

Mechanistically, osteocytes sense strain through fluid flow in the canalicular network. When strain exceeds a set point, osteoblasts are activated (bone formation); when strain falls below a lower set point, osteoclasts are activated (bone resorption). This can be modelled as:

$\frac{d\rho}{dt} = B(\varepsilon - \varepsilon_0)$

where $\rho$ is bone density, $\varepsilon$ is the local strain,$\varepsilon_0$ is the homeostatic strain set point, and $B$ is a remodelling coefficient. This explains why astronauts lose bone in microgravity and why exercise increases bone density.

Osteoporosis risk: Bone mineral density (BMD) is the primary determinant of fracture risk. The relationship between BMD and fracture strength is approximately quadratic: $\sigma_{\text{ult}} \propto \rho^2$. A 10% decrease in BMD leads to a ~20% decrease in strength, roughly doubling fracture risk.

Derivation: Lever Mechanics and Joint Forces

Step 1 — Mechanical advantage: Most human joints operate as third-class levers, where the muscle inserts between the fulcrum (joint) and the load. The mechanical advantage is:

$MA = \frac{d_{\text{effort}}}{d_{\text{load}}}$

For most human joints, $MA < 1$, typically 0.1–0.3. This means muscles must generate forces much larger than the external load, but this arrangement provides speed advantage: the hand or foot moves much faster than the muscle shortens.

Step 2 — Elbow joint during bicep curl: Consider holding a mass $m$ in the hand. The biceps inserts at distance $d_b \approx 4$ cm from the elbow joint, while the load is at distance $d_L \approx 35$ cm. From torque balance:

$F_{\text{biceps}} \times d_b = F_{\text{load}} \times d_L$

$F_{\text{biceps}} = \frac{d_L}{d_b} \times mg = \frac{35}{4} \times mg \approx 8.75 \times mg$

For a 5 kg weight: $F_{\text{biceps}} = 8.75 \times 49 = 429$ N! The joint reaction force is even larger:

$F_{\text{joint}} = F_{\text{biceps}} - mg = (8.75 - 1) \times mg = 7.75 \times mg \approx 380 \text{ N}$

(The joint force is compressive, directed into the joint.)

Step 3 — Hip joint during walking: During the single-support phase of walking, the body weight must be balanced by the hip abductor muscles (gluteus medius and minimus). From the free-body diagram of the pelvis:

$F_{\text{abd}} \times d_{\text{abd}} = W \times d_W$

where $d_{\text{abd}} \approx 5$ cm (moment arm of abductors) and$d_W \approx 15$ cm (moment arm of body weight about the hip). This gives:

$F_{\text{abd}} = 3W, \quad F_{\text{hip}} = F_{\text{abd}} + W \approx 4W$

The hip joint experiences ~4 times body weight during normal walking, increasing to 5–8 times during stair climbing and running. This enormous force on a surface area of ~20 cm$^2$explains why the hip is a common site for osteoarthritis and joint replacement.

Derivation: Inverted Pendulum Model for Walking

Step 1 — Cost of transport: The metabolic cost of transport (COT) quantifies locomotion efficiency:

$\text{COT} = \frac{P_{\text{metabolic}}}{m \times v}$

where $P_{\text{metabolic}}$ is metabolic power (W), $m$ is body mass, and $v$ is velocity. For humans, COT $\approx 3.3$ J/(kg$\cdot$m) for walking and ~4.0 J/(kg$\cdot$m) for running.

Step 2 — Inverted pendulum: During walking, the stance leg acts as an inverted pendulum. The centre of mass (CoM) vaults over the stance foot, exchanging kinetic energy (KE) and gravitational potential energy (PE):

$KE = \frac{1}{2}mv^2, \quad PE = mgh$

where $h$ is the vertical displacement of the CoM (~3 cm during walking). At mid-stance, the CoM is at its highest (minimum KE, maximum PE). At double-support, it is at its lowest (maximum KE, minimum PE). The energy recovery is:

$R = 1 - \frac{\Delta W_{\text{actual}}}{\Delta KE + \Delta PE}$

At the preferred walking speed (~1.3 m/s), $R \approx 65\%$ — walking is remarkably efficient due to this pendular energy exchange. The optimal walking speed can be estimated from the condition that the centripetal acceleration at mid-stance equals gravity:

$\frac{v^2}{L_{\text{leg}}} = g \quad \Rightarrow \quad v_{\text{max}} = \sqrt{gL_{\text{leg}}}$

For $L_{\text{leg}} = 0.9$ m: $v_{\max} = \sqrt{9.81 \times 0.9} \approx 3.0$ m/s. The preferred walking speed is about 40–50% of this maximum, minimising the COT.

Derivation: Spring-Mass Model for Running

Step 3 — Running as bouncing: Unlike walking, running involves a flight phase. The leg acts as a linear spring during stance. The spring-mass model gives the ground reaction force as:

$F_{\text{ground}} = k_{\text{leg}} \times \Delta L_{\text{leg}}$

where $k_{\text{leg}}$ is the effective leg stiffness ($\approx 10$–20 kN/m) and $\Delta L$ is the leg compression during stance. The peak vertical ground reaction force during running can be estimated from the duty factor $D$ (fraction of stride in stance):

$F_{\text{peak}} \approx \frac{mg}{D}$

At a running speed of 4 m/s, $D \approx 0.35$, giving$F_{\text{peak}} \approx 2.9 \, mg$. At sprinting speeds (10 m/s), $D \approx 0.20$, giving $F_{\text{peak}} \approx 5 \, mg$. The elastic energy stored and returned by the Achilles tendon during running is:

$E_{\text{elastic}} = \frac{1}{2}k_{\text{tendon}} \Delta L_{\text{tendon}}^2 \approx 35 \text{ J per step}$

This recovers up to 50% of the mechanical energy per step, making human running metabolically cheaper than it would otherwise be.

Muscle Fibre Types and Energetics

Skeletal muscle contains different fibre types with distinct biophysical properties, each optimised for different mechanical demands:

• Type I (slow oxidative): $v_{\max} \approx 1$ L$_0$/s, high fatigue resistance, $F_0/A \approx 20$ N/cm$^2$. Dominant myosin isoform: MHC-I (slow ATPase). Efficiency $\eta \approx 25\%$. Used for posture, walking.
• Type IIa (fast oxidative-glycolytic): $v_{\max} \approx 4$ L$_0$/s, moderate fatigue resistance. Intermediate properties. Used for sustained running.
• Type IIx (fast glycolytic): $v_{\max} \approx 8$ L$_0$/s, low fatigue resistance, $F_0/A \approx 30$ N/cm$^2$. Used for sprinting, jumping.

The efficiency of muscle contraction depends on the shortening velocity. Mechanical efficiency is defined as:

$\eta = \frac{P_{\text{mechanical}}}{P_{\text{metabolic}}} = \frac{F \times v}{\dot{E}_{\text{ATP}} \times \Delta G_{\text{ATP}}}$

Maximum efficiency (~25%) occurs at approximately $v \approx 0.2 \, v_{\max}$, slower than the velocity for maximum power output ($v \approx 0.3 \, v_{\max}$). This is because at higher velocities, more ATP is consumed per cross-bridge cycle due to rapid detachment and reattachment without completing the full power stroke.

The total metabolic cost of muscle activation includes: (1) cross-bridge cycling (~70% of total cost), (2) Ca$^{2+}$ pumping back into the sarcoplasmic reticulum via SERCA (~25%), and (3) Na$^+$/K$^+$-ATPase activity (~5%). Even during isometric contraction ($v = 0$, zero external work), metabolic cost is substantial because cross-bridges are still cycling (attaching, generating force, detaching) — this is the “isometric heat” discovered by Hill.

Biomechanical Scaling Laws

Geometric scaling: If an animal scales isometrically (maintaining the same proportions), then for a characteristic length $L$:

• Mass: $M \propto L^3$
• Muscle cross-section: $A \propto L^2$, so $F_{\max} \propto L^2 \propto M^{2/3}$
• Stress: $\sigma = F/A_{\text{bone}} \propto M^{2/3}/M^{2/3} = \text{const}$

This means that if animals were geometrically similar, peak bone stress would be independent of size. However, larger animals actually experience higher stresses because they cannot scale bone thickness as quickly as body mass increases. The solution: larger animals adopt more upright postures (reducing bending moments) and move with relatively straighter legs — an elephant cannot gallop.

McMahon's elastic similarity: Thomas McMahon proposed that animals scale to maintain elastic similarity (same degree of bending under body weight), which predicts $L \propto M^{1/4}$ and $D_{\text{bone}} \propto M^{3/8}$, in better agreement with empirical data than geometric similarity.

Applications of Musculoskeletal Biophysics

• • Prosthetic limbs: Modern prosthetics use carbon fibre running blades that store and return elastic energy, mimicking the spring-mass function of biological tendons. The stiffness is tuned to the user's body weight and activity level.
• • Sports biomechanics: Understanding the force-velocity curve explains why explosive movements (jumping, throwing) require different training than endurance activities. Peak power occurs at ~1/3 of maximum velocity and ~1/3 of maximum force.
• • Rehabilitation engineering: The length-tension relationship determines the optimal limb position for strength testing. Joint angles are standardised for reproducible strength measurements.
• • Bone fracture fixation: Internal fixation devices (plates, screws, intramedullary nails) must share load with the healing bone. Too stiff a fixation causes stress shielding (bone resorption by Wolff's law); too flexible allows excessive motion.
• • Joint replacement: Hip and knee replacements must withstand$\sim 10^8$ loading cycles over a 20-year lifetime. The contact stress between bearing surfaces must remain below the fatigue limit of the material (~20 MPa for polyethylene).

Tendon and Cartilage Biomechanics

Tendon mechanics: Tendons are composed primarily of type I collagen fibres arranged in a hierarchical structure. Their stress-strain curve has three distinct regions:

• Toe region (0–2% strain): Collagen fibre crimp straightens out. Low stiffness.
• Linear region (2–6% strain): Fibres are straight and resist loading. $E \approx 1.5$ GPa.
• Failure region (>6% strain): Micro-tears accumulate. Ultimate tensile strength $\approx 100$ MPa.

Tendons are viscoelastic: their stiffness increases with loading rate, and they exhibit hysteresis (energy loss per cycle $\approx 5$–10%). The low hysteresis is critical for elastic energy storage during locomotion. The energy returned is:

$\text{Resilience} = \frac{E_{\text{returned}}}{E_{\text{stored}}} \approx 93\%$

Articular cartilage: A biphasic material consisting of a solid collagen-proteoglycan matrix saturated with interstitial fluid. Under load, the fluid is squeezed out, and the matrix deforms — a process governed by the biphasic theory:

$\sigma_{\text{total}} = \sigma_{\text{solid}} - p_{\text{fluid}}$

The fluid pressurisation provides ~90% of load support during the first few seconds of loading (e.g., heel strike during walking), protecting the solid matrix from excessive stress. The permeability of cartilage ($k \approx 10^{-15}$ m$^4$/(N$\cdot$s)) determines the rate of fluid flow and hence the creep behaviour. In osteoarthritis, proteoglycan loss increases permeability, reducing fluid pressurisation and exposing the collagen network to higher stresses — a vicious cycle of degeneration.

The coefficient of friction in a healthy synovial joint is extraordinarily low: $\mu \approx 0.001$–0.01, lower than ice on ice ($\mu \approx 0.03$). This is achieved through a combination of boundary lubrication (lubricin on the cartilage surface), fluid film lubrication (squeezed-out interstitial fluid), and weeping lubrication (fluid exuded from loaded cartilage).

• • A.V. Hill (1938): Discovered the force-velocity relationship and the heat of muscle contraction, earning a share of the 1922 Nobel Prize. The Hill equation remains the standard model for muscle mechanics.
• • A.F. Huxley (1957): Proposed the cross-bridge model that provides a molecular explanation for Hill's force-velocity curve. His 1957 paper is one of the most cited in biophysics.
• • Julius Wolff (1892): A German anatomist who formalised the observation that bone remodels in response to mechanical loading. His law remains a foundational principle in orthopaedic biomechanics.
• • Thomas McMahon (1984): Pioneer in biomechanical scaling, showing how body size determines locomotion mechanics. His analysis explained why running track surfaces with tuned compliance improve performance (the Harvard “fast track”).