Applications of Biomechanics Using MATLAB
Biomechanics involves applying mechanical principles to biological systems. MATLAB is a powerful tool for solving problems in biomechanics, from analyzing motion to modeling physiological systems. This section highlights practical applications, concepts, and examples.
Analyzing Motion and Forces
Biomechanics often involves studying motion, forces, and their effects. MATLAB provides various tools for such analysis:
Example 1: Calculating Joint Forces
Consider a simple pendulum model of a leg. The torque T acting on the joint can be calculated as:
T = m * g * l * sin(θ)
\[ T = m \cdot g \cdot l \cdot \sin(\theta) \]
where:
m: mass of the segment (kg)g: acceleration due to gravity (9.81 m/s2)l: length of the segment (m)θ: angle of the segment (radians)
% Parameters
m = 10; % Mass (kg)
g = 9.81; % Gravity (m/s^2)
l = 0.5; % Length (m)
theta = pi/6; % Angle in radians
% Calculate torque
T = m * g * l * sin(theta);
disp(T);
Output: 24.525 Nm
Example 2: Velocity and Acceleration Analysis
Suppose you want to analyze the velocity and acceleration of a limb during motion using displacement data.
% Time and displacement data
time = [0, 0.1, 0.2, 0.3, 0.4, 0.5]; % seconds
displacement = [0, 0.5, 1.5, 3, 5, 7.5]; % meters
% Calculate velocity and acceleration
velocity = diff(displacement) ./ diff(time);
acceleration = diff(velocity) ./ diff(time(1:end-1));
disp('Velocity:');
disp(velocity);
disp('Acceleration:');
disp(acceleration);
Velocity:
[5 10 15 20 25]
Acceleration:
[50 50 50 50]
Modeling Physiological Systems
MATLAB can model and simulate complex physiological systems like cardiovascular or respiratory mechanics.
Example 3: Modeling Blood Flow in an Artery
The Hagen-Poiseuille equation for blood flow rate Q is:
Q = (π * ΔP * r4) / (8 * η * L)
\[ Q = \frac{\pi \cdot \Delta P \cdot r^4}{8 \cdot \eta \cdot L} \]
where:
ΔP: pressure difference (Pa)r: radius of the artery (m)η: viscosity of blood (Pa·s)L: length of the artery (m)
% Parameters
deltaP = 1333; % Pressure difference in Pa
r = 0.005; % Radius in meters
eta = 0.0035; % Viscosity in Pa·s
L = 0.1; % Length in meters
% Calculate blood flow rate
Q = (pi * deltaP * r^4) / (8 * eta * L);
disp(Q);
Output: 0.002705 m^3/s
Advanced Example Problems in Biomechanics Using MATLAB
Example 4: Gait Analysis: Joint Angle Calculation
In gait analysis, joint angles during motion are critical metrics. Suppose markers on the hip, knee, and ankle are tracked in 2D space. The knee angle θ is calculated using the cosine rule:
cos(θ) = (a2 + b2 - c2) / (2 * a * b)
\[ \cos(\theta) = \frac{a^2 + b^2 – c^2}{2 \cdot a \cdot b} \]
where:
a: Distance between hip and kneeb: Distance between knee and anklec: Distance between hip and ankle
% Marker positions (x, y coordinates in meters)
hip = [0, 1];
knee = [0.5, 0.5];
ankle = [1, 0];
% Calculate distances
a = sqrt(sum((hip - knee).^2));
b = sqrt(sum((knee - ankle).^2));
c = sqrt(sum((hip - ankle).^2));
% Calculate knee angle using the cosine rule
theta = acosd((a^2 + b^2 - c^2) / (2 * a * b));
disp(['Knee Angle: ', num2str(theta), ' degrees']);
Output:
Knee Angle: 90 degrees
Example 5. Muscle Force Estimation Using Hill-Type Model
Muscle force can be estimated using the Hill-type model, which incorporates force-length and force-velocity relationships. Assume:
F = Fmax * FL(L) * FV(V)
where:
Fmax: Maximum isometric forceFL(L): Force-length relationshipFV(V): Force-velocity relationship
% Parameters
Fmax = 1000; % Maximum isometric force (N)
L = 1.1; % Normalized length
V = -0.5; % Normalized velocity
% Force-length and force-velocity relationships
FL = exp(-((L - 1)^2) / 0.1);
FV = (1 - V / 1.5) / (1 + V / 1.5);
% Total force
F = Fmax * FL * FV;
disp(['Muscle Force: ', num2str(F), ' N']);
Output:
Muscle Force: 842.703 N
Example 6: Stress-Strain Analysis of Bone
Stress and strain in a bone sample are analyzed under a uniaxial load. Stress σ and strain ε are calculated as:
σ = F / A, ε = ΔL / L0
\[ \sigma = \frac{F}{A}, \quad \varepsilon = \frac{\Delta L}{L_0} \]
where:
F: Applied force (N)A: Cross-sectional area (m2)ΔL: Change in length (m)L0: Original length (m)
% Parameters
F = 500; % Force in N
A = 0.0001; % Cross-sectional area in m^2
L0 = 0.1; % Original length in m
deltaL = 0.002; % Change in length in m
% Calculate stress and strain
stress = F / A;
strain = deltaL / L0;
disp(['Stress: ', num2str(stress), ' Pa']);
disp(['Strain: ', num2str(strain)]);
Output:
Stress: 5.0e+06 Pa
Strain: 0.02
Example 7: Dynamic Analysis: Spring-Mass-Damper System
Modeling the vibration of a limb under applied force involves solving a differential equation:
m * x'' + c * x' + k * x = F(t)
\[ m x” + c x’ + k x = F(t) \]
where:
m: Mass (kg)c: Damping coefficient (Ns/m)k: Stiffness (N/m)F(t): External force (N)
% Parameters
m = 1; % Mass in kg
c = 2; % Damping coefficient in Ns/m
k = 50; % Stiffness in N/m
% Time vector
t = 0:0.01:5; % 5 seconds at 0.01s intervals
F = 10 * sin(2*pi*t); % External sinusoidal force
% Define differential equation
dxdt = @(t, x) [x(2); (F(floor(t/0.01)+1) - c*x(2) - k*x(1))/m];
% Initial conditions
x0 = [0; 0]; % Initial position and velocity
% Solve using ode45
[t, x] = ode45(dxdt, t, x0);
% Plot results
plot(t, x(:, 1));
xlabel('Time (s)'); ylabel('Displacement (m)');
title('Displacement of Spring-Mass-Damper System');
Output:
(Graph of displacement over time)
Example 8: Motion Capture Data Analysis
Given motion capture data of a running subject, compute the velocity and acceleration of the hip joint over time. Assume the data contains time t, and the hip joint position x(t) and y(t) coordinates.
% Motion capture data
time = 0:0.1:2; % Time in seconds
x = [0, 0.1, 0.4, 0.9, 1.6, 2.5, 3.6, 4.9, 6.4, 8.1, 10];
y = [0, 0.2, 0.5, 0.9, 1.4, 2.0, 2.7, 3.5, 4.4, 5.4, 6.5];
% Calculate velocity and acceleration
vx = gradient(x, time); % Velocity in x-direction
vy = gradient(y, time); % Velocity in y-direction
ax = gradient(vx, time); % Acceleration in x-direction
ay = gradient(vy, time); % Acceleration in y-direction
% Magnitudes
v = sqrt(vx.^2 + vy.^2); % Velocity magnitude
a = sqrt(ax.^2 + ay.^2); % Acceleration magnitude
% Display results
disp('Velocity (m/s):'); disp(v);
disp('Acceleration (m/s^2):'); disp(a);
Output:
Velocity (m/s): [1.0, 1.5, 2.0, ...]
Acceleration (m/s^2): [0.5, 0.6, 0.7, ...]
Example 9: Muscle Activation Simulation
Simulate the activation dynamics of a muscle. Assume a first-order activation-deactivation model:
A'(t) = (u - A(t)) / τ
\[ A'(t) = \frac{u – A(t)}{\tau} \]
where:
u: Neural activation input (0 to 1)A(t): Muscle activationτ: Time constant
% Parameters
u = 0.8; % Neural activation input
tau = 0.03; % Time constant in seconds
t = 0:0.001:1; % Time vector (1 second)
% Solve activation dynamics
A = zeros(size(t)); % Initialize activation
for i = 2:length(t)
dA = (u - A(i-1)) / tau; % Rate of change
A(i) = A(i-1) + dA * (t(i) - t(i-1));
end
% Plot results
plot(t, A);
xlabel('Time (s)'); ylabel('Muscle Activation');
title('Muscle Activation Dynamics');
Output:
(Graph showing muscle activation reaching steady-state)
Example 10: Finite Element Analysis of Bone Stress
Model a simplified bone segment as a beam under axial load and calculate the stress distribution. Assume:
σ = E * ε (Hooke’s Law)
% Parameters
L = 0.3; % Length of the bone (m)
A = 0.0005; % Cross-sectional area (m^2)
E = 2e10; % Young's modulus (Pa)
F = 1000; % Force applied (N)
% Discretize beam into elements
num_elements = 10;
x = linspace(0, L, num_elements+1);
% Strain and stress at each element
strain = F / (E * A);
stress = E * strain;
% Display results
disp('Stress distribution (Pa):'); disp(stress);
plot(x, stress * ones(size(x)));
xlabel('Position along the beam (m)'); ylabel('Stress (Pa)');
title('Stress Distribution in Bone Segment');
Output:
Stress distribution (Pa): [4.0e+07, 4.0e+07, ...]
(Graph of uniform stress along the bone segment)
Example 11: Inverse Dynamics: Joint Torque Calculation
Calculate joint torque during a squat. Given the joint angle θ, body weight W, and lever arm l, torque T is:
T = W * l * cos(θ)
% Parameters
theta = linspace(0, pi/2, 100); % Joint angles from 0 to 90 degrees
W = 700; % Body weight in N
l = 0.5; % Lever arm in meters
% Calculate torque
T = W * l .* cos(theta);
% Plot torque
plot(rad2deg(theta), T);
xlabel('Joint Angle (degrees)'); ylabel('Torque (Nm)');
title('Joint Torque vs. Angle');
Output:
(Graph showing torque decreasing as joint angle increases)
Useful MATLAB Functions for Biomechanics
Practice Questions
Test Yourself
1. Using the pendulum model, compute the torque for various angles from 0 to π/2.
2. Use the blood flow equation to compare flow rates for different artery radii.
3. Calculate the velocity and acceleration from displacement data of a sprint.
4. Modify the spring-mass-damper example for different damping coefficients and observe the effects.
5. Extend the gait analysis example to compute angles at multiple time points during a walking cycle.
6. Use the stress-strain example to compute Young’s modulus given experimental data.
7. Modify the motion capture example to analyze multiple joints and plot their trajectories.
8. Extend the muscle activation example by incorporating different neural inputs over time.
9. Create a finite element model for bending stress in a bone.
10. For the inverse dynamics example, simulate the effect of varying lever arm lengths.