## Biomechanics and Biostatistics

### I. Newtonian Mechanics

1. Vectors and scalars

a. Magnitude and direction (orientation)

b. Direction specified by

◦ Angle

◦ **Useful values: sine (30 degrees) = cosine (60 degrees) = 0.5, sine (45 degrees) = cosine (45 degrees) ≈ 0.7, sine (60 degrees) = cosine (30 degrees) ≈ 0.9**

◦ Resolved into components along x, y, and z axes

c. Examples: velocity, acceleration, force, moment

• Scalar

a. Magnitude only

b. Examples: mass, time, temperature, speed

2. Mass properties

• Mass: property of a physical body that determines its resistance to changes in velocity (acceleration) and the strength of the gravitational attraction to other bodies

a. Units: kilograms (kg)

• Density: amount of mass per unit area, or per unit volume

a. Units: kg/m^{2} (per unit area) or kg/m^{3} (per unit volume)

a. Symbol: Greek letter rho, ρ

• Center of mass (COM): unique, imaginary point at which all the mass of an object can be considered to be concentrated.

a. The COM of the human body is located within the pelvis/lower abdomen (anterior to S2) during standing and gait **(Fig. 12.2a, b)**

b. COM location depends on the distribution of mass (e.g., relative position of body segments) **(Fig. 12.2c)**

**c. Since there are equal amounts of mass on each side of the COM in a given plane, the COM is the point of balance.**

• Mass moment of inertia: distribution of the mass of an object about a given axis

a. Mass multiplied by length squared (I_{0} = r^{2}* m), where I_{0} is the mass moment of inertia, r^{2} is the distance of the mass from the axis, and m is the mass.

b. Resistance to rotation about an axis

**c. Moment of inertia is the rotational analog to mass**

**Fig. 12.2** **(a,b)** The center of mass (COM) of the human body is located within the pelvis/lower abdomen (anterior to S2) during standing and gait. **(c)** COM location depends on the distribution of mass (e.g., relative position of body segments).

• Polar moment of inertia: measure of the difficulty to turn a cross section about an axis perpendicular to it (J = r^4)

a. Greater polar moment of inertia increases a beam’s resistance to torsion

• Area moment of inertia: property of a cross section that characterizes its deflection under loading or I_{A} = r^4

a. Greater area moment of inertia requires more stress to deflect a beam

3. Velocity and speed

• Velocity is a vector quantity equal to distance/time along a direction.

• Speed is a scalar quantity equal to distance/time.

• Units: meters/second (meters per second)

4. Acceleration

• Acceleration is a vector quantity.

• Magnitude is equal to speed/time or distance/time^{2}.

• Units: meters/second^{2}

• Example: acceleration due to gravity, *g*, is 9.81 m/s^{2}

5. Angular velocity (ω)

• ω = *v*/*r*, where *v* is the tangential velocity of an object and *r* is the distance to the axis of rotation

• Units: radians/second (360 degrees/s = 2*PI radians/s)

**• Some joints are capable of very high angular velocities; for example, the shoulder can internally rotate at ~9,000 deg/s or 1500 rpm during a baseball pitch**

6. Angular acceleration (α)

• α = ω **/** time

• Units: radians/s^{2} or degrees/s^{2}

7. Force = mass * acceleration (Newton’s second law)

**• Layman’s description: a push or a pull**

• Force is a vector (magnitude and direction).

• Units: newtons (N) = kg * m/s^{2}

8. Moment (torque): force * perpendicular distance **(****Fig. 12.3)**

**• Layman’s description: rotational force**

• Perpendicular distance is called moment arm or lever arm.

• Units: newton-meters

• Moment is a vector (magnitude and direction).

**• In biomechanics, moment is often computed about a joint’s center of rotation.**

9. Work (*W*): force and the displacement it causes (work = force * distance)

• Energy: ability to perform work

• Units: joules

10. Power (*P*): energy per unit time

*• P* = force * velocity

*• P* = moment * angular velocity

• Units: joules/second or watts

**• During jogging, the power burst of the ankle joint at push-off is ~800 watts. For comparison, the power consumption of a smart phone during a phone call is ~3 watts.**

11. Newton’s laws

• First law (inertia): If net force is zero, body will not move or will move with constant velocity

• Second law: Force is equal to mass multiplied by acceleration (*F* = *m** *A*)

**• The rotational analog of newton’s second law is M = I*** α,

**where**α

*M*is the moment,*I*is the mass moment of inertia, and**is angular acceleration.**

a. Example: weight = mass * acceleration due to gravity (*w* = *m** *g*)

• Third law (action-reaction): For every action there is an equal and opposite reaction

a. Example: ground reaction force and weight **(****Fig. 12.4)**

### II. Application of Newtonian Mechanics

1. Statics: actions of forces and moments on rigid objects in a system in static equilibrium **(****Fig. 12.4)**

• Sum of all moments = 0

2. Dynamics: bodies accelerating and related forces and moments

• Sum of all forces = mass * acceleration

• Sum of all moments = moment of inertia * angular acceleration

3. Analysis

• Definitions of forces **(****Fig. 12.5)**

a. Normal force is perpendicular to surface it acts on.

b. Tangential force is parallel to surface it acts on.

c. Compressive force shortens a body in the direction of the force.

d. Tensile force lengthens a body in the direction of the force.

• Free-body diagram: used to solve statics and dynamics problems

a. Draw only one body that contains unknowns being solved.

b. Draw all known forces and moments at their points of application.

c. Draw unknown forces and moments including those due to bodies that are in contact.

d. Do not include internal forces that originate and terminate within the free body.

• Example statics problem: determine muscle-tendon force of the biceps when holding a ball **(****Fig. 12.6)**.

### III. Biomechanics

1. Definitions

• Biomechanics: study of structure and function of living organisms using principles of mechanics

• Kinematics: study of motion without consideration of causes of motion

• Kinetics: study of motion and its causes (forces, moments)

• Kinesiology: study of human motions/movements

2. Overview of joints and their characteristics

• Degrees of freedom

a. Number of translational and rotational motions a joint possesses out of a total possible of six (three x, y, and z rotations and three x, y, and z translations)

◦ **Examples: The hip joint has three rotational degrees of freedom, the wrist has two rotational degrees of freedom (flexion-extension, and radial-ulnar deviation), and the patella has six degrees of freedom.**

b. Rolling and sliding: joints roll and slide to maintain congruence (e.g., knee joint and knee total joint)

c. Mechanical approximations of human joints: joints are modeled as mechanical joints with varying degrees of freedom **(Fig. 12.7)**

• Friction: force that acts in the opposite direction to movement

a. Coefficient of friction is the ratio of the force of friction and the normal force. Examples: 0.002 to 0.04 for human joints, metal on ultrahigh molecular weight polyethylene (UHMWPE) is 0.05 to 0.15

### IV. Biomechanics of Individual Joints

1. Shoulder

• Combination of glenohumeral and scapulothoracic motion

a. Abduction: 120 degrees of glenohumeral motion and 60 degrees of scapulothoracic motion at a 2:1 ratio; total abduction is 165 degrees

• Arthrodesis: 15 to 20 degrees abduction, 20 to 25 degrees forward flexion, 40 to 50 degrees internal rotation

2. Elbow

• Range of motion (ROM): 150 degrees of flexion, 0 degrees of extension, 90 degrees of pronation, 90 degrees of supination

a. Functional ROM is 30 to 130 degrees (extension/flexion), 50 degrees (pronation/supination)

**Fig. 12.7** The classification of joints by shape. The *arrows* indicate the direction in which the skeletal elements can move around the axis or axes of the joint. Amphiarthroses (not shown here) are “stiff” because their mobility is greatly restricted by the shape of their articular surfaces and by tight ligaments (examples are the proximal tibiofibular joint and sacroiliac joint). **(a)** Plane/gliding joint. The only movement allowed is a translation (sliding) of one member on the other (example: vertebral facet joint). **(b)** Hinge joint. This joint has one axis of motion, resulting in two primary movements (example: parts of the elbow joint). **(c)** Ball-and-socket joint. This type of joint has three mutually perpendicular axes of motion, resulting in six primary movements (example: hip joint). **(d)** Saddle joint. This is a biaxial joint with four primary movements (example: the carpometacarpal joint of the thumb). **(e)** Pivot joint. This is a uniaxial joint with two primary movements (example: the proximal radioulnar joint). **(f)** Ellipsoid joint. The only movement allowed is a translation (sliding) of one member on the other (example: vertebral facet joint).

a. Unilateral: 90 degrees of flexion

b. Bilateral: 110 degrees of flexion for hand to reach mouth, 65 degrees of flexion for perineal hygiene

3. Hand/wrist

• Wrist ROM: 65 degrees of flexion, 70–90 degrees of extension, 20 degrees of radial deviation, 35 degrees of ulnar deviation

a. Functional ROM: 10 degrees of flexion, 35 degrees of extension, 0 degrees of radial deviation, 15 degrees of ulnar deviation

• Wrist arthrodesis: 10 to 20 degrees of extension; if bilateral, then fuse contralateral wrist at 0 to 10 degrees of palmar flexion or do a total wrist arthroplasty

• Hand ROM: metacarpophalangeal (MCP) joint is 100 degrees of flexion and 60 degrees of abduction/adduction; proximal interphalangeal (PIP) joint is 110 degrees; distal interphalangeal (DIP) joint is 80 degrees of flexion.

• Hand joint arthrodesis: finger MCP 20–40 degrees of flexion, PIP 40–50 degrees of flexion, DIP 0–5 degrees of flexion; thumb MCP 25 degrees of flexion, interphalangeal (IP) 20 degrees of flexion.

4. Hip

• Ball-and-socket joint

• ROM: 115 degrees of flexion, 30 degrees of extension, 50 degrees of abduction, 30 degrees of adduction, 45 degrees of internal rotation, 45 degrees of external rotation

• Joint reaction force can be three to six times body weight

• Trendelenburg gait is essentially a reduction in joint reaction force and abductor moment by shifting weight over the affected hip. This gait pattern compensates for weak hip abductors.

a. A cane in the opposite hand can produce an additional moment to reduce the hip abduction moment.

• Arthrodesis position is 25–30 degrees of flexion, 0 degrees of abduction/rotation.

5. Knee

• Flexion and extension involve rolling and sliding. Posterior rollback maximizes flexion.

• “Screw home mechanism”: tibia externally rotates during last 15 degrees of extension

• ROM: 0 degrees of extension, 130 degrees of flexion

a. At full extension there is minimal rotation; at 90 degrees of flexion can achieve 45 degrees of external rotation and 30 degrees of internal rotation

b. Require 110 degrees of flexion to rise from a chair after total knee arthroplasty (TKA)

• Patella slides 7 cm caudally during full flexion

• Stabilizers: anterior cruciate ligament (ACL)/posterior cruciate ligament (PCL) for anterior and posterior

• Arthrodesis: 0 to 7 degrees of valgus, 10 to 15 degrees of flexion

a. Anatomic axis of the femur is along the shaft.

b. Mechanical axis of the femur is the femoral head center to the knee center.

c. Anatomic axis of the tibia is along the shaft.

d. Mechanical axis of the tibia is the center of the plateau to the center of the ankle.

• Patella: increases lever arm to increase extension, loss of 30% extension power after patellectomy

• Ankle joint (tibiotalar) (also known as talocrural joint)

a. ROM: 25 degrees of dorsiflexion, 35 degrees of plantarflexion, 5 degrees of rotation

b. Arthrodesis: neutral dorsiflexion, 5–10 degrees of external rotation, 0–5 degrees of hindfoot valgus

• Subtalar joint (talus-calcaneus)

a. ROM: 5 degrees of pronation, 20 degrees of supination; functional ROM is 6 degrees

• Transverse tarsal joint (talus–navicular, calcaneal–cuboid)

a. Inversion/eversion

• Foot

b. Three arches

◦ Medial longitudinal

◦ Lateral longitudinal

◦ Transverse