Biomaterials and Biomechanics

Tensile stress-strain properties

Tensile testing applies an axial pulling force F to a specimen of original cross-sectional area A_0 and original gauge length L_0. Engineering stress is \sigma = F/A_0, and engineering strain is \varepsilon = \Delta L/L_0, which is dimensionless and often quoted as a percentage. In the initial linear region the material obeys Hooke's law, \sigma = E\varepsilon, and the slope E is the Young's modulus, the measure of stiffness. Rearranging the same law gives the strain \varepsilon = \sigma/E that a known stress produces in a material of known modulus. The stress at which permanent (plastic) deformation begins is the yield strength, the maximum stress on the curve is the ultimate tensile strength, and the area under the curve up to fracture is the toughness. Brittle materials such as ceramics fracture after very little plastic strain, so their curves are steep and short. Ductile metals such as titanium show a long plastic region before necking and fracture, and compliant polymers have low modulus with very large extensibility. Comparing curves, the steepest initial slope indicates the highest modulus, and a curve that ends early indicates a brittle material. The stiffness of biological tissues spans many orders of magnitude. Cortical bone is in the gigapascal range, tendon and ligament are in the hundreds of megapascals or less, cartilage is in megapascals, and soft organs are in kilopascals, so the decreasing order of modulus is bone, cartilage, liver, and lung. Tensile strength also depends strongly on material class, and fibre-reinforced composites can exceed bone, which in turn exceeds bone cement and polymer foams. Worked example: a rod of 2 cm diameter carries 2 kN and stretches by 0.1% of its length. Then A = \pi(0.01)^2 = 3.14\times10^{-4}\ \text{m}^2, \sigma = 2000/3.14\times10^{-4} \approx 6.37 MPa, and E = \sigma/\varepsilon \approx 6.37 GPa.

Biodegradable polymer degradation

A biodegradable polymer is broken down in the body by hydrolysis of its backbone linkages or by enzymatic cleavage, and the degradation is tracked by mass loss, strength loss and molecular weight decline. The percentage degradation is taken as the loss of dry mass relative to the initial dry mass, \%\ \text{degradation} = (W_0 - W_t)/W_0 \times 100. A water-insoluble polymer network becomes soluble in vivo when its crosslinks are cleaved, which releases water-soluble chains, or when backbone cleavage creates polar, hydrophilic end groups and shorter fragments. Cleaving side groups to form non-polar groups does not help, because it lowers water affinity. Enzymes such as lysozyme can degrade crosslinks and backbone bonds under physiological conditions, so enzymatic degradation can release soluble chains in the same way as hydrolysis.

• crosslink — a covalent or physical bridge joining two polymer chains; its cleavage frees chains. e.g. Removing bridges between two chains lets the network dissolve.
• bulk erosion — water penetrates the whole device, chains cleave throughout its volume, molecular weight falls first, and mass is lost only when small fragments diffuse out.
• surface erosion — degradation is confined to the exterior, so mass is lost layer by layer while the molecular weight of the remaining polymer stays nearly constant.

Mechanical strength is often modelled empirically as a function of time, and in a logarithmic form \sigma = \sigma_0 - b\ln t the strength at one time can be obtained from the strength at another time, because \sigma_0 cancels in the difference. Worked example: a scaffold of 50 mg dry mass that drops to 45 mg has lost (50-45)/50 \times 100 = 10\% of its mass.

Blood rheology

Blood is a suspension of red blood cells in plasma, which behaves as a Newtonian fluid. The non-Newtonian character of whole blood arises from the cells, which are deformable biconcave discs. Under shear, viscous stresses deform the cells and orient them so that their long axes lie along the streamlines, and this ordered orientation reduces the apparent viscosity at higher shear rates. The dynamic viscosity of whole blood is of the order of a few centipoise, and one centipoise equals 1 mPa·s, so viscosity must be converted to SI units before it is used in a formula. The Reynolds number for flow in a circular vessel is Re = \rho v D/\mu, where v is the mean velocity and D is the lumen diameter. Values below roughly 2000 indicate laminar flow in straight tubes. Worked example: blood with \rho = 1000 kg/m^3, v = 0.20 m/s, D = 4 mm and \mu = 0.001 Pa·s gives Re = 1000 \times 0.20 \times 0.004/0.001 = 800, which is laminar.

Surface characterization techniques

Biomaterial surfaces are characterized by techniques that each reveal a different property. Scanning electron microscopy images surface morphology and topography, mainly through low-energy secondary electrons, which give a three-dimensional appearance of the surface features. A backscattered electron detector collects high-energy electrons that are elastically scattered back from the specimen. Their yield increases with the mean atomic number of the irradiated region, so heavier elements appear brighter, giving compositional contrast between areas of different chemistry rather than a topographic image. X-ray photoelectron spectroscopy measures the elemental composition and chemical state of the outermost few nanometres of a surface. Fourier-transform infrared spectroscopy identifies functional groups from their characteristic vibrational absorption bands. Atomic force microscopy scans a sharp tip over the surface to map height, and the roughness is then computed, most commonly as the arithmetic mean deviation R_a = \frac{1}{N}\sum_{i}|z_i - \bar{z}|.

Pulse wave velocity in arteries

A pressure pulse travels along an artery at a speed that depends on how stiff the vessel wall is. For a thin-walled elastic tube filled with blood, the Moens-Korteweg relation gives the pulse wave speed as c = \sqrt{\dfrac{E\,t}{\rho D}}, where E is the wall modulus, t is the wall thickness, \rho is the blood density and D is the internal diameter. The wave speed rises with wall stiffness and wall thickness and falls with lumen diameter and fluid density. A dimensional check confirms the form, because E/\rho has units of m^2/s^2 and t/D is dimensionless, so c comes out in m/s. Stiffening of arteries with age therefore raises the pulse wave velocity. Worked example: with E = 2 MPa, t = 0.5 mm, D = 10 mm and \rho = 1000 kg/m^3, c = \sqrt{(2\times10^{6})(5\times10^{-4})/(1000\times 0.01)} = \sqrt{100} = 10 m/s.

Dimensions of mechanical quantities

Every mechanical quantity can be expressed in the base dimensions mass M, length L and time T, and the dimensions follow from the defining equation of the quantity. Force has dimensions MLT^{-2} and shear stress, a force per area, has ML^{-1}T^{-2}. Energy and work have ML^{2}T^{-2}. Newton's law of viscosity \tau = \mu\,(du/dy) gives dynamic viscosity as stress divided by velocity gradient, whose dimension is T^{-1}, so [\mu] = ML^{-1}T^{-1}. When a quantity is defined by an integral, the integrand's dimension is multiplied by the dimension of the differential. For example, a second derivative of position with respect to arc length has dimension L^{-1}, its square has L^{-2}, and multiplying by ds leaves L^{-1} for the integral, so a bending energy written as an integral of this kind has a constant of dimension ML^{3}T^{-2}. Kinematic viscosity, \nu = \mu/\rho, has dimensions L^2T^{-1}, which provides a quick check on any viscosity expression.

Elastic moduli relationships

For an isotropic, linear elastic material, the four elastic constants are linked, so only two are independent. Young's modulus E, shear modulus G and Poisson's ratio \nu satisfy G = \dfrac{E}{2(1+\nu)} and equivalently E = 2G(1+\nu). Knowing any two of these lets the third be found directly. Poisson's ratio of an isotropic material lies between -1 and 0.5, and soft biological tissues that are nearly incompressible have \nu close to 0.5. Bulk modulus K is related by E = 3K(1-2\nu). Worked example: a tissue with E = 6 kPa and \nu = 0.25 has G = 6/(2 \times 1.25) = 2.4 kPa.

Temperature-responsive polymer LCST

LCST stands for lower critical solution temperature, the temperature below which a polymer is fully miscible with its solvent and above which the solution separates into phases. Poly(N-isopropylacrylamide), with an LCST close to body temperature, is a common example used in smart drug delivery. The mechanism is a competition between two interactions. Below the LCST, the hydrophilic segments of the chain form hydrogen bonds with water molecules, which keeps the polymer dissolved. Above the LCST, the hydrogen bonds weaken and hydrophobic segments associate with one another, expelling water and causing phase separation. The entropy gained by releasing ordered water drives this transition, which is why the solubility of such polymers falls as temperature rises.

Tissue deformation under load

Most soft tissues are close to incompressible, so their volume is conserved during deformation. For a homogeneous deformation with principal stretches \lambda_1, \lambda_2 and \lambda_3, the condition is \lambda_1\lambda_2\lambda_3 = 1. If the axial stretch is \lambda_z, the two lateral stretches of a symmetric specimen are both \lambda_z^{-1/2}, so a cylinder stretched axially becomes thinner laterally. Worked example: a 20% axial stretch gives \lambda_z = 1.2 and lateral stretch 1/\sqrt{1.2} \approx 0.913, which reduces the diameter by about 9%. In a simple shear test, the shear stress is \tau = F/A, the shear strain is \gamma = \tan\theta for a deflection angle \theta, and for a linear isotropic solid the shear modulus is G = \tau/\gamma.

Surface wettability and contact angle

When a drop of liquid rests on a solid, the contact angle \theta is measured through the liquid between the solid surface and the tangent to the liquid-vapour interface at the three-phase contact line. Young's equation relates it to the interfacial tensions as \gamma_{SV} = \gamma_{SL} + \gamma_{LV}\cos\theta.

• hydrophilic — a surface with a water contact angle below 90^\circ, which is wetted by water. e.g. A surface with \theta = 40^\circ spreads water readily.
• hydrophobic — a surface with a water contact angle above 90^\circ, which resists wetting. e.g. A surface with \theta = 110^\circ forms rounded drops.

Surfaces with angles above roughly 150^\circ are called superhydrophobic. Hydrophilicity of an implant surface generally promotes protein adsorption and cell attachment, which is why surfaces are often modified to be more wettable.

Bone implant failure

Bone implants can fail through several mechanisms that are distinct from infection. Stress shielding occurs when a stiff implant carries load that would normally pass through the bone, so the unloaded bone resorbs and loses density, and the implant eventually loosens. Aseptic loosening is the loss of the bone-implant bond in the absence of infection, often driven by micromotion at the interface or by wear debris. Fretting fatigue is progressive fatigue damage produced by small-amplitude rubbing between contacting surfaces, which can nucleate cracks in the implant. Osseointegration, the direct structural and functional connection between bone and implant without intervening soft tissue, is the desired outcome and is therefore a sign of successful fixation rather than a failure mode.

Coronary stent design requirements

A coronary stent is a metallic scaffold that is placed permanently inside an artery and stays in contact with flowing blood, so its surface must resist thrombus formation. A stent must therefore be thromboresistant, with surface chemistry and coatings that limit platelet adhesion. The stent is also subjected to pulsatile pressure and vessel motion with every heartbeat, so it must resist fatigue fracture under many millions of cycles. Excessive proliferation and accumulation of smooth muscle cells, together with extracellular matrix deposition, produce neointimal hyperplasia and restenosis, which narrow the lumen again. Good stent design therefore favours rapid endothelialization over smooth muscle accumulation.

Prosthetic heart valves

Mechanical prosthetic heart valves were developed as replacements for diseased native valves, and their designs use a moving occluder to permit one-way flow. The ball and cage valve, introduced in 1960, uses a ball that moves within a cage, and the bi-leaflet valve, introduced in 1977, uses two hinged leaflets. These designs are distinct from general-purpose industrial valves. A globe valve is a flow-control valve used in pipelines, and a swing check valve is a standard one-way industrial valve, so neither is a named prosthetic heart valve design. Mechanical valves are durable but require lifelong anticoagulation because blood flows over a foreign surface.

Shoulder joint muscle mechanics

A joint in static equilibrium obeys the moment balance, so the sum of moments of all forces about the pivot is zero. For a muscle acting on a bone lever, only the component of the muscle force perpendicular to the lever arm produces a moment, so the moment is M = T\,d\sin\alpha when the muscle makes an angle \alpha with the lever. At the shoulder, the ball and socket joint acts as the pivot for the arm, and the weight of the arm produces an opposing moment equal to its weight times the horizontal distance from the joint. The muscle tension needed to hold the arm is then the weight moment divided by the muscle moment arm. Worked example: a weight of 40 N acting at 20 cm from the joint, opposed by a muscle attached 5 cm from the joint at a right angle, requires T = 40\times 20/5 = 160 N.

FTIR hydrogen bonding peaks

In infrared spectroscopy each functional group absorbs at a characteristic wavenumber, and hydrogen bonding perturbs these bands. Formation of hydrogen bonds on alcohol groups weakens the O-H bond and spreads the O-H stretching vibration over a range of energies, so the O-H stretching band near 3200 to 3500 cm^{-1} broadens and shifts to lower wavenumber. A free O-H group gives a sharp band, whereas a hydrogen-bonded O-H gives a broad band. The carbonyl stretch near 1700 cm^{-1} and the C-H stretches near 2800 to 3000 cm^{-1} arise from different bonds and are not the bands that signal alcohol hydrogen bonding.

Hip joint biomechanics

The hip joint transmits the body weight from the pelvis to the femoral head, and in analysis the force at the femoral head is treated as a joint reaction force that must be balanced by muscle forces and lever arms. The moment of a force about a point equals the force multiplied by the perpendicular distance from the point to the line of action of the force. For a vertical force, the perpendicular distance is the horizontal distance from the point to the force line. Where a prosthesis is inclined, the lever arms are found from the geometry, using sine and cosine of the inclination angles to obtain the perpendicular distances. Worked example: a vertical reaction of 500 N whose line of action lies 4 cm horizontally from a reference point produces a moment of 500 \times 0.04 = 20 N·m about that point.

Projectile motion kinematics

A projectile launched from height h with speed u at angle \theta above the horizontal, with no air resistance, has horizontal and vertical components x = u\cos\theta\,t and y = h + u\sin\theta\,t - \tfrac{1}{2}gt^2. Eliminating time gives the trajectory y = h + x\tan\theta - \dfrac{g x^2}{2u^2\cos^2\theta}. To find the launch speed from the horizontal range, the trajectory is set to y = 0 at the range, which gives a quadratic in u^2 when h is nonzero. Launching from an elevation increases the range compared with launching from ground level, because the projectile has more time to travel. Worked example: for h = 0, u = 10 m/s and \theta = 45^\circ, the range is u^2\sin 2\theta/g = 100/10 = 10 m.

Bioactive glass composition

Bioactive glasses are silicate glasses that bond to bone. The classic composition 45S5, also called Bioglass, contains 45 wt% SiO_2, 24.5 wt% Na_2O, 24.5 wt% CaO and 6 wt% P_2O_5, with a calcium to phosphorus molar ratio of roughly 5:1. The glass itself does not contain hydroxyapatite. Hydroxyapatite is a crystalline calcium phosphate that forms on the glass surface when it contacts body fluid, through a layer of silica-rich gel that precipitates a calcium phosphate layer. The phosphate content and the calcium to phosphorus ratio are therefore the key compositional features to remember.