BME400 Biomechanics Assignment Sample NUIG Ireland

BME400 Biomechanics is a required course for all undergraduate students in the Biomedical Engineering program at McGill University. The course covers the principles of mechanics as they apply to biological tissues and organs. Students learn about the structure and function of cells, tissues, and organs, and how these properties are affected by mechanical loading. In addition, students study the mechanics of movement and learn how to analyze movement patterns using motion-capture technology.

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In this section, we are describing some assigned briefs. These are:

Assignment Brief 1: Derive the Windkessel model to relate vessel compliance to aortic blood pressure.

The Windkessel model is a classic model of arterial (and venous) physiology that relates vessel compliance to blood pressure. The model consists of two “systems”, the elastic arteries and the muscle- contraction-controlled vessels, which are in series. The arterial system includes all of the large elastic arteries and delivers blood at a constant rate to the muscle-controlled vessels. The muscle-controlled vessels include all of the small arteries and arterioles, and they control the flow of blood through resistance.

To derive the Windkessel model, we start with some basic assumptions: 

• The heart pumps blood out at a constant rate 
• Arteries are fluid-filled elastic tubes (they can stretch and recoil) 
• The walls of the arteries are uniform throughout 
• The cross-sectional area of an artery stays constant as it stretches 
• Arteries are inversely proportional to their lengths 
• All of the blood in the body is contained within the arteries

Now that we have our assumptions, we can start deriving the equations that describe the Windkessel model. Our first equation will relate vessel compliance to blood pressure.

Vessel compliance (C) is a measure of a vessel’s ability to stretch and recoil. It is defined as the change in volume (ΔV) divided by the change in pressure (ΔP):

C = ΔV/ΔP

We can rearrange this equation to solve for ΔV:

ΔV = C*ΔP

Now, we can substitute the definition of vessel compliance into our equation to relate blood pressure to vessel stretch.

ΔV = (ΔV/ΔP)*ΔP

ΔV = ΔP/C

So, as blood pressure increases, vessel stretch also increases. We can see from this equation that vessels with high compliance (low C) will stretch more than vessels with low compliance (high C).

Now that we have our first equation, we can derive our second equation, which relates vessel compliance to arterial blood pressure.

To do this, we will start with the definition of arterial blood pressure:

P = F/A

Where P is arterial blood pressure, F is the total force acting on the vessel wall, and A is the cross-sectional area.

Assignment Brief 2: Calculate the compliance of a porcine aorta.

The compliance of a porcine aorta is difficult to calculate without extensive knowledge of the material properties and geometry of the aorta. However, assuming that the porcine aorta is similar in material properties and geometry to other mammalian aortas, we can use the formula for Young’s Modulus (E) to calculate its compliance.

The Young’s Modulus for common materials such as steel, aluminium, and rubber can be found in many engineering textbooks or online. For example, the Young’s Modulus for steel is about 200 GPa, aluminium is about 70 GPa, and rubber is about 3 GPa. Therefore, we can calculate the compliance of a porcine aorta as follows:

C = 1/E

For a steel aorta, C would be 0.005; for an aluminium aorta, C would be 0.014; and for a rubber aorta, C would be 0.333.

Assignment brief 3: Analyse the flow conditions required for the efficient closure of heart valves.

Four heart valves ensure the blood flows in the correct direction through the heart chambers: the tricuspid valve, pulmonary valve, aortic valve, and mitral valve. For these valves to close efficiently, certain flow conditions must be met.

The leaflets of the tricuspid and mitral valves are forced together by the pressure gradient between the left ventricle and atrium during systole. This closing force is aided by tethering strands called chordate tendineae, which attach to papillary muscles in the ventricle walls. To prevent regurgitation (backflow), these leaflets must seal tightly along their edges.

Valve closure is also assisted by fluid vortices called vortex rings, which are created by the inflow of blood into the ventricle. These vortex rings travel from the atrium towards the valve opening, and as they approach, they help to push the leaflets together and seal the valve.

The aortic and pulmonary valves work differently than the tricuspid and mitral valves. The aortic valve has three cusps (leaflets) that are forced together by the pressure gradient between the left ventricle and the aorta during systole. The pulmonary valve also has three cusps, which are forced together by the pressure gradient between the right ventricle and pulmonary artery during systole.

Both the aortic and pulmonary valves are assisted by the Bernoulli Principle, which states that when fluid flow increases, the pressure inside the fluid decreases. This decrease in pressure helps to push the cusps of the valve together and seal the valve.

Assignment Brief 4: Explain the conditions required for the non-Newtonian flow of blood in the vasculature. Derive an equation for non-Newtonian blood flow.

Non-Newtonian blood flow can be derived from a general equation for non-Newtonian fluid flow, which is given by:

Where: 

D=shear rate 

η=viscosity 

τ=absolute time) 

γ=kinetic energy per unit volume) (1)

In the context of blood flow, the kinetic energy per unit volume is related to the blood velocity and hence can be written as: 

γ=u2/2 (2)

Where u is the blood velocity. Substituting equation (2) into equation (1) gives: 

Which can be rearranged to give an expression for non-Newtonian blood flow: 

Where Q is the volumetric flow rate, A is the cross-sectional area, and μ is the blood viscosity.

The above equation shows that the non-Newtonian flow of blood is determined by the shear rate, viscosity, and volumetric flow rate. The shear rate is a measure of the rate at which the blood flow changes concerning distance, and is given by: 

D=∂u/∂y (3)

Where u is the blood velocity and y is the distance. The viscosity of blood is a measure of its resistance to flow and is given by: 

μ=τ/D (4)

Where τ is the absolute time. The volumetric flow rate is given by: 

Q=Au (5)

Where A is the cross-sectional area and u is the blood velocity.

From equation (5), it can be seen that the volumetric flow rate is directly proportional to the blood velocity. Therefore, for the volumetric flow rate to increase, the blood velocity must also increase. This can be accomplished by either increasing the diameter of the vessel or by increasing the heart rate.

Assignment Brief 5: Derive equations for fracture and fatigue failure of bone. Calculate the fracture toughness of bone.

The most common type of bone fracture is a stress fracture, which occurs when the bone is unable to withstand the stress placed on it. The first step in calculating the fracture toughness of bone is to determine the maximum stress that the bone can withstand without failure. This can be done by conducting a compression test on a sample of bone and measuring the amount of force required to cause failure.

Once the maximum stress is known, the fracture toughness can be calculated using the following equation:

Ft = (σf-σm)/(KIC)^0.5

where σf is the maximum stress, σm is the modulus of elasticity, and KIC is the fracture toughness.

The modulus of elasticity is a measure of the stiffness of the bone and is given by:

σm = E/((1+v)(1-2v))

where E is Young’s modulus and v is Poisson’s ratio.

The fracture toughness, KIC, is a measure of the amount of energy required to cause fracture and is given by:

KIC = Y/sqrt(πa)

where Y is the yield strength and a is the crack length.

The above equation can be used to calculate the fracture toughness of bone for any given material. However, it should be noted that different bones have different fracture toughness values.

Assignment Brief 6: Explain the Hill equation for muscle contractility. Derive the sliding-filament muscle contractility model.

The Hill equation is a mathematical model that describes the relationship between muscle force and the number of motor units recruited. It was developed by A. V. Hill in 1938.

The equation takes into account the fact that a muscle’s force-generating capacity depends on the number of active motor units, as well as on the firing frequency of those motor units. The equation can be used to predict how much force a muscle can generate given its current level of activation.

The sliding filament theory is a model of muscle contraction that explains how muscles generate force. The theory was first proposed by H. E. Huxley in 1954, and it was later refined by A. V. Hill and J. Duchenne de Boulogne.

The theory states that muscle contraction is the result of the sliding of two types of filaments past each other. The filaments are made up of protein molecules called myosin and actin.

Myosin filaments are attached to the Z-disk, while actin filaments are attached to the M-line. When a muscle contracts, the myosin filaments slide past the actin filaments. This action pulls the Z-disk closer to the M-line, and this is what causes the muscle to shorten.

The sliding filament theory explains how muscles generate force, but it does not explain how they move. To move, muscles must generate a force that is greater than the force of gravity. This can be accomplished by either increasing the muscle’s force-generating capacity or by changing the lever arm.

Assignment Brief 7: Determine the stress state of metal at yield and analyze the performance of a cardiovascular stent.

The stresses that metal experiences at yield depend on the type of material and the loading conditions. For many metals, the predominant stress at yield is tensile, meaning that the metal is being stretched. Other important stresses include shear stress and compressive stress.

When a metal experiences yield stress, its ability to resist further deformation or failure depends on several factors. The most important of these is the nature of the microstructure of the metal. Metals with high-strength microstructures, such as martensitic steels, can often resist large amounts of deformation before experiencing failure. In contrast, metals with lower-strength microstructures, such as ductile steels, will usually fail at much lower levels of deformation.

The second important factor is the orientation of the metal’s grains. Metals with grains that are oriented in the same direction as the applied force will usually fail at lower stresses than those with randomly oriented grains. This is because the forces can act along the length of the grains, rather than being resisted by the grains.

The third factor is the amount of cold work present in the metal. Cold metals worked will usually have higher yield stress than those that have not been cold worked. This is because the cold working process strengthens the metal by introducing defects into the microstructure.

The fourth factor is the presence of impurities in the metal. Impurities can act as sites for cracks to nucleate, and they can also make the metal more susceptible to corrosion.

The fifth factor is the temperature of the metal. Metals that are being cooled from high temperatures will usually have lower yield stresses than those that are at room temperature. This is because the high temperatures make the metal more ductile.

The sixth factor is applied stress. Metals that are under high stresses will usually have lower yield stresses than those that are under low stresses. This is because the high stress can cause the metal to flow plastically.

All of these factors must be considered when determining the yield stress of a metal.

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