Biomechanics

1
Biomechanics

• The objective of this next set of lectures is to
  give a basic introduction to biomechanics the
  study of force relationships of different systems
  of the human body.
• Biomechanics is be used to
• To help understand how certain physiological
  systems function (e.g. cardiovascular system)
• To model these systems to aid in the design of
  prosthetic devices (e.g. artificial vessels)

2
Mechanical Forces

• Forces
• The physical influence that produces a change in
  a physical quantity (i.e. force mass x
  acceleration).
• Forces are vectors and thus have both magnitude
  and direction.
• Forces can be resolved along different directions
  and are usually resolved along the axes of thr
  Cartesian coordinate system (using Pythagoras
  theorem)

Vx V cos q Vy V sin q
Note q is measured positive counterclockwise in
these equations
3
Moments

• Moment
• The moment of a force about a point or axis is a
  measure of its tendency to cause rotation.
• A moment is the cross-product of the force and
  its distance vector about the point of rotation
  (i.e. M F x r)
• The scalar definition of a moment is the product
  of the force magnitude and its perpendicular
  distance from the point of rotation to the line
  of action of the force

Mo F d
where, d r sin q
4
Static Equilibrium

• Static equilibrium of a body (or any part of it)
  which is currently stationary will remain
  stationary if the resultant forces and moments
  acting on the body are zero.
• This leads to three equations (in 2D Cartesian
  space)

S Fx 0 S Fy 0 S M 0
5
Static Equilibrium Example

• Russell Traction Rig

6
Anthropometrics
Anthropometric data provides physical information
(weight, length, etc.) of body segments of a
typical individual (expressed in terms of
individual body weight and height).
7
Static Equilibrium Example
8
Cardiac Cycle

• The cardiac cycle can be modeled as a
  thermodynamic system. Lets look at a P-V plot of
  the one of the ventricles

P
1) Ventricular Filling
Win
2) Atrial Contraction
3) Ventricular Contraction
4) Ventricular Ejection
5) Ventricular Relaxation
Wout
V
End systolic volume
End diastolic volume
9
Cardiac Cycle

• Ventricular Filling
• ventricle muscle is completely relaxed and
  pressure is low
• pressure in atria exceeds that of the ventricle
  due to the continuous flow of blood into the
  atria from the circulation
• forward pressure causes AV (atrioventricular)
  valves to open
• Atrial Contraction
• ventricular filling occurs rapidly at first as
  the accumulated blood in the atria flows into the
  relaxed ventricles
• atrial contraction occurs at the end of the
  ventricular filling phase, immediately before
  ventricular contraction
• normally, atrial contraction increases
  ventricular filling by 10-30, but at high heart
  rates, available time for ventricular filling is
  reduced and atrial contraction becomes more
  important
• end of diastole

10
Cardiac Cycle

• 3) Ventricular Contraction
• beginning of systole
• ventricular pressure increases rapidly to a level
  greater than in the atria
• AV valves close and semilunar valves are still
  closed, resulting in an ISOVOLUMETRIC contraction
• Ventricular Ejection
• pressure in the ventricle exceeds that of the
  circulation (aorta and pulmonary arteries) and
  the semilunar valve open ejecting blood into the
  circulation
• muscle is still contracting through this phase
  and there is a slight increase in ventricular
  pressure
• end of systole

11
Cardiac Cycle

• 5) Ventricular Relaxation
• beginning of diastole
• ventricular pressure drops and this results in
  the closure of the semilunar valves preventing
  blackflow into the heart
• AV valves are still closed, resulting in an
  ISOVOLUMETRIC relaxation
• when ventricular pressure drops below the atrial
  pressure, the AV valves open and the ventricles
  start filling again

12
Modeling the Cardiac Cycle

• Standard thermodynamic assumptions (for pumps)
• - neglect Dke and Dpe
• - adiabatic, isentropic, steady flow, steady
  state

P
Win
P2 systolic pressure P1 atrial filling
pressure V2 end diastolic volume V1 end
systolic volume (V2 V1) is often referred to
as the stroke volume (SV)
P2
Wout
P1
V
V1
V2
13
Modeling the Cardiac Cycle
negative since we must add work to keep the heart
pumping
14
Fluid Properties

• Fluid Rheology
• Recall a shear stress versus velocity gradient
  (shear rate) plot for a fluid

Casson
Bingham
pseudoplastic
Newtonian
15
Fluid Properties

• Continuity Equation
• mass in mass out (assuming no accumulation)

for an incompressible fluid, r is a constant
Laminar Flow at low velocities, fluids tend to
flow with no lateral mixing and adjacent fluid
layers slide past one another like playing cards
(from lamina layer)
16
Fluid Properties

• Turbulent Flow
• lateral mixing between adjacent layers (i.e. no
  distinct layers)

Reynolds Number measure of turbulence and is
defined as the ratio of inertial forces to
viscous forces of the fluid
where D is the diameter of the pipe (or
vessel) for Re lt 2000, pipe flow is laminar
17
Blood

• Consists of cells and plasma
• Cells
• numerous cell types
• erythocytes (RBC) dominate and make up about 95
  of cellular content
• other cells are immune system cells (WBC) and
  platelets
• WBCs 0.1, platelets 4.9
• Plasma
• liquid component of blood
• contains salts, sugars, proteins, amino acids,
  fats, etc.
• protein content of blood is primarily albumin
  (then followed by fibrinogen)

18
Blood Properties

• Plasma
• a Newtonian fluid with a viscosity between 1.16
  and 1.35 mPas (cP) at 37C
• water has a viscosity of 0.69 cP at 37C
• increase in plasma viscosity over water due to
  the presence of globular proteins
• Whole Blood
• exhibits non-Newtonian behaviour due to presence
  of cells and displays both a yield stress and
  shear thinning (pseudoplastic) properties
• behaves like a Casson fluid (which was originally
  developed to account for a two-phase system)

19
Blood Properties

• yield stress is due to the presence of cells,
  primarily the RBCs called hematocrit (volume
  percentage that is composed of RBCs) and
  fibrinogen
• another version of this equation is

• where
• H hematocrit
• CF fibrinogen concentration (g/100 mL)
• aa a protein dependent parameter
• mp plasma viscosity

20
Blood Properties

• fibrinogen is considered
• since this protein interacts with RBCs
• RBCs in saline behave a Newtonian fluid
  however, fibrinogen tends to reversibly bind
  RBCs together forming a weak gel
• hematocrit is considered
• since RBCs tend to stick when pushed together
  (same as above)
• normal H 0.35 to 0.50

21
Effect of Vessel Radius

• In reasonably large vessels (radius gt 1 mm) blood
  viscosity is independent of radius however, the
  viscosity decreases rapidly at lower radii and
  this is called the Fahraeus-Lindquist Effect
• concentration and velocity of RBCs are greater
  at the vessel centre-line which leaves a small
  space near the vessel wall occupied only with
  plasma
• axial accumulation occurs because of the flow
  stream profile as the plasma imparts a spin to a
  RBC caught between adjacent flow layers of
  differing velocities
• the higher velocity tends to pull the cells
  towards the centre-line and the result is that
  the viscosity is lowest at the vessel wall
• In the small vessels this means that the overall
  viscosity of blood has decreased
• plasma has moved to the wall where the shear
  forces are the greatest while the cell-rich
  centre has the lowest shear forces
• another consequence is that branched vessels skin
  plasma from the parent vessel which lowers the
  hematocrit in smaller vessels

22
Fahraeus-Lindquist Effect
23
Hemodynamics

• Shear Stress at Vessel Wall
• one of the most important variables as tw
  indicates how much force is being exerted on the
  vessel wall
• in healthy vessels, tw is low ( 15-20 dynes/cm)
  and is not harmful to RBCs or platelets
• excessively high levels of wall shear (due to
  athersclerotic plaque or artificial heart valves)
  may damage RBCs (hemolysis), the endothelium, or
  platelets
• platelets may also be activated by prolonged
  exposure to high shear stresses or rapid
  increases in shear stress to initiate clot
  formation

24
Hemodynamics

• Blood Flow in Arteries
• for the most part, blood flow is laminar under
  resting conditions (i.e. 60-70 bpm)

Vessel Radius cm Re (mean)
proximal aorta 1.5 1,500
femoral artery 0.27 180
left main coronary artery 0.43 270
right main coronary artery 0.10 233
terminal arteries 0.05 17

• turbulent flow can occur when blood velocity
  increases due top exercise but only in the large
  vessels

25
Hagen-Poiseuille Expression

• Consider the steady flow of a constant density
  fluid in a fully developed flow field through a
  horizontal pipe. If we were to look at the forces
  acting on an element of the fluid concentric with
  the axis of the pipe

The force pushing the element of fluid through
the tube is
26
Hagen-Poiseuille Expression
This force is resisted by a friction force (shear
force)
Since there is no net momentum and steady flow,
the sum of these forces are zero
27
Hagen-Poiseuille Expression
at the wall, rR and ttw
Loosing some formal convention
but Dpp2-p1, which is negative, so, if we adopt
a convention that Dp is a pressure drop
28
Hagen-Poiseuille Expression
Recall for a Newtonian fluid
29
Hagen-Poiseuille Expression
umax occurs at r0
30
Hagen-Poiseuille Expression
The flow rate in the pipe can also be determined
or,
Hagen-Poiseuille Expression
This implies that the resistance to flow is
proportional to
31
Hagen-Poiseuille Expression
This development assumes 1) incompressible
fluid 2) no slip at wall 3) tube is straight,
rigid and cylindrical with constant
radius 4) Newtonian fluid 5) flow is steady and
fully developed
32
Bernoulli Equation
Is derived from a consideration of the energies
involved mechanical work kinetic
energy potential energy For an
incompressible fluid Using the law of
conservation of energy and ignoring frictional
losses
Bernoulli Equation
33
Bernoulli Equation
This development assumes 1) incompressible
fluid 2) no viscous losses 3) mechanical energy
at any point is constant
34
Poiseuille or Bernoulli?
Poiseuilles equation can be used to model flow
but only for large vessels (i.e. arteries) where
the vessels do not change much in diameter. In
smaller vessels and the capillaries, where the
shear is low, blood must be treated as a
non-Newtonian fluid and Poiseuilles equation
does not apply. If there is a constriction or
dilation in the vessel, then Bernoullis equation
(as Poiseuille is not valid) even though there
will be errors introduced by not accounting for
blood viscosity.
35
Hemodynamics
Within a given organ, blood vessels are arranged
in both series and parallel
Each vessel imparts a resistance R to flow,
determined by the length and radius of the vessel
(e.g. Poiseuille). For vessels in parallel
The total resistance is the sum of the
reciprocals of the individual vessel resistances
36
Hemodynamics
Example For three vessels in parallel with
Thus, the total resistance of a network of
parallel vessels is less than the resistance of
the vessel having the lowest resistance. Therefor
e, a parallel arrangement of vessels greatly
reduces resistance to blood flow. This explains
why capillaries which have the largest resistance
to blood flow due to their small diameter,
constitute only a small portion of the total
vascular resistance
37
Hemodynamics
For vessels in series
The total resistance is the sum of the individual
vessel resistances
Example For five vessels in series with
38
Hemodynamics
Poiseuilles equation indicates that the
resistance to flow is Therefore, the smaller the
vessel, the greater the resistance to flow.
This means that changes in large artery
resistance will have little impact on the total
resistance. However, changes in small artery
resistance will have a large impact and are the
principal vessels regulating blood flow.
Estimates of Total Arteriolar and Capillary
Resistances
arterioles capillaries
radius mm 15 4
individual vessel resistance dynes/cm2 15 x 107 3 x 1010
number of units 107 1010
total resistance dynes/cm2 15 3
39
Critical Stenosis

• Stenosis
• narrowing of an artery that results in a
  significant reduction in blood flow
• often leads to angioplasty to remove the blockage

Total Resistance (RT)
where, RL is the large vessel resistance RS is
the small vessel bed resistance
40
Critical Stenosis
Typically, RL represents about 0.1 of RT without
a blockage If the radius of the vessel is
decreased by 50 by the presence of a blockage,
then (assuming Poiseuille flow)
and
thus,
and
Therefore, a decrease in vessel radius by 50
leads to increase in total resistance of only 1.5
41
Critical Stenosis
What if the radius of the vessel is decreased by
75?
now,
Therefore, a decrease in vessel radius by 75
leads to increase in total resistance of
25.5 This would cause a clinically significant
decrease in blood flow and would require
intervention. In practice, angioplasty is not
performed unless there is a gt70 decrease in
vessel radius.
42
Wave Propagation
In our previous discussion we have assumed that
the vasculature was comprised of rigid vessels
with (predominantly) Poiseuille flow in
them. The assumption of rigid vessel walls poses
a problem when we consider the pulsatile nature
of blood flow (remember the changes in blood flow
between systole and diastole). In a rigid system
of vessels, the pressure waveform at any point in
the circulation would be similar and synchronous
with that of the aortic root (start of the aorta)
but with a scaled down amplitude (due to vessel
resistance).
43
Wave Propagation
However, it turns out that the pressure waveform
in the vasculature is neither similar nor
synchronous throughout the circulation. This is
due to the elasticity of the vessel walls. When
blood is pumped from the heart (systole), the
pressure rises in the large arteries and they
expand. When the pressure falls (diastole) they
contract again so that the flow rate through the
small peripheral vessels does not immediately
fall to zero. Essentially, the arteries serve to
store some blood during systole and eject it
during diastole to keep the blood flowing.
44
Wave Propagation
The bolus of fluid ejected during each
contraction causes a pressure wave in the vessel
wall as the bolus moves downstream.
tension
relaxation
The speed of the pressure pulse (c) is governed
by
where, E is the elastic modulus of vessel
wall t is the vessel wall thickness r is the
vessel radius r is the density of blood
45
Reflection and Transmission of the Pulse
At every point where the properties (or
dimensions) of the artery changes, there is a
site of partial reflection of the pressure
pulse. Throughout the arterial system, there are
many potential sites for reflections but the most
obvious and potent discontinuities are the
junctions between vessels (i.e. bifurcations).
Remember that there is blood flow associated with
the propagation of the pressure pulse and thus
the reflected portion of the wave causes some
flow reversal (i.e. backflow).
46
Reflection and Transmission of the Pulse
What governs how much of the pressure pulse will
be transmitted versus reflected is the ratio of
the vessels impedances (Z)
where, r is the density of blood c is the speed
of the pulse A is the undisturbed area of the
vessel
or, we can also use the vessels admittance (a)
47
Reflection and Transmission of the Pulse
The reflectance (G) and transmission (T)
coefficients of the pressure pulse at the
junction are
and
where
In the case of an end-to-end anastomosis, a20
48
Impedance Matching
This is important as if a vessel is to be
replaced, the characteristics of the pressure
pulse propagation must be maintained in order to
ensure proper blood flow. For a properly matched
end-to-end anastomosis (graft)
and
This occurs when the sum of the admittances of
the daughter vessels is equal to the admittance
of the parent vessel
It is unlikely that any vascular graft will have
the same elastic modulus (E) as a blood vessel,
thus the wall thickness to diameter ratio must
be compensated for to ensure a proper impedance
match.