physics in medicine ph3708 dr r.j. stewart. scope of module cardio-vascular system –fluid flow in...
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Physics in Medicine
PH3708
Dr R.J. Stewart
Scope of Module
• Cardio-vascular system– Fluid flow in pipes, circulation system, pressure
• Membranes– Osmosis and solute transport
• Transmission of electrical signals– Nerves, ECG
• Optical Fibres and Endoscopy
Scope of Module
• Ultrasound– Imaging and Doppler measurements
• Radioisotope imaging and radiology
• X-ray generation and imaging
• NMR imaging
Module Resources
• Web Page:– http://www.rdg.ac.uk/physicsnet/units/3/ph3708/ph3708.htm
• Books:– Good general books:
“Physics of the Body”, Cameron, Skofronick and Grant “Medical Physics”, J.A. Pope
– Other more specialised books are given in the unit description and will be referred to where necessary
Cardiovascular System
• Physics of the Body, Cameron, Skofronick and Grant, Ch. 8
• In considering the circulation of blood, one essentially considers the flow of a viscous fluid through pipes of different diameters
• Define:– Viscosity: arises from frictional forces associated
with the flow of one layer of liquid over another
Viscosity
• Consider a circular cross section pipe:– Flow through pipe due to pressure difference– Assume: flow at walls of pipe = 0, maximum in
the centre (arrows in figure represent velocity)– Frictional force per unit area, F, proportional to
the velocity gradient
Fdv
dr
Viscosity
F
x
)(rv
Viscosity
• The slower moving fluid outside the central (shaded) region exerts a viscous drag across the cylindrical surface at radius r. For a length Δx of pipe the area of surface is 2πrΔx. The force points in the opposite direction to the direction of fluid motion and is of magnitude
2πrΔx η |dv/dr|
2r
2a
Volume Flow Rate
• The average flow from the heart is the stroke volume (the volume of blood ejected in each beat) x number of beats per second. This is ~ 60 (ml/beat) x 80 (beats/min) = 4800 ml/min
Volume Flow Rate
• Poiseulle’s Equation– Volume flow rate, Q, related to pressure
difference P, length l and radius a by:
l
a
P1 P2
P= P1 - P2
Qa
lP
4
8
Volume Flow Rate
• Often convenient to define a resistance, R to flow, such that P=QR
P1 P2 P3
R1 R2 R3
P= P1 + P2 + P3
=QR1+QR2+QR3
=QRR=R1+R2+R3
Series Parallel
R1,Q1
R2,Q2Q=Q1+Q2
=P/R1+P/R2
=P/RR=1/R1+1/R2
Resistance R
• The resistance decreases rapidly as a increases R = ΔP/Q = 8 l η / πa4 The units of R are Pa m-3 s A narrowing of an artery leads to a large increase in the resistance to blood flow, because of 1/ a4 term.
Volume Flow Rates
• Effect of restrictions and blockages:– Series, whole flow is reduced/stopped– Parallel, flow partially reduced, increased in
other parts of the network
Transport System
• A closed double-pump system:
SystemicCirculation
LungCirculation
Left side of heart
Right side of heart
Transport System
• Structure of the Heart
Inferior vena cava(from lower body)
Superior vena cava(from upper body)
Aorta
Transport System
• Branching of blood vessels– Ateries branch into arterioles, veins into
venulesArteries
Arterioles
Capillaries
VenulesVeins
Heart
Transport System
• Capillaries– Fine vessels penetrating
tissues
– Main route for gas/nutrient exchange with tissues
– About 190/mm2 in cut muscle surface
– Sphincter muscles (S) control flow
Transport System
• Blood is in capillary bed for a few seconds
• 1Kg of muscle has a volume of about 106 mm3 (density of muscle ~1gm/cm3 or 1000 Kg/m3 ), hence there are about 190km of capillaries with a surface area of ~12 m2 assuming a typical capillary is 20μm in diameter.
Pressures
• Large pressure variations throughout the system (note 1 kPa = 7.35 mm Hg)– 17 kPa (125 mmHg) after left ventricle– 2 kPa (15 mm Hg) after systemic system– 3.4 kPa (25 mmHg) after right ventricle
Blood pressure monitor on arm measures 120 mmHg systole and 80 mmHg diastole for a healthy young person
Pressure
Pressure
• Effect of gravity on pressure– Density of blood ~ 1.04x103 kg/m3
– Distance heart-head~ 0.4 m– Heart-feet ~ 1.4 m– P = gh
9.3 kPa
13.3 kPa
26.7 kPa
13.3 kPa13.1 kPa 13.2 kPa
Pressure
• Consequences– Varicose veins
• Normally (e.g., during walking) muscle action helps return venous blood from the legs
• One-way valves in leg veins to prevent backward flow
• Defective valves means pooling of blood in leg veins
Pressure
• Acceleration– Consider upward acceleration, a - augments gravity– effective gravity = a+g– Pressure difference = (a+g)h
• Pressure at head reduced.
• E.g., a = 3g
• Pheart-head = 1.04x103 x4gx0.4 = 16 kPa
• Pressure from heart = 13.3 kPa head receives no blood - Blackout!
Rate of blood flow
• Blood leaves heart at ~ 30 cm/s
• In capillaries, flow slows to ~ 1mm/s– Surprising - continuity should imply higher
flow– Recall individual capillaries only ~20m in
diameter, but very many hence total cross section equivalent to a tube 30 cm in diameter using estimate of 225 x 106 capillaries in body
Effect of Constrictions
• Bernoulli effect– Narrowing of tube gives increased velocity, but
reduced pressure
• Increasing velocity at obstruction leads to a transition from laminar to turbulent flow
Effect of Constrictions
• Transition from laminar to turbulent flow characterised by Reynold’s Number, K
Flo
w r
ate
Pressure
Lam
inar
Turbulent
Qc
– Critical velocity Vc = Qc/A
– Vc = K/R
– For many fluids, K ~1000
– e.g, in the aorta (R~1cm), Vc ~ 0.4m/s
Effect of Constrictions
• Apparent that one can get a rapid increase in flow as a function of pressure in the laminar region, but relatively slow in turbulent region– During exercise, 4-5 time increase in blood
flow required– Obstructed vessel may not be able to deliver
• Chest pains and heart attack!
Further Reading
• All in Physics of the Body, Cameron, Skofronick and Grant, Ch. 8,
• Measurement of blood pressure– Section 8.4
• Physics of heart disease– Section 8.10
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