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 
x

dv
F 
dr
Viscosity
v(r )
F
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:
a 4
Q
P
8l
a
P1
P2
l
P= P1 - P2
Volume Flow Rate
• Often convenient to define a resistance, R
to flow, such that P=QR
Series
Parallel
R1
R2
R3
P1
P2
P3
P= P1 + P2 + P3
=QR1+QR2+QR3
=QR
\R=R1+R2+R3
R1,Q1
R2,Q2
Q=Q1+Q2
=P/R1+P/R2
=P/R
\1/R=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:
Left side of heart
Lung
Circulation
Right side of heart
Systemic
Circulation
Transport System
• Structure of the Heart
Aorta
Superior vena cava
(from upper body)
Inferior vena cava
(from lower body)
Transport System
• Branching of blood vessels
– Ateries branch into arterioles, veins into
venules
Arteries
Arterioles
Heart
Capillaries
Veins
Venules
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
9.3 kPa
P = rgh
13.3 kPa
13.1 kPa
13.3 kPa 13.2 kPa
26.7 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 = r(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 ~20mm 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
Flow rate
– Critical velocity
Vc = Qc/A
Qc
– Vc = K/rR
– For many fluids,
K ~1000
– e.g, in the aorta
(R~1cm), Vc ~ 0.4m/s
Pressure
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