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