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Physics in Medicine

Physics in Medicine. Dr R.J. Stewart. 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:

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Physics in Medicine

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  1. Physics in Medicine Dr R.J. Stewart

  2. 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

  3. 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 Viscosity

  4. 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

  5. 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

  6. a P1 P2 l DP= P1 - P2 Volume Flow Rate • Poiseulle’s Equation • Volume flow rate, Q, related to pressure difference DP, length l and radius a by:

  7. R1 R2 R3 R1,Q1 DP1 DP2 DP3 R2,Q2 Volume Flow Rate • Often convenient to define a resistance, R to flow, such that DP=QR Series Parallel DP= DP1 + DP2 + DP3 =QR1+QR2+QR3 =QR \R=R1+R2+R3 Q=Q1+Q2 =DP/R1+DP/R2 =DP/R \1/R=1/R1+1/R2

  8. 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.

  9. 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

  10. Left side of heart Systemic Circulation Lung Circulation Right side of heart Transport System • A closed double-pump system:

  11. Transport System • Structure of the Heart Aorta Superior vena cava (from upper body) Inferior vena cava (from lower body)

  12. Transport System • Branching of blood vessels • Ateries branch into arterioles, veins into venules Arteries Arterioles Heart Capillaries Veins Venules

  13. 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

  14. 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.

  15. 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

  16. Pressure

  17. 9.3 kPa 13.3 kPa 13.2 kPa 13.3 kPa 13.1 kPa 26.7 kPa Pressure • Effect of gravity on pressure • Density of blood ~ 1.04x103 kg/m3 • Distance heart-head~ 0.4 m • Heart-feet ~ 1.4 m • DP = rgh

  18. 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

  19. 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 • DPheart-head = 1.04x103 x4gx0.4 = 16 kPa • Pressure from heart = 13.3 kPa \head receives no blood - Blackout!

  20. 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

  21. 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

  22. Turbulent Qc Flow rate Laminar Pressure Effect of Constrictions • Transition from laminar to turbulent flow characterised by Reynold’s Number, K • Critical velocity Vc = Qc/A • Vc = Kh/rR • For many fluids,K ~1000 • e.g, in the aorta (R~1cm), Vc~ 0.4m/s

  23. 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!

  24. 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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