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1 AgoraLink Agora for Life Science Technologies Linköpings Universitet Kurs i Fysiologisk mätteknik Biofluidflöden Fysiologisk mätteknik

2 Anatomy of the heart

3 The complex myocardium structure right ventricle left ventricle Muscle fibers are obliquely oriented to optimize the contractility of the heart

4 Simulated strain patterns visualized for the left ventricle Illustrates the complex muscle architecture and contraction pattern of the heart

5 Heart valves The normal heart valves open and close properly in each cycle. A stenotic heart valve is too rigid and narrow to let enough blood to pass. An insufficient heart valve cannot close properly, allowing blood to flow backwards.

6 Artificial valves Biological artificial valve Valve repair Mechanical artificial valve

7 Blood vessels The blood vessels have a complex branching structure

8 Arteries / Veins The discovery of blood circulation is attributed to William Harvey ( ). He found that arteries brought blood to peripheral parts of the body and veins returned it to the heart. The later was observed by occluding a vein and finding that its blood was stocked distally to the occlusion.

9 Blood vessel branching throughout the circulatory system

10 The bifurcation of a blood vessel into two branches can be treated as an optimization problem From: YC Fung. Biodynamics. Circulation. Springer, 1984

11 Radius of arteries with increasing distance from the heart From: YC Fung. Biodynamics. Circulation. Springer, 1984 The aortic area deceases exponentially with the distance from the heart, (x). A A e 0 Bx R 0, A 0 = area of aorta, R 0 = Aorta radius, x = distance from the heart

12 Blood pressure related to total cross section area of vessels Pressure and cross-sectional area in different parts of the circulatory system lung body Cross-sectional area 4-6 cm 3-5m 8-10 cm Flow velocity cm/s 0,1 mm/s 0-5 cm/s

13 Pressure pulse-wave propagation and flow velocity Pressure wave velocity about 5 m/s Pressure waves Flow velocity about 0.5 m/s

14 Pulse propagation c P A c A dp Eh da R

15 Pressure curve change throughout the arterial system Characteristic impedance: Z c A and Z P Q Transportation work per time unit: dw dt pq p Z

16 Tapered blood vessel If we can ignore pressure losses p Z const. c p const. Z const A p increases as A decreases

17 Pressure curve change in a dog From: YC Fung. Biodynamics. Circulation. Springer, 1984

18 Pressure curve change in a human

19 Aortic/arterial elasticity damps pressure pulsations

20 Windkessel model A three element windkessel model of the arterial system as presented by Westerhof From: Westerhof et al

21 Windkessel models From: Westerhof et al

22 Characteristic impedance for different windkessel models From: Westerhof et al

23 Predicted pressure and flow curves From: Westerhof et al

24 Data on different vessel segments

25 A 0 element hydraulic model of the arterial system Each element describes the inertance, resistance and compliance of a certain arterial segment

26 A 17 element hydraulic model of the arterial system

27 Ancient models of the circulatory system The circulation system as described in an Encyclopedia from the middle of the 1800th century.

28 Fluid dynamics Biofluids

29 Newtonian Fluid Shear stress v y Shear rate v y v h Force v da lb h Illustration of the concept of viscosity. A fluid film of thickness h covers the plane surface. A board on top of the film moves with a constant velocity of v. The detailed figure shows the velocity profile of the fluid film.

30 Red blood cells The red blood cell is approx 7 x μm Density: 1060 kg/m 3 Viscosity Ns/m 1cp = Ns/m

31 Blood viscosity and shear rate Viscosity Haematocrit = the rate of blood cell components in (%) of blood volume. Blood viscosity as a function of shear rate for different haematocrit levels and compared to plasma and physiological saline solution shear rate

32 Blood viscosity and haematocrit Relative viscosity D = oo The blood viscosity as a function of the haematocrit for different vascular diameters

33 Blood viscosity and vessel diameter Viscosity (cp) Blood viscosity decreases with vessel diameter due to the Fåhraeus-Lindqvist effect Vessel diameter (mm)

34 The Fåraeus-Lindquist effect low haematocrit high haematocrit Large and small blood vessels for low and high haematocrit. For the larger vessels the viscosity increases as the haematocrit increases (the blood gets thicker ) In the small vessels, blood cells are lined up in the vessel. The fluid between the blood cells can be expected to be a cylinder with uniform velocity. If more blood cells are put into this cylinder the resistance to flow is not expected to increase. This might be interpreted as if the viscosity does not change.

35 Fluid mechanics

36 Laminar Flow Laminar flow in free space (a). The flow lines are parallel with equal velocity. Laminar flow next to a wall (b). The velocity next to the wall is zero. At higher velocities the flow is turbulent (c). The flow lines are not approximately parallel.

37 Continuity equation The continuity equation for cross-sections 1 and Q Av 1 1 Av V aorta = 0.5 m/s, A aorta = 5 cm, v kap = 0.1 mm/s gives A kap =.5 m

38 The Reynolds number The Reynolds number defines if a flow is laminar or turbulent. If Re < 000 the flow is laminar. Re d v mean

39 The Poiseuille s formula Laminar, stationary flow Resistance R p p R dr r r R p p Q L 8 ) ( ) ( L 4 ) ( 4 1 R L 8 Q p p R f

40 Turbulent Flow p1 p Re > 000 p p L R v 1 f m f V m = mean velocity f f = friction factor

41 Flow in Aorta d = 0.05 cm v = 1 m/s = Ns/m Re = 8000 => turbulent flow

42 Bernoulli s equation Loss-less, stationary flow. Describes a relation for the energy per unit volume. section section 1 p 1 1 v 1 gh 1 p 1 v gh pressure energy kinetic enargy potential energy

43 Flow through a stenosed valve left ventricle stenosed valve The Bernoulli equation 1 1 p v gh p v gh can be reduced to p v 1 p 1 ( p v 1 p ) The valve area can be calculated by the relation A =Q/v => Q A ( p p ) 1

44 Contraction of flow at a coarctation coarctation contraction coefficient velocity coefficient orifice coefficient

45 The impulse theorem control volume inlet outlet F F x x lkv t dm dt dm dt ( v v1 x v x x ) dm dt 1 v 1 x F is the total force on the control volume

46 Navier-Stokes Equations in the direction i: ) ( x v x v x v x p F x v v x v v x v v t v p t i i i i i i i i i i v g v v v 1 x v i Transient term Convective term Force term Pressure term Viscous term This equation is not able to solve analytically. Its importance increased dramatically when it became possible to solve it numerically using the finite element method (FEM).

47 Examples of blood flow simulations using the FEM

48 Examples of blood flow simulations using the FEM

49 Examples of simulations using the FEM

50

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