cardiovascular fluid flow

7/29/2019 Cardiovascular Fluid Flow

1/25

Cardiovascular Circulation:Pum s Pi es and Fluids in Nature

ChE3200

pr ng

Prof. Victor Breedveld


2/25

Cardiovascular Circulation Discovered by William Harvey (1578-1657)

ump ear , p pes ar er es, ve ns

and fluid (blood)

o y s ma n ranspor mec an sm:- Oxygen (lungs organs)

-

- Heat (organs body extremities)

- Nutrients

Failure is leading cause of death

- Waste

Goal: Engineering Analysis


3/25

Cardiovascular Circulation: Numbers

Blood volume: 5.2 l

Flow rate: 5 l/min (rest) 30 l/min (strenuous exercise) Blood velocity in aorta: 0.3 m/s (rest)

in capillary veins: 0.4 mm/s (rest)

Combined len th of blood vessels: 100 000 km

Heartbeat rate: ca 70/min (rest) 200/min (max)

. Power output of heart: 1.3 W (rest) 8 W (exercise)

Efficiency of heart: ca. 10%


4/25

Pressure

- maximum: systolicpressure (120 mm Hg - gauge)

- minimum: diastolicpressure (80 mm Hg - gauge)

Systolic pressure must be high enough to overcome gravity+ friction of piping

vertical distance heart head (zhh) is important!!!

Psys,min

= blood

g zhhExam les :

- humans: moderate (120 mm Hg, zhh ~ 0.4 m)

- horses: higher(180 mm Hg, zhh ~ 1 m)

- - , hh - snake: ground low

climbing tree high

solution: snake heart close to head

- underwater animals: low (blood - water)


5/25

Cardiovascular Circulation: Schematics

Simple design

(reptiles, fish)

More complex design

(mammals, birds)

Heart

Lungs/

Gills

Lungs/

Gills

Bodyheart

Left

Body

ear

Double pumping action (booster pump)

friction of capillary vessels in lungs and body


6/25

Design of Heart: Close-up

Two ventricles/atria,

separated by valves.

Ventricles have thick walls

Left ventricle thicker walls

than right


7/25

Design of Heart: Muscle Contraction

Why are walls of heart ventricles thick?

ore musc e more power

(left thicker: higher pressure)

,

the ventricle volume by

ca. 80%

Muscles are most efficient

Sphere modeldrastic (~ 10%)

Thick wall design enables efficient reduction

of ventricle volume


8/25

Design of Heart: Role of atria

Muscles cannot actively expand, only relax refilling ventricles is slow

Atria contract to provide filling pressure for ventricles

Heart stroke two hases:

- contraction of atria (gentle)

- contraction of ventricles (hard)

Engineering analog:

supercharger on pump


9/25

Pipes: arteries, veins, capillaries

Contradiction:

Efficient pumping requires large pipes (friction),

e c en exc ange o ea an gases requ res narrow

pipes.

Solution:

Different pipe sizes in network;use capillaries

,

*Arteries (heartcapillaries):

hi h ressure thick muscular walls flexible

*Veins (capillaries heart):lower pressures, thin muscular walls, less flexible

*


10/25

Downsizing Pipes

What is the most efficient way to go from arteries

to capillaries?

Cost function of blood vessels (Murray, 1926):

Total cost = frictional cost + metabolic cost

= Q P + C1 a2

L-~ 2 1 -

(cost function)/a = 0 aopt ~ Q1/3


11/25

Downsizing Pipes (2) Qn+1

Qn

an+1

Bifurcation:

one vessel splits into two

sma er ones equa s ze

- Qn+1 = 0.5Qn (mass conservation)

- Optimizing total cost function

an+1

= 0.794 an

and = 37.5

- rom aorta a0 = . cm to

capillary (510-4 cm) we would need:

-

. .


12/25

Regulating Flow

Overall flow: Heart rate

Distribution:Adapt to needs of body

Organ Fraction of flow[%]

Flow/organ mass [ l/(kgmin)]

.

Kidneys 22 4.0

Liver 13 0.85

Brain 14 0.55

Heart muscle 5 0.8

er musc e .(in exercise) 75 0.55

Skin 4 0.08

(max.

vasodilation)

- 1.2


13/25

Regulating Flow (2)

How to regulate flow distribution in piping network?

Control friction of different paths!!!

Hagen-Poiseuille (laminar flow):

4(take logarithm on both sides)

log [Q]= log [P/L (a4)/(8) ](differentiate, P and L constant)

Q/Q ~ 4a/a

ecreas ng p pe ame er y a a = - . resu sin 40% decrease in flow rate (Q/Q = -0.40).

c en regu a ory mec an sm:blood vessels can expand and contract.


14/25

Pulsatile Flow

How to characterize/analyze blood flow?

Reynolds numberRe is not suitable for dimensional

analysis of transient, pulsatile flow.

New dimensionless group:Womersley number: Wo = a (/)

,: blood viscosity and density

a: radius of blood vessel


15/25

Flexible Arteries: Surviving incompressibility

Blood is incompressible

reducing volume of circulatory system

where does blood go without bursting the vessels?

Solution: flexible arteries to dampen pressure peaks


16/25

Flexible Arteries (2)

(lipid/cholesterol deposition)

High blood pressure,

Increased risk of breakage of pipe walls


17/25

Blood: Composition

Plasma (90% water, 7% proteins, 1% inorganic material)- Newtonian fluid with viscosity 1.2 mPa.s

Cells- Red cells (erythrocytes) oxygen transport

- White cells (leukocytes, various types)

immune system

-

Volume ratio

600 : 1 : 1

Red cells dominate the blood flow properties


18/25

Blood: Red cells

Disks (diameter 7.6m, thickness 2.8m)

onu - e s ape

allow deformation with minimum increaseof membrane area + efficient transport of O2

a e up - o oo vo ume ema ocr


19/25

Blood: Viscosity

Blood is shear-thinning fluid:

viscosity becomes lower at higher flow rates

Power-law model describes blood pretty accurately


20/25

Blood: Viscosity (2)

Why is blood shear-thinning?

Red blood cell properties

- aggregation (aggregates break at high shear lower)

- deformation (cells can deform to slip past each other and

- shape (ellipsoids can orient in flow)


21/25

Blood: Viscosity (3)

Combined effects


22/25

Blood: Sickle Cell Anemia

Disease of hemoglobin (oxygen transport protein):

inside the red sickle cells

cells become stiff andcan no longer deform

viscosity increases and


23/25

Blood: Sickle Cell Anemia (2)


24/25

Blood: Sickle Cell Anemia (3)

Bodys strategy for survival:

red blood cells


25/25

. , ,

circulatory systems, Oxford University Press, 1992.

-. . , , , .

Y.C. Fung, Biomechanics: mechanical properties of living

tissues, S rin er-Verla , 1981.

K.H. McDonald III, Sickle cell anemia as a rheologic

disease, The American Journal of Medicine 70, 288-298,

1981.

cardiovascular fluid flow

Documents