Download Fluid flow, circulation, cardiac biophysics

2012. 09. 25.
Fluid flow, circulation,
cardiac biophysics
19. September 2012.
Tamás Huber
Main properties of fluids
• Fluid is a substance that can flow.
• Fluids are either liquids or gases (states of matter: solid, liquid, and
gas).
• Liquid: A state of matter in which the molecules are relatively free to
change their positions with respect to each other but restricted by
cohesive forces so as to maintain a relatively fixed volume
• Gas: a state of matter in which the molecules are practically unrestricted
by cohesive forces. A gas has neither definite shape nor volume.
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Hydrostatic pressure in a liquid
The hydrostatic pressure is independent of the form of the container and is
proportionally to the height (h) and the density (ρ) of the fluid column.
Pressure is always perpendicular to the surface of an object.
p=
×
=
× ×
=
× × ×
Pascal’s law: Any external pressure applied to an enclosed fluid at rest is
transmitted undiminished throughout the liquid and onto the walls of the
containing vessel.
Fluids are uncompressible:
p = F1/A1 = F2/A2.
F1 « F2
If the height of the fluid's surface above the bottom of the five vessels is the
same, in which vessel is the pressure of the fluid on the bottom of the vessel
the greatest ?
The pressure at a given depth does not depend upon the shape of the vessel containing the
liquid or the amount of liquid in the vessel.
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Archimedes' principle
Any object completely or partially submerged in a fluid is buoyed up by a
force whose magnitude is equal to the weight of the fluid displaced by the
object.
A crane lowers an iron container into sweet water (water = 1000 kg/m3) which is
suspended on a rope. The container has a mass of 0.5 tons and its density is 7850
kg/m3. What is the tension in the rope?
Vsubmerged = m/iron
T= G-Fbuoyant= mg - fluid*g*Vsubmerged
T= 4905 – 625 = 4280 N
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Continuity equation
Fluids are incompressible, the intensity of current is constant in both
position and time.
The cross section of the tube (A) is inversely proportional to the velocity of
flow (v).
IV 
V A  v  t

 Av
t
t
Iv = Q = A* v = constant, stationary flow
Bernoulli’s law
Upon flow in a curved tube:
the potential energy:
mgh1 = mgh2
p1 V + mgh1 + (mv 12/2) = p2 V + mgh2+ (mv22/2)
p1 + ρgh1 + (ρv 12/2) = p2 + ρgh2 + (ρv 22/2)
p1 
  v 12
  v 22
   g  h1  p 2 
   g  h 2  const .
2
2
The general form of Bernoulli’s low:
Static pressure
Hydrostatic pressure
Dynamic pressure
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Water flows from a large pressurized container into the open air. The pressure
difference p is measured between the cross sections A1 and A2.
p
g
pa

h
A2
A1
A3
p
A1 = 0,8 m2, A2 = 0,2 m2, A3 = 0,4 m2, h = 0,9 m, ρ = 103 kg/m3,
pa = 105 N/m2, p = 0,5 · 105 N/m2, g = 10 m/s2.
Calculate
a) the velocities v1, v2 and v3,
b) the pressures p1, p2, p3 and the pressure „p” in the tank above the water surface.
b) p1= 1,1*105 Pa; p2=6*104 Pa; p3= 105 Pa; p=1,04*105 Pa
Bernoulli’s law:
p1 
Continuity equation :
 2


v1  p2  v22  p3  v32
2
2
2
v1  A1  v2  A2  v3  A3  v1 
a)
v2  A2
A1
p

 


 2  2

p
2
p1  v1  v2  p2  p1  p2  v12  v22 
2
2
2
2
2
2 p
2p
v
 v12  v22 
 v22  v12


2
1

A22 
 A2   1 A 
2p
v 2  A2
 v22  2 2 2  v22 1  22  


A1
 A1 
2p
2p
1
2p

v22 



 v2 
2
A22
A22



A
2
1 2
1  2  1  2 
A1
A1
A1 

2
1
2p
 A22 
 1  2 
 A1 
a) v 1=2,58 m/s; v2=10,33 m/s; v3=5,17 m/s
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b)
I.:
pa  p3
II.:
p2 
III.:
p  gh  p3 
 2



v2  p3  v32  p2  p3  v32  v22
2
2
2
2
 2

v3  p  p3  v32  gh
2
2
b) p1= 1,1*105 Pa; p2=6*104 Pa; p3= 105 Pa; p=1,04*105 Pa
Laminar flow of real fluids
Newton’s law of friction
F    A 
Ns
m2
F
Viscosity (kinematic):
Viscosity (dynamic):
 
v
h
=/
 Pa  s
Viscosity depends on:
• Quality of material
• Concentration
• Temperature (↑temp , η ↓)
• Pressure
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=
∙
∙
A Newtonian fluid with a dynamic viscosity of 0.41 Pas and a density of 820 kg/m3 flows
through a 25 mm diameter pipe with a velocity of 2.4 m/s. Is this flow laminar or
turbulent?
R = (2.4*820*12.5*10-3) / 0.41 = 60
Laminar flow
When water is running in a round tube of radius 3 cm at a flow velocity of 2.2 m/s, is this
flow laminar or turbulent? Assume that the kinematic viscosity of water is 9.11*10-7 m2/s.
∙
 =  /  =
R = (2.2*0.03) / 9.11*10-7 = 72448
Turbulent flow
Osborne Reynolds (1842-1912)
The world's longest continuously running laboratory experiment
(Thomas Parnell, University of Queensland, 1927)
The pitch has a viscosity approximately 230
billion (2.3×1011) times that of water.
http://smp.uq.edu.au/content/pitch-drop-experiment
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Circulation
The blood circulation is maintained by a
pressure difference called blood pressure.
The origine of this pressure difference is the pump function of the heart.
The HAGEN-POISEUILLE law in a tube of circular cross section:
R 4 p
Q
,
8 l
If the radius of a tube decreases a greater pressure difference is required
to maintain the previous flow rate.
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ANEURYSM, the devil’s circle.
A positive feedback.
Increased diameter of a weak part of the blood vessel
A1
V1
A2
p1
A2 >A1 (continuity
equation)
V2 < V1
A increases
V2
(Bernoulli’s
law)
V1
p1
A1
p2
p2 > p1
Positiv feedback
p increases
v decreases
Continuity equation
v  A  constant
Bernoulli’s law
p+
1
2
 v2  constant
Blood composition I.
Blood cells:
- red blood cells, also called RBCs or
erythrocytes (4-5 million/ 1 mm³ of blood,
diameter approx. 7-8 μm, thickness 2-3 μm).
- white blood cells, also called leukocytes
(4000-10000/ 1 mm³ of blood,
granulocytes, monocytes, lymphocytes).
- platelets, also called thrombocytes (150400 thousand/ 1 mm³ of blood).
Hematocrit (hct, ) is the proportion of blood
volume that is occupied by red blood cells.
Normalvalue: 0.4-0.5.
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Blood composition II.
Blood plasma:
- Approx. 90% watercontent.
- Mineral ions (Na+, K+, Ca2+,Cl-,HCO3-)
- Organic molecules (glucose, aminoacids,
carbamide and uric acid)
- Plasma proteins:
albumins
globulins
fibrinogen
Blood serum is blood plasma without fibrinogen or the other clotting
factors.
Circulatory System I.
The circulatory system:
• consists of the heart and the blood vessels
(arteries, capillaries and veins)
• closed system (the blood can not escape)
Function:
• carry oxygen and nutrients to tissues.
• carry away the products of metabolism.
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2012. 09. 25.
Circulatory system II.
Vessel
type
Diameter
Total crosssectional
area
(cm2)
Aorta
25 mm
2.5
Artery
4 mm
20
Arteriole
30 µm
40
Capillary
8 µm
2500
Venule
20 µm
250
Vein
5 mm
80
Vena cava
30 mm
8
Ratio of total
Average
Flow rate
blood volume
pressure
(m/s)
(%)
(Hgmm/kPa)
100/13
15
0.33
96/12.7
85->30/
11.3->4
5
30->10/
4->1.3
0.0003
10/1.3
59
5/0.66
0.006
0/0
0.22
Physical parameters in different parts
of the circulatory system
Flow rate
Tot al c rosssectiona l area
Pressure
Aorta
Arteries Arterioles Capillaries
Veins
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Cardiac biophysics
The heart muscle
•
•
•
•
•
„brick” shaped cells (20 µm X 100 µm)
Usually contains 1 central nucleus
Striated
Contains contractile proteins (actin & myosin)
End to end junctions (intercalated disc: electric
synapse)  fast propagation of the action
potential from cell to cell
• Excitability: pacemaker function, automacy

(  nerves (skeletal muscle))
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Pulmonary and systemic circulation
(functional and structural separation)
Pulmonary circulation:
• Heart-lung
(right ventricule – lung – left atrium)
• O2 uptake from the lung
• low pressure
Systemic circulation:
• Heart - body
(left ventricule – body – right atrium)
• O2 to the body
• High pressure
Structure of the human heart
Aorta
A. pulmonalis
Left atrium
Bulbus
aortae
Aorta
valve
Mitral
(bicuspid)
valve
Right atrium
Tricuspid
valve
Left ventricle
Right ventricle
Septum
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Cardiac cycle
Systole (contraction)
•Isovolumetric contraction
• Ejection
0.3 s
Diastole (relaxation)
• Isovolumetric relaxation
• Rapid ventricular filling
• Diastasis ( occurs just before contraction and
0.8 s
(HR:72/min.)
0.5 s
during which little additional blood enters the
ventricle)
Pressure – volume diagram
Pressure (kPa)
Aortic valve closing
systole
ejection
Aortic valve opens
120 Hgmm = 16 kPa
P=~15 kPa
diastole
isovolumetric
relaxation
Area!
systole
isovolumetric
contraction
~ 10 Hgmm = 1-2 kPa
80
diastole 140
ventricular filling
Volume (ml)
V=140-80=60ml
Work = (15*103) Pa x (60*10-6)m3 = 0.9 J = 900 mJ (/contraction)
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The work of the heart
• Static component = p * ΔV
• Dynamic component = ½ m * v2
Work = p * ΔV + ½ m * v2
~98%
~2%
Work = 15x103 N/m2 * 60x10-5 m3 + ½ 0.07kg * (0.5 m/s)2
= 0.9 + 0.0175
= ~ 0.92 Joule
The static (volumetric) component dominates over the dynamic.
Cardiac performance
Cardiac output: the volume of blood pumped out in
each minute.
stroke volume
(the amount of blood pumped out in
one contraction (~60-70 ml))
Depends on:
• preload
• afterload
• contractility
CO = HR x SV
cardiac output expressed in l/min
(normal ~5 l/min)
the number of beats per minute (~70-80/min.)
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Cardiac preload
• the load to which the cardiac muscle is
subjected before shortening.
• the initial stretching of the cardiac myocytes
before the contraction.
• altered end-diastolic pressure and volume.
 preload  sarcomere length 
Sarcomere length – tension relationship
 preload  sarcomere length 
 sarcomere length  tension, force ?
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Sarcomere length – isometric tension
Myosin
Actin filaments
Gordon AM, Huxley AF, Julian FJ. The variation in isometric tension with sarcomere length in vertebrate muscle fibres. J Physiol. 1966 May;184(1):170-92.
Force development during muscle
contraction
Tension or force
The final, resultant force
passive force
active force
Muscle length
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Measuring the CO
• Non-invasive (do not enter the body (circulatory system))
– Transoesophageal echocardiography
– 2D echocardiography (Doppler US)
– MRI
– Arterial pulse contour analysis (based on following the
pressure pulsation)
• Invasive (a part of the body is entered, as by puncture or
incision)
– The Fick principle
– The dilution technique
The Fick principle
the volume of blood
flowing through an
organ in a minute
Q
M
VA
the number of moles of a
substance added to the blood
by an organ in one minute
the venous and
arterial
concentrations of
that substance.
To measure the blood flow through an organ that adds substances to,
or removes substances from, the blood.
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2012. 09. 25.
Determination of stroke volume
according to Fick's principle
a.
The quantity of O2 taken up during one ventilation cycle (inspiration +
expiration) is equivalent to the quantity of O2 used for the oxygenisation
of blood during this period.
b.
Inspired air contains 21% O2. Expired air contains 16% O2.
The difference is 5%.
c.
Since the average volume of one inspiration is 500 ml, 25 ml O2 was
absorbed in the blood.
d.
The O2 contents of arterial and venous blood are 20% and 12%,
respectively. The difference is 8%. That is, 8% of volume of blood
flowing through the lungs during one ventilation cycle (x) is 25 ml.
Therefore, x=(100/8)*25=312.5 ml.
e.
Since there are ~4 cardiac cycles for each ventilation cycle, the stroke
volume is ~ 78 ml.
Dilution technique
cc.
• Dye dilution
— A known amount of dye (indocyanine green, lithium) is injected
into the pulmonary artery
— its concentration is measured at the periphery.
— CO can be calculated from the injected dose, the under curve
area and its duration (Short duration  high CO).
time
• Thermodilution
— Small amount of cold saline (5-10 ml ) injected through the
port of a pulmonary artery catheter.
— Temperature changes are measured by a distal thermistor.
(e.g.: PiCCO Monitoring)
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2012. 09. 25.
PiCCO Monitoring
Pulse Contour Cardiac Output
• A combination of transpulmonary thermodilution and
arterial pulse contour analysis.
• Able to
a) assess cardiac function
b) assess volume status
c) evaluate treatment e.g. inotropes (an agent that
alters the force or energy of muscular contractions)
•
Measuring the CO with PiCCO
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2012. 09. 25.
The End!
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