fluid mechanics: principles and medical applications 1. · 2019-10-11 · fluid mechanics:...
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Fluid mechanics: principles and medical applications 1.
Ferenc PetákProfessor, head
Department of Medical Physics and Informatics
University of Szeged2019
Medical physics 1.
Medical aspects of fluid mechanics
• Water takes about ~2/3 of human body• ~1/3 extracellular, continuously
flowing• Blood circulation
• Arterial system• capillaries• Venous system
• Lymphatic flow• Flow of other body fluids
• aqueous humour, urine, gall liquid, etc.
• ~10 000 liter gas cyclic flowing in the lungs ~20 000x
• Upper, conducting and peripheral airways
Lecture contentMedical fluid mechanics 11. What states of matter are encountered in the human body, and how
can we describe their properties?• 3 states of matter• Definitions, units and measurements of basic quantities: pressure, volume,
density, volumetric flow rate, volumetric flux
2. What are the basic laws for fluid flow in vital processes?• How the blood flow rate changes in the vasculature?
• Continuity equation• What are the pressure-consequences of dynamic blood/air flow in?
• Bernoulli-law
3. What are the medical aspects of laminar and turbulent fluid flow?• Laminar and turbulent flow in the vascular and pulmonary systems• Blood pressure measurement concepts (Riva-Rocci and Korotkoff methods,
oscillometry approach)
1. Three states of matter
Solid Liquid Gas
Cohesion forces
Strong forces, ordered
arrangement
Attracting, small-distant strong
molecular forces
No intermolecular interactions(ideal gas)
Movement of atoms, molecules
Vibration Moving around, mobile structure
Continuous, random (Brownian motion)
Shape, space-filling Defined shape
Conforms to the shape of its container
Move freely in the entire container
Compressibility Close and strong repelling forces -incompressible Compressible
Fluids
1. Basic quantities, units, measurements Pressure (P): force (F) expressed on a unit area (A)P:= F/A [N/m2 = Pa] [Tor] [mmHG] [cmH2O] [PSI]Measurement, piezo resistive transducer: electrical resistance of a metal or semiconductor changes with deformation (Ohm’s law)
Volume (V): space occupation [m3] [Liter]Solid, liquidMeasurement: direct, water displacement GasMeasurement : direct (bell spirometer), indirect (integration of gas flow)
𝑅 𝜌𝑙𝐴
: specific resistance
Density (): mass for unit volume = m / V [kg/m3] [g/cm3]
Volumetric flow rate (flow rate): volume of flowing fluid per unit timeI: = dV / dt [m3/s] [l/s] [l/perc]Measurement:• Thermodilution (cardiac output VA)• US Doppler (blood flow in large vessels)• Pneumotachography, turbine, US (gas flow in airways)In a pipe:I = dV/dt = As/dt = Avdt/dt = Av
Volumetric flux rate (flux): Volumetric flow rate per unit areaJ := dI /dA [m/s] [l/s/m2] [l/min/m2]
1. Basic quantities, units, measurements
Lecture contentMedical fluid mechanics 11. What states of matter are encountered in the human body, and how
can we describe their properties?• 3 states of matter• Definitions, units and measurements of basic quantities: pressure, volume,
density, volumetric flow rate, volumetric flux
2. What are the basic laws for fluid flow in vital processes?• How the blood flow rate changes in the vasculature?
• Continuity equation• What are the pressure-consequences of dynamic blood/air flow in?
• Bernoulli-law
3. What are the medical aspects of laminar and turbulent fluid flow?• Laminar and turbulent flow in the vascular and pulmonary systems• Blood pressure measurement concepts (Riva-Rocci and Korotkoff methods,
oscillometry approach)
2. Basic laws for fluid flowContinuity equationConservation of mass for stationery flow:
m1 = 1V1 = 1A1x1 = 1A1v1tm2 = 2V2 = 2A2x2 = 2A2v2tm1 = m21A1v1= 2A2v2
A1v1= A2v2
constant:
• Volumetric flow rate is constant at any site in the tube (I = Av)• Volume flowing per unit time is the same at any cross-section of the tube
V1 x1
2. Continuity equation – medical aspects
Vessel Diameter(cm)
Length(cm)
Number of branches
ATotal(cm2)
v(cm/s)
aorta 2,4 40 1 4,5 23
arteries 0,4 15 160 20 5
arterioles 0,003 0,2 5,7ꞏ107 400 0,25
capillaries 0,0007 0,07 1,2ꞏ1010 4500 0,022
venules 0,002 0,2 1,3ꞏ109 4000 0,025
veins 0,5 15 200 40 2,5venae cavae 3,4 40 2 18 6
Medical Biophysics. Sándor, Damjanovich, Judit, Fidy, János, Szöllősi (2007)Medicina Könyvkiadó Zrt.
Blood flow rate inversely proportional to the total cross sectional area of the vessels
• Cross-sectional area is the smallest in aorta and vena cava
• Blood flow is much smaller in the capillaries (~1000x) than in the aorta
Gas exchange
• Cross sectional area is decreasing in the venous system
blood flow increases
2. Continuity equation – medical aspects
Conservation of energy for stationary fluid flow
𝐸 , ∆𝑚𝑔ℎ
𝐸 ,12 ∆𝑚𝑣
𝐸 , ∆𝑚𝑔ℎ
𝐸 ,12 ∆𝑚𝑣
Potential energy:
Kinetic energy:
2. Basic laws for fluid flow - Bernoulli-law
Δm mass flows between sites (1) és (2) in time Δt• Height changes between h1 to h2 change in potential energy• Velocity changes (continuity) change in kinetic energy
Work done by the fluid at (1) and (2):
𝑊 𝐹 𝑠 𝑝 𝐴 𝑣 ∆𝑡 𝑝 ∆𝑉 𝑝 ∆
𝑊 𝐹 𝑠 𝑝 𝐴 𝑣 ∆𝑡 𝑝 ∆𝑉 𝑝 ∆
𝑊 𝑊 𝑊∆𝑚𝜌 𝑝 𝑝
V1 = V2m1 = m2
F and s opposite directions
V
V
2. Basic laws for fluid flow - Bernoulli-lawConservation of energy for stationary fluid flow
Conservation of energy: 𝑊 ∆𝐸 ∆𝐸 ∆𝐸∆ 𝑝 𝑝 ∆𝑚𝑔 ℎ ℎ ∆𝑚 𝑣 𝑣 · 𝜌
𝑝 𝜌𝑔ℎ12 𝜌𝑣 𝑝 𝜌𝑔ℎ
12 𝜌𝑣
Bernoulli-law
𝑝 𝜌𝑔ℎ12 𝜌𝑣 constant
2. Basic laws for fluid flow - Bernoulli-lawConservation of energy for stationary fluid flow
𝒑 𝝆𝒈𝒉𝟏𝟐 𝝆𝒗𝟐 constant
Staticpressure
Hydrostaticpressure
Dynamicpressure
Total pressure
Staticpressure
Hydrostaticcpressure
Dynamicpressure+ +
Daniel Bernoulli1700-1782
=
2. Basic laws for fluid flow - Bernoulli-law
2. Bernoulli-law - applications
𝒑 𝝆𝒈𝒉𝟏𝟐 𝝆𝒗𝟐 c
https://opentextbc.ca/physicstestbook2/chapter/bernoullis‐equation/
Total pressure
Staticpressure
Hydrostaticpressure
Dynamicpressure+ + =
• Airplane wing profile• Sail boat• Downforce (F1 car)• Bunsen burner• Vaporizer• Medical suction
Venturimask
Medicalvacuum
Pitot-tubespirometer
2. Bernoulli-law - applications
𝒑 𝝆𝒈𝒉𝟏𝟐 𝝆𝒗𝟐 c Total
pressureStatic
pressureHydrostatic
pressureDynamicpressure+ + =
Venturi-effect
2. Bernoulli-law - applications
𝒑 𝝆𝒈𝒉𝟏𝟐 𝝆𝒗𝟐 c Total
pressureStatic
pressureHydrostatic
pressureDynamicpressure+ + =
Dynamic airway collapse:• forced expiration• cough
https://www.mednote.dk/index.php/Mechanics_of_breathing
2. Bernoulli-law - applications
𝒑 𝝆𝒈𝒉𝟏𝟐 𝝆𝒗𝟐 c Total
pressureStatic
pressureHydrostatic
pressureDynamicpressure+ + =
A. Fluid mechanics of a stenosis. The pressure drop across a stenosis can be predicted by the Bernoulli equation. It is inversely related to the minimum stenosis cross-sectional area and varies with the square of the flow rate as stenosis severity increases.
Stenosis
Duncker DJ et al. Prog Cardiovasc Dis. 2015 ; 57(5): 409–422
Aorta aneurism
Blood flow pattern with pressure distribution in abdominal aortic aneurysm (AAA), (a) without stent graft (SG), (b) with stent graft (SG)
Mohammad NF et al. Proc. Intern Conf. ApplDesign in Mech Eng
2. Bernoulli-law - applications
𝒑 𝝆𝒈𝒉𝟏𝟐 𝝆𝒗𝟐 c Total
pressureStatic
pressureHydrostatic
pressureDynamicpressure+ + =
Lecture contentMedical fluid mechanics 11. What states of matter are encountered in the human body, and how
can we describe their properties?• 3 states of matter• Definitions, units and measurements of basic quantities: pressure, volume,
density, volumetric flow rate, volumetric flux
2. What are the basic laws for fluid flow in vital processes?• How the blood flow rate changes in the vasculature?
• Continuity equation• What are the pressure-consequences of dynamic blood/air flow in?
• Bernoulli-law
3. What are the medical aspects of laminar and turbulent fluid flow?• Laminar and turbulent flow in the vascular and pulmonary systems• Blood pressure measurement concepts (Riva-Rocci and Korotkoff methods,
oscillometry approach)
3. Laminar/turbulent flowThe Reynolds number predicts if a flow is laminar or turbulent
𝑅𝑒𝐼𝑛𝑒𝑟𝑡𝑖𝑎𝑙 𝑓𝑜𝑟𝑐𝑒𝑠𝑉𝑖𝑠𝑐𝑜𝑢𝑠 𝑓𝑜𝑟𝑐𝑒𝑠
𝑅𝑒𝑣𝐷𝜌
𝑣𝐷𝜗
v: velocity D: diameter: density: dynamic viscosity𝜗: kinematic viscosity
𝑚𝑠 𝑚 𝑘𝑔
𝑚𝑁𝑠𝑚
𝑚𝑠 𝑚 𝑘𝑔
𝑚𝑘𝑔𝑚𝑠𝑠 𝑚
1
Rekrit ~ 2300
𝑣2320 · 0.04 𝑃𝑜𝑖𝑠𝑒
1.06 𝑔/𝑐𝑚3 · 2 𝑐𝑚 43 𝑐𝑚/𝑠
Critical flow rate in the aorta:
3. Lamináris/turbulens áramlás
𝑅𝑒𝐼𝑛𝑒𝑟𝑡𝑖𝑎𝑙 𝑓𝑜𝑟𝑐𝑒𝑠𝑉𝑖𝑠𝑐𝑜𝑢𝑠 𝑓𝑜𝑟𝑐𝑒𝑠
𝑅𝑒𝑣𝐷𝜌
𝑣𝐷𝜗
• Heart valves• Ascending aorta• Stenosis• Bifurcations• Aneurisms• Larynx• Bronchoconstriction
Re 2300 Laminar flow Re 2300 Turbulent flow
Physiological:• Most of the blood vessels• Airways
The Reynolds number predicts if a flow is laminar or turbulent
Auscultation method
Nyikolaj SzergejevicsKorotkov, 1905
3. Application – blood pressure measurement
Scipione Riva-Rocci1896
• Inflatable narrow cuff• Return of pulse by palpation• Estimation of MAP
• Inflatable wide cuff• Return of pulse by
auscultation• Estimation of systolic and
diastolic blood pressure
Theory for auscultation method3. Application – blood pressure measurement
Origin of Korotkoff-sounds
cavitationwall detachment turbulence
+ other theories and combinations
3. Application – blood pressure measurement
The oscillometry approach
Liu, J et al. Annals of Biomedical Engineering 41(3), 2012
Analysis of the oscillatory component of cuff pressure• Mean arterial pressure (MAP):
• Maximum of pulse amplitude• Systolic and diastolic pressures:
• Empirical assessments
3. Application – blood pressure measurement