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