AME 60676 Biofluid & Bioheat Transfer 3. Body thermo-fluid-mechanics.

AME 60676Biofluid & Bioheat Transfer

3. Body thermo-fluid-mechanics

Objectives

• Overview of blood behavior as it flows in blood vessels:– Normal (physiologic) flow conditions– Pathophysiologic conditions– Blood flow past biomedical devices

• Overview of the mechanics and mechanical properties of cardiovascular structures

Outline

1. Blood characteristics2. Viscous behavior3. Pressure-flow relationships for non-

Newtonian fluids4. Blood vessel mechanics5. Cardiac muscle mechanics6. Heart valve mechanics

Heart valve mechanicsCardiac muscle mechanicsBlood vessel mechanics

1. Blood characteristics

Non-Newtonian modelsViscous behaviorBlood characteristics

Blood elements

Plasma ( 55%) cellular elements ( 45%)

water (92%) proteins (7%) others (1%)

albumins (60%) globulins fibrinogen

90% of all proteins

blood

erythrocytes (RBCs)

leukocytes (WBCs)

thrombocytes (platelets)

Heart valve mechanicsCardiac muscle mechanicsBlood vessel mechanicsNon-Newtonian modelsViscous behaviorBlood characteristics

Blood elements

• Erythrocytes (red blood cells)– Involved in transport of O2 and CO2

– Biconcave disc shape (maximizes surface-to-volume ratio)

– Volume: 85 – 90 mm3

– Lifespan: 125 days– Concentration: 5 M/mm3 of whole blood

8 m

1 m2 m


Blood elements

• Erythrocytes (red blood cells)

– Production (erythropoiesis)• in bone marrow• requires hemoglobin and iron

– Destruction• destroyed by macrophages• iron recycled in bone marrow into new hemoglobin


Blood elements

• Leukocytes (white blood cells)– Involved in phagocytosis and immune responses– Phagocytes

• Removal of foreign bodies• Neutrophils, eosinophils, basophils

– Immunocytes• lymphocytes

– Concentration: 4k – 11k/mm3 of whole blood


Blood elements

• Leukocytes (white blood cells)

– Neutrophils• Most abundant leukocyte• Key role in body’s defense against

bacterial invasion• Neutrophil leukocytosis

(> 7500/mm3) results from inflammation, necrosis

Neutrophil and erythrocytes (McGraw-Hill)


Blood elements


– Eosinophils• Role in allergic response, defense

against parasites• Detoxification of foreign proteins

Eosinophil, neutrophil, erythrocytes and platelets

(University of Texas)


Blood elements


– Basophils• Releases histamine in area of

tissue damage– To increase blood flow– To attract other leukocytes

Basophil, erythrocytes and platelets (University of Texas)


Blood elements


– Lymphocytes• Release of antibody molecules• Disposal of antigen• B lymphocytes (20%)

– Produced in bone marrow– Antibody molecules synthesis (humoral

immunity)

• T lymphocytes (80%)– Produced in thymus gland– Attack virus-infected cells or regulate other

immune cells (cell mediated immunity)

Lymphocyte, erythrocytes and platelets (University of Texas)


Blood elements

• Thrombocytes (platelets)– Involved in blood clotting

(formation of mechanical plugs in hemostatic response to injury)

• Adhesion, secretion, aggregation, fusion

– Size: 1-2m in diameter– Normal count: 250,000/mm3 of

whole blood

Platelets and erythrocytes(University of Texas)


2. Viscous behavior

Heart valve mechanicsCardiac muscle mechanicsBlood vessel mechanicsNon-Newtonian modelsBlood characteristicsViscous behavior

Plasma

• Density: 1035 kg/m3

• Dynamic viscosity: 0.0012 kg/(m.s)


Whole blood

• Density: ρ = 1060 kg/m3 (water: 1000 kg/m3)

• Dynamic viscosity: = 0.003 – 0.004 kg/(m.s)(water: 0.001 kg/(m.s))

• Non-Newtonian fluid: : strain rateT: temperatureH: hematocrit (volume percent of blood occupied by formed elements; normal: 40% - 45%)

, ,T H


Whole blood

• Nonlinear behavior at low shear rates

• Existence of a minimum yield stress

Shear stress normalized to plasma viscosity vs. rate of shear (Whitmore, 1968)

Square root of shear stress normalized to plasma viscosity vs. square root of shear rate

(Whitmore, 1968)

• Casson’s fluid behavior:

y cK

1.53 2.0p

yield stressyield stress


Whole blood

• Apparent viscosity:

• Apparent viscosity increases as shear rate decreases

• Asymptotic behavior as >>1

2

2.01.53app p

Apparent viscosity vs. shear rate(Whitmore, 1968)

0.035

1.53 2.0p


Whole blood

• Effect of hematocrit– Blood viscosity increases

with hematocrit

– Blood viscosity decreases with shear rate

– RBC structure (flexible membrane) minimizes viscosity Apparent blood viscosity normalized to

plasma viscosity vs. hematocrit (Whitmore, 1968)


Whole blood

• Effect of plasma protein content– Globulin tends to increase

blood viscosity

– Albumin tends to lower blood viscosity

– Fibrinogen lowers blood viscosity but to a lower extent due to low concentration Apparent blood viscosity vs. shear rate

for different plasma protein contents (Flow properties of blood, 1960)


Whole blood

• Yield stress– Mainly dependent on

fibrinogen content and hematocrit

CF: fibrinogen content (g / 100mL)

H: hematocrit (%)

– Normal value: 0.7 dyne/cm2

– Effect on capillary blood flow

Flow initiated when:

Yield stress vs. fibrinogen content and hematocrit (Merill et al., 1965)

0.1 0.5y FH C

2w y

R p

L


Fahraeus-Lindqvist effect

• Definition: apparent viscosity of blood increases with the increase in vessel diameter

• Migration of the RBCs from vessel wall to center of the vessel (due to BL effect, spinning)

• Existence of 2 flow regions:– cell-free plasma region near the wall– core region near the centerline of the vessel



• Cell-free marginal layer modelvessel wallred blood cells

flow direction

core region cell-free region

rz1 1,

2 2,

R

4

1 2 8

R pQ Q Q

L

1

1

2

1 4 1app

R

From local flow analysis:

In large vessels:

In small vessels: O R

R



• Sigma effect– For small vessel diameters (capillaries), continuum

hypothesis is not valid– Need to discretize the velocity profile in different

concentric layers

R

24

18i

R pQ Q

L R

2

1app R


• More aggregates at lower shear rates

• N cells in a rouleaux will cause a greater disturbance than N individual cells rouleaux increase apparent viscosity

Rouleaux and aggregates

Rouleaux of human red cells(Fung, 1993)


• Under high shear rates:– blood aggregate size ↓– viscosity ↓– RBC deformation becomes more evident– RBCs elongate and line up along flow direction

Red blood cell deformation

RBC deformation under shear stress (Caro, 1978)

t = 10 N.m-2 t = 300 N.m-2


3. Pressure-flow relationships for non-Newtonian fluids

Heart valve mechanicsCardiac muscle mechanicsBlood vessel mechanicsViscous behaviorBlood characteristics Non-Newtonian models

Power law fluid

• n = 1: Newtonian fluid• n < 1: pseudoplastic

(shear-thinning) behavior

• n > 1: dilatant (shear-thickening) behavior

nK K: flow consistency index (Pa.sn)n: behavior index (dimensionless) 0.75 for human whole blood


Power law fluid

• Velocity profile:

• Volume flow rate:

nK K: flow consistency index (Pa.sn)n: behavior index (dimensionless) 0.75 for human whole blood

1 1 1

1 2

n n n

n nn pu r R r

n KL

13

3 1 2

nn R R p

Qn KL

02

RQ r u r dr


Bingham plastic

y K y: yield stressK: viscosity coefficient


• Radius at which yield stress is attained:

2 2

4yp

u r R r R rKL K

2 yc

LR

p

uc

u(r)

Rc


Bingham plastic



44 2 24 1

18 3 3

y yR pQ

KL R p L R p L

0

2 2 c

c

R R

RQ r u r dr r u r dr

uc

u(r)

Rc


Casson’s fluid(preferred whole blood model)



• Radius at which yield stress is attained:

2 2 3 2 3 22 2 2

1 4

4 3 2y y pp

u r R r R r R rK L K K L

2 yc

LR

p


Casson’s fluid(preferred whole blood model)



4 1 24

2

2 2 21 4 161

8 21 3 7y y yR p

QK L R p L p L R p L

0

2 2 c

c

R R

RQ r u r dr r u r dr


Whole blood model: summary

• Non-Newtonian behavior (Casson’s fluid)

• Apparent viscosity is high at low shear rates (RBC aggregation)

• Shear thinning behavior (as shear rate , apparent viscosity )


Hemolysis

• Blood cell damage (destruction or activation)• Depends on:

• Shear stress magnitude• Exposure time

• RBC hemolysis:– Hemoglobin release in

plasma (anemia)• Platelet activation:

– Adherence of activated platelets to subendothelialstructures (thrombus formation)

Shear stress vs. exposure time for hemolysis of RBCs and platelet

destruction


4. Blood vessel mechanics

Heart valve mechanicsCardiac muscle mechanicsNon-Newtonian models Viscous behaviorBlood characteristics Blood vessel mechanics

Blood vessel components

• Elastin fibers: all vessels but capillaries and venules

• Collagen fibers: stiffer but slack

• Smooth muscle

• Endothelial cells


Blood vessel components• Elastin + collagen function: to

maintain a steady tension within the vessels to act against the transmural pressure

• Smooth muscle function: to provide an active tension by means of contraction under physiological control (vascular resistance regulated primarily in smaller arteries and arterioles)

• Endothelial cells: sensors, no mechanical function


Blood vessel behavior

• Structure contribution to material behavior

• Stress-strain behavior– Individual components: linear elastic materials– Individual fiber combination: bilinear curve– Tissue-level: nonlinear behavior

collagen fiberselastin fibers


Blood vessel behavior• Elastic modulus

– Elastin fiber: 106 dynes/cm2

– Collagen fiber: 109 dynes/cm2

– Smooth muscle: depends on muscle state• Typical stress-strain curve for blood vessel:

collagen

elastin

collagen + elastin collagen fiber icollagen fiber i+1

collagen fiber i+2


Material characterization considerations

• Testing protocol requirements:– Preconditioning over several loading-unloading

cycles (compensate for hysteresis)– Physiological environment

• Controlled ionic content, moisture, temperature, …

• Test on native blood vessels:– long segments to neglect end-effects– compensate for effect of smooth muscle

relaxation/contraction



• Assumptions:– Knowledge of p, R, t E

2pRE

t R

2 2 21 1 1 1 2

2 2 2 22 1 2 1

1 1 1p R p R RrE

u urR R R R

Thin-walled tube

t/R < 0.1Thick-walled tube

t/R > 0.1

2

1 22 22 1 2

2inc

R R pE

R R R

2

2p

RE p

R


increasing E


• Assumptions:– Homogeneity, isotropy, compressibility

• Non homogeneous material (different layers, different components)

• Anisotropic material:

information reported in Poisson’s ratio • Incompressible material ( = 0.5)

longitudinalradial circumferential


Residual stresses• Arterial wall tissue is not stress-free at zero

transmural load


Residual stresses

• Magnitude described in terms of opening angle of sector shape

• Residual strains computed by measuring circumferential lengths of intima and adventia in uncut and cut configurations

• Function of:– Anatomical location– Species– Location of radial cut


5. Cardiac muscle mechanics

Heart valve mechanicsBlood vessel mechanicsNon-Newtonian models Viscous behaviorBlood characteristics Cardiac musle mechanics

Material characterization

• Diastolic and systolic contractile properties

• Orthotropic material: force-deformation behavior stiffer along fibers than in transverse direction

intercalated disk (sectioned)

mitochondrianucleus

cardiac muscle cell

contractile fibers

intercalated disk



• Fiber orientation varies along wall thickness

Tests performed on papillary muscles (longitudinal fiber orientation)

intercalated disk (sectioned)

mitochondrianucleus

cardiac muscle cell

contractile fibers

intercalated disk



• Non linear exponential stress-strain behavior• Effects of contractile properties:

– Muscle contraction muscle shortening

shortening velocity

afterload

maximal isometric

force

Vmax



• Non linear exponential stress-strain behavior• Effects of contractile properties:

– Effect of preload

shortening velocity

afterload

Vmax

Increasing preload


6. Heart valve mechanics

Cardiac muscle mechanicsBlood vessel mechanicsNon-Newtonian models Viscous behaviorBlood characteristics Heart valve mechanics

Motivations

• Mechanical forces influence the remodeling of live tissues

• Changes in the mechanical environment result in pathological responses (e.g., valve calcification)

• Mechanical environment is essential for the design of implants and medical devices

• Tissue engineered heart valve


Hemodynamic Environment

Aortic valve Mitral valve

Flow visualization (polymeric valve) Ultrasound Doppler Velocimetry (native valve)


Mechanical EnvironmentAortic valve Mitral valve

aorta

left ventricle

pressuretensile stressbending

stress

Diastolic forces Mitral valve leaflet motion

Fluid shear stress, bending stress, tensile stress,pressure, chordal tension, muscular contraction


flow

aorta

left ventricle

bending stress

tensile stress

Mechanical EnvironmentAortic valve Mitral valve

Systolic forces Mitral valve leaflet motion

Fluid shear stress, bending stress, tensile stress,pressure, chordal tension, muscular contraction


AV Leaflet Structure• Tri-leaflet structure• Tri-layered structure:

– Fibrosa: circumferentially aligned collagen fibers– Spongiosa: loose, watery connective tissue

containing mainly GAG– Ventricularis: radially aligned elastin fibers

intertwined with collagen fibers• Cell population:

– Endothelial cells– Interstitial cells:

• Myofibroblasts: prominent stress fibers and α-SMA• Fibroblasts: synthetic phenotype with synthetic and

secretory organelles• Smooth muscle cells

Aortic valve leaflet structure

Ventricularis

Spongiosa

Fibrosa

left ventricle

aortaendothelial cells

interstitial cells

Dissected and unfolded aortic valve

L R N

coronary ostia

leaflets left ventricle

aorta


AV Leaflet Mechanical Response

• Leaflet tissue behaves as other soft tissues:• Anisotropic: Complex structural characteristics

lead to directional response

• Non-homogeneous: material response varies throughout the leaflet

• Non linear stress-strain relationship

• Viscoelastic: hysteresis, creep, stress relaxation, loading rate dependence



• Unstressed state: disorganized collagen fibers in unstressed state

• Stressed state: elastin and collagen fibers orientation in circumferential direction

Collagen fiber orientation in AV leaflet (small angle scattering) (Billiar and Sacks, 2000)

25

35

45

55

65

Stress-free state p = 4 mmHg



• Anisotropic material

• Greater stiffness in direction of collagen fibers

• Non-linear stress-strain relationship (exponential relationship)

Mean stress-strain response of AV cusps under equibiaxial tension (Billiar and Sacks, 2000)


AV Mechanical Forces

0

0.1

0.2

0.00 0.20 0.40 0.60 0.800 0.2 0.4 0.6 0.80

0.2

0.1

20

19

18

17

16

a)

b)

stra

inle

afle

t le

ng

th (

mm

)

time (s)

0

0.1

0.2

0.00 0.20 0.40 0.60 0.800 0.2 0.4 0.6 0.80

0.2

0.1

2020

1919

1818

1717

1616

a)

b)

stra

inle

afle

t le

ng

th (

mm

)

time (s)

Stretch waveform (Thubrikar, 1990)

0 0.2 0.4 0.6 0.8 1-10

-5

0

5

10

15

aortic surface

wall shear stress

(dyn/cm2)

time (s)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0

20

40

60

80

100

ventricular surface

wall shear stress

(dyn/cm2)

time (s)

Shear stress waveform (Sucosky, 2009)

0

0.1

0.2

0.00 0.20 0.40 0.60 0.80

Time (s)

Str

ain

0 0.2 0.4 0.6 0.80

0.2

0.1

20

19

18

17

16

a)

b)

stra

inle

afle

t le

ng

th (

mm

)

time (s)

0

0.1

0.2

0.00 0.20 0.40 0.60 0.80

Time (s)

Str

ain

0 0.2 0.4 0.6 0.80

0.2

0.1

2020

1919

1818

1717

1616

a)

b)

stra

inle

afle

t le

ng

th (

mm

)

time (s)


AV Leaflet Mechanobiology:Effects of Shear Stress

• Effects of steady shear stress on collagen and GAG synthesis

• Effects of steady shear stress on cathepsin L expression and activity

Ratio of sGAG

content relative to

control

Ratio of collagen

synthesis relative to

control

1.5

1

0.5

1.8

0.9

0

control

Shear stress (dyn/cm2) Shear stress (dyn/cm2)1 9 25 40 80 1 9 25 40 800

control

Cathepsin L activity (gelatin zymography)Cathepsin L expression

(Western blot)

25 kDa

25 kDa

cont

rol

shea

r

Cathepsin gelatin zymography

Cathepsin gelatin zymography + E-64

cont

rol

shea

r

25 kDa

Den

sito

met

ric

fold

uni

ts

(% o

f con

trol

)

control shear

*150

100

50

0

Xing, 2004

Platt et al., 2006


AV Leaflet Mechanobiology:Effects of Shear Stress

Fresh tissue Ventricular surface

Aortic surface

Effect of altered shear stress

Fresh tissue Ventricular surface

Aortic surface

Effect of normal shear stress

Fold

incr

ease

of e

xpre

ssio

n pe

r cel

l vs.

fres

h

Fold

incr

ease

of e

xpre

ssio

n pe

r cel

l vs.

fres

h

0

2

4

6

8 VCAM-1 ICAM-1 BMP-4TGF-beta1

0

2

4

6

8 VCAM-1 ICAM-1BMP-4 TGF-beta1

*

***

Normal condition Altered condition

ventricular surface

aortic surface

Altered condition Normal condition

ventricular surface

aortic surface

Sucosky et al., 2009 Arterioscler Thromb Vasc Biol


AV Leaflet Mechanobiology:Effects of Stretch

• Collagen content increased in presence of cyclic stretch

• Cyclic stretch alters fiber composition

0

10

20

30

40

50

60

70

80

90

100

Fresh Static Stretched

Composition of Collagen Fibers in Aortic Valve

Perc

enta

ge o

f par

ticul

arty

pe o

f col

lage

n fib

er (%

)

Increasing Fiber maturity

Colla

gen

amt (

ug/m

g dr

y tis

sue

wei

ght)

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

Am

ount

(ug

/mg

dry

tissu

e w

eigh

t)

*

* * p<0.05

Fresh Static Stretched Balachandran et al., 2006



Static Stretched

Fresh 70 bpm, 48h 120 bpm, 48h

A

V

A

50 μm

A: aortic surface; V: ventricular surface; blue: nucleus; red: MMP-2

• Stretch increases MMP-2 expression

• Stretch increases MMP-2 activityFresh S-48h S-120h 70b/ 70b/ 120b/ 120b/

48h 120h 48h 120h

72 kDa pro-MMP267 kDa MMP2



120 h static 70 bpm, 120 h 120 bpm, 120 h

48h static 120 bpm 48 h70 bpm 48 h

A

V

A

100 μm

A

V

A

V

A

V

A

• Cyclic stretch induces cell proliferation

A: aortic surface; V: ventricular surface; blue: nucleus; red: proliferating cellsCardiac muscle mechanicsBlood vessel mechanicsNon-Newtonian models Viscous behaviorBlood characteristics Heart valve mechanics

MV Leaflet Structure

• Bi-leaflet structure• Tri-layered structure:

– Atrialis: elastin fibers– Spongiosa: GAG– Ventricularis: collagen fibers

Mitral valve leaflet structure

Ventricularis

Spongiosa

Atrialis

left ventricle

left atrium

endothelial cells

Dissected and unfolded mitral valve

anterior marginal chord

anterior strut chord

commissuralchord

posteriormarginal

chord

posterior intermediatechord

basal posteriorchord

papillary muscle

anterior leaflet

commissureposterior

leaflet

Chord structureelastin layer

inner collagen core

outer endothelial layer


MV Leaflet Mechanical Response

• Unstressed state: disorganized collagen fibers in unstressed state

• Stressed state: elastin and collagen fibers orientation in circumferential direction

Collagen fiber orientation in MV leaflet (small angle scattering) (Sacks et al., 2003)

25

35

45

55

65

radial direction

circumferential direction



• Major principal stretch: radial direction• Early leaflet closure: rapid rise• Valve closure: plateau (collagen fiber locking)• Valve opening: principal stretch drops down to strain free

levels

In vitro assessment of stretch levels on anterior MV leaflet (marker technique)

1

1.1

1.2

1.3

1.4

1.5

0 0.1 0.2 0.3 0.4 0.5

Pri

nci

pal

str

etch

Time (s)

Major principal stretch Minor principal stretch

0.9

1

1.1

1.2

1.3

1.4

1.5

Major Minor

Str

ec

h

Principal stretch

Posterior leaflet

Anterior leaflet

Comparison of the principal stretches on

anterior and posterior MV leaflets



• Early leaflet closure: small p large • Leaflet stiffening: large p small (collagen

fiber locking)

MV5 NNat r=s=0

Areal strain1.0 1.1 1.2 1.3 1.4 1.5 1.6

Tra

ns-

mitr

al p

ress

ure

(m

mH

g)

0

20

40

60

80

100

Physiologic load-strain curve of the MV anterior leaflet

collagen fiber configuration


AME 60676 Biofluid & Bioheat Transfer 3. Body thermo-fluid-mechanics.

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Transcript of AME 60676 Biofluid & Bioheat Transfer 3. Body thermo-fluid-mechanics.