school of biomechanic nsaaaa

1

Biomechanics of Blood Vessels

Prof.dr. Mirko A. Rosi}

The cardiovascular system is composed of:

-pump (the heart)

-a series of distributing and collecting

tubes (blood vessels)

-an extensive systems of thin vessels that

permit rapid exchange (molecular transport)

between the tissues and vascular channels

(capillaries)

Blood from vena cava, entering the right ventricle via the right atrium is pumped through the pulmonary

arterial system. The blood than passes through the lung capillaries. The oxygen-reach blood returns via the

pulmonary veins to the left atrium, where it is pumped from the ventricle to the periphery, thus completing

the cycle.

2

The heart as a pump actually consists of two

pumps in series:

-one pump propels blood through the lungs for

exchange of O2 and CO2 (the pulmonary

circulation)

-the other pump propels the blood to all other

tissues of the body (systemic circulation)

The flow of blood through the heart is

unidirectional – achieved by the appropriate

arrangement of flap valves.

Although the cardiac output is intermittent, continuous flow to the body tissues occurs by distention

of the aorta and its branches during ventricular contraction (systole), and by elastic recoil of the

walls of the large arteries with forward propulsion of the blood during ventricular relaxation (diastole).

Blood moves rapidly through the aorta and its branches, which narrow and their walls become thinner as they

approach the periphery.

The aorta is predominantly elastic structure, but the peripheral arteries become more muscular until at the

arterioles muscular layer predominates.

3

In the large arteries frictional resistance is relatively small and pressure are only slightly less than in the aorta.

The small arteries offer moderate resistance to blood flow. This resistance reaches a maximal level in the

arterioles, which are sometimes referred to as stopcocks of the vascular system

The pressure drop is greatest across the terminal segment of the small arteries and arterioles.

Adjustment in the degree of contraction of the circular muscle of these small vessel permits regulation of tissue

blood flow and aids in the control of arterial blood pressure.

In addition to the reduction in pressure along the arterioles, there is a change from a pulsatile to a steady flow

The pulsatile arterial blood flow caused

by the intermitent ejection of blood from

the heart, is damped at the capillary

level by a combination of two factors:

- distensibility of the large

arteries

-frictional resistance in the

small arteries and arterioles

Hiperthyroidism – Graves’ disease – elevated basal metabolism associated with arteriolar vasodilatation -

reduction in arteriolar resistance diminishes the damping effect on the pulsatile pressure and is manifested as

pulsatile flow in the capilaries, as observed in the finger nailbed of patients with this ailment

4

In the large arteries frictional resistance is relatively small and pressure are only slightly less than in the aorta.

The small arteries offer moderate resistance to blood flow. This resistance reaches a maximal level in the

arterioles, which are sometimes referred to as stopcocks of the vascular system

The pressure drop is greatest across the terminal segment of the small arteries and arterioles.

Adjustment in the degree of contraction of the circular muscle of these small vessel permits regulation of tissue

blood flow and aids in the control of arterial blood pressure.

Many capillaries arise from each arteriole. The total cross-sectional area of the capillary bed is very large,

despite the fact that the cross-sectional area of each individual capillary is much less than that of arteriole.

As a result, blood flow velocity becomes quite slow in the capillaries, analogous to the decrease in velocity

of flow in the wide regions of a river.

Because capillaries consist of short tubules

with walls that are only one cell thick and

because flow velocity is low, conditions in

capillaries are ideal for the exchange of

diffusible substances – molecular transport –

between blood and tissue.

Note that the velocity of blood flow and the cross-sectional area at each level of the

vasculature are essentially mirror images

5

Between the aorta and the capillaries the number of vessels increases about 3 billion-fold, and the total

cross-sectional areas increases about 500-fold.

The volume of blood in the systemic circulation is greatest in the veins – 67%. In contrast, blood volume in

the pulmonary vascular bed is abot equally divided among the arterial, capillary and venous vessels.

The cross-sectional area of the venae cavae is larger than that of the aorta. Therefore, the velocity of flow is

slower in the venae cavae than that in the aorta

In the normal intact circulation the total volume of

blood is constant, and an increase in the volume of

blood in one area must be accompained by a decrease

in another.

However, the distribution of the circulating blood to

the different regions of the body is determined by the

the cardiac output and by the contractile state of the

blood vessels of these regions.

The circulatory system is composed of conduits

arranged in series and in parallel. This arrangement

has important implications in terms of RESISTANCE

FLOW and PRESSURE in the blood vessels

6

The definition of precise mathematical terms for the pulsatile blood flow through the cardiovascular

system is fraught with problems:

- the heart is complicated pump and many physical and chemical factors affect its behavior

- the blood vessels are multibranched, and their elasticity and various regulatory mechanisms

allows continuous variation in their dimensions.

- the blood itself is not a simple, homogeneous solution but insted a complex suspension of

cells, platelets and lipid globules dispersed in a colloidal solution of proteins.

7

Despite complexity it is possible to gain insight into the dynamics of the cardiovascular system

by applying the elementary principles of fluid mechanics as they pertain to simple hydraulic system

Jean Louis Marie Poiseuille

1799-1869Georg Ohm

1789-1854

Isaac Newton

1643-1727

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velocity – sometimes called “linear velocity” refers to the rate of displacement of fluid with respect to

time and it is expressed in units of distance per unit of time ( cm/sec )

In a tube with varying cross-sectional dimensions, velocity – V, flow – Q and cross-sectional area – A are

related by equation:

V = Q / A

For any given flow, the ratio of the velocity depends only on the inverse ratio of the respective areas that is:

Va / Vb = Ab / Aa

Q = 10 ml/sec

in sections a, b and c

V = Q / A

Velocity of blood flow and the cross-sectional area at each level of the vasculature are

essentially mirror images

9

In the specific portion of a hydraulic system in which the total energy remains virtually constant, changes

in velocity may be accompanied by appreciable changes in measured pressure.

Pressure probe insert tangentially (P2,4,6) to the direction of flow – lateral or static pressure (Ps),

if the opening of the probes face upstream – total pressure =Ps + Pd (P1,3,5) where Pd = dynamic pressure

component that is affected by the kinetic energy of the flowing fluid and may be calculated from the equation:

Pd = qV2 / 2

q = density of the fluid

If the midpoint of segments A,B, and C are at he same hydrostatic level, ten the corresponding total pressure

(P1,P3,and P5) will be equal.

Because of differences in cross-sectional area along the system, the concomitant velocity changes alter the Pd

Two general conclusions:

1. As velocity decreases, the dynamic component (which is affected by the kinetic energy of

the flowing fluid) becomes a less significant component of the total pressure.

2. In narrowed section of a tube, the dynamic component (Pd) increases significantly,

because the flow velocity is associated with a large kinetic energy.

10

The peak velocity of flow in the ascending aorta is about (150cm/sec). Because the Pd is significant factor

of Ptot, the measured pressure may vary significantly, depending on he orientation of he pressure probe.

In the descending aorta the peak velocity is substantially less than that in the ascending aorta and even

lesser velocity have been recorded in still more distal arterial sites.

In most arterial locations, the Pd will be a negligible fraction of the Ptot, and the orientation of the probe

will not significantly influence the measured pressure.

At the site of constriction, high flow velocity is associated with a large kinetic energy, and therefore

the Pd may increase significantly. Hence, the Ps would be reduced correspondingly.

Two pressure transducers inserted into the left heart of a patient with aortic stenosis. The transdicers were

located on the same catheter and were 5 cm apart.

A. When booth transducers were well within the left ventricle they both recorded the same pressure.

11

B. When the proximal transducer was positioned in the aortic valve orifice, the Ps recorded during ejection

was much less than that recorded by transducer in the ventricular cavity (much greater V in the narrowed

valve orifice – conversion some potential energy to kinetic energy)

C. When proximal transducer was in the aorta he pressure difference was even more pronounced

because substantial energy was lost through friction (viscosity) as blood flowed eapidly through the

narrow orifice

The reduction of Ps in the region of the narrowed aortic valve orifice may influence the coronary blood flow in

patients with aortic stenosis. The intial segments of coronary blood vessels are thus oriented at right angles to

the direction of blood flow through the aortic valves. Therefore the Ps is that component of Ptot that propels the

blood through the two major coronary arteries.

Durin the ejection phase of the cardiac cycle the Ps is diminished by the conversion of potential energy to

kinetic energy.

The pronounced drop in Ps during cardiac ejection may contribute to the tendency for patients with

severe aortic stenosis to experience angina pectoris which can lead to sudden death !!!

12

The most fundamental low that governs the flow of fluids through cylindrical tubes was derived

empirically by Poiseuille.

This low applies only to steady, laminar flow of newtonian fluids through cylindrical tubes

steady flow = absence of variations of flow in time

(i.e., a nonpulsatile flow)

laminar flow is he type of motion in which the fluid

moves as a series of individual layers, with each lauer

moving at a different velocity from its neighboring leyers.

At the most basic level, Poiseuille’s law describes the flow

of fluids through cylindrical tubes in terms of flow, pressure

the dimension of the tube and viscosity of liquid in the tube.

The flow (Q) of fluid through a tube connecting two reservoirs R1 and R2 is proportional to the

difference between the pressure, Pi at the inflow end and the pressure Po at the outflow end of the tube

13

The flow (Q), of the fluid through a tube is inversely proportional to the length (l) and viscosity ( ) and

is directly proportional to the fourth power of the radius (r)

η

Viscosity has been defined by Newton as the ratio of shear stress to the shear rate of the fluid

The bottom plate (the bottom of a large basin) is stationary, and the upper plate moves along the upper

surface of fluid.

The shear stress “ “ is defined as the ratio of F to A (F is the force applied to the upper plate in the direction of

its motion along the upper surface of the fluid and A is the area of the upper plate that is in contact with the

fluid)

The shear rate is du/dy where “u” is the velocity of a minute fluid element in the direction parallel to the

motion of that fluid element above the bottom stationary plate, and “y” is the distance of that fluid element

above the bottom stationary plate.

For a plate that travels with constant velocity “U” across the surface of homogenous fluid, the velocity

profile of the fluid will be linear. The fluid layer in contact with the upper plate will adhere to it and therefore

will move at the same velocity as the plate.

Each minute element of fluid between the plates will move at a velocity “u” that is proportional to its

distance “y” from the lower plate. Therefore the shear rate will be U/Y

1 dyn sec / cm2 = 1 poise

14

In summary, for the steady, laminar flow of a newtonian fluid through cylindrical tube, the flow (Q)

varies directly as the pressure difference (Pi-Po) and the fourth power of the radius (r) of the tube,

and it varies inversely as the length of the tube and viscosity ( ) of the fluid. Thus the full

statement of Poiseuille’s law is: η

where /8 is the constant of proportionality

In electrical theory. Ohm’s law states that the resistance (R), equals the ratio of voltage drop (E) to

current flow (J). Similarly, in fluid mechanics, the hydraulic resistance (R), may be defined as the

ratio of pressure drop (Pi-Po) to flow (Q).

For the steady, laminar flow of a newtonian fluid through cylindrical tube, the physical components of

hydraulic resistance may be appreciated by rearranging Poiseuille’s law to give the hydraulic resistance

equation:

Thus, when Poiseuille’s law applies, the resistance to flow depends only on the dimensions of the tube

and on characteristics of the fluid.

Because resistance varies inversely as the fourth power of the radius of the tube, the principal determinant

of the resistance to blood flow through any individual vessel within the circulatory system is the caliber

of the vessel. Even small changes in radius alter resistance greatly.

The most important factor that leads to a change in vessel caliber is the contraction of the circular smooth

muscle cells in the vessel wall.

15

The resistance per unit length of vessel ploted against the vessel diameter

The resistance is highest in the capillaries (diameter 7µm) and it diminishes as the vessels increase in

diameter on the arterial and venous sides of the capillaries.

Nevertheless, it is the arterioles, not capillaries, that have the greatest resistance of all the different varieties

of blood vessels that lie in series with one another (there are fare more capillaries than arterioles, and total

resistance across the many capillaries is much less than total resistance across the fewer arterioles). The

resistance in single capillary is much greater than that in single arteriole.

Furthermore, arterioles have a tick coat of circulatory arranged smooth muscle fibers, which can vary the

lumen radius

The capillaries throughout the body are in most instance parallel elements, except for the renal vasculature

(in which the peritubular capillaries are in series with the glomerular capillaries) and the splanhnic

vasculature (in which the intestinal and hepatic capillaries are aligned in series with each other)

The pressure drop across the entire system (Pi-Po), consist of the sum of the pressure drops across each of the

individual resistance

The flow (Q) throufh any given cross-section must equal the flow through any other cross-section

The hydraulic resistance (R1,R2 and R3) arranged in series

It is evident that (from the definition of resistance) for rsistance in series, the total resistance (Rt) of

entire system equals the sum of the individual resistance

16

The hydraulic resistance (R1,R2 and R3) arranged in parallel

For resistance in parallel, the Pi and Po are the same for all tubes. The total flow (Qt) equals the sum of the

flows through the individual parallel element

For resistance in parallel the reciprocal of the Rt equals the sume of the reciprocals of the individual

resistance

A simpler way of stating this relation is to use the term CONDUCTANSE (which can be defined as the

reciprocal of resistance.

For tubes in parallel, the total conductance is the sum of the individual conductance's.

R1=R2=R3

1/Rt=3/R1

Rt = R1/3

The Rt must be less

than that of any

The fluid elements in any given lamina remain in that

lamina as the fluid progresses along the tube.

The velocity at the centre of the stream is maximal,

the longitudinal velocity profile is that of a paraboloid

In turbulent flow the elements of the fluid move

irregularly in axial, radial, and circumferential

directions. Vortices frequently develop.

Whether turbulent or laminar flow will exist in a tube may be predicted on the basis of a dimensionless number

called Reynold’s number (Nr).

q = density of the fluid; D = the tube diameter; v = mean velocity; = viscosity

>2000 – laminar flow < 3000 turbulent flow

η

17

When turbulent flow exist within the cardiovascular system

it may be detected during a physical examination as murmur

In severe anemia, functional murmurs (not caused by structural

abnormalities) are frequently detectable (reduced viscosity

and increased velocity – high cardiac output)

Thrombi are much more likely to develop in turbulent flow

(valvular heart disease, artificial valves). The thrombi may be

dislodged and occlude a crucial blood vessels.

A considerably greater pressure is required to force a given flow of fluid through the same tube when

the flow is turbulent than when it is laminar. Hence, to produce a given flow, a pump such as the heart

must do considerably more work if turbulence develops.

An external force was applied to a plate floating

on the surface of a liquid in a large basin

= (du/dy) η

Shear stress equals the product of the viscosity

and shear rate

Hence, the greater the rate of flow, the greater

is the shear stress (i.e., the viscous drag) that

the liquid exerts on the walls of the container in

which it flows.

For the precisely the same reasons, the rapidly

flowing blood in a large artery tends to pull the

endothelial lining of the artery along with it. This

force, the viscous drag, is proportional to the

shear rate of the layers of blood near the wall

In patients with hypertension, the subentothelial layers of vessels tend to degenerate locally, and small

regions of the endothelium may lose their normal support. The viscous drag on the arterial wall may cause

a tear between a normal supported and an unsupported region of endothelial lining. (dissecting aneurysm)

It occurs most often in the proximal portions of the aorta (high velocity of blood flow).

The at the vessels wall also influences many other functions:

- permeability of the vascular walls to large molecules

- biosynthetic activity of the endothelial cells

- integrity of the formed elements in the blood

- coagulation of the blood

An increase in shear stress is also an effective stimulus for the release of NO from the vascular endothelial

cells; NO is a very potent vasodilatator.

18

The various endothelial and non-endothelial factors may directly or indirectly cause vasodilatation.

The crescent-shaped thickenings proximal to the

capillary beds represent the arterioles

Shematic diagram of the vessels composing

the circulatory system

The arterioles, the terminal components of arterial

systems, are high-resistance vessels that regulate the

distribution of flow to the various capillary beds.

Because of their elasticity, the aorta, the pulmonary

artery, and their major branches form a system of

channels capable of handling considerable volume.

These two features of the arterial systems – its

elastic conduits and high resistance terminals – are

also shared by fluid systems called hydraulic filters

which tend to dampen fluctuations in flow.

The main advantage of hydraulic filtering in the

arterial system is that it converts the intermittent

output of the heart to a steady flow through the

capillaries.

19

Because the heart pumps intermittently, the entire stroke volume is discharged into the arterial system

during systole.

During diastole, the elastic recoil of the arterial walls converts potential energy into capillary blood flow.

If the arterial walls were rigid, capillary flow would not occur during diastole.

Hydraulic filtering minimizes the workload of the heart. More work is required to pump a given flow

intermittently than steadily. The more effective the hydraulic filtering, the less work is required.

Relationships between pressure and flow for tree hydraulic systems in which the overall flow and

resistances are equal

20

Relationship between myocardial oxygen consumption and stroke volume in an anesthetized dog whose

cardiac output could be pumped by the left ventricle either through the aorta or through a stiff plastic

tube to the peripheral arteries.

The oxygen consumption reflect the energy expenditure in the body.

Rigid conduits in a hydraulic system create the need for more energy to pump fluid through the system.

Pressure-volume relationship for aortas

obtained at autopsy from human in

different age groups.

A good way to appreciate the elastic properties of

the arterial wall is to consider the static pressure –

volume relationship for the aorta.

The aortic compliance at any point on the curve is

represented by the slope, dV/dP at that point.

As a people age, the pressure-volume curves of

their arterial system shift downward, and the

slopes of these curves diminish. Thus, for any

pressure above about 80 mmHg, the compliance

decreases with age.

This change in compliance is a manifestation of

the increased rigidity (atherosclerosis) of the

system caused by progressive changes in the

collagen and elastin contents of the arterial walls

21

Pulsatile changes in diameter, measured ultrasonically, in 22-year-old and 63-year-old man

The ultrasound imaging technique show that the increase in diameter of the aorta produced by

each cardiac contraction is much less in elderly persons than in young persons

Effects of age on the elastic modulus (Ep) of the

abdominal aorta in agroup of 61 human subjects

where P is the aortic pulse pressure (i.e., the change in aortic pressure during a cardiac cycle);

D is the mean aortic diameter and D is the maximal change in aortic diameter during the cardiac cycle.

The fractional change in diameter ( D/D) of

the aorta during the cardiac cycle reflects the

change in aortic volume.

Thus Ep is inversely related to compliance,

which is the ratio of V to P.

Consequently, the increase in elastic modulus

and decrease in compliance both reflect the

stiffening of the arterial walls.

Elastic modulus

22

Mirko Rosic, Suzana Pantovic, Vladimir Rankovic,

Zdravko Obradovic, Nenad Filipovic, Milos Kojic

EVALUATION OF DYNAMIC RESPONSE

AND BIOMECHANICAL PROPERTIES OF

ISOLATED BLOOD VESSELS

J. Biochem. Biophys. Methods 70 (2008) 966-972

Figure 1. Schematic diagram of the System for Biomechanical and FunctionalTissue Investigations (1. perfusion fluid; 2. peristaltic pump; 3. AD converter andamplifier; 4. organ bath; 5. metal tube or blood vessel; 6. transducer; 7. water bath;8. oxygen bottle; 9. PC; 10. Camera;11. Two-way tap and H0 and H1, hydrostaticlevels.)

MATERIALS AND METHODS

23

( )−= − 21 1

b xy b e

b1 has units of pressure, and

b2 has units of time-1

Figure 2. A: The pressure-time relation.

T= 1/ b2

For t=5T (t=time) we consider that the

plateau is reached, because in this case:

− −= ≈/ 5 0t Te e ≈ 1y b, and

�Experimentally recorded dependence of pressure on time was fitted

using an exponential mathematical function:

Pressure-time relations


• To define dynamic properties of the blood vessels, we introduced a

parameter of the comparative pressure dynamics (Pd). This parameter is

defined as the integral of the difference between two fitted curves,

normalized with respect to the applied hydrostatic pressure (H):

( ) ( ){ }∞

− − = − − − ∫ 1 1

0

2 21

1 1db ab x b x

P b e b e dxH

b2a is the coefficient of the first curve (control curve) and b2b is the

coefficient of the second curve (test curve).


24

The calculated Pd is the area between the test and control curves (figure right).

Positive value of Pd indicates the shift of the test curve to the left and faster development of maximal pressure (alternate steady state).

Negative value of the Pd shows the shift of the test curve to the right and slower development of maximal pressure.

The solution of previous equation is:

−=

×2 21

2 2

b ad

a b

b bbP

H b b

Figure 2. B: Three exponential curves and the

corresponding positive (Pd+) and negative (Pd-)

values of the parameter Pd.


�Experimentally recorded dependence of blood vessel diameter on time was

fitted using an exponential mathematical function:

( )−= − 21 1

b xy b e

Diameter-time relations


b1 has units of diameter, and

b2 has units of time-1

25

−=

×2 21

2 2

b ak

a b

b bbD

D b b

Positive values of Dk indicate faster development of maximal diameter (alternate steady state), while

negative values indicate slower development of maximal diameter.

The parameter of comparative distensibility Dk

( ) ( ){ }∞

− − = − − − ∫ 1 1

0 0

2 21

1 1kb ab x b x

D b e b e dxD

b2a is the coefficient of the first curve (control curve) and b2b is the coefficient of the second

curve (test curve).


Stress –strain relation

σδ

= tt t

t

RP

σ t

tP

tR

δt

is stress [kPa]

is pressure

is blood vessel radius

is blood vessel wall thickness

ε−

= 0

0

mt mt

m

R R

R

εtmtR

0mR

is circular strain

is blood vessel mean radius

is blood vessel mean radius at the time zero (original mean radius);

and the right lower index t indicates the values at time t.


26

Assuming that the blood vessel wall is incompressible, we can determine the wall thickness for the

current mean radius mtR

δ π = − −

2 0t mt mt

VR R

L

where L - length of the isolated blood vessel and

V0 is the original volume of the blood vessel wall

( )π δ δ= −0 0 0 0 0V R L

with the lower index „0“ indicating the original values


Shear stress evaluation

The wall shear stress (in kPa) can also be calculated, from the following

equation:

µτ

π=

3

4t

t

Q

R

τ tµ

tR

where is shear stress (in kPa)

is viscosity in poise

Is radius (in cm)


Q is flow in ml/s

27

Rat, A.abdominalis b1

b2

5T

KRS (Krebs-Ringer solution)

62.36 ± 0.223.03 ± 0.35 1.66 ± 0.109

KRS+L-arginine (1mM)

62.54 ± 0.263.16 ± 0.426 1.58 ± 0.13


61.37 ± 0.613.36 ± 0.08 1.49 ± 0.017


62.19 ± 0.314.2 ± 0.122 * 1.19 ± 0.017*

Table 1. Effects of increasing molar concentration of L-arginine in rat abdominal

aorta on b1 and b2 coefficients; and on time (s) within which the maximal pressure

is developed (taken as 5T). Values are represented as mean ± S.E.M. Asterisk (*)represent significantly different value. (One-way analysis of variance, p<0.01)

RESULTS

Abdominal aorta of the rat Pd

KRS+L-arginine (1mM) / KRS 0.014 ± 0.001 *



Table 2. Effect of increasing molar concentration of L-arginine in rat abdominal

aorta on the parameter of comparative pressure dynamics (Pd). Values are

represented as mean ± S.E.M. Asterisk (*) represent significantly different value. (One-way analysis of variance, p<0.01)

RESULTS

28

Abdominal aorta of the

ratb1

b2

5T

KRS 2.23±0.0005 6.34±0.2 * 0.789±0.03 *

KRS+L-arginine (10mM) 2.22±0.029 5.3±0.32 * 0.943±0.056*

Table 3. Effects of L-arginine (10mM) in rat abdominal aorta on b1 and b2

coefficients, and on time (s) within which the maximal diameter is developed

(taken as 5T). Values are represented as mean ± S.E.M. Asterisk (*) represent significantly different value. (Student′s t test, p<0.05)

RESULTS

Figure 3. Stress / Strain curves in the presence of L-arginine (10mM) and in the

absence of L-arginine. *Significantly different from the stress value under control

conditions; P<0.05. The coefficients of correlation are marked as R2. Dots

represent experimental results and curves are obtained by polynomial functions.

RESULTS

29

Figure 4. Shear stress/Strain curves in the presence of L-arginine (10mM) and in the

absence of L-arginine. *Significantly different from the shear stress value under control

conditions; P<0.05. The coefficients of correlation are marked as R2. Dots representexperimental results and curves are obtained by polynomial functions.

RESULTS

In summary, our results demonstrate the influence of the test drug (L-

arginine) on dynamic responses and biomechanical properties of the

isolated blood vessel.

First of all, in the presence of the test drug some relaxation (increase in

diameter) of blood vessel occurs, development of pressure is faster after

hydrostatic pressure was applied, while, at the same time, change in

blood vessel diameter is slower.

Although L-arginine relaxes smooth muscle in the blood vessel wall, the

wall becomes more rigid when the hydrostatic pressure was applied.

Furthermore, after the hydrostatic pressure was applied, the shear stress

in the presence of the test drug approaches the values of the shear

stress in the absence of the test drug, besides the fact that the test drug

itself initially increased the blood vessel diameter.

30

I hope he was not

boring too much !

school of biomechanic nsaaaa

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