1. Mechanical Properties and Active Remodeling of Blood Vessels
Gross anatomy of systemic and pulmonary circulation
Microscopic structure
Mechanical properties and testing
Wave propagation
Pseudoelasticity
Residual stress
Three-dimensional testing
Arterioles
Capillaries
Veins
Remodeling

2. Anatomy and Structure of Blood Vessels
Gross anatomy of systemic and pulmonary circulation
Systemic blood vessels: aorta and other large elastic arteries, smaller muscular arteries and arterioles, capillaries, small and large veins, vena cavae
Pulmonary blood vessels: arteries, capillaries and veins
Microscopic structure: intima, media and adventitia
intima: mainly endothelial cells and basal lamina, some other stuff
media: boundary is elastic membrane, smooth muscle cell layers divided by elastic lamellae with helical patterns of collagen fibers and elastin fibers, ground substance, vasa vasorum

3. Blood Vessel Microstructure
Microscopic structure: intima, media, adventitia (cont’d)
aorta: high elastin content damps flow oscillations
arterioles: high smooth muscle content for regulating resistance
capillaries: intima only
veins: high collagen content and thin walls, stiff but collapsible, appear soft because they operate at low end of stress-strain curve, contain >70% blood
Vascular forces usually born by elastin and collagen fiber; vessel elasticity affects capacitance, resistance, and pulse propagation and is influenced by local composition, h/r, muscle tone and surrounding environment; endothelium has many important functions including nutrient and waste transport, regulation of vessel tone and growth through secretion of paracrine and autocrine factors, involvement in coagulation.

4. Mechanical Properties and Testing
Blood vessels exhibit nonlinear material properties, hysteresis, preconditioning, stress relaxation and creep, anisotropy; properties vary with type, location, environment and current state; various tests account for one or more of these properties and may avoid them to simplify a given study; uniaxial tests, biaxial tests, 3D testing
Wave propagation
Wave equation, wavespeed - the Moens-Korteweg formula

5. Mechanical Properties and Testing
Pseudoelasticity
W: Strain energy functions
Residual stress
No-load state is not stress free
Three-dimensional testing
Study layer properties
Arterioles
Smooth muscle creates active tension

6. Mechanical Properties and Structure
Capillaries
Mesentery: tunnel-in-a-gel; elasticity determined by surrounding tissue; no collapse
Alveoli: no surrounding tissue so much more compliant, easily collapses
Veins
Remodeling

7. Systemic Circulation

8. Systemic Arterial Tree
Schema of the canine systemic arterial tree
(from Nichols WW and O’Rourke MF, 1990)

9. Circle of Willis
The Circle of Willis and nearby vessels of the cerebral vasculature. Saccular aneurysms often occur at bifurcations in and near the circle, particularly in the anterior portion (from CIBA Medical Education Division).

10. Vasa Vasorum
Figure 8.2:5 in textbook. Photograph of the pulmonary artery of a rabbit, showing the vasa vasorum in the wall.

11. Elastic Artery Structure
Schematic cross-section showing the three layers – intima, media and adventitia – and their primary constituents (from Rhodin JAG, 1979)

12. Muscular Artery Structure
Schematic cross-sections of typical muscular artery showing the three layers – intima, media and adventitia – and their primary constituents (from Rhodin JAG, 1979)

13. Musculo-Elastic Fascicle
Schema of the “musculo-elastic fascicle” that was proposed by Clark & Glagov. E denotes elastin, Ce denotes smooth muscle cells, and F denotes collagen bundles which exist between the elastin sheets (from Clark & Glagov, 1985)

14. Wall composition: Canine arteries
From Fischer GM & Llaurado JG, % H20 is per wet weight, whereas protein is per unit dry weight; c:e denotes collagen to elastin ratio

15. Preconditioning Behavior
First Piola-Kirchhoff stress P11 versus stretch 1 of passive bovine coronary artery: Cyclic responses during uniaxial testing – note preconditioned behavior after 15 cycles and the diminishing hysteresis (data from Humphrey JD, et al, 1996).

16. Tangent Modulus
Text figure 8.3:1. Plot of the Young’s modulus (tangent modulus, dT/d) vs. the tensile stress (T) in a specimen of thoracic aorta of the dog in a loading process. Part (a) shows a power law for small T and an approximate straight line for T greater than 20 kPa. Part (b) shows the straight-line representation for various segments of the aorta.

17. Stress Relaxation
Uniaxial stress relaxation (first Piola-Kirchhoff stress P11) of a passive bovine coronary artery: (data from Humphrey JD, Salunke N, Tippett B, 1996).

18. Reduced Relaxation Function
Figure 8.3:3 textbook
Normalized relaxation function G(t) for circumferential segments of arteries. Mean ± s.d. (n = 10).

19. Reduced Creep Function
Figure 8.3:5 in textbook. A typical creep curve, plotted as a reduced creep function J(t). Dog carotid artery.

20. Vessel Test Apparatus
Typical experimental set-up from which a majority of the data on arterial wall behavior has been obtained. Cyclic inflation tests at a fixed axial extension are easily performed with this set-up (from Takamizawa K & Hayashi K, 1987).

21. Test equipment for vessel torsion, stretch, and inflation
Figure 8.4:1 in textbook. Test equipment for torsion, longitudinal stretching, and circumferential inflation of blood vessels. The torque is measured by the differential air pressure in the jets of an impinging on a little flag, and holding in a null position.

22. Wave Propagation in Tubes
Assume: infinitely long tube
circular cylinder
elastic tube
homogeneous, incompressible, inviscid fluid
Conservation of Mass:
Equation of Motion:
Assume A depends on transmural pressure alone
(ignore tube’s inertia)
internal pressure
external pressure

23. Wave Propagation in Blood Vessel

24. Wave Propagation in Blood Vessel

25. Wave Propagation in Blood Vessel

26. Approaches for the determination of the mechanical properties of the blood vessels
1. To understand the structural, mechanical and physiological characteristics of blood vessels:
1) The material is non-homogeneous, non-elastic, and anisotropic;
2) The stress-strain relationship is non-linear;
3) The mechanical properties depend on the type, location, state, and environment of blood vessels.
2. To make assumptions to simplify the problem:
1) The material in the vessel wall is homogeneous, pseudo-elastic, and incompressible.
2) The shape of the vessels is cylindrical.

27. Mechanical Properties of Blood Vessels
3. Select test methods based on the geometry, structure and function of the vessels and carry out experiments:
1) Uniaxial test: For a 1-layered model. A vessel strip can be tested by stretching in one direction.
2) Biaxial test: For a 1-layered model. A vessel should be cut open longitudinally and stretched in x and y direction.
3) Inflation test: 1-D test for a 1-layered model.
4) Bending test: For vessel strips, a two-layered model.
4. To select theoretical models for the stress-strain relationship and carry out theoretical analysis:
1) Linear stress-strain relation: Hooke’s law: s = Ee
2) Nonlinear stress-strain relation: Pseudo-strain-energy function with exponential or polynomial expressions.

28. Example: Two-Dimensional Inflation Test
1 Experiments:
1) Loading and unloading pressure-diameter curves are determined by carrying out inflation and deflation tests, respectively. Pressure and diameter (external) are needed for the calculation of stress and strain as described below.
2) Reference parameters for the calculation of stress and strain are measured at the zero-stress state of the blood vessels.

29. Example: Two-Dimensional Inflation Test
2 Analysis:
1) Calculation of stresses (Kirchhoff):
where S and Szz are Kirchhoff’s stresses in the circumferential and longitudinal directions, respectively, P is blood pressure, rI and ro are internal and external radii, h is vessel wall thickness, and  and z are stretch ratios in the circumferential and longitudinal directions, respectively.
2) Calculation of strain (Green):
where E and Ezz are Green strains in the circumferential and longitudinal directions, respectively.

30. Example: Two-Dimensional Inflation Test
3) Derivation of the stress-strain relationship using a strain-energy function:
where 0 is the material density of the artery (mass per unit volume), W is the strain energy per unit mass, 0W is the strain energy per unit volume, and E* and Ez* are reference strains at a physiological pressure in the circumferential and longitudinal directions, respectively.
4) Determination of the constants a1, a2 and a4 of the strain energy function by using a non-linear least square fitting method. These constants are used to describe the stiffness of the arteries

31. Pseudoelasticity
Figure 8.6:1 from textbook. The effect of the material constants a1, a2, on the shape of the stress-strain curves. The scales of the coordinates are so chosen that all curves pass through the point S*, E*. The curves pass through the origin if Ezz = 0; otherwise they do not.

32. Residual Strain
The “opening-up” (right) of originally unloaded intact arterial ring (left) following a radial cut (from Fung, 1984)

33. Zero-Stress State
Light micrographs of arterial cross-sections fixed in a 120 mmHg (panel A), a no load (panel B), and a radially-cut, stress-free configuration (panel C). Note the increased waviness in the internal elastic lamina (black) in the unloaded ring, which is consistent with the existence of compressive residual stresses in the inner wall in this configuration (from Fung YC & Liu SQ, 1992).

34. Referring Strain to the Stress-Free State
Schema of the mappings from a cut, stress-free configuration, to an unloaded intact configuration, to a “loaded” configuration

35. Opening Angles A
Opening-up of inner (panel B) and outer (panel C) unloaded arterial rings taken from a single (panel A) arterial cross-section – the dashed line in panel A shows where the original cut was made to separate the ring into concentric rings (from Vossoughi J et al, 1993). B C

36. Nonhomogeneous Residual Stress
Opening angle in the rat aorta as a function of location from the aortic root (0%) to the aorto-iliac bifurcation (100%) (from Liu SQ & Fung YC, 1988)

37. Residual Stress Distributions
Calculated distributions of the transmural residual Cauchy stress (trr, t and tzz) through the wall of an unloaded, intact elastic rabbit artery (see text for numerical values of the vessel geometry and properties)

38. Effects of Residual Stress
Calculated distributions of the transmural Cauchy stress, with and without the inclusion of residual stress: - pressure is 120 mmHg and axial stretch,  or  alone, is (Chuong CJ & Fung YC, 1986). Note reductions in the magnitudes and gradients of the stresses when residual stress is included.

39. Bending Test System
Figure 8.8:1 in text. A sketch of Fung’s apparatus for measuring the strains in an arterial specimen subjected to bending.

40. mesentery: tunnel-in-a gel lung: sheet (parking garage)
Capillaries
mesentery: tunnel-in-a gel
lung: sheet (parking garage)

41. Vessel Remodeling during Hypertension
Figure 8.13:2 in textbook. Response of the pulmonary arterial structure to a step increase of pulmonary blood pressure.

42. Effects of Smoking
Mean stress-strain curves of pulmonary arteries: (A) and (B) circumferential and longitudinal stress-strain curves in 2-month smoke-exposed and control rats, respectively; (C) and (D) 3-month smoke-exposed and control rats.
(Liu and Fung)

43. Blood Vessels: Summary of Key Points
Blood vessels form arterial and venous networks in the systemic and pulmonary circulations
Vessel walls have an intima, media and adventitia
Composite tissue structure affects vessel properties
Vessel mechanics can be affected by surrounding tissue
Vessels are nonlinear, anisotropic, viscoelastic and exhibit preconditioning behavior
Biaxial testing is used to measure anisotropic properties
Blood vessels propagate pulse waves along their walls
Blood vessels have residual stress in the no-load state
Blood vessel structure, mechanics and residual stress can change (remodel) with changes in blood pressure