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Vascular Mechanics

Philip B. Dobrin

10.1002/cphy.cp020303

Source: Supplement 8: Handbook of Physiology, The Cardiovascular System, Peripheral Circulation and Organ Blood Flow

Originally published: 1983

Published online: January 2011

Full Article on Wiley Online Library



Abstract

The sections in this article are:

1 Behavior of Arteries in Vivo
2 Circumferential Properties of Arteries
2.1 Relaxed Arteries
2.2 Contracted Arteries
3 Longitudinal Properties of Arteries
3.1 Elastic Characteristics
3.2 Intimal Shear Stresses
4 Radial Properties of Arteries
4.1 Elastic Characteristics
4.2 Mechanics and Wall Nutrition
5 Special Aspects of Vascular Mechanics
5.1 Baroreceptors
5.2 Arteries of Hypertensive Subjects
5.3 Atherosclerotic Arteries
6 Multidirectional Properties of Arteries
6.1 Analytic Assumptions
6.2 Incremental Analysis
6.3 Finite‐Deformation Analysis
Figure 1. Figure 1.

Diagram of arterial segment illustrating circumferential (θ), longitudinal (z), and radial (r) directions. Strains and stresses in θ and z directions are tensile because tissue is stretched in these directions with pressurization. Strains and stresses in z direction also are tensile because vessels are elongated at in situ length. Strains and stresses in r direction are compressive because wall becomes thinner with pressurization.

From Dobrin 72
Figure 2. Figure 2.

A: simultaneous pressure (P) and diameter (D) recordings of right common carotid artery of 68‐yr‐old male patient during operation. Catheter‐tip manometer inserted through side branch into artery recorded P; a noncontact photoelectric gauge with 2 photocells measured D. Recordings illustrate close correspondence between changes in P and D. B: 3 P‐D loops from 64‐yr‐old male patient during operation. Upper limbs correspond to rising phases of arterial pressure cycle. Lower limbs correspond to falling phases. Note dicrotic notch in lower limb. As pulse pressure increased from 30 mmHg (left) to 99 mmHg (right), width of hysteresis loop increased. As mean pressure and pulse pressure increased, curvilinear character of P‐D curve also became evident.

From Busse et al. 45
Figure 3. Figure 3.

Size and composition of components of vascular tree. End., endothelium; Ela., elastin; Mus., smooth muscle; Fib., fibrous connective tissue—i.e., collagen.

From Burton 43
Figure 4. Figure 4.

Free‐body diagram of cylindrical segment of blood vessel at equilibrium illustrating circumferential forces. PT, transmural pressure; σθ, circumferential stress; l, vessel length; h, wall thickness; di, internal diameter.
Figure 5. Figure 5.

Distribution of stresses (σ) across artery wall in circumferential (σθ), longitudinal (σz), and radial (σr) directions. Data computed assuming a homogeneous artery wall. Stresses are greatest at lumen and decline curvilinearly across wall thickness; θ and z stresses are tensile and remain positive at all points across wall; σr is compressive and plotted with a negative sign. Unlike σθ and σz, σr vanishes at outer edge of wall. At lumen σr, is equal in magnitude to transmural pressure; σθ and σz are 10–20 times larger in magnitude. Data are for transmural pressure of 150 cmH2O in relaxed vessel, but proportional values have been reported for both contracted and relaxed vessels over a wide range of pressures 78,195,215.

From Vaishnav et al. 232, by permission of the American Heart Association, Inc
Figure 6. Figure 6.

Stress‐strain relations of dog carotid artery after exciting vascular muscle with supramaximal doses of norepinephrine (NE) and after inactivating vascular muscle with supramaximal doses of potassium cyanide (KCN). Difference between NE and KCN curves gives length‐active stress relationship for vascular muscle. Behavior after treatment with KCN gives stress‐strain relationship for connective tissue. Strain (Δr/r0) is the fractional increase in circumferential length, a measure of circumferential deformation.

From Dobrin and Rovick 80
Figure 7. Figure 7.

Isometric and isobaric contraction of vascular smooth muscle. A: polygraph record of artery 1st undergoing isometric contraction [pressure (PT) was elevated just enough to maintain vessel diameter constant]. Maintaining isometric contraction required 150‐mmHg increase in PT, which then was reduced to preexcitation levels to permit isobaric contraction. This caused 25% reduction in diameter {Δd). B: summary of pressure‐strain data for 160 arteries, where strain (Δd/d0) is a measure of circumferential deformation. Means ± SE for 16 vessels relaxed and then excited at 1 of 10 pressures between 10 and 275 mmHg. Data show curvilinear pressure‐strain relationship of relaxed vessel and more linear curve of contracted vessel. Horizontal distances between relaxed and contracted data points (e.g., ΔPT) represent isometric contractions and reflect unimodal length‐active stress curve of vascular muscle. Vertical distances (e.g., Δd) represent isobaric contractions. Both isometrically and isobarically contracted vessels tend to fall along a single pressure‐strain curve, indicating equivalence of the 2 modes of contraction.

From Dobrin 69
Figure 8. Figure 8.

Elastic modulus for a relaxed and a contracted cylindrical segment of dog carotid artery. A: circumferential (θ) elastic modulus plotted as function of θ strain after exciting muscle with NE and after inactivation of muscle with KCN. Activating muscle increased elastic modulus at all but largest strains. B: identical data plotted as function of PT. Paradoxically, activating the muscle decreased wall elastic modulus. Arrows point to vessel strains at 100 mmHg in relaxed and contracted states; activating the muscle caused constriction to smaller strains at each pressure, producing decreased modulus when plotted as function of PT.

From Dobrin and Rovick 80
Figure 9. Figure 9.

A: pressure‐radius curves for dog carotid artery in relaxed, pretreatment state (Pre), after muscle was excited with NE, and after metabolic poisoning with KCN. Relaxed vessel (Pre, KCN) has markedly biphasic pressure‐diameter curve and stiffens at 75–100 mmHg. Muscularly active vessel has decreased dimensions at low pressures and has static pressure‐radius hysteresis. Pressure‐radius coordinates, however, are equal for relaxed and contracted arteries at 300 mmHg 80. These distension data were used to compute elastic‐modulus data in Fig. 6. B: stress‐strain curves computed by Gow 110 from data in A. Strain computed with respect to 2 values: 0.078 cm for contracted artery and 0.115 cm for relaxed artery; slopes are proportional to vessel stiffness. Thus computed, relaxed vessel is stiffer than constricted vessel at all except smallest dimensions. Note that computing strains with 2 separate reference values causes identical radii exhibited by the relaxed and contracted artery at 300 mmHg in A to correspond to markedly different strains (highest data points in B). Therefore apparently equivalent strains do not correspond to comparable real dimensions. C and D: radii (r) in A were plotted as function of strain (ɛ). In C, ɛ was computed with respect to 2 separate reference values using Gow's method 110. Each • was associated with 2 absolute radii. Also each Δr was associated with larger Δɛ for NE‐constricted vessels than for KCN‐relaxed vessel. In D, ɛ was computed with respect to 1 reference value. Each • and Δɛ was associated with 1 r and 1 Δr for both NE‐constricted and KCN‐relaxed vessels.
Figure 10. Figure 10.

Free‐body diagram of cylindrical segment of blood vessel at equilibrium illustrating logitudinal forces. , longitudinal traction force; re and ri, external and internal radii, respectively; σz, longitudinal stress.
Figure 11. Figure 11.

Longitudinal stress (σz) in relaxed cylindrical segment of dog carotid artery plotted as function of circumferential strain. Zero strain indicates vessel circumference of relaxed, excised, totally unloaded vessel. Data show how presence of traction maintains vessel at relatively constant length and σz; σz is the sum of the stress due to traction () and that due to pressure (). Interaction between these components results in almost constant net σz up to large circumferential strains and high pressures. Constancy of longitudinal force tends to keep vessel length constant.

Adapted from Dobrin and Doyle 79, by permission of the American Heart Association, Inc
Figure 12. Figure 12.

Longitudinal extension ratios (λ) for 16 dog carotid arteries (means ± SE). Lengths between identifiable branches were measured in situ immediately after death with neck flexed or extended to 4 positions. Extension ratios were computed by dividing length in situ at each neck position by length of excised, retracted, unloaded vessel.
Figure 13. Figure 13.

Histologic sections of dog carotid artery fixed while loaded uniaxially in radial (r) direction with stresses equivalent to 0, 50, 100, and 150 mmHg. Elastin fixes poorly. However, elastic lamellae are held by fixation of adjacent soft tissues, preventing retraction of lamellae into corrugated configuration. Because load is applied uniaxially it is equal at each point through the wall thickness. To analyze distribution of tissue stiffness, media was divided conceptually into 3rds and lamellae in each 3rd were counted (numbers right of specimens). Although lamellae are distributed nonuniformly across wall thickness, number of lamellae in each 3rd remains approximately constant during loading. Therefore radial elastic modulus is essentially uniform across thickness of media. If it were not, more compressible regions of wall would appear to gain lamellae, while other less compressible regions would appear to lose a commensurate number of lamellae.
Figure 14. Figure 14.

Poisson's ratios in an elastic body representing arterial wall. A: body subjected to uniaxial load in longitudinal (z) direction. This produces incremental strain in the z direction (Δɛz) and causes a narrowing strain in the circumferential (Δɛθ) and the radial directions (Δɛr). B: same elastic body subjected to uniaxial load in the θ direction. This produces incremental strain in the θ direction (Δɛθ) and narrowing strains in the z (Δɛz) and r directions (Δɛr). These strains are used to compute Poisson's ratios (Eqs. 20,21).

From Dobrin 72
Figure 15. Figure 15.

Data from dog thoracic aorta in situ that illustrate static anisotropy. Eθ, Ez, and Er are moduli in these directions; λθ and λz are extension ratios in θ and z directions. Top panels, means ± SE for longitudinal extension ratios (λz; left to right): 1.45 ± 0.04, 1.56 ± 0.02, and 1.51 ± 0.02. Bottom panels, means ± SE for λθ (left to right): 1.46 ± 0.02, 1.58 ± 0.02, and 1.48 ± 0.02. Eθ, Ez, and Er are not equal.

From Patel et al. 188, by permission of the American Heart Association, Inc
Figure 16. Figure 16.

Incremental viscoelastic moduli vs. frequency. E′θ, E′z, and E′r are storage, or dynamic elastic moduli, in these directions; E″θ, E″z, and E″r are corresponding loss, or viscous moduli. Vertical bars to right are average SE for each curve; symbols identify appropriate curves.

From Patel et al. 189, by permission of the American Heart Association, Inc
Figure 17. Figure 17.

Experimental evaluation of exponential and polynomial expressions of strain‐energy density. A and B: comparison of stress‐strain relationships from exponential strain‐energy function given by Eq. 34 and stress expressions given by Eqs. 39 and 40. Symbols defined in upper left corners. A: circumferential (θ) stress‐strain data. B: longitudinal (z) stress‐strain data. C and D: comparison of stress‐strain relationships from polynomial strain‐energy function given by Eq. 33 and stress expressions given by Eqs. 37 and 38. C: z stress‐strain data. D: θ stress‐strain data. Both exponential and polynomial expressions agree well with experimental data. Line V is a single value of strain, whereas line H is a single value of stress. Line V does not apply to all vessels, but line H does. This argues for referencing data to a common stress, rather than to a common strain.

From Fung et al. 101


Figure 1.

Diagram of arterial segment illustrating circumferential (θ), longitudinal (z), and radial (r) directions. Strains and stresses in θ and z directions are tensile because tissue is stretched in these directions with pressurization. Strains and stresses in z direction also are tensile because vessels are elongated at in situ length. Strains and stresses in r direction are compressive because wall becomes thinner with pressurization.

From Dobrin 72


Figure 2.

A: simultaneous pressure (P) and diameter (D) recordings of right common carotid artery of 68‐yr‐old male patient during operation. Catheter‐tip manometer inserted through side branch into artery recorded P; a noncontact photoelectric gauge with 2 photocells measured D. Recordings illustrate close correspondence between changes in P and D. B: 3 P‐D loops from 64‐yr‐old male patient during operation. Upper limbs correspond to rising phases of arterial pressure cycle. Lower limbs correspond to falling phases. Note dicrotic notch in lower limb. As pulse pressure increased from 30 mmHg (left) to 99 mmHg (right), width of hysteresis loop increased. As mean pressure and pulse pressure increased, curvilinear character of P‐D curve also became evident.

From Busse et al. 45


Figure 3.

Size and composition of components of vascular tree. End., endothelium; Ela., elastin; Mus., smooth muscle; Fib., fibrous connective tissue—i.e., collagen.

From Burton 43


Figure 4.

Free‐body diagram of cylindrical segment of blood vessel at equilibrium illustrating circumferential forces. PT, transmural pressure; σθ, circumferential stress; l, vessel length; h, wall thickness; di, internal diameter.



Figure 5.

Distribution of stresses (σ) across artery wall in circumferential (σθ), longitudinal (σz), and radial (σr) directions. Data computed assuming a homogeneous artery wall. Stresses are greatest at lumen and decline curvilinearly across wall thickness; θ and z stresses are tensile and remain positive at all points across wall; σr is compressive and plotted with a negative sign. Unlike σθ and σz, σr vanishes at outer edge of wall. At lumen σr, is equal in magnitude to transmural pressure; σθ and σz are 10–20 times larger in magnitude. Data are for transmural pressure of 150 cmH2O in relaxed vessel, but proportional values have been reported for both contracted and relaxed vessels over a wide range of pressures 78,195,215.

From Vaishnav et al. 232, by permission of the American Heart Association, Inc


Figure 6.

Stress‐strain relations of dog carotid artery after exciting vascular muscle with supramaximal doses of norepinephrine (NE) and after inactivating vascular muscle with supramaximal doses of potassium cyanide (KCN). Difference between NE and KCN curves gives length‐active stress relationship for vascular muscle. Behavior after treatment with KCN gives stress‐strain relationship for connective tissue. Strain (Δr/r0) is the fractional increase in circumferential length, a measure of circumferential deformation.

From Dobrin and Rovick 80


Figure 7.

Isometric and isobaric contraction of vascular smooth muscle. A: polygraph record of artery 1st undergoing isometric contraction [pressure (PT) was elevated just enough to maintain vessel diameter constant]. Maintaining isometric contraction required 150‐mmHg increase in PT, which then was reduced to preexcitation levels to permit isobaric contraction. This caused 25% reduction in diameter {Δd). B: summary of pressure‐strain data for 160 arteries, where strain (Δd/d0) is a measure of circumferential deformation. Means ± SE for 16 vessels relaxed and then excited at 1 of 10 pressures between 10 and 275 mmHg. Data show curvilinear pressure‐strain relationship of relaxed vessel and more linear curve of contracted vessel. Horizontal distances between relaxed and contracted data points (e.g., ΔPT) represent isometric contractions and reflect unimodal length‐active stress curve of vascular muscle. Vertical distances (e.g., Δd) represent isobaric contractions. Both isometrically and isobarically contracted vessels tend to fall along a single pressure‐strain curve, indicating equivalence of the 2 modes of contraction.

From Dobrin 69


Figure 8.

Elastic modulus for a relaxed and a contracted cylindrical segment of dog carotid artery. A: circumferential (θ) elastic modulus plotted as function of θ strain after exciting muscle with NE and after inactivation of muscle with KCN. Activating muscle increased elastic modulus at all but largest strains. B: identical data plotted as function of PT. Paradoxically, activating the muscle decreased wall elastic modulus. Arrows point to vessel strains at 100 mmHg in relaxed and contracted states; activating the muscle caused constriction to smaller strains at each pressure, producing decreased modulus when plotted as function of PT.

From Dobrin and Rovick 80


Figure 9.

A: pressure‐radius curves for dog carotid artery in relaxed, pretreatment state (Pre), after muscle was excited with NE, and after metabolic poisoning with KCN. Relaxed vessel (Pre, KCN) has markedly biphasic pressure‐diameter curve and stiffens at 75–100 mmHg. Muscularly active vessel has decreased dimensions at low pressures and has static pressure‐radius hysteresis. Pressure‐radius coordinates, however, are equal for relaxed and contracted arteries at 300 mmHg 80. These distension data were used to compute elastic‐modulus data in Fig. 6. B: stress‐strain curves computed by Gow 110 from data in A. Strain computed with respect to 2 values: 0.078 cm for contracted artery and 0.115 cm for relaxed artery; slopes are proportional to vessel stiffness. Thus computed, relaxed vessel is stiffer than constricted vessel at all except smallest dimensions. Note that computing strains with 2 separate reference values causes identical radii exhibited by the relaxed and contracted artery at 300 mmHg in A to correspond to markedly different strains (highest data points in B). Therefore apparently equivalent strains do not correspond to comparable real dimensions. C and D: radii (r) in A were plotted as function of strain (ɛ). In C, ɛ was computed with respect to 2 separate reference values using Gow's method 110. Each • was associated with 2 absolute radii. Also each Δr was associated with larger Δɛ for NE‐constricted vessels than for KCN‐relaxed vessel. In D, ɛ was computed with respect to 1 reference value. Each • and Δɛ was associated with 1 r and 1 Δr for both NE‐constricted and KCN‐relaxed vessels.



Figure 10.

Free‐body diagram of cylindrical segment of blood vessel at equilibrium illustrating logitudinal forces. , longitudinal traction force; re and ri, external and internal radii, respectively; σz, longitudinal stress.



Figure 11.

Longitudinal stress (σz) in relaxed cylindrical segment of dog carotid artery plotted as function of circumferential strain. Zero strain indicates vessel circumference of relaxed, excised, totally unloaded vessel. Data show how presence of traction maintains vessel at relatively constant length and σz; σz is the sum of the stress due to traction () and that due to pressure (). Interaction between these components results in almost constant net σz up to large circumferential strains and high pressures. Constancy of longitudinal force tends to keep vessel length constant.

Adapted from Dobrin and Doyle 79, by permission of the American Heart Association, Inc


Figure 12.

Longitudinal extension ratios (λ) for 16 dog carotid arteries (means ± SE). Lengths between identifiable branches were measured in situ immediately after death with neck flexed or extended to 4 positions. Extension ratios were computed by dividing length in situ at each neck position by length of excised, retracted, unloaded vessel.



Figure 13.

Histologic sections of dog carotid artery fixed while loaded uniaxially in radial (r) direction with stresses equivalent to 0, 50, 100, and 150 mmHg. Elastin fixes poorly. However, elastic lamellae are held by fixation of adjacent soft tissues, preventing retraction of lamellae into corrugated configuration. Because load is applied uniaxially it is equal at each point through the wall thickness. To analyze distribution of tissue stiffness, media was divided conceptually into 3rds and lamellae in each 3rd were counted (numbers right of specimens). Although lamellae are distributed nonuniformly across wall thickness, number of lamellae in each 3rd remains approximately constant during loading. Therefore radial elastic modulus is essentially uniform across thickness of media. If it were not, more compressible regions of wall would appear to gain lamellae, while other less compressible regions would appear to lose a commensurate number of lamellae.



Figure 14.

Poisson's ratios in an elastic body representing arterial wall. A: body subjected to uniaxial load in longitudinal (z) direction. This produces incremental strain in the z direction (Δɛz) and causes a narrowing strain in the circumferential (Δɛθ) and the radial directions (Δɛr). B: same elastic body subjected to uniaxial load in the θ direction. This produces incremental strain in the θ direction (Δɛθ) and narrowing strains in the z (Δɛz) and r directions (Δɛr). These strains are used to compute Poisson's ratios (Eqs. 20,21).

From Dobrin 72


Figure 15.

Data from dog thoracic aorta in situ that illustrate static anisotropy. Eθ, Ez, and Er are moduli in these directions; λθ and λz are extension ratios in θ and z directions. Top panels, means ± SE for longitudinal extension ratios (λz; left to right): 1.45 ± 0.04, 1.56 ± 0.02, and 1.51 ± 0.02. Bottom panels, means ± SE for λθ (left to right): 1.46 ± 0.02, 1.58 ± 0.02, and 1.48 ± 0.02. Eθ, Ez, and Er are not equal.

From Patel et al. 188, by permission of the American Heart Association, Inc


Figure 16.

Incremental viscoelastic moduli vs. frequency. E′θ, E′z, and E′r are storage, or dynamic elastic moduli, in these directions; E″θ, E″z, and E″r are corresponding loss, or viscous moduli. Vertical bars to right are average SE for each curve; symbols identify appropriate curves.

From Patel et al. 189, by permission of the American Heart Association, Inc


Figure 17.

Experimental evaluation of exponential and polynomial expressions of strain‐energy density. A and B: comparison of stress‐strain relationships from exponential strain‐energy function given by Eq. 34 and stress expressions given by Eqs. 39 and 40. Symbols defined in upper left corners. A: circumferential (θ) stress‐strain data. B: longitudinal (z) stress‐strain data. C and D: comparison of stress‐strain relationships from polynomial strain‐energy function given by Eq. 33 and stress expressions given by Eqs. 37 and 38. C: z stress‐strain data. D: θ stress‐strain data. Both exponential and polynomial expressions agree well with experimental data. Line V is a single value of strain, whereas line H is a single value of stress. Line V does not apply to all vessels, but line H does. This argues for referencing data to a common stress, rather than to a common strain.

From Fung et al. 101
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Article Title: Vascular Mechanics
 

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Philip B. Dobrin. Vascular Mechanics. Compr Physiol 2011, Supplement 8: Handbook of Physiology, The Cardiovascular System, Peripheral Circulation and Organ Blood Flow: 65-102. First published in print 1983. doi: 10.1002/cphy.cp020303