Comprehensive Physiology Wiley Online Library

Mechanics of Vascular Smooth Muscle

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Abstract

The sections in this article are:

1 Uses of Mechanical Studies and Analytical Framework
1.1 Purposes and Scope of Chapter
1.2 Analytical Framework of Classic Muscle Mechanics
2 Preparations and Normalization
2.1 Criteria for Preparations Used in Mechanical Studies
2.2 Suitable Tissue Preparations and Procedures
2.3 Definitions, Units, and Symbols
3 Mechanics of Vascular Smooth Muscle Tissues
3.1 Parallel Elastic Component and Mechanical Hysteresis
3.2 Series Elastic Component
3.3 Static Properties of the Contractile Component
3.4 Dynamic Properties of the Contractile Component
4 Applicability of Tissue Mechanics to Cells and the Contractile Component
4.1 Relationships Between Cell and Tissue Mechanics
Figure 1. Figure 1.

Common representations of mechanical properties of muscle. A: generalized apparatus for mechanical measurements. When the lower stop is in place, measurement is isometric (constant length). Activation of solenoid portion of lower stop provides a quick release, converting isometric response into isotonic (constant load) contraction. B: isometric force as a function of length in relaxed (dotted curve) and maximally stimulated (dashed curve) muscle. Difference between these curves (solid curve) reflects active force development. Maximal force (F0) is developed at the optimal length for force development (L0). C: comparison of isotonic and isometric contractions. Upper left curve is isometric force when no shortening can occur. Lower left curves show that the amount of shortening and shortening velocity increase as load is diminished. This dependence is plotted (right curve) as the force‐velocity relationship, with an extension of the data to situations in which the contracting muscle is forcibly stretched. D: if isometrically contracting muscle is subjected to quick release, active shortening is preceded by elastic recoil (partially obscured by inertial oscillations). From this recoil the elastic extension of contractile material and other elements in series with the contractile material can be plotted as load‐extension curve.

Figure 2. Figure 2.

Analogue models of muscle. Simple two‐component model is shown on the left. A: three‐component system arranged as in Maxwell element (a). B: model with the three components in Voigt element (b) configuration. C and D: more complicated four‐component models. sec, Series elastic component; pec, parallel elastic component; cc, contractile component.

From Simmons and Jewell
Figure 3. Figure 3.

Relationships between tension development by pig carotid artery strips and tissue geometry. A: graph showing records of mechanical tension elicited by norepinephrine (10−6 M) vs. oxygen partial pressure in the solution. Arrows placed at points that reflect the minimal solution Po2 that could sustain full force development. Numbers above the arrows are strip thicknesses in centimeters. B: graph showing best fit for the data obtained, in which the limiting solution Po2 is plotted against the square of one‐half tissue thickness (a′), with α′ used to indicate that tissue geometry was corrected because the strips were not infinite sheets. Equation shows the theoretical relationship between critical Po2 for cell in the center of strip (Po2cell) and the solution Po2 in an infinite sheet (Po2s), assuming diffusion limitation of O2 is responsible for the mechanical tension reduction. M, metabolic rate; a′, one‐half tissue thickness (in infinite sheet); D, diffusion coefficient for oxygen.

Adapted from Pittman and Duling
Figure 4. Figure 4.

Stress‐strain behavior of various types of materials. A: extension of pure elastic material is proportional only to stress. B: addition of viscous drag leads to hysteresis and time‐dependent properties. C: tissues usually show complex behavior as indicated by the hysteresis loop. Studies of smooth muscle often involve an isotonic release from a fixed length (horizontal arrow) with accompanying length changes that are a function of time (creep). Alternatively, tissue may be initially stretched to some fixed length, and isometric force declines with time (vertical arrow, stress‐relaxation).

From Alexander . Reprinted from Federation Proceedings
Figure 5. Figure 5.

Estimation of passive elasticity of smooth muscle. A: change in force (Fm) and length in stimulated skeletal muscle released to a shorter, constant length is indicated in the top two graphs. Corresponding length changes in the parallel elastic component (pec), series elastic component (sec), and contractile component (cc) are indicated below. ΔLm, change in muscle length; ΔLpec, change in length of parallel elastic component; ΔLsec, change in length of series elastic component; ΔLcc, change in length of contractile component. B: hanges in force (F) observed after step length (L) changes of unstimulated pig carotid media tissue possessing some degree of tone. Three protocols for length changes are indicated in curves a, b, and c, in which the final length was always 9.0 mm. Response of unstimulated smooth muscles upon release was similar to that of stimulated skeletal muscle in A. Minimal force observed at 9.0 mm varied, with Fa > Fb > Fc. C: force‐length curves (a, b, and c) were obtained using the protocols (a, b, and c, respectively) depicted in B. Curve d was obtained using protocol c after overnight equilibration in calcium‐free solution. Force on the ordinate is expressed as a fraction of maximal active force at optimal length for force development (L0). Protocol c appears to reflect true passive length‐force properties for this tissue. F0, maximal active force.

A from Simmons and Jewell ; B and C from Herlihy and Murphy , by permission of the American Heart Association, Inc
Figure 6. Figure 6.

Suggested model for arrangement of dense bodies (solid circles) connected by relatively inelastic 10‐nm intermediate filaments (straight lines) in a smooth muscle cell. Stretching causes an increase in fiber length (A to B), with axial consolidation of network seen in cross section (a to b) as a result of reorientation of the filaments linking dense bodies. Results are based on electron micrographs of extracted taenia coli fibers.

From Cooke and Fay
Figure 7. Figure 7.

Isotonic release of contracting skeletal muscle to low constant load. Length of the series elastic component (sec) changes very rapidly because of elastic recoil, with further changes in muscle length reflecting shortening of the contractile component (cc). Fm, muscle force; ΔLm, change in muscle length; ΔLsec, change in sec length; ΔLcc, change in cc length.

From Simmons and Jewell
Figure 8. Figure 8.

Length‐force curves for pig carotid media. Solid lines give experimental data that Herlihy and Murphy obtained for active force developed as a fraction of optimal tissue length, for the PEC, and for extension of the SEC. Dashed line and upper abscissa represent correction for length of contractile system after shortening against SEC. Dotted extensions of this corrected curve are linear extrapolations to zero force. PEC, parallel elastic component; SEC, series elastic component; F, force; F0, maximal force; L, length; L0, optimal length for force development.

From Murphy , with permission of S. Karger AG, Basel
Figure 9. Figure 9.

Isometric force (lever fixed) and isotonic shortening (against preload equal to passive force) in spontaneously contracting rat portal vein. A: muscle length was 3 mm and passive force (and preload) was 4 × 10−4 N. B: muscle was stretched to 3.8 mm, with increase in passive force (and preload) to 35 × 10−4 N. C: vertical arrows indicate isometric force development at two lengths, and horizontal arrows show isotonic shortening. D: Maxwell and Voigt (II) analogue models. F, force; L, length.

From Johansson . Reprinted from Federation Proceedings
Figure 10. Figure 10.

Effect of wall thickness on geometrical artifacts in simple tissue preparations. Illustrated vessel ring has ri:r0 ratio of 0.5. Were such a ring opened, tissue would assume trapezoidal shape if no outside stresses were imposed. Because mounting procedures tend to yield a tissue with more rectangular section, resulting stresses produce a variation in fiber length (or in stress distribution) whose maximal extent can be estimated from graph. ri, Inner radius; r0, outer radius; Δ, variation; α, (outer circumference – inner circumference)/2; b, mean circumference.

Figure 11. Figure 11.

Comparison of length‐force relationships in skeletal and smooth muscle. Solid line, sarcomere length‐force data from electrically stimulated single frog semitendinosus fibers . Dashed line indicates minimal estimate for force (F) development in skinned semitendinosus fibers when stimulated by Ca2+ at short lengths . Solid circles represent more realistic estimates . Frog skeletal muscle data were normalized to sarcomere length of 2.1 μm. In smooth muscles L0 represents length at which maximal isometric force (F0) was obtained. Open circles, pig carotid media ; open squares, cat duodenyl circular ; stars, dog trachealis .

From Murphy , with permission of S. Karger, AG, Basel
Figure 12. Figure 12.

Force‐velocity data. A: curve according to Hill's equation, which is (P + a) (V + b) = (P0 + a)b, where P is load on muscle, P0 is maximal isometric force, V is shortening velocity, a is a constant with dimensions of force, and b is a constant with dimensions of length. B: curve acording to Aubert's equation, which is P = (P0 + F) exp (–V/B) – F, where F and B are constants. Original notation using P and P0 instead of F and F0 is retained in this figure. There are various ways of determining constants of these equations, but if data are rather scattered the best method is to make use of corresponding linear equations. C: curve according to Hill's equation, (P0P)/V = (P/b) + (a/b). D: curve acording to Aubert's equation, loge (P + F) = loge (P0 + F) – (V/B). To obtain values for a and b of the Hill equation, the best straight line is drawn through a plot of (P0P)/V against P. Constants can then be obtained as shown from slope of the line and its intercept on force axis. See Reference to obtain values for B and F of Aubert's equation (B and D). Note that in C the high force values tend to have too big an influence in determining position of the line, whereas in D it is high‐velocity values that dominate. Because both equations have three constants, the curve‐fitting procedure involves loss of three degrees of freedom from data (i.e., equivalent of three data points). If there are only three data points, then curves drawn according to either equation must pass through these three points. It follows that judgments about fitting the data by the equations can be made only if there are substantially more data points.

From Simmons and Jewell
Figure 13. Figure 13.

Force‐velocity data for pig carotid media. Solid lines were obtained by the quick‐release method for initial shortening velocity as function of load in electrically stimulated tissues. Power output (curve starting at origin) was obtained as product of force/cross‐sectional area and velocity . Dashed line represents results obtained from potassium‐stimulated tissues (J. T. Herlihy, unpublished observations). L0, optimal length for force development.

From Murphy , with permission of S. Karger AG, Basel
Figure 14. Figure 14.

Three‐dimensional representation of dynamic length‐force‐velocity phase space of contractile component. Data were obtained using rat gracilis anticus muscle at 17.5°C. L′, calculated length of the contractile component after correction for the series elastic component; L'0, calculated optimal length for force development; F, force; F0, maximal force.

From Bahler et al.
Figure 15. Figure 15.

Tentative three‐dimensional diagram illustrating force‐velocity curves of contractile elements in vascular smooth muscle at different muscle lengths with maximal (full lines) and submaximal (broken lines) activation of contractile machinery. Heavy line in the zero velocity plane represents the relationship between length and active force at maximal contraction. Dotted lines below the zero velocity (or force‐length) plane illustrate possible relationships between force and lengthening velocity when contracted muscle is stretched by forces greater than maximal force. V, velocity; L, length; F, force.

From Johansson
Figure 16. Figure 16.

Force‐velocity plots at beginning of isotonic movements in cat soleus muscle. Plotted values of velocity were those measured 15 m/s after release of isometrically contracting muscle to load corresponding to scale on ordinate. Initial length of muscle was constant. Velocity at which muscle lengthens under an imposed load is not constant but varies with load, stimulation frequency, and other factors. Stimulation frequencies (pulses/s, designated by pps) are indicated to the left of each curve.

From Joyce and Rack
Figure 17. Figure 17.

Different arrangements of six axially oriented cells anatomically in series (small solid boxes) comprising a constant fraction of tissue cross‐sectional area. A: cells linked in series. B and C: series and parallel linkages. D: cells all in parallel. Thin lines indicate stiff force‐transmitting structures. Microscopic examination would not necessarily distinguish these arrangements in tissue. Table gives effect of model on tissue force developed (FT)/cell force developed (FC), tissue shortening (FT)/cell shortening (SC), and tissue shortening velocity (VT)/cell shortening velocity (VC). Critical (tissue) segment indicates minimal tissue segment length (relative units) at which force per cross‐sectional area remains independent of overall tissue length.

From Driska and Murphy , with permission of S. Karger AG, Basel


Figure 1.

Common representations of mechanical properties of muscle. A: generalized apparatus for mechanical measurements. When the lower stop is in place, measurement is isometric (constant length). Activation of solenoid portion of lower stop provides a quick release, converting isometric response into isotonic (constant load) contraction. B: isometric force as a function of length in relaxed (dotted curve) and maximally stimulated (dashed curve) muscle. Difference between these curves (solid curve) reflects active force development. Maximal force (F0) is developed at the optimal length for force development (L0). C: comparison of isotonic and isometric contractions. Upper left curve is isometric force when no shortening can occur. Lower left curves show that the amount of shortening and shortening velocity increase as load is diminished. This dependence is plotted (right curve) as the force‐velocity relationship, with an extension of the data to situations in which the contracting muscle is forcibly stretched. D: if isometrically contracting muscle is subjected to quick release, active shortening is preceded by elastic recoil (partially obscured by inertial oscillations). From this recoil the elastic extension of contractile material and other elements in series with the contractile material can be plotted as load‐extension curve.



Figure 2.

Analogue models of muscle. Simple two‐component model is shown on the left. A: three‐component system arranged as in Maxwell element (a). B: model with the three components in Voigt element (b) configuration. C and D: more complicated four‐component models. sec, Series elastic component; pec, parallel elastic component; cc, contractile component.

From Simmons and Jewell


Figure 3.

Relationships between tension development by pig carotid artery strips and tissue geometry. A: graph showing records of mechanical tension elicited by norepinephrine (10−6 M) vs. oxygen partial pressure in the solution. Arrows placed at points that reflect the minimal solution Po2 that could sustain full force development. Numbers above the arrows are strip thicknesses in centimeters. B: graph showing best fit for the data obtained, in which the limiting solution Po2 is plotted against the square of one‐half tissue thickness (a′), with α′ used to indicate that tissue geometry was corrected because the strips were not infinite sheets. Equation shows the theoretical relationship between critical Po2 for cell in the center of strip (Po2cell) and the solution Po2 in an infinite sheet (Po2s), assuming diffusion limitation of O2 is responsible for the mechanical tension reduction. M, metabolic rate; a′, one‐half tissue thickness (in infinite sheet); D, diffusion coefficient for oxygen.

Adapted from Pittman and Duling


Figure 4.

Stress‐strain behavior of various types of materials. A: extension of pure elastic material is proportional only to stress. B: addition of viscous drag leads to hysteresis and time‐dependent properties. C: tissues usually show complex behavior as indicated by the hysteresis loop. Studies of smooth muscle often involve an isotonic release from a fixed length (horizontal arrow) with accompanying length changes that are a function of time (creep). Alternatively, tissue may be initially stretched to some fixed length, and isometric force declines with time (vertical arrow, stress‐relaxation).

From Alexander . Reprinted from Federation Proceedings


Figure 5.

Estimation of passive elasticity of smooth muscle. A: change in force (Fm) and length in stimulated skeletal muscle released to a shorter, constant length is indicated in the top two graphs. Corresponding length changes in the parallel elastic component (pec), series elastic component (sec), and contractile component (cc) are indicated below. ΔLm, change in muscle length; ΔLpec, change in length of parallel elastic component; ΔLsec, change in length of series elastic component; ΔLcc, change in length of contractile component. B: hanges in force (F) observed after step length (L) changes of unstimulated pig carotid media tissue possessing some degree of tone. Three protocols for length changes are indicated in curves a, b, and c, in which the final length was always 9.0 mm. Response of unstimulated smooth muscles upon release was similar to that of stimulated skeletal muscle in A. Minimal force observed at 9.0 mm varied, with Fa > Fb > Fc. C: force‐length curves (a, b, and c) were obtained using the protocols (a, b, and c, respectively) depicted in B. Curve d was obtained using protocol c after overnight equilibration in calcium‐free solution. Force on the ordinate is expressed as a fraction of maximal active force at optimal length for force development (L0). Protocol c appears to reflect true passive length‐force properties for this tissue. F0, maximal active force.

A from Simmons and Jewell ; B and C from Herlihy and Murphy , by permission of the American Heart Association, Inc


Figure 6.

Suggested model for arrangement of dense bodies (solid circles) connected by relatively inelastic 10‐nm intermediate filaments (straight lines) in a smooth muscle cell. Stretching causes an increase in fiber length (A to B), with axial consolidation of network seen in cross section (a to b) as a result of reorientation of the filaments linking dense bodies. Results are based on electron micrographs of extracted taenia coli fibers.

From Cooke and Fay


Figure 7.

Isotonic release of contracting skeletal muscle to low constant load. Length of the series elastic component (sec) changes very rapidly because of elastic recoil, with further changes in muscle length reflecting shortening of the contractile component (cc). Fm, muscle force; ΔLm, change in muscle length; ΔLsec, change in sec length; ΔLcc, change in cc length.

From Simmons and Jewell


Figure 8.

Length‐force curves for pig carotid media. Solid lines give experimental data that Herlihy and Murphy obtained for active force developed as a fraction of optimal tissue length, for the PEC, and for extension of the SEC. Dashed line and upper abscissa represent correction for length of contractile system after shortening against SEC. Dotted extensions of this corrected curve are linear extrapolations to zero force. PEC, parallel elastic component; SEC, series elastic component; F, force; F0, maximal force; L, length; L0, optimal length for force development.

From Murphy , with permission of S. Karger AG, Basel


Figure 9.

Isometric force (lever fixed) and isotonic shortening (against preload equal to passive force) in spontaneously contracting rat portal vein. A: muscle length was 3 mm and passive force (and preload) was 4 × 10−4 N. B: muscle was stretched to 3.8 mm, with increase in passive force (and preload) to 35 × 10−4 N. C: vertical arrows indicate isometric force development at two lengths, and horizontal arrows show isotonic shortening. D: Maxwell and Voigt (II) analogue models. F, force; L, length.

From Johansson . Reprinted from Federation Proceedings


Figure 10.

Effect of wall thickness on geometrical artifacts in simple tissue preparations. Illustrated vessel ring has ri:r0 ratio of 0.5. Were such a ring opened, tissue would assume trapezoidal shape if no outside stresses were imposed. Because mounting procedures tend to yield a tissue with more rectangular section, resulting stresses produce a variation in fiber length (or in stress distribution) whose maximal extent can be estimated from graph. ri, Inner radius; r0, outer radius; Δ, variation; α, (outer circumference – inner circumference)/2; b, mean circumference.



Figure 11.

Comparison of length‐force relationships in skeletal and smooth muscle. Solid line, sarcomere length‐force data from electrically stimulated single frog semitendinosus fibers . Dashed line indicates minimal estimate for force (F) development in skinned semitendinosus fibers when stimulated by Ca2+ at short lengths . Solid circles represent more realistic estimates . Frog skeletal muscle data were normalized to sarcomere length of 2.1 μm. In smooth muscles L0 represents length at which maximal isometric force (F0) was obtained. Open circles, pig carotid media ; open squares, cat duodenyl circular ; stars, dog trachealis .

From Murphy , with permission of S. Karger, AG, Basel


Figure 12.

Force‐velocity data. A: curve according to Hill's equation, which is (P + a) (V + b) = (P0 + a)b, where P is load on muscle, P0 is maximal isometric force, V is shortening velocity, a is a constant with dimensions of force, and b is a constant with dimensions of length. B: curve acording to Aubert's equation, which is P = (P0 + F) exp (–V/B) – F, where F and B are constants. Original notation using P and P0 instead of F and F0 is retained in this figure. There are various ways of determining constants of these equations, but if data are rather scattered the best method is to make use of corresponding linear equations. C: curve according to Hill's equation, (P0P)/V = (P/b) + (a/b). D: curve acording to Aubert's equation, loge (P + F) = loge (P0 + F) – (V/B). To obtain values for a and b of the Hill equation, the best straight line is drawn through a plot of (P0P)/V against P. Constants can then be obtained as shown from slope of the line and its intercept on force axis. See Reference to obtain values for B and F of Aubert's equation (B and D). Note that in C the high force values tend to have too big an influence in determining position of the line, whereas in D it is high‐velocity values that dominate. Because both equations have three constants, the curve‐fitting procedure involves loss of three degrees of freedom from data (i.e., equivalent of three data points). If there are only three data points, then curves drawn according to either equation must pass through these three points. It follows that judgments about fitting the data by the equations can be made only if there are substantially more data points.

From Simmons and Jewell


Figure 13.

Force‐velocity data for pig carotid media. Solid lines were obtained by the quick‐release method for initial shortening velocity as function of load in electrically stimulated tissues. Power output (curve starting at origin) was obtained as product of force/cross‐sectional area and velocity . Dashed line represents results obtained from potassium‐stimulated tissues (J. T. Herlihy, unpublished observations). L0, optimal length for force development.

From Murphy , with permission of S. Karger AG, Basel


Figure 14.

Three‐dimensional representation of dynamic length‐force‐velocity phase space of contractile component. Data were obtained using rat gracilis anticus muscle at 17.5°C. L′, calculated length of the contractile component after correction for the series elastic component; L'0, calculated optimal length for force development; F, force; F0, maximal force.

From Bahler et al.


Figure 15.

Tentative three‐dimensional diagram illustrating force‐velocity curves of contractile elements in vascular smooth muscle at different muscle lengths with maximal (full lines) and submaximal (broken lines) activation of contractile machinery. Heavy line in the zero velocity plane represents the relationship between length and active force at maximal contraction. Dotted lines below the zero velocity (or force‐length) plane illustrate possible relationships between force and lengthening velocity when contracted muscle is stretched by forces greater than maximal force. V, velocity; L, length; F, force.

From Johansson


Figure 16.

Force‐velocity plots at beginning of isotonic movements in cat soleus muscle. Plotted values of velocity were those measured 15 m/s after release of isometrically contracting muscle to load corresponding to scale on ordinate. Initial length of muscle was constant. Velocity at which muscle lengthens under an imposed load is not constant but varies with load, stimulation frequency, and other factors. Stimulation frequencies (pulses/s, designated by pps) are indicated to the left of each curve.

From Joyce and Rack


Figure 17.

Different arrangements of six axially oriented cells anatomically in series (small solid boxes) comprising a constant fraction of tissue cross‐sectional area. A: cells linked in series. B and C: series and parallel linkages. D: cells all in parallel. Thin lines indicate stiff force‐transmitting structures. Microscopic examination would not necessarily distinguish these arrangements in tissue. Table gives effect of model on tissue force developed (FT)/cell force developed (FC), tissue shortening (FT)/cell shortening (SC), and tissue shortening velocity (VT)/cell shortening velocity (VC). Critical (tissue) segment indicates minimal tissue segment length (relative units) at which force per cross‐sectional area remains independent of overall tissue length.

From Driska and Murphy , with permission of S. Karger AG, Basel
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R. A. Murphy. Mechanics of Vascular Smooth Muscle. Compr Physiol 2011, Supplement 7: Handbook of Physiology, The Cardiovascular System, Vascular Smooth Muscle: 325-351. First published in print 1980. doi: 10.1002/cphy.cp020213