smooth muscle modeling
TRANSCRIPT
Smooth Muscle Modeling
Activation and contraction of contractile units insmooth muscle
Sae-Il Murtada
Licentiate Thesis No. 103, 2009
KTH School of Engineering Sciences
Department of Solid Mechanics
Royal Institute of Technology
SE-100 44 Stockholm, Sweden
TRITA, HFL-0475
ISSN 1654-1472
ISRN KTH/HLF/R-09/03-SE
”Science... never solves a problem without creating ten more.”
- George Bernard Shaw
Preface
The research presented in this licentiate thesis was carried out at the Department of Solid
Mechanics, Royal Institute of Technology (KTH) between April 2006 and October 2009. The
work was financially supported by the Swedish Research Council (grant 2005-6167) which is
sincerely acknowledged.
I would like to thank Professor Gerhard A. Holzapfel, for his fruitful discussions, support
and guidance which has kept me on the right track during the last three years. I would like
to thank him for introducing me to the field of biomechanics and giving me an opportunity
to work with a very interesting and challenging topic.
I would like to express my gratitude to Dr. Martin Kroon for his valuable advices and
knowledge of mechanics and thermodynamics. He has always been available for my questions,
which often resulted in very long but productive discussions.
I would also like to thank my colleagues at the Department of Solid Mechanics for making
the time here more pleasant and enjoyable.
Finally I would like to extend my gratitude to my mother Kil Ok, my sisters Yun-Ha
and Sung-Ha and my brother-in-law Daniel for their advices and encouragement, and my
girlfriend Jessica for her support and love.
Stockholm, October 2009
Sae-Il Murtada
Contents
Introduction and motivation 9
Summary of the papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
A Paper 15
A.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
A.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
A.3 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
A.3.1 A brief on smooth muscle physiology . . . . . . . . . . . . . . . . . . . 19
A.3.2 Chemical model for smooth muscle contraction . . . . . . . . . . . . . 21
A.4 A new mechanochemical model for smooth muscle contraction/relaxation . . 25
A.4.1 Chemical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
A.4.2 Mechanical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
A.4.2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
A.4.2.2 Strain-energy function . . . . . . . . . . . . . . . . . . . . . . 28
A.4.2.3 Evolution law . . . . . . . . . . . . . . . . . . . . . . . . . . 30
A.4.3 Linearization of the mechanochemical model . . . . . . . . . . . . . . 32
A.4.4 Thermodynamical aspects . . . . . . . . . . . . . . . . . . . . . . . . . 33
A.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
A.5.1 Fitting the chemical model . . . . . . . . . . . . . . . . . . . . . . . . 35
A.5.2 Fitting the mechanical model . . . . . . . . . . . . . . . . . . . . . . . 36
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SMOOTH MUSCLE MODELING
A.5.3 Evolution of the internal state variable urs . . . . . . . . . . . . . . . . 39
A.5.4 Convergence of the mechanochemical model . . . . . . . . . . . . . . . 39
A.6 Discussion and summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
B Paper 49
B.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
B.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
B.3 Background of the chemical model . . . . . . . . . . . . . . . . . . . . . . . . 52
B.4 Mechanical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
B.4.1 A model for perfectly aligned contractile units . . . . . . . . . . . . . 55
B.4.2 Contractile fibers oriented with dispersion . . . . . . . . . . . . . . . . 57
B.4.3 Evolution law for urs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
B.5 Specific choice of the density function ρ . . . . . . . . . . . . . . . . . . . . . 62
B.6 Model calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
B.6.1 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
B.6.2 Defining the rate constants . . . . . . . . . . . . . . . . . . . . . . . . 65
B.6.3 Estimating the dispersion parameter χ . . . . . . . . . . . . . . . . . . 65
B.6.4 Estimating the material parameters . . . . . . . . . . . . . . . . . . . 67
B.7 Numerical example, artery contraction . . . . . . . . . . . . . . . . . . . . . . 70
B.7.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
B.7.2 Maintaining the circumferential stretch for an increase in pressure . . 72
B.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
B.9 Appendix: Uniaxial tension of SM tissue . . . . . . . . . . . . . . . . . . . . . 76
B.9.1 Kinematics and stresses . . . . . . . . . . . . . . . . . . . . . . . . . . 76
B.9.2 Orientation density function ρ′ in the deformed configuration . . . . . 77
B.9.3 Conditional probability density function . . . . . . . . . . . . . . . . . 79
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
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Introduction and motivation
Smooth muscle cells are one of three muscle cell types in the human body. They can be found
in blood vessels, airways, bladder, intestines, stomach and other hollow organs. Smooth
muscle cells can also be found in hair follicles and in the iris. The smooth muscle cells are
spindle-shaped cells and are organized into muscle layers, see Fig. 1. The main function
of smooth muscle cells is to contract and relax, and by doing so the smooth muscle cells
can regulate and control the size of organs, and thereby the flow of its content such as the
diameter of air and blood vessels. When the smooth muscle does not function properly either
through an infection or disease, it can result in pathological conditions such as atherosclerosis,
incontinence or asthma. One of the key steps to understand the rise of these critical conditions
is to understand the behavior of smooth muscle cells, from activation to contraction and
deactivation to relaxation.
Smooth muscle contraction is regulated through intracellular calcium concentration; the
main source is from the extracellular fluids. When certain ion-channels in the smooth muscle
cell membrane opens, the intracellular calcium concentration in smooth muscle increases due
to the higher calcium concentration in the extracellular space which leads to activation and
contraction. The intracellular calcium behavior has a characteristic behavior after activation
where it first increases to a maximum value and then slowly drop to a lower steady-state
value. The calcium concentration in the extracellular space is much higher, about 10.000
times higher than the mean intracellular calcium concentration during activation, and is
thereby not affected significantly by the changes in the intracellular calcium behavior. When
the smooth muscle is deactivated, the intracellular calcium concentration decreases and the
muscle relaxes.
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SMOOTH MUSCLE MODELING
The contracting mechanism in muscle consists of interactions between myosin motors and
actin protein filaments. The myosin motors are organized into thick filaments which have the
ability to walk along thin actin filaments causing the thin and thick filaments to slide relative
each other. In smooth muscle the thick and thin filaments are organized into a contractile
unit consisting of thin filaments and one thick filament. This organization of filaments allows
a contraction of the contractile filament unit. The contractile filament units in smooth muscle
are organized in a preferred direction, with a certain dispersion of orientation [10] but in a
smooth manner with no visible striations, as in skeletal muscle.
1
Intima
Media
Adventitia
Endothelial cell
Internal elastic lamina
Smooth muscle cell
Collagen �brils
Elastic �brils
Elastic lamina
External elastic lamina
Bundles of collagen �brils
Helically arranged �ber�
reinforced medial layers
Composite reinforced by
collagen �bers arranged
in helical structures
Figure 1: Diagrammatic model of the general structure of a healthy elastic artery composedof three layers: intima (I), media (M), adventitia (A); adapted from [5].
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When intracellular calcium concentration increases in smooth muscle, the myosin motors
are activated through a chain of chemical reactions resulting in a sliding between the thin
and thick filaments which cause muscle contraction. During activation and contraction, the
myosin can be organized into four different states which is captured by, for example, the Hai
and Murphy model [3] through a system of ordinary differential equations. Two of these states
are load-bearing states of the myosin, which can be used to couple the chemical activation to
the mechanical contraction.
The goal of this licentiate thesis is to propose a new mechanical model of smooth muscle
which uses the currently existing theory of smooth muscle in a continuum-based framework
enabling the implementation of the model into a more complex boundary-value problem such
as a spherical bladder or a tube-like artery. The proposed mechanical model is coupled to
an activating chemical model such as the Hai and Murphy model [3] to connect the chemical
activation with the mechanical contraction. The mechanical model is formulated on the
basis of the underlying structural information available of the smooth muscle contractile
filaments. The model is simple and has just a few material parameters, all with a clear
biophysical interpretation. The model should be able to predict realistic data of smooth
muscle contraction.
Summary of the papers
The licentiate thesis consists of two scientific papers:
Paper A: S. Murtada, M. Kroon and G. A. Holzapfel. A calcium-driven mechanochemical
model for prediction of force generation in smooth muscle (submitted for publication). In this
paper, layers of smooth muscle are modeled as contractile fibers in series surrounded by an
elastic matrix through a new coupled mechanochemical model. The chemical model consists
of the latch-state Hai and Murphy model [3] which describes the fraction of myosin heads in
four different states whereas two states are load-bearing states. The fraction of the states
are coupled through rate constants, whereas the activating rate constant can be coupled to
the external calcium concentration. The fractions of the two load-bearing states are used to
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SMOOTH MUSCLE MODELING
couple the Hai and Murphy model to the proposed mechanical model which is characterized
by a strain-energy function and an evolution law for the relative sliding between the thick and
thin filaments. The proposed model has four material parameters and satisfies the second
law of thermodynamics. The mechanical material parameters are fitted against isometric and
isotonic quick-release experiments of guinea-pig taenia coli smooth muscle [1].
Paper B: S. Murtada, M. Kroon and G. A. Holzapfel. Modeling the dispersion effects
of contractile fibers in smooth muscles (submitted for publication). In the second paper, the
effects of contractile fibers with a dispersion of orientation is studied through a model based on
the coupled mechanochemical model presented in paper A. The chemical model is described
through the latch-state model by Hai and Murphy [3], where the activating rate constant
is coupled to the internal calcium concentration, which varies with time after activation.
The orientation dispersion of the contractile fibers is implemented in a similar manner as
documented in Gasser et al. [2] by introducing a contractile fiber density function which varies
with orientation. The density function is fitted against experimental data of myofilament
networks in swine carotid media [10]. The contractile fibers are modeled by a similar model as
described in Paper A. The network of contractile fibers are embedded in a passive surrounding
matrix modeled through a strain-energy function that consists of isotropic and anisotropic
parts, according to Holzapfel [4]. These parts represent the elastin and the collagen fibers.
The passive material parameters are fitted against uniaxial tension experiments of swine
carotid media [6] and the active material parameters are fitted against isometric contractions
of swine carotid media performed at different stretches [7, 8, 9]. Together with the estimated
material parameters and the proposed model, numerical studies of arterial vessel contractions
are simulated.
Future work
Since early 1970 several experiments of smooth muscle contraction have been conducted
performing isometric contraction and quick-release experiments of different types of smooth
muscle. However, many of these experiments are not conducted to be used to fit biomechanical
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BIBLIOGRAPHY
models. Vital information such as cross-section, initial load and details of chemical activation
are frequently missing, resulting in uncomplete experiments. In addition, with the rise of
new coupled models describing chemical activation to mechanical contraction more specific
and detailed experiments are required. There are currently no available data of chemical
activation and mechanical contraction of smooth muscle strips cut out in different directions.
With a working model an additional possible next step could be to implement the model
into a finite element program to solve more complex boundary-value problems. With a tool
like this available, large-scale effects that are not visible in a two-dimensional solution could
be revealed.
Bibliography
[1] Arner, A. 1982 Mechanical characteristics of chemically skinned guinea-pig taenia coli.
Eur. J. Physiol. 395, 277–284.
[2] Gasser, T. C., Ogden, R. W. & Holzapfel, G. A. 2006 Hyperelastic modelling of arterial
layers with distributed collagen fibre orientations. J. R. Soc. Interface 52, 2617–2660.
[3] Hai, C. M. & Murphy, R. A. 1988 Cross-bridge phosphorylation and regulation of latch
state in smooth muscle. J. Appl. Physiol. 254, C99–106.
[4] Holzapfel, G. A. 2000 Nonlinear solid mechanics. A continuum approach for engineering.
John Wiley & Sons, Chichester.
[5] Holzapfel, G. A., Gasser, T. C., Ogden, R. W., 2000. A new constitutive framework
for arterial wall mechanics and a comparative study of material models.J. Elasticity 61,
1–48.
[6] Kamm, K. E., Gerthoffer, W. H., Murphy, R. A. & Bohr, D. F. 1989 Mechanical prop-
erties of carotid arteries from DOCA hypertensive swine. Hypertension 13, 102–109.
[7] Rembold, C. M. & Murphy, R. A. 1990 Muscle length, shortening,myoplasmic [Ca2+]
and activation of arterial smooth muscle. Circ. Res. 66, 1354–1361.
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SMOOTH MUSCLE MODELING
[8] Rembold, C. M. & Murphy, R. A. 1990 Latch-bridge model in smooth muscle: [Ca2+]i
can quantitatively predict stress. Am. J. Physiol. 259, C251–C257.
[9] Rembold, C. M., Wardle, R. L., Wingard, C. J., Batts, T. W., Etter, E. F. & Murphy,
R. A. 2004 Cooperative attachment of cross bridges predicts regulation of smooth muscle
force by myosin phosphorylation. Am. J. Physiol. Cell Physiol. 287, C594–C602.
[10] Walmsley, J. G. & Murphy, R. A. 1987 Force-length dependence of arterial lamellar,
smooth muscle, and myofilament orientations. Am. J. Physiol. Heart Circ. Physiol. 253,
H1141–H1147.
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