Biomech Model Mechanobiol (2010) 9:749–762 DOI 10.1007/s10237-010-0211-0
ORIGINAL PAPER
A calcium-driven mechanochemical model for prediction of force generation in smooth muscle Sae-Il Murtada · Martin Kroon · Gerhard A. Holzapfel
Received: 20 August 2009 / Accepted: 9 March 2010 / Published online: 31 March 2010 © Springer-Verlag 2010
Abstract A new model for the mechanochemical response of smooth muscle is presented. The focus is on the response of the actin–myosin complex and on the related generation of force (or stress). The chemical (kinetic) model describes the cross-bridge interactions with the thin filament in which the calcium-dependent myosin phosphorylation is the only regulatory mechanism. The new mechanical model is based on Hill’s three-component model and it includes one internal state variable that describes the contraction/relaxation of the contractile units. It is characterized by a strainenergy function and an evolution law incorporating only a few material parameters with clear physical meaning. The proposed model satisfies the second law of thermodynamics. The results of the combined coupled model are broadly consistent with isometric and isotonic experiments on smooth muscle tissue. The simulations suggest that the matrix in which the actin–myosin complex is embedded does have a viscous property. It is straightforward for implementation into a finite element program in order to solve more complex boundary-value problems such as the control of short-term changes in lumen diameter of arteries due to mechanochemical signals. Keywords Biomechanics · Calcium · Smooth muscle contraction · Kinetic model · Mechanical model · Mechanochemical S.-I. Murtada · M. Kroon · G. A. Holzapfel Department of Solid Mechanics, School of Engineering Sciences, Royal Institute of Technology (KTH), Osquars Backe 1, 100 44 Stockholm, Sweden G. A. Holzapfel (B) Institute of Biomechanics, Center of Biomedical Engineering, Graz University of Technology, Kronesgasse 5-I, 8010 Graz, Austria e-mail:
[email protected]
1 Introduction Vascular smooth muscle (SM) cells are key constituents in the vascular system responsible for the control of short-term changes in lumen diameter due to various chemical, mechanical, and neural signals and long-term changes in extracellular matrix turnover (collagen synthesis due to cyclic stretch), see Li et al. (1998). The middle layer of an artery, for example, contains SM cells that are embedded in the extracellular matrix; SM cells are covered by a thin basement-type membrane so that up to 50% of the SM volume is due to this investing connective tissue (Humphrey 2009). Polarized light microscopy has shown that collagen and SM cells in the media are consistently circumferentially and coherently aligned (Canham et al. 1989; Finlay et al. 1995). Arteries such as the aorta may consist of 40–70 alternating layers of SM and elastic lamina that represent the discrete structural and functional units. Arterial SM is partially contracted in its homeostatic state that gives the wall a certain amount of tautness—the basal tone. The primary roles of the contraction/relaxation of vascular SM are to change the stiffness of large arteries or to decrease/increase the lumen in medium and small arteries (Milnor 1990) to regulate blood flow, for example. It seems that SM also affects the residual stresses and strains in vascular walls. In particular, the work of Rachev and Hayashi (1999) shows that the incorporation of the basal muscular tone in a model further reduces (in parallel to residual stress) the computed stress gradients in the wall, which is in agreement with the experimental findings documented by Matsumoto and Hayashi (1996). However, a detailed quantification is still missing. The relationship between the chemical activation and the state of mechanical contraction/relaxation is still not well understood. A quantification of the biomechanics of SM
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contraction is of utmost importance to better understand vascular physiology; SM-related diseases such as atherosclerosis, asthma, and incontinence; and to develop reliable, stable, and conclusive models of the activated muscles to capture the complete coupled process of SM contraction. One of the oldest and most common approaches to model muscle contraction is based on Hill (1938) who initially considered muscle in tetanized condition. The model consists of a hyperbolic relationship such as (v + b)(P + a) = b(P0 + a), relating the active force P to the velocity v during a quick-release (isotonic) experiment. The muscle has first been isometrically contracted reaching a (constant) maximum tension of P0 in tetanized condition. Two phenomenological constants a and b are determined by fitting the model to experiments. Another frequently used model is the three element Hill model documented by Fung (1970). It describes the mechanical behavior of a heart (striated) muscle sarcomere and consists of a contractile element (dashpot) in series with an elastic element, which represents the contractile unit. A parallel elastic element is added to represent the passive surrounding. Based on the three element Hill model, other models have been developed, see, for example, Hatze (1977), Wexler et al. (1997), Ettema and Meijer (2000), Lloyd and Besier (2003), Lichtwark and Wilson (2005). Since contractile filaments behave similarly in SM as in striated muscle, the model by Hill has also been used to capture SM contraction, see, for example, Gestrelius and Borgström (1986), Yang et al. (2003a), Zulliger et al. (2004), which are now briefly summarized. Gestrelius and Borgström (1986) proposed a model in which the series element in the Hill model is replaced by two elements, one representing the elasticity of the cross-bridges and the other representing the external elasticity. Yang et al. (2003a) presented an integrated electrochemical and mechanochemical model of SM contraction. The mechanochemical part couples the model by Hai and Murphy (1988) with a mechanical model based on Hill’s model. The biomechanical model is similar to the one by Gestrelius and Borgström (1986) except that the external elastic series element has been replaced by a viscoelastic series element and that the inner elastic element behaves with an exponential length–force relationship. A different approach to model arterial mechanics considering the influence of SM was proposed by Zulliger et al. (2004). Their model uses a pseudo strain-energy function consisting of non-mechanical related (phenomenological) parameters describing the mechanical properties. The strain energy is divided into a passive part that describes the collagen and the elastic surrounding, and an active part taking care of the vascular SM. Stålhand et al. (2008) used a similar version of Hai and Murphy (1988) to define the concentration of the chemical states; the rates k1 and k6 are functions of the calcium concentration and the stretch. The thermodynamically consistent model of Stålhand et al. (2008) includes a chemical state law
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due to Hai and Murphy (1988) and a model for SM contraction that reduces to the model by Yang et al. (2003a,b) in the limit of linear, small deformations. The mentioned models are different in several aspects but they do have features in common: they all consider the sliding filament theory and the cross-bridges are modeled as elastic. The benefit to use a strain energy-based approach to model SM contraction as, for example, in Stålhand et al. (2008), is that the model can easily be combined with other hyperelastic models and that it allows a description of SM contraction in a more complex environment with the potential of an implementation into a finite element program. Two other noticeably and more recent models not focusing on vascular SM are by Bates and Lauzon (2007) and Bursztyn et al. (2007). The objective of this paper is to develop a new model for SM contraction that is able to: (i) couple the biochemistry with the mechanics and thereby focus on the structure of the contractile units, i.e., the actin–myosin complex (and not on the extracellular matrix); (ii) incorporate only a few material parameters, each of which has a clear physical meaning; (iii) demonstrate the consistency of the model by comparing with experimental data obtained from activating SM cells. In particular, for the description of the kinetics of myosin phosphorylation and cross-bridge interactions with the thin filaments in SM, the model by Hai and Murphy (1988) is adopted. For the description of the relative sliding between actin and myosin filaments and the related forces, we base our approach on Hill’s three-component model. We postulate a strain-energy function and an evolution law that captures the general characteristics of SM contraction.
2 Background 2.1 A brief on smooth muscle physiology The smooth muscle (SM) is an important physiological component in several organs such as blood vessels, airways, intestines, and urinary bladder. Smooth muscle tissue is comprised of aggregates of muscle cells that are linked together. Inside the muscle cells, mechanical contraction is accomplished by the contractile apparatus that consists of actin (thin) and myosin (thick) filaments. Figure 1 illustrates the multi-scale smooth muscle histology from an artery: from the three layers to the contractile apparatus. The relative sliding between actin and myosin filaments is responsible for SM contraction. The relevant motor protein is myosin. The myosin monomer consists of two coiled heavy chains, each with a distinctive head and tail region. The head region is the part where attachments with the actin occur (these attachments are also known as cross-bridges). The tail region is the part of the myosin which can attach to other myosin monomers. Below each myosin head are two
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Fig. 1 Multi-scale smooth muscle histology, from left to right: threelayered artery → layers of smooth muscle cells → smooth muscle cell cytoplasm → contractile apparatus. The basic unit of the contractile
apparatus is the myosin/actin unit to the very right. These contractile units are arranged in series within cells and separated by dense bodies, symbolized by gray spheres
light chains (also known as regulatory light chains) whose main function is to regulate phosphorylation of the myosin head, which is a crucial step in SM contraction. Several myosin monomers can join together in different ways and form thick filaments that constitute one of the main components in the contractile apparatus of muscle cells. The myosin monomer only walks in one direction and, depending on how the myosin monomers organized in the thick filament, the contraction behavior will differ. It will also affect the arrangements of the surrounding thin filaments. In skeletal muscle, for example, the myosin molecules are organized to form a bipolar structure. SM observations indicate that myosin molecules are organized into a side-polar thick filament (see, for example, Craig and Megerman 1977; Cooke et al. 1989). The myosin heads are then evenly distributed along the thick filament with empty spots only on the sides. The organization of actin in SM thereby differs from the organization of actin in the skeletal muscle. When compared to striated muscle, SM has a much higher ratio of actin to myosin. In SM, the actin–myosin complexes are generally considered not to be organized in a distinct structure of sarcomeres; however, there may still exist an underlying organized structure (Herrera et al. 2005) although the precise relative arrangement of actin and myosin within the smooth muscle awaits discovery. SM cells are spindle-shaped and are typically 100 μm long with a diameter of about 5 μm except near the nucleus where they are slightly thicker. Activation of muscle contraction can be initiated through different mechanochemical signals. The primary determinant of muscle contractility is intracellular free calcium Ca2+ (other factors include local concentrations of O2 , CO2 , NO, ET-1, PGI2 ); increases in intracellular calcium concentration leads to contraction (600–800 nM in fully contracted SM) while decreases lead to relaxation (100 nM in resting SM); see Somlyo and Somlyo (1992), Horowitz et al. (1996). The intracellular calcium concentration is regulated by calcium channels which can open and close, to enable Ca2+ to flow into the cell. The calcium channels are located on the cell membrane and can be triggered to open through signaling molecules such as histamine, endothelin, and angiotensin or by nerve impulses, i.e., change of electrical potential. The
increase of calcium ion concentration inside the muscle cell may also be a consequence of an internal triggering of the sarcoplasmic reticulum which contains a reservoir of calcium ions. The external calcium concentration is mainly constant at all times whereas the intracellular calcium ion concentration is increased upon muscle activation when the ion channels open; thereafter, the intracellular calcium concentration reaches a peak value and subsequently drops again. The relationship between the external and intracellular calcium concentration is, however, complex. In SM, the intracellular calcium concentration can bind to the calcium binding protein calmodulin (CaM), which in turn binds to and activates myosin light-chain kinase (MLCK). MLCK plays an important role in myosin lightchain (MLC) phosphorylation which is the fundamental step of SM activation. MLC phosphorylation and de-phosphorylation may be seen as the resulting outcome of the two smooth muscle-related enzymes, the phosphorylating MLCK and the de-phosphorylating myosin light-chain phosphatase (MLCP), respectively. MLC phosphorylation enables the attachment of cross-bridges to actin filament binding sites that leads to contraction. SM contraction varies with time; starting at 5–100 ms after the initial stimulus and taking up to 10 s (Singer and Murphy 1987; Somlyo and Somlyo 1992) or even minutes (Arner 1982; Rembold and Murphy 1990), depending on the type of activation and SM. More details on biomechanical aspects of smooth muscle physiology is considered in the recent educational text by Humphrey (2009). 2.2 Chemical model for smooth muscle contraction by Hai and Murphy (1988) The so-called sliding filament theory and the related chemical framework have been well accepted since Huxley (1957) introduced the first version which he developed for striated muscle contraction. The sliding filament theory is the explanation for how muscles produce force (or contract). It explains that muscle contraction occurs as a result of the relative sliding between actin and myosin filaments, and this sliding is caused by cycling cross-bridges. The cross-bridges consist of myosin heads that are attached to binding sites
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on the actin filaments. Several chemical models based on the sliding filament theory have been proposed since 1957. The most common chemical model for striated muscle contraction appears to be due to Huxley and Simmons (1971) where elastic cross-bridges are modeled through two different states; an attached state and a free state that are connected through two rate parameters which are dependent on the contraction. Hai and Murphy (1988) developed a chemical (kinetic) model for SM contraction, which we adopt in the present work. It may be seen as an extended version of the (classical) Huxley–Simmons model. The kinetic model by Hai and Murphy (1988) was developed to describe the crossbridge interactions with the thin filament in SM in which calcium-dependent myosin phosphorylation is the only regulatory mechanism. In the Hai and Murphy model, each myosin head acts independently. The myosin heads may be in four different (functional) states, two non-force generating (detached) states (A and B), and two force generating (attached) states (C and D), compare with Fig. 2. State A describes the free actin and free de-phosphorylated myosin, where de-phosphorylated myosin denotes that the myosin regulatory light chains are not phosphorylated. The myosin phosphorylation is then catalyzed by a complex of calcium (Ca2+ ), calmodulin (CaM), and MLCK, leading to State B. At State B, MLCK has phosphorylated the myosin regulatory light chains but the myosin heads are not yet attached to the actin filament. Thereby through ATP binding and hydrolysis, ATP transforms to adenosine diphosphate (ADP) and inorganic phosphate, i.e., orthophosphate (Pi ), which causes the myosin head to straighten. The rate of transition from the State A to State B is characterized by a rate parameter k1 , which represents the phosphorylation of the myosin regulatory light chains. Conversely, the rate of transition from the State B to State A represents the de-phosphorylation of myosin and can be related to the MLCP activity and expressed by the rate parameter k2 . The first attached state is C where the (phosphorylated) myosin head is attached to a binding site on the actin filament. Then, Pi is released causing a move of the myosin head and due to the attachment to the actin filament, this results in a force-related power stroke followed by a release of ADP (see e in Fig. 2). ATP can then bind to the attached (phosphorylated) myosin head causing the head to release from the actin (see f in Fig. 2), and the myosin head returns back to State B. The cycle B → C → (e → f →) B is the crossbridge cycle and can be described through the rate parameters k3 and k4 representing the attachment and detachment of the fast cycling phosphorylated cross-bridges, respectively (see Fig. 2). The rate parameter k4 describes the transition between C → (e → f →) B. The myosin regulatory light chains are always phosphorylated in State B and C, indicated by the phosphate P. The States B and C with the rate
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parameters k3 and k4 can be compared to the two-state Huxley-Simmons model. Note that e and f in Fig. 2 are only included to better illustrate the kinetics of the cross-bridge cycle; they are not considered in the original model by Hai and Murphy (1988) or in the present analysis. Some myosin heads may remain attached to actin and generate force when de-phosphorylated and this is described as the latched State D. The de-phosphorylation and phosphorylation of the attached myosin heads are described through the rate parameters k5 and k6 , respectively. The latched cross-bridges in State D may slowly detach back to State A. The detachment rate of the latched myosin heads is then characterized through k7 . The fractions of myosin in States A, B, and C are demonstrated by biochemical data, whereas the fractions of myosin in State D are based on mechanical and energetic data. The kinetics of the fractions of myosin in the different states are described through a set of ordinary differential equations expressed here in a matrix form as ⎤ ⎡ −k1 k2 n˙ A ⎢ n˙ B ⎥ ⎢ k1 −(k2 +k3 ) ⎥ ⎢ ⎢ ⎣ n˙ C ⎦ = ⎣ 0 k3 n˙ D 0 0 ⎡ ⎤ nA ⎢ nB ⎥ ⎥ ×⎢ ⎣ nC ⎦ , nD ⎡
0 k4 −(k4 + k5 ) k5
⎤ k7 ⎥ 0 ⎥ ⎦ k6 −(k6 +k7 ) (2.1)
where the superimposed dot denotes the material time derivative. Since n i are fractions, they satisfy the constraint 4 i=1 n i = 1, with n i ≥ 0 and, consequently, n i ≤ 1. For instance, the rate of State A fraction is expressed as n˙ A = −k1 n A + k2 n B + k7 n D . In the model by Hai and Murphy (1988), all cross-bridges are considered initially detached and de-phosphorylated (State A) in relaxed tissues (n A = 1, n B = n C = n D = 0), and by fitting the model behavior to experimental data, the rate parameters can then be determined. It is assumed that the affinities of MLCK and MLCP for detached and attached cross-bridges are similar and, therefore, we set k1 = k6 and k2 = k5 . Note that k1 and k6 are the only two rate parameters which are Ca2+ -regulated in the model. Hai and Murphy (1988) fit the model to experiments performed on swine carotid arteries (Singer and Murphy 1987) and on tracheal SM (Kamm and Stull 1985). The rate parameters may differ significantly depending on the method of activation and the muscle type. It is not certain if the cross-bridges act uniformly or not, but it appears to be an irregular procedure. The cross-bridges may cycle and move with the same rate but some cross-bridges may start prior to others resulting in a different force acting on each cross-bridge.
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Fig. 2 Structure of the model by Hai and Murphy (1988): A Myosin is de-phosphorylated. The intracellular calcium Ca2+ increases and form a complex of Ca2+ , calmodulin (CaM) and myosin light-chain kinase (MLCK). B Myosin is phosphorylated through the Ca2+ -CaM-MLCK complex and an ATP. An additional ATP transforms through hydrolysis to adenosine diphosphate (ADP), inorganic phosphate (Pi ) and energy, which causes the myosin head to straighten; C the (phosphorylated) myosin head attaches to the actin filament. B → C then releases Pi , which, in e, activates the force-related power stroke followed by release of ADP; f ATP can then bind to the attached (phosphorylated)
myosin head causing the head to release from the actin and through ATP hydrolysis return back to State B; D the myosin is de-phosphorylated while remaining attached (latched state). The relationships between the fractions n A , n B , n C , n D of different states are described by the rate parameters ki , i = 1, . . . 7, where k1 = k6 and k2 = k5 . The cycle B → C → (e → f →) B is the cross-bridge cycle, described through the States B, C and the rate parameters k3 , k4 , where k4 describes the average rate between C → e → f → B. In States B and C, the myosin are phosphorylated, which is indicated by the phosphate symbol P
3 A new mechanochemical model for smooth muscle contraction/relaxation
phase). The kinematics of a basic contractile unit is defined and an appropriate strain-energy function is postulated that is able to characterize the energy stored in the contractile units and the surrounding matrix. The related evolution law for the relative sliding between the myosin and actin filaments is proposed. Finally, we discuss the convergence and the thermodynamical aspects of the proposed model.
The smooth muscle cells are modeled to be oriented with their longest axes along the contractile direction and evenly distributed along the depth and height of the muscle tissue. In addition, we assume that the muscle cells contract uniformly and act as a single unit. The SM cells consist of active and passive components. The contractile unit constitutes the active component and consists of two thin filaments (actin) and one thick filament (myosin) in a side-polar structure with cross-bridges attaching these filaments (see the zoom up in Fig. 3). The contractile units are modeled to be arranged in series, separated by dense bodies forming long contractile fibers inside the SM tissue. The surrounding ‘material’ constitutes the passive component such as the nucleus and the cytoskeleton, and the extracellular plexus of elastin and collagen as well as an aqueous ground substance matrix including proteoglycans. They are modeled to act in parallel to the contractile units and are represented by a hyperelastic material in which the contractile units are embedded, see Fig. 3. In this section, we start with the chemical model (state at a fixed mechanical phase) and continue with a detailed description of the mechanical model (state at a fixed chemical
3.1 Chemical model Smooth muscle contraction is activated through an increase of the intracellular calcium concentration Ca2+ (inside the muscle cell) that originates from a flow of calcium ions from the extracellular space (outside the muscle cell) through calcium channels. The regulation of the calcium channels are not described here. In fact, we assume that channels are permanently open and that internal calcium concentration is proportional to the external calcium concentration. As a basis, we use the model by Hai and Murphy (1988) and relate the parameter k1 to the concentration of the calcium– calmodulin complex [CaCaM] and to the half-activation K CaCaM of the CaCaM-dependent phosphorylation rate parameter (Yang et al. 2003a). We define a relationship between [CaCaM] and the external calcium concentration [Ca2+ ]. From this assumption, [Ca2+ ] can be used to trigger
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Fig. 3 Spindle-shaped smooth muscle cells assembled of active and passive components: the contractile unit constitutes the active component, i.e., two thin (actin) filaments and one thick (myosin) filament in a side-polar structure with cross-bridges as shown in the close up on the bottom right; the surrounding matrix constitutes the passive component
surrounding matrix (passive component) contractile filament (active component) dense bodies
k1 as (Michaelis–Menten kinetics) k1 =
[CaCaM]2 2 [CaCaM]2 + K CaCaM
,
[CaCaM] = α[Ca2+ ], (3.1)
where α > 0 is a positive constant. Hence, k1 correlates with the external calcium concentration. Relation (3.1)2 was used as the simplest functional dependency between [CaCaM] and Ca2+ . By using Eq. (3.1) together with the Hai and Murphy model, α can be determined by fitting the results from the Hai and Murphy model for different values of external calcium concentration [Ca2+ ], as will be shown later on. 3.2 Mechanical model We base the mechanical model on the three-element Hill model.
mechanical stiffness. The thin and thick filaments are considered to be much stiffer than the cross-bridges and are hence taken to be rigid. The change in the length of a contractile unit, say 2r , is caused by the relative sliding in the unloaded configuration u rs between thick (myosin) and thin (actin) filaments and the displacement u cb due to the elastic elongation of cross-bridges (the index ‘rs’ stands for ‘relative sliding’ while ‘cb’ means ‘cross-bridge’. Hence, the total relative sliding between the thick and thin filament in the loaded configuration is the difference between u rs and u cb . Note that u rs is taken to be positive for contraction, whereas u cb is positive for extension. The length l of the deformed unit can then be expressed as l = L − 2r,
λf =
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(3.2)
see Fig. 4. From (3.2), the stretch λf = l/L of a contractile unit is
3.2.1 Kinematics Based on studies of the structure of SM cells (Herrera et al. 2005), we adopt a basic contractile unit with length L in the (unloaded) reference configuration, where it is relaxed; see Fig. 4. The thin filaments in the contractile unit are considered to be longer than the thick filament providing a ceratin overlap L m . During contraction, the thin filaments slide relative to the thick filament and cause the length of the contractile unit to decrease to a length l in the (loaded) current configuration. The overlap L m between the thin and thick filaments is considered to be constant (usually the overlap depends on the stretch), and the active stress does not dependent on the muscle length. Effects such as the optimal muscle length is thereby not included in the model. The cross-bridges are located with a constant distance h apart and not restricted to attach to certain binding sites on the thin filaments. The attached cross-bridges both phosphorylated and de-phosphorylated are considered as elastic with the same
2r = u rs − u cb ,
L − u rs + u cb l = . L L
(3.3)
3.2.2 Strain-energy function We assume the existence of a strain-energy function, say , which we define per unit reference volume. In addition, we postulate that the strain energy stored in the SM tissue consists of two parts: (i) the energy stored in the network of contractile units (active component), say a ; (ii) the energy stored in the surrounding matrix (the passive components of the SM cells and the intermixed fibrous components), say p . Thus, = a + p .
(3.4)
Phosphorylated and de-phosphorylated attached crossbridges are approximated to have the same stiffness E f with unit force per length. The force over a single contractile unit
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Fig. 4 Basic contractile unit in the unloaded reference configuration (left) and the loaded current configuration (right): the length of the unit is L in the reference configuration, while it is l in the related current configuration. The overlap between the thin and thick filament L m is
can be expressed as Pf /Nf , where Nf is the number of contractile units per unit area with respect to the reference configuration and Pf is the (averaged) first Piola–Kirchhoff stress over the contractile units subsequently called active stress. The total elastic stiffness of a contractile unit is then the sum of the contributions from all attached cross-bridges. The displacement u cb may thus be expressed as u cb =
Pf Nf
, (3.5) Lm Ef h where n C and n D are the concentrations related to the attached States C and D, respectively, and, again, h is the distance between each cross-bridge. From Eqs. (3.3)2 and (3.5), the active stress Pf can then be written as
Fig. 5 Average forces acting on a single thin (actin) filament governing the evolution law for u¯ rs . The rate u˙¯ rs is proportional to the difference between the driving force Pc /Nf in the actin filament and the reaction force Pf /Nf due to the resistance from the surrounding matrix
(n C + n D )
Pf = μa (n C + n D )(λf + u¯ rs − 1),
(3.6)
where μa = L L m E f Nf / h is treated as a parameter of the active material and u¯ rs = u rs /L is the normalized relative sliding between the thick and thin filaments. Basically, the stress relation (3.6) is decomposed into a chemical part (n C + n D ) and a mechanical part μa (λf + u¯ rs − 1). Physics relates Pf to the energy stored in the network of contractile units a (Holzapfel 2000) so that Pf =
considered constant during contraction. The difference between the relative sliding of thick and thin filaments and the displacement due to the elastic elongation of cross-bridges is 2r
∂a . ∂λf
where (3.8) and (3.9) have been used. 3.2.3 Evolution law
(3.7)
Thus, by integration of Pf , the energy stored in the network of contractile units is μa a = (3.8) (n C + n D ) (λf + u¯ rs − 1)2 . 2 By introducing a unit vector a0 , describing the direction of a contractile unit, the stretch λf of a contractile unit can be described through the fourth invariant I4 of the right Cauchy-Green tensor C and a0 , i.e., I4 = a0 · Ca0 = λ2f .
Hence, the strain energy (3.8) stored in the contractile unit can then be re-defined. The strain energy p = μp (I1 − 3) stored in the surrounding (passive) matrix is simply assumed to follow the hyperelastic neo-Hookean model (see, for example, Holzapfel 2000), where μp is the shear modulus of the passive matrix material and I1 = trC is the first principal invariant. Hence, the strain-energy function (3.4) of SM tissue can be provided as μa = (n C + n D ) ( I4 + u¯ rs − 1)2 + μp (I1 −3) , (3.10) 2
(3.9)
Now, we consider an evolution law for the normalized relative sliding u¯ rs between the thick and thin filaments, which works as an internal state variable. The contractile unit is set to behave like an active dashpot during contraction where the rate u˙¯ rs is proportional to the resulting force acting upon it, i.e., the difference between the driving force Pc /Nf from the cross-bridges and the force Pf /Nf due to the resistance from the surrounding matrix, see Fig. 5. Thus, Pc Pf − . u˙¯ rs ∼ Nf Nf
(3.11)
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Note that the cross-bridges in State D (latched state) are assumed to not work against the driving cross-bridges in State C during contraction. The entities Pc and Pf should be understood as average first Piola–Kirchhoff stresses. Dividing by Nf , number of contractile units per unit area, gives the corresponding average forces acting on a single contractile unit. During contraction, the driving force Pc /Nf is related to the force arising from the driving power-stroke movements of the attached cross-bridges. In the model by Hai and Murphy, this force would relate to the attached phosphorylated cross-bridges of State C, i.e., Pc /Nf ∼ L m / hn C . Thus, we choose Pc = κn C , where κ is treated as a positive material parameter related to the driving force per cross-bridge. The attached cross-bridges in the latched State D are considered to be passive during contraction and do not contribute or resist contraction. The attached cross-bridges are, however, considered to resist extension. When Pf increases to a value larger than the driving force κn C , u¯ rs is assumed to halt until Pf reaches a value higher than κ(n C + n D ). The normalized relative filament sliding u¯ rs will then change proportionally to the difference of κ(n C + n D ) and Pf . The evolution law for the normalized relative filament sliding u¯ rs can then be summarized as ηu˙¯ rs = Pc − Pf ,
(3.12)
version of the model and a state vector according to z = [n A n B n C n D u¯ rs ]T . Since we only use the linearized model in isometric cases, the stretch λf of the contractile units is treated as a constant. The mechanochemical model may then be expressed in the form z˙ = f(z), where f(z) is a nonlinear vector-valued function, which is based on the model by Hai and Murphy (1988) and the evolution law (3.12), (3.13) for the normalized relative filament sliding u¯ rs . The chemical model is (already) linear, and by linearizing the evolution law with respect to u¯ rs , a linearized version of the mechanochemical model is obtained. Thus, in the vicinity of some time t0 and the associated state z0 , the kinetic behavior of the system, including the chemical and mechanical systems, can be expressed as
∂f
z˙ ≈ f(z0 ) + · (z − z0 ) = f(z0 ) + K (z − z0 ) , ∂z z=z0 (3.14) with the system matrix ⎡ k2 0 k7 −k1 ⎢ −(k +k ) k 0 k ⎢ 1 2 3 4 ⎢ ⎢ k3 −(k4 +k5 ) k6 K =⎢ 0 ⎢ ⎢ −(k6 +k7 ) 0 k5 ⎣ 0 0
with for Pf < κn C , Pc = κn C , for κn C ≤ Pf ≤ κ(n C + n D ), Pc = Pf , Pc = κ(n C + n D ), for Pf > κ(n C + n D ),
0
(κ −μa λf )/η
−μa λf /η
⎤
0
⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦
0 0 0 −μa (n C +n D )/η
(3.15) (3.13)
where η is a positive material parameter and Pf is according to (3.7). In Fig. 5, the three (average) forces assumed to act on a single actin filament are indicated. Thus, the driving force Pc /Nf is balanced by the reacting/opposing force Pf /Nf and by ηu˙¯ rs /Nf , where ηu˙¯ rs /Nf may be interpreted as an internal friction in the cross-bridge cycling process. The mechanical model of the SM tissue is then described by the strain-energy function = (n C , n D , I4 , u¯ rs ), particularized in Eq. (3.10), together with the evolution law for u¯ rs , i.e., Eqs. (3.12) and (3.13). Four material parameters (μa , μp , η, κ) have to be determined for the mechanical model when considering that the contractile units (active component) are embedded in a surrounding matrix (passive component) considered to be a neo-Hookean material. Note that all (mechanical) material parameters do have a clear physical meaning.
and λf = λf + u¯ rs − 1. The five eigenvectors and associated eigenvalues of the system matrix are denoted as s1 , . . . , s5 , and 1 , . . . , 5 , respectively, and the solution to the linearized problem (3.14) may be expressed as z=
5
ci exp( i (t − t0 )) + bi si ,
(3.16)
i=1
where ci and bi are constants. The five eigenvalues i of the system matrix define the convergence of the system and may be converted to the time constants τi = −1/ i . 3.4 Thermodynamical aspects In this section, we explore some of the thermodynamical implications of the proposed mechanochemical model. Thus, we consider a piece of SM tissue with a reference volume and treat the volume as an open system with an internal energy e per unit volume. The change of e for this system is then (Holzapfel 2000)
3.3 Linearization of the mechanochemical model
e˙ = Pmech + Q + Pchem ,
It is of interest to study the convergence of the proposed mechanochemical model and especially to compare the convergence of the chemical (sub)model with the mechanical (sub)model. We, therefore, introduce a linearized
where Pmech = P : F˙ is the rate of internal mechanical work (P is the first Piola–Kirchhoff stress tensor and F the deformation gradient), Q is the addition of heat energy per unit time to the system, and Pchem is the increase in chemical
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(3.17)
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energy per unit time. The latter term accounts for the flux of chemical substances across the boundary of the system. For conventional engineering materials, the last term is normally of no concern, but for biological tissues, it is vital. The Clausius–Planck inequality then takes on the form Dint = Pmech + Pchem − e˙ + θ s˙ ≥ 0,
(3.18)
where Dint is the internal dissipation, θ is the absolute temperature, and s is the entropy per unit reference volume. The Helmholtz free energy H is now introduced. This entity includes the strain energy , as introduced in the previous section, but could also include other types of free energies, for example, chemical energy. Using the Legendre transformation H = e − θ s, the Clausius–Planck inequality may be recast into ˙ H − θ˙ s ≥ 0. Dint = Pmech + Pchem −
∂H ˙ ∂H ˙ ∂H ˙ :F+ : ξ. u¯ rs + ∂F ∂ u¯ rs ∂ξ
(3.20)
Inserting this result into the Clausius–Planck inequality yields ˙H Dint = Pmech + Pchem − ∂H ˙ ∂H ˙ : ξ ≥ 0, = Pchem − u¯ rs − ∂ u¯ rs ∂ξ
Pc −
∂H ∂ u¯ rs
∂a ∂H ˙ − u˙¯ rs = ηu˙¯ rs + u¯ rs ∂λf ∂ u¯ rs = ηu˙¯ 2rs ≥ 0.
(3.23)
Hence, the proposed model is thermodynamically consistent provided that η > 0.
4 Results The model was fitted to experimental data performed on intact SM obtained from guinea pig taenia coli (Arner 1982). That study was primarily chosen due to its detailed description of both the chemical and the mechanical experiments. Note that Arner (1982) performed experiments on both skinned and intact SM.
(3.19)
Subsequently, we ignore thermal effects (θ˙ = 0) such that the last term in (3.19) vanishes. In the proposed constitutive model, there are essentially two mechanical internal variables: the deformation gradient F and the normalized relative sliding u¯ rs between actin and myosin filaments. The chemical state may, in principle, be characterized by a state variable vector ξ , which will hold the variables n A , n B , n C , and n D plus other variables necessary to characterize the chemical state. Thus, H = H (F, u¯ rs , ξ ), and the time derivative of H then becomes ˙H =
(3.21)
where we have made use of the definition P = ∂/∂F. We emphasize that muscle contraction is completely dependent on an influx of chemical energy Pchem , but in general, both Pchem and ∂H /∂ξ : ξ˙ are unknown (or they are at least very difficult to identify and measure). However, the active contraction power in the contractile units is Pc u˙¯ rs , and this power must be taken from the chemical energy. The inequality Pc u˙¯ rs ≤ Pchem − ∂H /∂ξ : ξ˙ must, therefore, hold. The mechanical part of the model will be thermodynamically consistent if the inequality ∂H ˙ ∂H ˙ ˙ Pc u¯ rs − (3.22) u¯ rs = Pc − u¯ rs ≥ 0 ∂ u¯ rs ∂ u¯ rs is fulfilled. In the present model, we have the relationships Pc = ηu˙¯ rs + Pf = ηu˙¯ rs + ∂a /∂λf and ∂a /∂λf = ∂a /∂ u¯ rs = ∂H /∂ u¯ rs so that (3.22)2 takes on the (entropy) inequality form
4.1 Fitting the chemical model Arner (1982) activated intact SM tissues with different external calcium concentrations. From these results, a relationship between the external calcium concentration [Ca2+ ] and the normalized steady state active stress (in percent) was obtained, see the experimental data in Fig. 6. The proposed chemical model was fitted to experimental results by Arner (1982) using Eq. (3.1) together with the Hai and Murphy model. For a certain value of external calcium concentration, the rate parameter k1 was obtained through Eq. (3.1), which was used in the Hai and Murphy model to acquire the steady state value of (n C + n D ). Consequently, the parameter α of Eq. (3.1)2 was fitted by comparing (n C + n D )/(n C + n D )max with the experimental active stress Pf /Pmax in Arner (1982) for different values of external calcium ion concentration. The related plot of the model results can be seen in Fig. 6 in addition to the fraction chemical states plotted as a function of time at an external calcium concentration of 2.5 mM. The half-activation constant for the CaCaM-dependent phosphorylation rate K CaCaM was set to 1.78·10−7 M, which is according to Yang et al. (2003a), and the constant α in Eq. (3.1)2 was estimated to α = 35 · 10−6 . For an external calcium concentration of 2.5 mM and by assuming fully opened calcium ion channels, the calcium related rate parameter k1 was calculated to be 0.1946. The values for the other rate parameters were set to k6 = k1 , k2 = k5 = 0.5, k4 = 0.1, k3 = 4k4 , and k7 = 0.01, which is according to a swine carotid SM (Hai and Murphy 1988). No recordings of rate parameters for bladder smooth muscle were found. 4.2 Fitting the mechanical model The mechanical experiments that are used to identify the model parameters can be divided into two types, i.e.,
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(a)
Fig. 6 Relationship between the active stress Pf /Pmax in percent of the maximum active stress and the external calcium concentration [Ca2+ ]. The active stress in percent is equivalent to (n C + n D )/(n C + n D )max in percent and is obtained through Eq. (3.1) and the Hai and Murphy
isometric and isotonic (Arner 1982). In the isometric contraction experiments, the muscle strips are mounted in a glass cup where one end is fixed and the other end is attached to a force transducer and maintained at fixed length. Through comparison of the maximal isometric stress and the maximal measured isometric force given by Arner (1982), the average cross-section of the specimens was estimated to a value of 0.07 mm2 . The pre-stretch of the specimens was estimated to a value of λ = 1.008 by using the given preload of P0 = 3 mN and assuming a hyperelastic neo-Hookean passive matrix in a uni-axial boundary condition, i.e., P0 = ∂p /∂λ and I1 = λ2 + 2/λ. From the isotonic quickrelease experiment (Arner 1982), the time-dependent stretch behavior is obtained for different afterloads, i.e., different constant loads working against the sudden stretch change during the quick release. From the stretch behavior information, the elastic recoil is extracted, i.e., the sudden change of stretch due to the elasticity of the cross-bridges and stretch rate. After setting the afterload equal to the active force, the length of the muscle strip remains constant and the elastic recoil and the stretch rate are zero. To simulate the isometric contraction experiment, the model was initially pre-loaded with the corresponding prestretch (λ = 1.008), as in the experiment, and then activated by assuming ‘maximal flow’ through the calcium ion channels from an external source of calcium with a concentration of [Ca2+ ] = 2.5 mM (corresponding to k1 = 0.1946 in the Hai and Murphy model); see Fig. 7 for the simulated active stress results and the related experimental data. Through several isotonic quick-release simulations with different afterloads, the related behavior of isotonic stretch λf of the contractile units versus time may be obtained, as illustrated in Fig. 8. The plot on the top right within Fig. 8 is a zoom up of the encircled region for an afterload of 50 kPa. The visualized
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(b)
model (a); fraction chemical states solved from the Hai and Murphy model as function of time at an external calcium concentration of 2.5 mM using the rate parameters k1 = k6 = 0.1946, k2 = k5 = 0.5, k4 = 0.1, k3 = 4k4 , and k7 = 0.01, with n A +n B +n C +n D = 1 (b)
Fig. 7 Active stress Pf behavior during isometric contraction of smooth muscle obtained through Eq. (3.6). The contraction is activated by setting the external calcium concentration [Ca2+ ] = 2.5 mM and allowing maximally flow through the calcium channels
stretch–time curve starts with a very small negative slope, and this slope decreases toward zero in a monotonic manner. The dotted curve therein indicates the characteristic experimental response according to Arner (1982).From these results, the elastic recoil λf , defined to be the old stretch minus the new stretch, is obtained for different afterloads, and is compared with the experimental data (see Fig. 9). By fitting the model to the experimental data (presented in Figs. 7 and 9), the parameters were estimated to be μa = 4.5 · 106 Pa, μp = 0.9 · 106 Pa, η = 60 · 106 Pa s, and κ = 0.93 · 106 Pa. 4.3 Evolution of the internal state variable u¯ rs The internal state variable u¯ rs quantifies the average normalized relative sliding between the thick and thin filaments.
A calcium-driven mechanochemical model for prediction
Fig. 8 Behavior of isotonic stretch λf of the contractile units versus time for different afterloads (starting from 0 at the bottom to 50 kPa at the top, with an increment of 10 kPa). The plot on the top right is a zoom up of the encircled region for an afterload of 50 kPa. The dotted curve therein indicates the characteristic experimental response according to Arner (1982). No suitable experiments were available for comparison
Fig. 9 Elastic recoil λf after isotonic quick-release as a function of the afterload. Intact smooth muscle is isometrically activated. Comparison of model results with experiments (Arner 1982)
In Fig. 10, the evolution of u¯ rs is demonstrated for the isometric test, followed by an isotonic contraction. During the isometric regime, the state variable u¯ rs goes from 0 up to 0.0584. After 200 s, the isotonic regime starts. During a typical isotonic quick-release experiment, u¯ rs increases up to a value of about 0.1, see Fig. 10. 4.4 Convergence of the mechanochemical model The convergence of the chemical and the mechanical parts of the proposed that mechanochemical model can to some extent be quantified in terms of the time constants of the linearized system, as defined in Eqs. (3.14), (3.15). More specifically with convergence, we mean here how fast/slow each of the (sub)models in the linearized system converge to their steady-state values for a constant calcium level.
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Fig. 10 Evolution of the normalized relative sliding u¯ rs between actin and myosin filaments (contraction): isometric behavior is displayed for t = 0–200 s and the isotonic contraction is displayed for t = 200– 400 s. The isotonic contraction is performed at an afterload of 50 kPa. No suitable experiments were available for comparison
Under isometric conditions and in the vicinity of some state z0 , the convergence of the system may be quantified by the time constants τ1 to τ5 . During the simulation of the isometric test, the linearized version of the proposed model was analyzed. For each time step, z0 was taken as the current state, and the evolution of the eigenvalues i could then be computed. Four eigenvalues i of the system matrix K were negative ( 1 = −1.99, 2 = −0.66, 4 = −0.084, and 5 = −0.054 at mechanical steady-state), and one was zero ( 3 ). The eigenvalues 1 , . . . , 4 are constants and relate to the chemical model (note that the chemical system is linear), while 5 relate to the mechanical model. From the four negative eigenvalues, the time constants τi = −1/ i were then computed to be τ1 = 0.92 s, τ2 = 1.51 s, τ4 = 11.86 s and τ5 = 18.45 s at mechanical steady-state. The evolution of these time constants for each of the submodels is illustrated in Fig. 11. The solid lines relate to τ1 , τ2 and τ4 , while the dashed curve relate to the remaining time constant τ5 of the mechanical model, which reached a steady-state behavior within about 50 s during the isometric simulation. In Eq. (3.16), an analytical solution to the linearized problem is provided. The eigenvalues and the associated eigenvectors govern the convergence rate of the chemical state variables and the internal state variable. The eigenvectors in the four first terms of (3.16) have only non-zero components (that evolve with time), whereas the fifth eigenvector is s5 = [0; 0; 0; 0; 1]. Since τ5 is always the largest time constant, the mechanical system will converge slower than the chemical system. Figure 11 also illustrates that, initially, the mechanical model is considerably slower than the chemical one. The convergence of the mechanical model depends strongly on the two state variables n C and n D , and as these
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Fig. 11 Evolution of the time constants for the chemical and the mechanical (sub)models: the three horizontal solid lines are related to the convergence of the chemical model indicated by the constants τ1 , τ2 , τ4 (τ3 → ∞); the dashed curve is related to the mechanical model which reaches a steady-state behavior within about 50 s during the isometric simulation
two state variables increase so does the convergence of the mechanical model, as seen in the initial part of the plot in Fig. 11. The time constant related with the mechanical model decreases to a value of 18.45 s. Thus, the mechanical model remains slower than the chemical one throughout the domain of analysis, but at steady-state, the convergence rate of the two systems is of the same order.
5 Discussion and summary We proposed a new constitutive model for the coupled chemical and mechanical behavior of SM tissue that is an important constituent in the vascular system. We analyzed the thermodynamic properties of the proposed mechanochemical model and concluded that it fulfils the second law of thermodynamics, which is responsible for the direction of the energy transfer process. The model by Hai and Murphy (1988) was adopted to describe the chemical link between the calcium ion concentration and the muscle activation. The chemical model considers the rate parameters between four different states as average constant values. The rate parameters k1 and k6 , which describe the phosphorylation rate of myosin (by MLCK), are set as a constant value. Most likely these rate parameters vary with time due to their relation with the intracellular calcium concentration. In addition, in reality, the channels are not permanently open so that the intracellular calcium concentration varies with time after activation, which in turn would also affect the rate parameter k1 . The values for k1 and k6 used in the present analysis can be seen as a steady-state value because they were estimated by comparing to the steady-
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state value of the active stress for a specific external calcium concentration. Due to these conditions, a simple linear relationship between the concentration of CaCaM and external calcium concentration was chosen. The rate parameters k2 and k5 describe the de-phosphorylation rate of myosin by MLCP. In the literature, it has been discussed whether or not the activity of MLCP is connected to the calcium concentration and/or to the nitric oxide, which is not considered in the present model. The rate parameters k3 and k4 describe the cross-bridge cycle. These two parameters can be compared to the rate parameters of the two-state Huxley–Simmons model. In cardiac and skeletal muscles, the sarcomere lengths are coupled with the calcium sensitivity (Endo 1972; Hibberd and Jewell 1982). With increasing calcium concentration, the sarcomere contracts, and with increasing contraction, and thereby shorter sarcomere length, the calcium sensitivity increases. By approximating the rate parameters as average constant values and not as a function of the deformation might affect the contractile behavior, especially the rate parameter k1 . This may also help to explain why the model did not manage to simulate the initial behavior of the isometric contraction (see Fig. 7). By assuming that the latched state exists, after a certain contraction,the cross-bridges in the latched state will reach a maximal length leading to a detachment from their binding sites. This is described through the rate parameter k7 which could, in principle, also be seen as a function of the contraction. Note that there have also been studies in the literature that have found SM contraction to be independent of both calcium phosphorylation and myosin phosphorylation (Sato et al. 1992; Walsh et al. 1996). The rate parameters may in fact be functions of the calcium ion concentration and the contraction. For example, k3 and k4 have been proposed to be a function of the contraction, see the studies on airway smooth muscles (Fredberg et al. 1999; Wang et al. 2008). In general, the main difficulty with the rate parameters is to understand their behavior and how they depend on the contraction or other parameters. Strictly speaking, the model of the cross-bridge kinetics by Hai and Murphy (1988) is only valid for isometric contraction. It is one of our model assumptions that the chemical model may be applied to cases where the isometric requirement is not fully met. For the mechanical part of the mechanochemical model, a new constitutive model based on Hill’s threecomponent model approach was proposed. In the mechanical model, we assumed that the contractile units (active component) are embedded in a surrounding matrix (passive component). The model includes one internal state variable that describes the contraction of the contractile units. The model is characterized by a strain-energy function and an evolution law. The mechanical model has four material parameters by considering the passive component as a neo-Hookean material. All (mechanical) material parameters have a clear physical meaning. For the derivation of the strain-energy function
A calcium-driven mechanochemical model for prediction
(3.8) of the network of contractile units, we have made the simplest possible assumption namely that the stiffness E f of the cross-bridges is constant, i.e., independent of stretch. This assumption enabled a straightforward expression for a . Alternative forms need to be investigated. When compared to the experimental data obtained from tests on intact SM from guinea pig taenia coli, the mechanochemical model was able to match the data. This pertained to both isometric and isotonic tests. To identify the stiffness of the cross-bridges and the passive surrounding, the model behavior of the elastic recoil and u¯ rs were studied. However, it should be noted that experimental data were only available for the elastic recoil and not for u¯ rs . Some deviations from the experimental data could, however, be detected, which are now briefly discussed. By comparing the predicted model results with the data from the isometric contraction test at fixed [Ca2+ ], the model showed a (slightly) slower initial active stress increase than the experimental data, compare with Fig. 7. It appeared that this mismatch could not be corrected by any other set of (mechanical) model parameters. As illustrated in Fig. 11, the convergence of the mechanical model depends on the number of attached cross-bridges, and if that number increases too slowly, then the mechanical model is not able to produce force at a rate fast enough to match the experiments. Thus, the mismatch in the isometric contraction test might be caused by the low constant value of k1 in the chemical model. In addition, the experimental data from the elastic recoil experiments showed a somewhat nonlinear decay while the model predicted essentially a linear behavior (Fig. 9). This suggests that the surrounding matrix does not comply with the neo-Hookean model, as used here. Above all, there is most likely a kind of viscous property present in the surrounding matrix, which was not accounted for by the current model. Further evidence for such a viscous property may be appreciated by considering the initial stretch behavior in quick-release experiments as performed, for example, by Arner (1982) (see Fig. 3 therein). The characteristic relationship between the stretch λf of the contractile units and the time can thereby be divided into three parts (see the dotted curve in the top right plot of Fig. 8): (i) a distinct change in the stretch, i.e., the elastic recoil; (ii) a short-time ‘damping response’; (iii) a steady-state shortening which is thought to be the actual isotonic contraction of the contractile units. The isotonic contraction of the contractile units occurs since the afterload opposing the contraction is set to a lower constant value, which allows the cross-bridges to attach further along the thin filament and contribute to further contraction. The model is able to predict the elastic recoil and the subsequent isotonic contraction but, as can be seen from the zoom up in Fig. 8, the second step, i.e., the short-time ‘damping response’ (dotted curve in the plot), cannot be recovered by the proposed model.
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As a further consequence, the mechanochemical model developed for the response of the actin–myosin complex is not able to predict the profiles of the stretch rates (often referred to as velocity), as presented by, for example, Arner (1982) (see Figs. 4 and 5 therein). In particular, the stretch rates predicted by the model underestimate the experimentally obtained values. It is interesting to note that the incorporation of a viscous component in the (passive) strain-energy function p allows the prediction of the experimentally obtained stretch rates. Smooth muscle cells like several other cells have a complex cytoskeleton that may also contribute to the overall mechanochemical response. How the contractile units and cytoskeleton interact with each other is still an open question; however, the cytoskeleton may also contribute to a viscoelastic behavior. The proposed calcium-driven mechanochemical model is able to predict force generation in SM. It is straightforward to implement it into a finite element program and to study more complex problems in vascular mechanics such as the changes in lumen diameter of vessels which occur due to SM activation. Acknowledgments The authors wish to thank Professor Anders Arner from the Karolinska Institutet in Stockholm for the constructive discussions. Financial support for SM was provided through a Project grant (# 2005–6167) from the ‘Swedish Research Council’ (VR). This support is gratefully acknowledged.
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