It has been shown, in animal models, that gastrointestinal tract (GIT) motility is influenced by temperature; nevertheless, the basic mechanism governing thermal GIT smooth muscle responses has not been fully investigated. Studies based on physiologically tuned mathematical models have predicted that thermal inhomogeneity may induce an electrochemical destabilization of peristaltic activity. In the present study, the effect of thermal cooling on human colonic muscle strip (HCMS) contractility was studied. HCMSs were obtained from disease-free margins of resected segments for cancer. After removal of the mucosa and serosa layers, strips were mounted in separate chambers. After 30 min, spontaneous contractions developed, which were measured using force displacement transducers. Temperature was changed every hour (37, 34, and 31°C). The effect of cooling was analyzed on mean contractile activity, oscillation amplitude, frequency, and contraction to ACh (10−5 M). At 37°C, HCMSs developed a stable phasic contraction (∼0.02 Hz) with a significant ACh-elicited mean contractile response (31% and 22% compared with baseline in the circular and longitudinal axis, respectively). At a lower bath temperature, higher mean contractile amplitude was observed, and it increased in the presence of ACh (78% and 43% higher than the basal tone in the circular and longitudinal axis, respectively, at 31°C). A simplified thermochemomechanical model was tuned on experimental data characterizing the stress state coupling the intracellular Ca2+ concentration to tissue temperature. In conclusion, acute thermal cooling affects colonic muscular function. Further studies are needed to establish the exact mechanisms involved to better understand clinical consequences of hypothermia on intestinal contractile activity.
- human colonic smooth muscle strips
- thermal cooling
- postoperative ileus
- thermochemomechanical coupling
- mathematical model
it is well known that temperature can influence the contractile activity of smooth muscle, acting at different levels (33, 47). Gastrointestinal tract (GIT) motility in animal models seems to be influenced by temperature: cooling of isolated rat GIT smooth muscle preparations rapidly induced a tonic contractile response inversely proportional to the temperature (46).
During major abdominal surgery, both deep anesthesia and operating theatre low temperature can significantly alter the thermal state of the tissue for a long period, probably playing a role in the pathogenesis of postoperative ileus (POI) (15, 32, 40, 70). POI is a multifactorial disease (4, 61), being the result of many interacting factors such as surgical trauma, local inflammatory reactions to intestinal manipulation, overactivation of the sympathetic nervous system due to several autonomic, hormonal, and behavioral responses to stressful regional stimuli, and low temperature (11, 22, 32, 41, 57). More recently, it has been shown that tissue damage and inflammatory cytokine release into the circulation are important factors determining the severity of POI, which most likely can contribute to the impaired motility of the nonmanipulated intestine (69). Less investigated is the role that temperature variations during surgery determine POI in humans.
Such a cooling-induced contraction appears not to be linked to a neural process, involving receptor activation and the release of neurotransmitters or mediators, but a myogenic response (47). Nevertheless, the basic mechanism governing thermal GIT smooth muscle responses has not been fully investigated in experimental measurements in in vivo and in vitro models. Studies based on physiologically tuned mathematical models have predicted that thermal cooling may induce an electrochemical destabilization of peristaltic activity (29).
The objective of the present study was to assess the effect of thermal cooling on human colonic smooth muscle strip (HCMS) contractility and to analyze the different responses of both circular and longitudinal smooth muscles (1). We statistically quantified basal tone, contraction frequency, and amplitude variations. Moreover, we proposed a thermochemomechanical model for intestine smooth muscle contraction reproducing the experimental recorded thermal effects by extending the mechanochemical formulation proposed by Stålhand et al. (64).
MATERIALS AND METHODS
Muscle specimens were obtained from the healthy margins of colonic resections from 10 patients with adenocarcinoma of the colon (6 men and 4 women, age range: 53–72 yr) treated at the Campus Bio-Medico University of Rome between September 2012 and May 2013. None of the patients had a history of colonic motility or a neuromuscular or collagen disorder; specimens found with diverticula were excluded. Hemicolectomy was performed in all patients, and a specimen from the resected colon was obtained at a distance from the area involved by the carcinoma.
Specimens were immediately taken to the laboratory in oxygenated, chilled Krebs solution containing (in mM) 116.6 NaCl, 21.9 NaHCO3, 1.2 KH2PO4, 5.4 dextrose, 1.2 MgCl2, 3.4 KCl, and 2.5 CaCl2. The experimental protocols were approved by the Ethics Committee of Campus Bio-Medico University of Rome, and written informed consent was obtained from all individuals before surgery.
After removal of the mucosa and submucosa layers, colonic smooth muscle was cut into small strips (10 mm long by 2 mm wide, weight: 0.7 ± 0.3 g) by sharp dissection, identifying the circular and longitudinal orientation according to the position of the taenia coli. Strips were mounted in separate 10-ml muscle chambers as previously described (6). Strips were initially stretched to 20 g of load to bring them near conditions of optimum force development and equilibrated for an additional 30 min after continuous perfusion with oxygenated Krebs solution. The solution was equilibrated with a gas mixture containing 95% O2 and 5% CO2 −5 M ACh) to analyze ACh-induced contraction variations during thermal cooling. Temperature values were established on the basis of intraoperative colonic temperature measurements during surgery. A preliminary histopathological evaluation of the dissected mucosa and submucosa was also conducted. To perform histopathological evaluation after the thermal cooling protocol, smooth muscle strips were fixed in buffered formalin for 12 h and embedded in paraffin with a melting point of 55–57°C. Three- to four-micrometer sections were cut and stained with hematoxylin-eosin stain for morphological analysis.
Calculated force amplitude and frequencies within the defined range of interest for each experiment were analyzed as both time variations and averages. Fast Fourier transform (FFT) analysis was also applied for a fine identification of the main contraction frequencies that varied due to both temperature and ACh effects. A Wilcoxon signed-rank test for paired observations (hereafter termed Wilcoxon test) was used to test the significance of the measurement variations between 31 and 37°C on the same tissue (P < 0.05). This choice was motivated by non-normal data distributions and by the low number of samples considering matched-paired observations, which makes the median more informative than the mean. Significant differences were therefore studied in terms of the medians of the groups. The following two different data normalization procedures were also implemented on mean values of the exerted force to gain a fine characterization of the statistical significance of the analysis: 1) specimen normalization according to the force exerted by the strip at 31°C without ACh (Norm 1) and 2) percent normalization according to the absolute minimum and maximum force exerted by the strip (Norm 2). A further level of analysis was also introduced considering paired measurements on the same strip. Principal component analysis (PCA) was finally performed to highlight segregation patterns and to identify data correlations. Means were calculated; means ± SE are reported unless means ± SD are specified.
Thermochemomechanical contraction model.
Electrochemical destabilizations of myogenic activity due to thermal inhomogeneities have been recently considered as a possible cofactor for the pathogenesis of POI through a physiologically based mathematical model (29). Accordingly, similar investigations have been performed applying the same reasoning to different tissues, thus building predictive mathematical models on experimental evidence (8, 9, 25). In this context, inhomogeneity and anisotropy are well known to induce complex electrical spiraling behaviors in many biological contexts, such as the heart (26) and brain (17, 42), affecting the action potential propagation velocity as well as the entire electrophysiological dynamics (7). Dysrhythmias, in particular, have been theoretical predicted and experimentally recorded in the stomach and intestine (2, 29, 36, 50, 65) by making similar observations to what is known in brain and heart tissues (18, 28, 55). It is well known that the contractile rhythm of intestinal smooth muscle is due to the pivotal role of interstitial cells of Cajal (ICCs), which form a network of cells between the circular and longitudinal muscle layers making contact with surrounding smooth muscle cells and acting as a pacemaker of spontaneous motility in the gut (33). Advanced experimental tools have been developed in this direction (24, 48, 49, 52), leading to a fine characterization of electrical propagating fronts. Noninvasive measurements are currently being investigated to extend the diagnostic analysis to a wider clinical audience (13, 62).
From a biomechanical point of view, characterizing the complete response of soft biological media is still an open problem, although there have been recent technological improvements (58, 59, 66), and these tools are fundamental for the tissue specimen tests performed in the present work (5). The theoretical formulation proposed here is based on a continuum thermodynamical framework for smooth muscle cell contraction (44, 63) and was applied to HCMS modeling. Such an approach has received growing attention due to its general statements and applicability in a wide range of smooth muscle tissue models (61, 64). Explicit derivation and details of model equations are given in the appendix.
Smooth muscle cells contract thanks to a complex chain of mechanical, electrochemical, and metabolic stimuli (34). The electrochemical states involve 1) depolarization of the cell membrane, 2) binding of Ca2+ to calmodulin, 3) phosphorylation of myosin chains, and 4) actin-myosin cross-bridge activation. The last phase is thought to be a continuous cycling with repetitive attaching of phosphorylated myosin to actin heads, thus generating a force-related power stroke. The active force and contraction, thus, are the result of the combined emerging effects of the ensemble of cross-bridge cycles. Moreover, as for all the excitable and deformable media of the human body, mechanical stimuli induce an important impact on the force generation (31, 56, 71). In particular, cell stretching can increase the phosphorylation rate and alter the tissue sensitivity to Ca2+ concentration.
In this work, we introduce a simplified thermodynamic relationship able to modify the ionic variables dynamics according to the thermal bath considered. In particular, a linear coupling between temperature and intracellular Ca2+ concentration was established as a first approximation. Such a procedure allowed us to set a complete feedback between Ca2+ dynamics and mechanical variables via stretch and, at the same time, the coupling between temperature gradients and stress variations via Ca2+. In detail, the intracellular Ca2+ concentration ([Ca2+]i) is no longer assumed constant but is linearly linked to the thermal bath temperature (T) according to the following equation: (1) where [Ca2+]io is a reference initial [Ca2+]i typical of normal tissue at 37°C, T0 is the reference tissue temperature, and α is a slope parameter. The simplified proposed relation states that the intracellular Ca2+, ruling the activation of the contraction process, varies according to the thermal state reached by the tissue, thus inducing different stretch and stress levels in HCMSs for different thermal baths.
The linear formulation adopted was based on the experimental protocol implemented (see materials and methods), where the stepwise variation of the thermal bath was interpolated after both increasing or decreasing law. Below, we show the agreement between this simplified modeling approach with the experimental evidence.
Statistical trend analysis.
Our data set was composed of a total of 253 samples obtained from 40 strips of human colon, with 165 samples analyzed without ACh and 88 samples in the presence of Ach. Due to the high variability of the observed frequencies, we used 120 samples for the evaluation of the average oscillation frequency in longitudinal-oriented HCMSs and 107 samples in circular-oriented HCMSs. A preliminary histological evaluation showed no necrotic signs or significant inflammatory infiltrates in the specimens examined.
Missing values were selected if the calculated period was >360 s. The smallest data set at 39°C confirmed the observations done for the other three temperatures and is shown in the figures as well.
Figure 1 shows a comparison of averaged normalized mean values in Norm 1 and oscillation amplitude and frequency of the force exerted with respect to temperature variations in the presence and absence of ACh and distinguishing between longitudinal- and circular-oriented HCMSs. As shown in Fig. 1A, a decrease of the temperature resulted in a clear increment of the mean value for the two strip orientations, which was more significant in the presence of ACh than in the absence of the cholinergic stimulus. In fact, at 37°C, HCMSs developed a stable phasic contraction with a visible ACh-elicited contractile response, 31% and 22% higher than the baseline in the circular and longitudinal axis, respectively. At 31°C, the absolute highest difference was observed with the ACh-elicited response equal to 78% and 43% compared with baseline in the circular and longitudinal axis, respectively. The Wilcoxon test between these two temperatures revealed P < 0.05 for the total data set (a complete list of statistical test results is shown in Table 1).
Thermal bath cooling also induced an observable decrease in the oscillation amplitude, as expected, with the exception of the longitudinal-oriented strips, as shown in Fig. 1B. In particular, the Wilcoxon test between 31 and 37°C for the oscillation amplitude resulted in significant differences considering all data (P < 0.005) and longitudinal-oriented strips (P < 0.05) in the presence of the ACh-elicited response. More pronounced global variation due to temperature setup is shown in Fig. 1C in terms of oscillation frequency. In this case, when the temperature was decreased, a decrement of the frequency (equivalently an increment of the period) was observed for both strip orientations. These differences (between 31 and 37°C) were statistically significant for the ACh-elicited response and for the basal tone considering all data (P < 0.05 and P < 0.005, respectively). A significant difference was also obtained for the basal tone frequency considering the circular-oriented strips (P < 0.05).
Figure 2 shows the comparison of median averaged normalized mean values in Norm 1 and oscillation amplitude and frequency of the force exerted with respect to temperature variations in the presence and absence of ACh for the full data set. The boxplot representation was adopted to show data distributions at each temperature and was motivated by the fact that the median gains more robustness with respect to the mean in the presence of outliers. Moreover, this representation was in line with the statistical test adopted characterizing the differences in the medians of two groups of paired observations. In this case as well, the partial observations done for the longitudinal- and circular-oriented strips were confirmed for the overall data set.
In summary, when the temperature was decreased, a reduction of the baseline, oscillation amplitude, and frequency was observed, whereas an increase of the ACh-elicited contractile response was registered.
Two-dimensional PCA scores and loading plots on the first two principal components are shown in Fig. 3. The information recovered was equal to the 60% of the total information gained from the data set. Temperature was positively correlated with oscillation amplitude and negatively correlated with normalized means and periods. The negative correlation with the mean was probably due to the dominating trend of the ACh-elicited response, in which the tone mean value decreased for temperature increments. The solid and open circles in Fig. 3 further distinguish between the different point distributions for ACh-free and ACh-stimulated HCMSs, highlighting that the differences were focused within period-mean directions.
Figure 4 shows two representative sequences of the adopted experimental protocol. In particular, the top traces in Fig. 4 show the force variation with respect to the sole temperature, whereas the bottom traces in Fig. 4 show the same variation comparing the coupled effects of ACh and temperature.
These trends were in agreement with the statistical analysis previously discussed. Figure 5 shows two representative examples of different active contraction at 34°C together with their corresponding FFT spectra. In Fig. 5A, the clear frequency peak was centered at 0.038 Hz (period: 26.4 s), highlighting a very active phase with an amplitude of ∼4 g, whereas in Fig. 5B, a wide range of the spectrum was invaded (period: 15.5–82.0 s) with a corresponding small oscillation amplitude of the recorded pattern at ∼1.2 g. These examples highlight the strong variability of the frequency response recorded in the HCMS specimen data set.
Interestingly, in a limited number of cases, we observed peculiar patterns similar to those known for other biological systems, i.e., cardiac electrical arrhythmias (28) and pancreatic β-cell electrical activity (53). Figure 6, left, shows the high variability of the phasic contractions in terms of amplitude, frequency, and waveform at three different temperatures. Figure 6, right, shows alternating patterns and burst-like behaviors varying between long and short oscillating phases. These observations unveil how critical our understanding of the overall complex dynamics of the intestine tissue is still.
The thermochemomechanical model presented in above was able to reproduce the experimental observed phenomena in terms of mean stress state varying with temperature. Figure 4C shows the time variation of mean stress by applying the same experimental thermal protocol in the model (31-34-37-34-31°C) compared with the basal tone recorded for representative tissue preparations. In particular, the model was able to catch the increasing of the tone as temperature lowered with a short accommodation time. Such a fast dynamic stabilization of mean stress allowed an optimal tuning of the model onto the desirable stress state. This feature underlines the wide range of applicability of the model toward different thermal protocols inducing different tissue stress regimes. Moreover, nonlinear Ca2+-temperature coupling would result in similar stress-temperature dynamic analysis for the mean stress state.
Finally, Fig. 7 shows the stress-strain response for five different Ca2+ concentrations fine tuned on different thermal baths. The resulting nonlinear relations changed according to Eq. 1 and highlight the physical hyperelastic nature of the model, which is typical of fiber reinforced biological media. The normalized stress at zero strain (stretch = 1) increased nonlinearly with the Ca2+ concentration, reaching a saturation level for Ca2+ concentration of ≈11 μM; when the stretch was increased into the tissue, the resulting normalized stresses similarly tended to the maximum allowable value but followed very different patterns. These features are very important for modeling applications, with more complex Ca2+ dynamics coupling slow and fast oscillations of the voltage membrane as well.
Data emerging from the present study show that lowering of temperature induced a significant stepwise increase of the tonic contractile response in human colonic smooth muscle, which was more significant in the presence of ACh than in the absence of the cholinergic stimulus. Moreover, cooling significantly decreased the frequency and amplitude of rhythmic contractions, which was more evident in circular than in longitudinal muscle layers. When temperature was restored up to 37°C, the tone, amplitude, and frequency were rapidly restored, suggesting that a morphological alteration during the cooling period does not occur, as reported in another similar experimental setup (37). This evidence was confirmed by the histological evaluation, in which no necrotic signs or significant inflammatory infiltrates were detected in the specimens examined.
Our results are in agreement with previous findings in which Mustafa et al. (47) reported that cooling of intestinal smooth muscle results in an increased force of contraction, with a significant reduction of the frequency and amplitude of the spontaneous activity. The authors (47) tested, in an animal model, a large scale of temperature (from 37 to 5°C) and showed a rapid and reproducible stepwise increase in the level of tone that was inversely proportional to the temperature. In the present study, temperature values were established on the basis of intraoperative colonic temperature measurements during surgery that ranged between 31 and 37°C and were in line with the animal model evidence. Moreover, Mustafa et al. studied the effect of cooling in various segments of the intestinal tract and showed the same response in the esophagus, stomach, duodenum, jejunum, and colon.
As stated above, in animal models, the contractile responses to cooling were not influenced by pretreatment with either muscarinic and histamine H1 antagonists or neural blockers (capsaicin and tetrodotoxin), suggesting that it is likely not mediated by a neurogenic response but by a myogenic mechanism (47). The same authors demonstrated that cooling affected Ca2+ homeostasis by interfering with processes of translocation of extracellular Ca2+. For this reason, the mathematical model formulated here takes into account the linear relationship between Ca2+ concentration and temperature. It is known that temperature affects both ionic conductances and rate constants modulating action potential propagation. We hypothesized that temperature could influence the chemomechanical response of intestinal muscle. Confirming this theory, animal studies and clinical trials have demonstrated that the duration of POI is significantly decreased in patients undergoing laparoscopic instead of open procedures for colon surgery, in which the abdominal organs are more exposed to manipulation and intraoperative thermal cooling (12, 35, 62). On the other hand, recent studies have shown that there are not significant clinical differences in POI outcomes using warmed humidified laparoscopic gas versus dry cold gas during laparoscopic colectomy and appendectomy (60, 75), hypothesizing that thermal changes after main abdominal surgeries may be transitory and that they don't permanently damage the tissue for longer periods.
In summary, these findings may provide several different applications considering the complex electrical activity of the gastrointestinal system (16). First, the experimental setup used in this investigation can be adopted to study the pharmacological effects of single or multiple drugs at both experimental and modeling levels to identify possible therapeutic approaches for POI, in particular exploring the role of ICCs. As mentioned above, increasing evidence confirmed the main role of ICCs as the pacemaker of spontaneous motility in the gut (38, 68). In a recent study (33) performed on an animal model, the authors examined the effects of temperature and metabolic inhibitors on the generation of pacemaker potentials recorded from ICCs in situ in the mouse small intestine. They showed that altering temperature over the range of 26–40°C influenced the frequency, half-width, and maximal change in voltage over time of pacemaker potentials without affecting the resting membrane potential and amplitude, showing that ICCs in situ have different temperature sensitivities.
Moreover, engineering and clinical applications rely on the experimental validation of new endoscopic robotic devices as well (21). It has been demonstrated that these devices can affect the local temperature of internal colon layers, thus leading to unexpected and undesirable effects similar to POI drawbacks. On this line, in particular, hyperelastic-based models are being studied for applications in medical robotics (20, 30), taking into account the internal irregularities of intestinal villi (3), intestinal edema (74), and viscoelasticity (10, 67). The effect of temperature may play an interesting role in most of these open problems. At the same time, investigating the role of intraoperative thermal cooling on intestinal motor function could also be explored to prevent POI modulating the operating theatre room temperature during surgery.
Limitations and future perspectives.
Limitations to the present study can be identified at both the experimental and modeling levels. From the experimental point of view. a better characterization of the recorded signal would be of great interest to explore the colonic contractile response upon a wider thermal range and different external stimuli could be applied, such as drugs or inhibitors of neural mechanisms (27). Particularly, external electrical stimulations could also be studied, as stretch-induced contractions, blocking the neural activity to better identify the role of ICCs in modulating the contractile response to cooling. Moreover, in vitro tissue studies have demonstrated that some inflammatory mediators, especially prostaglandins, can modulate the frequency of contractions in the stomach (50, 73); for this reason, use of selective inhibitors would help to better explore the mechanism responsible for the cooling contractile response of smooth muscle tissue. An extended data set for a wider range of thermal conditions would also help to unveil the contractile response of longitudinal- and circular-oriented human colonic strips. Other cumulative and coupled factors could also be studied, such as stretch-induced contractions of external electrical stimulations (19, 54).
From a modeling point of view, a forthcoming work in this direction will deal with realistic intestinal geometries with space-dependent dynamics of slow wave propagation and entrainment on the line of previous theoretical and experimental works (14, 39, 45, 72). More complex Ca2+ dynamics will be considered in the nonlinear coupling, with both mechanical quantities and temperature gradients. An extra level of coupling will consider the electrical activity of the tissue in conjunction with thermal and metabolic feedbacks. In particular, thermal heterogeneities will be introduced, analyzing the resulting stress and deformation state. Fiber reinforcement and dispersion will be considered as well to better match the high degree of anisotropy of the intestinal wall based on microscopic Ca2+ dynamics (43, 51, 70). A direct extension of such an approach should consist in analyzing thermoionic feedback on realistic reconstructed domains.
The International Center for Relativistic Astrophysics Network partially supported this work.
No conflicts of interest, financial or otherwise, are declared by the author(s).
A.A., A.G., M.P.L.G., R.A., M.C., and S.F. conception and design of research; A.A., S.C., M.D., and R.A. performed experiments; A.A., A.G., A.L., S.C., and M.D. analyzed data; A.A., A.G., M.P.L.G., S.C., M.D., M.C., and S.F. interpreted results of experiments; A.A., A.G., M.P.L.G., A.L., S.C., M.C., and S.F. drafted manuscript; A.A., A.G., M.P.L.G., R.A., M.C., and S.F. edited and revised manuscript; A.A., A.G., M.P.L.G., A.L., S.C., M.D., R.A., M.C., and S.F. approved final version of manuscript; A.G., A.L., and M.D. prepared figures.
The authors thank the Pathological Anatomy Laboratory at the University Campus Bio-Medico of Rome for support with the histological analysis.
A continuum thermodynamical framework was adopted here based on the additive decomposition of the deformation gradient (a filament translation and a stretch of myosin heads) (19, 44). Balance laws for the mechanical and electrochemical relationships were justified via the virtual power principle (23). Constitutive equations were derived by applying the dissipation inequality for a nonlinear hyperelastic material model, and the chemical state of myosin was modeled as a first-order kinetic (18, 43, 45). The coupling between the different models was also justified, and the effect of temperature was introduced.
The chemical state was based on Hai and Murphy (31) smooth muscle cell myosin phosphorylation and stress development, modeling a so-called latch state allowing for a basal tone level. It consists of four variables (n1, n2, n3, and n4) representing the different functional states of the smooth muscle thus connected via seven kinetic constants (with rates k1–k7). The resulting coupled system of first-order ordinary differential equations can be described as follows: (2) (3) (4) (5) where Σi = 14 ni = 1, ηi = Ai(λ − 1.12) + 1, for i = 1 . . . 4, varies according to tissue deformation [stretch (λ)], where η is a positive material fitting parameter and Ai is a chemical state parameter. These equations thus represent the direct effect of the mechanical deformation (stretch) onto the chemical variables of the Hai and Murphy model. In particular, the rate variables vary according to the strain level induced into the tissue. Ca2+ concentration and deformation feedbacks were introduced through the following equations: (6) where β is the current configuration, β0 is a reference configuration, and C0 and C1 are chemical state paramters. Model parameters are shown in Table 2.
Considering smooth muscle as composed of two parallel sections, a spring and a contractile element, we set β0 and β where the total length was measured. To reach such a state, an intermediate noncompatible configuration (βa) was passed through, in which the contractile element was separately activated to a length (λa) and the cross-bridge elasticity then acted via an additional stretch (λc). Such a structure formalized in the following multiplicative decomposition relation: (7) Upon these assumptions, we can defined the total Helmholtz strain-energy function associated with passive and active phases of the myocyte contraction as follows: Ψ = Ψ(λ, λa, ni, β, T). We thus considered the superposition of free energies in isothermal conditions as follows: (8) which was explicitly defined by applying the extended balance principle of mechanics (51). In particular, N(λa) represents the effective area overlap between the active filaments for generating the contractile force. Such an approach allowed us to define, in general terms, the evolutionary laws of λa, ni, β, and T, thus allowing us to measure the stress (P) for a given stretch λ as follows: (9) where B is the force work conjugate with beta, Pa is the active stress contribution with C = (f1n3 + f2n4)N(λa), where f1 is the friction of the phosphorylated cross-bridge and f2 is the friction of the attached cross, aij is [aij = aij(λa,λc,β)], and r is a time multiplier constrained by Eq. 2.
Specific forms of the following potential functions were selected according to Refs. 44 and 67: (10) (11) (12) (13) where q1 and q2 are fitting parameter, E1 is the stiffness of the phosphorylated cross-bridge, E2 is the stiffness of the attached cross-bridge. We can thus recover the active stress definition as Pa = −f1n3vN(λa) (where v is the cycling velocity of the cross-bridge) and [Ca2+]i as B = β.
Finally, total stress can be described as follows: (14) and active contraction evolution can be described as follows: (15) In case HCMS fiber orientations is considered, the Helmholtz free energy would modify its functional dependence with respect to the principal directions of activations. In particular, the introduction of fiber-based invariants would be necessary (see Ref. 32a).
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