# Virgin Passive Colon Biomechanics and a Literature Review of Active Contraction Constitutive Models

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Preparation of the Equibiaxial Tensile Test Specimens

#### 2.2. Test Protocol

#### 2.3. Virgin Passive Anisotropic Strain Energy Function Description

**F**, the isochoric deformation gradient is $\overline{\mathbf{F}}$ = ${J}^{-1/3}\mathbf{F}$, ${\overline{I}}_{1}$ = $\mathrm{tr}\overline{\mathbf{C}}$ is the first principal invariant of the isochoric right Cauchy–Green deformation tensor, ($\mathbf{C}$ = ${\overline{\mathbf{F}}}^{\mathrm{T}}\overline{\mathbf{F}}$), ${\mathbf{a}}_{0i}$ = ${\left\{{a}_{0ix},{a}_{0iy},{a}_{0iz}\right\}}^{\mathrm{T}}$ is the unit directional vector of the ith fiber family in the undeformed configuration and ${\overline{I}}_{4,{\mathbf{a}}_{0i}}$ = $\overline{\mathbf{C}}:{\mathbf{A}}_{0i}$ = ${\mathbf{a}}_{0i}\times \overline{\mathbf{C}}\xb7{\mathbf{a}}_{0i}$ includes the anisotropic invariants, which are quantitatively equal to the square of the respective fiber stretches ${\lambda}_{{\mathbf{a}}_{0i}}^{2}$. At low strain, collagen fibers are not stretched and do not store strain energy. Thus, the tissue mechanical behavior is solely due to the non-collagenous ground matrix and its isotropic response can be defined using the classical Neo-Hookean material model. At large strains, the collagen fibers resist the applied load and show a strong stiffening effect, which can be represented mathematically by an exponential function using the fiber-reinforced hyperelastic Holzapfel–Gasser–Ogden (HGO) model [39]:

#### 2.4. Numerical Simulations

## 3. Results

#### 3.1. Equibiaxial Tensile Experiment

#### 3.2. Intestinal Anisotropy

#### 3.3. Parameter Stability

#### 3.4. Numerical Simulations

#### 3.4.1. Equibiaxial Tension of a Linear Brick Element

#### 3.4.2. Equibiaxial Tension of an Intestine Specimen

## 4. Physiology of Intestinal Peristalsis

^{+}cell syncytium (SIP syncytium) connected via gap junctions [48]. Innervated with excitatory enteric motor neurons, the interstitial cells of Cajal (ICC) stimulate unique Ca

^{++}-activated Cl

^{−}channels or K

^{+}channels and generate excitatory electrical slow waves or currents to enhance SIP syncytium excitability and to initiate the rapid contraction of SMCs [49,50]. In contrast, platelet-derived growth factor receptor α-positive (PDGFRα

^{+}), the inhibitory neurotransmitter released from enteric motor neurons, expresses small conductance calcium-activated K

^{+}channel 3 (Ca

^{++}-activated SK3) channel to generate inhibitory currents, thereby reducing SIP syncytium excitability and reducing SMC contraction [49,51]. Most importantly, the level of free cytoplasmic or intracellular Ca

^{++}within SMC determines its cellular contraction–relaxation activity. In response to a slow influx of Ca

^{++}through leak channels, acetylcholine (AcH) neurotransmitter, H

^{+}, hormones, sarcoplasmic reticulum calcium spark, the stimulation of voltage-gated Ca

^{++}channels, ATPase, sacro/endoplasmic reticulum Ca

^{++}ATPase (SERCA), stromal interaction molecule (STIM) protein and Orai protein activities rapidly increase Ca

^{++}concentrations. Such processes are schematically shown by three green-dashed regions in Figure 9a. In any such cases, enhanced SMC contraction is depicted by a rapid spike of the membrane potential, known as action potential well above the threshold potential, which is counteracted by repolarization, i.e., outflow of K

^{+}ions from the activated K

^{+}channels (Figure 9a).

^{++}independent. Triggered by inositol 1,4,5-trisphosphate (IP3), Ca

^{++}activates the enzymatic domain of MLCK, which is mediated by the calmodulin (CaM) via CaM-4Ca

^{++}-dependent complex formation (Figure 9b). This CaM-4Ca

^{++}complex phosphorylates the Ser19 hydroxyl group at the neck of MLC20 to stimulate myosin Adenosine tri-phosphate (ATP) hydrolysis into Adenosine di-phosphate (ADP), inorganic phosphate (${\mathbf{P}}_{\mathrm{i}}$) and energy at its globular head. The inhibition of tropomyosin (Trop), calponin (CaP) and caldesmon (CaD) followed by the release of ${\mathbf{P}}_{\mathrm{i}}$ from the myosin head exposes the myosin binding sites to initiate myosin–actin cross bridging (Figure 9c). Powered by the chemical energy stored in the myosin’s head, actin–myosin slide over one another with the release of ADP to shorten the length between two dense bodies by $-\delta $. Sliding continues until a new ATP molecule is attached to the myosin head, which immediately detaches the myosin from the actin binding sites. At the end of this cycle, Ca

^{++}-independent MLCP dephosphorylates MLC20 [52] and inhibits CaM-4Ca

^{++}-activated ATPase activity [53]. Furthermore, the kinases from cyclic nucleotides inhibit incoming Ca

^{++}channel activities, phosphorylate MLCK to reduce its affinity for the activated CaM-4Ca

^{++}complex and finally decrease myosin phosphorylation [54]. Furthermore, they may stimulate Ca

^{++}uptake from the CaM-4Ca

^{++}complex and the cytosol to the sarcoplasmic reticulum and outside the SMC through Ca

^{++}/Na

^{+}or Ca

^{++}/H

^{+}exchanges (Figure 9c). Thus, reduced intracellular CaM-4Ca

^{++}MLCK activity and increased MLCP activity relaxes SMCs.

## 5. Constitutive Modeling of Large Intestinal SMC Contraction

^{++}concentration due to stimulated nerve action potential for which the conductance-based model by Hodgkin–Huxley is useful [63,64]. The voltage-dependent K

^{+}and Na

^{+}ion channels (action potential) and leakage (L) ion channel (small conductance for resting membrane potential) are regulated by small gates in the intestinal SMCs. Membrane potentials higher than the threshold potential open the Na

^{+}channel to depolarize the membrane and to activate voltage-gated Ca

^{++}channel for rapid inflow, followed by a rapid spike or action potential and the contraction. In a voltage clamp set up (Figure 10a), the Hodgkin–Huxley model uses electric circuit relationships and ODEs to measure ion currents (${I}_{i}$) through each ion channels (i = ${\mathrm{K}}^{+},{\mathrm{Na}}^{+},\mathrm{L}$) and compares the input/output currents at a desired membrane potential (${V}_{m}$) at any time t:

_{p}), actin attached phosphorylated myosin (AM

_{p}), also known as cross bridge, and actin attached unphosphorylated myosin (AM), also known as latch bridge. The model is governed by a first-order differential equations system:

_{p}) + ${n}_{3}$(AM

_{p}) + ${n}_{4}$(AM) = 1. This means that the total sum of the myosin fraction must remain constant. Here, (${k}_{1},{k}_{6}$) are the position-independent state transition rates of myosin phosphorylation, (${k}_{2},{k}_{5}$) are the position-independent rates of myosin dephosphorylation, (${k}_{4},{k}_{7}$) and (${k}_{3}$) are the position-dependent rates for myosin detachment and attachment, respectively. The interesting aspects of this models are as follows: (a) ${k}_{1}$ and ${k}_{6}$ are regulated by Ca

^{++}concentrations and (b) AM→A∼M is irreversible. Furthermore, a hyperbolic dependency of steady-state active stress on cross-bridge phosphorylation [65] and a direct correlation of myosin phosphorylation relative to the force generation and tissue shortening velocity have been found [66]. Singer et al. [67] further supported the Ca

^{++}-dependent latch bridged mechanical differences with phosphorylated cross-bridges and decomposed the total SMCs stress into active and passive stress: the latter by depleting extracellular Ca

^{++}. Kato et al. [68] followed Hai and Murphy and proposed a kinetic model to correlate the CaM-4Ca

^{++}-dependent activation of MLCK, the phosphorylation–dephosphorylation of myosin and isometric contractions in SMC at a steady state.

**M**is the direction vector of the contractile filaments in the undeformed configuration,

**C**is the right Cauchy–Green deformation, ${\lambda}_{\mathrm{e}}$ = ${\lambda}_{\mathrm{f}}/{\lambda}_{\mathrm{fc}}$ is the elastic stretching of the cross-bridges, ${\lambda}_{\mathrm{fc}}$ is the relative sliding between contractile filaments and ${\lambda}_{\mathrm{f}}^{2}$ = $\mathbf{MCM}$ is the tissue stretch along the contractile filament direction. The evolution of ${\lambda}_{\mathrm{fc}}$ is adopted as $\eta {\dot{\lambda}}_{\mathrm{fc}}$ = ${P}_{\mathrm{cc}}-tial{\psi}_{\mathrm{act}}/tial{\lambda}_{\mathrm{fc}}$, where η is the viscous damping coefficient associated with the cycling/contraction process and ${P}_{\mathrm{cc}}$ is the active stress due to attached cross-bridges causing muscle contraction. The process of the filament contraction development in AM

_{p}AM states can be accounted for by an active stress formulation:

_{p}and passive resisting strength of cross-bridges in state AM, respectively. To model taenia coli SMCs, ${\mu}_{\mathrm{f}}$ = ${\kappa}_{3}$ = 1.11 MPa, ${\kappa}_{4}$ = 0 (for monotonic contraction with inactive latch state) and η = 44.5 MPa, ${k}_{1}$ = ${k}_{6}$ = $0.17\text{}{\mathrm{s}}^{-1},{k}_{2}$ = ${k}_{5}$ = $0.5\text{}{\mathrm{s}}^{-1},{k}_{3}$ = $4{k}_{4},{k}_{4}$ = $0.11\text{}{\mathrm{s}}^{-1}$ and ${k}_{7}$ = $0.01\text{}{\mathrm{s}}^{-1}$ have been used for both isotonic and isometric behavior with an increase in Ca

^{++}= 2.5 mol/m

^{3}[69].

## 6. Intestinal Active Viscoelasticity

**F**, which are the first and second isotropic principal invariants and anisotropic principal invariants of fiber directions, ${\mathbf{a}}_{1}$ and ${\mathbf{a}}_{2}$, respectively. The material parameters fitting the viscous active electromechanical model for porcine intestine obtained from the biaxial tensile experiments in [70] are shown in II of Table 2.

## 7. Discussion

^{++}ions are undoubtedly essential for the numerical study of the intestine in vivo. There are few models of active contraction in colon tissue, which are reviewed in [64,69,70]. However, there are several other approaches to model SMC contraction, which have been mainly used for arteries [74,75]. Stålhand et al. [76] formulated a chemomechanical finite strain model for SMC contraction based on a spring element and a contractile element in parallel controlled by Ca

^{++}-dependent state variables, as proposed by Hai and Murphy. The one and two-dimensional structural constitutive models by Tan and De Vita [77] and the three-dimensional phenomenological constitutive model by Schmitz and Böl [78] show a good approximation of the passive, active and total arterial mechanics. Chen and Kassab [79] described the three-dimensional microstructural response of coronary artery differentiating into elastin, collagen and SMCs. Dependent on myosin phosphorylation and cross-bridge kinetics, a dynamic muscle model has been proposed by Gestrelius and Borgström to estimate active force generation in SMCs [80]. Recently, Brandstaeter et al. [81] modeled the pacemaker’s electrophysiology and SMCs’ contractility in gastric tissue using an electro-mechanical constitutive model. By a multiplicative decomposition of the deformation gradient $\mathbf{F}$ = ${\mathbf{F}}_{\mathrm{elas}}{\mathbf{F}}_{\mathrm{act}}$ with additional tissue incompressibility, ${J}_{\mathrm{act}}$ = 1, the active part ${\mathbf{F}}_{\mathrm{act}}$ is computed for SMC. Similarly to intestines, the gastric tissue also consists of circumferential and longitudinal SMCs; therefore, the active gastric mechanics presented in Brandstaeter et al. possess relevance in modeling other GI tracts in the future.

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Biaxial tensile test rig with a 40 mm × 40 mm intestine specimen. (

**a**) Color photo of undeformed configuration during daylight; (

**b**) black-and-white photo of undeformed configuration during polarized light setup and (

**c**) in deformed configuration. Four collagen fiber orientations are shown: longitudinal muscular (${\mathbf{a}}_{01}$) orthogonal to circumferential muscular (${\mathbf{a}}_{02}$) and submucosal (${\mathbf{a}}_{03},{\mathbf{a}}_{04}$) aligned at $\pm {30}^{\circ}$ with the longitudinal direction.

**Figure 2.**(

**a**) A linear brick FE showing fiber directions. (

**b**) FE mesh of the one quarter of the specimen No. 3. All units are in millimeter (mm).

**Figure 3.**Green-Lagrange strain heat map in the specimen measured in the ISTRA4D DIC software in (

**a**) undeformed configuration and (

**b**) deformed configuration for X-axis and (

**c**) for Y-axis. Red circle in (

**a**) represents the center of the specimen, and blue circle represents the X-axis designated by ISTRA4D software. In (

**b**,

**c**), a polygon is constructed as a field of interest to evaluate the average Green–Lagrange strain for material characterization. All units are in millimeters.

**Figure 4.**Fitted biaxial tensile stress–stretch curves of the porcine large intestine specimen 1–5 (

**a**–

**e**) in the longitudinal direction (blue) and the circumferential direction (red) for virgin passive anisotropy. (

**f**) The average curve is overlaid over the fitted curve and the anisotropic material parameters of the chosen HGO model are estimated for each set of curves and the average curve (Table 1).

**Figure 5.**Parameter stability check: three-dimensional surface plot of the eigenvalues (

**a**) ${\eta}_{1}$ and (

**b**) ${\eta}_{2}$ of the Hessian matrix as a function of biaxial stretches for the average virgin passive material parameters listed in Table 1.

**Figure 6.**(

**a**) Deformation of the linear brick element under equibiaxial tension with displacement $\Delta Z=-0.61$ mm of the top surface. The dashed geometry represents the undeformed configuration and the solid geometry represents the deformed configuration. (

**b**) Comparison of the Cauchy stress–stretch curves from the experimental data (Figure 4c) vs. simulation.

**Figure 7.**Displacement heat maps of the specimen No. 3 along the longitudinal and the circumferential directions from the ISTRA4D DIC software (

**a**,

**b**) and corresponding biaxial tensile simulation (

**c**,

**d**), respectively. The 20 mm × 20 mm × 1.25 mm computer model in (

**c**,

**d**) is equivalent to the specimen region inside the dotted square. All units are in millimeters.

**Figure 8.**Schema of large intestinal peristalsis inducing fecal transport between four haustral segments. Green arrows and circles with +ve sign represent the excitatory response for contraction, whereas red arrows and circles with −ve sign represent an inhibitory response for relaxation.

**Figure 9.**Schematic representation of processes involved in SMC contraction within the intracellular space. (

**a**) Intracellular Ca

^{++}is increased mainly by ion channels and excitation by ANS. Green and red arrows represent ion inflow and outflow, respectively. (

**b**) Increased four intracellular Ca

^{++}ions bind with each inactive CaM (brown) to form a CaM-4Ca

^{++}complex (green) that stimulates MLCK for MLC

_{20}activation. (

**c**) ATP and MLCK activity initiates cross-bridging between actin and myosin filaments, followed by power stroke and contraction ($-\delta $). Intracellular Ca

^{++}from the CaM-4Ca

^{++}complex is supplied back to the sarcoplasmic reticulum and extracellular matrix via ion channels. Green circles with +ve sign represent an excitatory response, whereas red circles with −ve sign represent an inhibitory response.

**Figure 10.**Schematic representation of (

**a**) Hodgkin–Huxley electrochemical model, membrane viewed as electric circuit and (

**b**) Hai and Murphy four-stage cross-bridge chemical kinetic SMC contraction model.

**Table 1.**Fitted anisotropic parameters of the porcine large intestine from biaxial tensile experiment data.

Specimen | ${\mathit{\mu}}_{0}$ (MPa) | ${\mathit{k}}_{1,\mathbf{Mus}}$ (MPa) | ${\mathit{k}}_{2,\mathbf{Mus}}$ | ${\mathit{k}}_{1,\mathbf{Sm}}$ (MPa) | ${\mathit{k}}_{2,\mathbf{Sm}}$ | ${\mathit{R}}_{\mathbf{L}}^{2}$ | ${\mathit{R}}_{\mathbf{C}}^{2}$ |
---|---|---|---|---|---|---|---|

1 | 0.001 | 0.0171 | 1.5827 | 0.0233 | 2.4303 | 0.9900 | 0.9878 |

2 | 0.001 | 0.0153 | 0.8525 | 0.0043 | 2.5866 | 0.9968 | 0.9944 |

3 | 0.001 | 0.0268 | 0.5080 | 0.0072 | 2.0980 | 0.9924 | 0.9980 |

4 | 0.001 | 0.0218 | 1.9014 | 0.0125 | 0.9415 | 0.9873 | 0.9904 |

5 | 0.002 | 0.0061 | 0.7490 | 0.0300 | 0.4768 | 0.9798 | 0.9846 |

Average curve | 0.0048 | 0.0133 | 1.2090 | 0.0136 | 1.2483 | 1.0000 | 1.0000 |

**Table 2.**Parameters for the porcine intestine [70].

I Electrical-chemical-mechanical constitutive parameters. Units in cm, s | ||||||||||||

LM: | ${k}_{\mathrm{LM}}$ = 10 | ${a}_{\mathrm{LM}}$ = $0.06$ | ${\beta}_{\mathrm{LM}}$ = 0 | ${\gamma}_{\mathrm{LM}}$ = 8 | ${\u03f5}_{\mathrm{LM}}$ = $0.15$ | ${\alpha}_{\mathrm{LM}}$ = 1 | ||||||

ICC: | ${k}_{\mathrm{I}}$ = 7 | ${a}_{\mathrm{I}}$ = $0.5$ | ${\beta}_{\mathrm{I}}$ = $0.5$ | ${\gamma}_{\mathrm{I}}$ = 8 | ${\u03f5}_{\mathrm{I}}$ = ${\u03f5}_{\mathrm{I}}\left(z\right)$ | ${\alpha}_{\mathrm{I}}$ = $-1$ | ||||||

LM: | ${D}_{\mathrm{LM},\phantom{\rule{4.pt}{0ex}}\mathrm{I}}$ = $0.3$ | ${D}_{\mathrm{LM}}$ = $0.4$ | ||||||||||

ICC: | ${D}_{\mathrm{I},\phantom{\rule{4.pt}{0ex}}\mathrm{LM}}$ = $0.3$ | ${D}_{\mathrm{I}}$ = $0.04$ | ||||||||||

II Viscoelastic-active electromechanical model parameters | ||||||||||||

${\kappa}_{0}$ | ${\mu}_{1}$ | ${\mu}_{2}$ | ${k}_{1,{\mathbf{a}}_{1}}$ | ${k}_{2,{\mathbf{a}}_{1}}$ | ${k}_{1,{\mathbf{a}}_{2}}$ | ${k}_{2,{\mathbf{a}}_{2}}$ | $\zeta $ | $\eta $ | ${\chi}_{\mathrm{iso}}$ | ${\chi}_{\mathrm{fiber}}$ | ${\chi}_{\mathrm{iso}}^{\mathbf{C}}$ | ${\chi}_{\mathrm{fiber}}^{\mathbf{C}}$ |

kPa | kPa | kPa | kPa | - | kPa | - | kPa s | kPa s | - | - | - | - |

5.5 | 1 | 1 | 55 | 56 | 20 | 29 | 0.125 | 0.09 | 1 | 5 | 3 | 12 |

^{−12}Farads per meter.

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## Share and Cite

**MDPI and ACS Style**

Bhattarai, A.; Horbach, A.J.; Staat, M.; Kowalczyk, W.; Tran, T.N.
Virgin Passive Colon Biomechanics and a Literature Review of Active Contraction Constitutive Models. *Biomechanics* **2022**, *2*, 138-157.
https://doi.org/10.3390/biomechanics2020013

**AMA Style**

Bhattarai A, Horbach AJ, Staat M, Kowalczyk W, Tran TN.
Virgin Passive Colon Biomechanics and a Literature Review of Active Contraction Constitutive Models. *Biomechanics*. 2022; 2(2):138-157.
https://doi.org/10.3390/biomechanics2020013

**Chicago/Turabian Style**

Bhattarai, Aroj, Andreas Johannes Horbach, Manfred Staat, Wojciech Kowalczyk, and Thanh Ngoc Tran.
2022. "Virgin Passive Colon Biomechanics and a Literature Review of Active Contraction Constitutive Models" *Biomechanics* 2, no. 2: 138-157.
https://doi.org/10.3390/biomechanics2020013