## Abstract

Irritable bowel syndrome afflicts 10–20% of the global population, causing visceral pain with increased sensitivity to colorectal distension and normal bowel movements. Understanding and predicting these biomechanics will further advance our understanding of visceral pain and complement the existing literature on visceral neurophysiology. We recently performed a series of experiments at three longitudinal segments (colonic, intermediate, and rectal) of the distal 30 mm of colorectums of mice. We also established and fitted constitutive models addressing mechanical heterogeneity in both the through-thickness and longitudinal directions of the colorectum. Afferent nerve endings, strategically located within the submucosa, are likely nociceptors that detect concentrations of mechanical stresses to evoke the perception of pain from the viscera. In this study, we aim to: (1) establish and validate a method for incorporating residual stresses into models of colorectums, (2) predict the effects of residual stresses on the intratissue mechanics within the colorectum, and (3) establish intratissue distributions of stretches and stresses within the colorectum in vivo. To these ends we developed two-layered, composite finite element models of the colorectum based on our experimental evidence and validated our approaches against independent experimental data. We included layer- and segment-specific residual stretches/stresses in our simulations via the prestrain algorithm built into the finite element software febio. Our models and modeling approaches allow researchers to predict both organ and intratissue biomechanics of the colorectum and may facilitate better understanding of the underlying mechanical mechanisms of visceral pain.

## 1 Introduction

The biomechanics of visceral organs govern the macroscopic mechanical organ stimuli and the local mechanical stresses and strains at sensory nerve endings [1,2]. Understanding and predicting these biomechanics will further advance our understanding of visceral pain and complement the existing literature on visceral neurophysiology [3–5]. Visceral pain is a leading complaint from patients with irritable bowel syndrome, a condition afflicting 10–20% of the world's population [6,7]. Conscious perception from the viscera usually present as discomfort and/or pain, which is not evoked reliably by noxious stimuli like heating, cutting, pinching, or piercing, but caused by mechanical distension of hollow visceral organs like the distal colon and rectum (colorectum) [3,8]. Clinically, patients with irritable bowel syndrome show increased sensitivity to colorectal distension and heightened perceptions of pain during normal mechanical bowel movements [8–10]. Consistent with the mechanical stimuli which evoke visceral pain, the vast majority (70–80%) of sensory afferent nerve endings that innervate the colorectum are mechanosensitive [3]. Thus, visceral pain has a unique biomechanical component.

Mechanosensitive afferents undertake neural encoding of mechanical stimuli in the colorectum, and most of these afferents have unmyelinated C-type axons with free nerve endings embedded within the colorectum [3]. Afferent endings exist in all sublayers of the colorectum except in the serosa [11,12], i.e., the mucosal, submucosal, circular muscular, myenteric plexus, and longitudinal muscular layers. The colorectum is thus heterogenous through the thickness, as recently characterized by us using both biaxial extension tests on layer-separated tissue patches [13,14] and nonlinear imaging via second harmonic generation [15]. These combined data reveal that the submucosal layer, containing dense collagen, is the primary load-bearing structure of the colorectum, along with the two muscular layers [14]. Interestingly, the submucosa and myenteric plexus (sandwiched by the two muscular layers) also present significantly higher densities of afferent nerve endings [12]. We contend that these regions within the colorectum contain both concentrated loads and high densities of afferent nerves not by mere coincidence, but that colorectal mechanotransduction and mechanonociception takes place by afferents strategically located in the submucosa and myenteric plexus. Thus, local mechanical stresses in focal regions of the colorectum could have profound impact on the mechanotransduction that underlies perception of visceral pain.

The local, intratissue mechanical stresses result not only from the mechanical properties of different sublayers of the colorectum but are also significantly affected by residual stresses widely present in healthy soft tissues under physiological conditions, e.g., the arterial walls [16–18]). Indeed, the existence of residual stresses in healthy arteries is long and well established [19–23]. Residual stresses tend to homogenize the stress distribution through the thickness of arteries in vivo [24–31]. Yet, no analogous understanding exists for the colorectum.

In separating the layers of the colorectum, we observed that square specimens changed their circumferential lengths from the complete composite configuration to the layer-separated configuration. We quantified these dimensional changes and our results indicated compressive and tensile circumferential residual stresses within the inner and outer composite layers, respectively, and with increasing magnitudes from the colonic to rectal segments. We further quantified the segment-specific residual stretches/stresses by completing opening-angle tests. We made longitudinal cuts in intact rings of colorectums from the colonic, intermediate, and rectal segments and quantified the resulting angles using optical methods [13]. However, we still don't know how residual stresses alter the total intratissue mechanical stresses within different regions of the colorectum, and thus, the biomechanical and mechanobiological roles of residual stresses on the intratissue mechanics of the colorectum remain unclear.

To better understand the underlying mechanisms of visceral pain by mechanotransduction in the colorectum, and establish potential treatment targets, we seek to predict local organ/tissue biomechanics, particularly mechanical stresses, from bulk mechanical colorectal distension and peristalsis by systematically incorporating residual stresses. To this end, in this study we aim to: (1) establish and validate a method for incorporating residual stresses in models of colorectums, and (2) predict the effects of residual stresses on the intratissue distributions of stretches and stresses within the colorectum, and (3) establish through-thickness and longitudinal, intratissue distributions of stretches and stresses within the colorectum under conditions mimicking those in vivo. To these ends, we developed two-layered, composite finite element (FE) models of the colorectum based on our experimental evidence and validated our approach against independent experimental data. We incorporated residual stretches/stresses within the individual layers of our FE simulations the colorectum using the prestrain algorithm in febio. The results of our FE simulations indicate that these residual stretches/stresses have a profound impact on the local (intratissue) mechanics, particularly through the thickness of the colorectum.

## 2 Materials and Methods

### 2.1 Experimental Evidence.

We previously completed pressure-diameter tests on the intact (complete composite) distal 30 mm of colorectums of mice (i.e., within the colonic, intermediate, and rectal segments) [14,32]. In completing these mechanical tests, we quantified the cylindrical geometry of the colorectums in the near-stress-free configurations (assumed as reference configurations) under a constant internal pressure of 10 mmHg [33], reported as mean±standard deviation (M±SD) in Table 1.

Segment | Outer radius (mm) | Inner radius (mm) | Inner thickness (mm) |
---|---|---|---|

Colonic (n = 6) | 1.80 ± 0.167 | 1.68 ± 0.191 | 0.0872 ± 0.029 |

Intermediate (n = 6) | 1.80 ± 0.297 | 1.65 ± 0.199 | 0.107 ± 0.026 |

Rectal (n = 6) | 1.80 ± 0.460 | 1.62 ± 0.219 | 0.113 ± 0.020 |

Segment | Outer radius (mm) | Inner radius (mm) | Inner thickness (mm) |
---|---|---|---|

Colonic (n = 6) | 1.80 ± 0.167 | 1.68 ± 0.191 | 0.0872 ± 0.029 |

Intermediate (n = 6) | 1.80 ± 0.297 | 1.65 ± 0.199 | 0.107 ± 0.026 |

Rectal (n = 6) | 1.80 ± 0.460 | 1.62 ± 0.219 | 0.113 ± 0.020 |

We previously also completed biaxial extension tests on the distal 30 mm of colorectums of mice (again the colonic, intermediate, and rectal segments) dissected into inner (mucosa and submucosa) and outer (muscular layers and serosa) composite layers [14]. In preparing the layer-separated specimens of colorectum, we observed that square specimens changed their circumferential lengths from the complete composite configuration (7 × 7 mm^{2}) to the layer-separated configuration, reported as M±SD in Table 2. The corresponding radial lengths of the layer-separated inner and outer composite specimens did not change length appreciably after dissection.

Segment | Inner Composite (mm) | Outer Composite (mm) |
---|---|---|

Colonic (n = 10) | 7.12 ± 0.028 | 6.83 ± 0.051 |

Intermediate (n = 11) | 7.15 ± 0.025 | 6.79 ± 0.054 |

Rectal (n = 11) | 7.14 ± 0.029 | 6.68 ± 0.049 |

Segment | Inner Composite (mm) | Outer Composite (mm) |
---|---|---|

Colonic (n = 10) | 7.12 ± 0.028 | 6.83 ± 0.051 |

Intermediate (n = 11) | 7.15 ± 0.025 | 6.79 ± 0.054 |

Rectal (n = 11) | 7.14 ± 0.029 | 6.68 ± 0.049 |

We also quantified the opening angles from the intact (complete composite) distal 30 mm of colorectums of mice (again the colonic, intermediate, and rectal segments) [13]. We cut open rings of composite colorectums 2 mm in length longitudinally (in the radial-axial plane) to form arcs and directly measured the subtended angles through the inner midpoint of the arcs, reported as M±SD in Table 3 [13].

### 2.2 Constitutive model for the Composite Colorectum.

*κ*is a nonphysical, positive penalty parameter used to enforce near incompressibility. We also used the multiplicative decomposition $F\xaf=J\u22121/3F$, and similarly the isochoric right Cauchy-Green tensor $C\xaf=J\u22122/3C$, where $C=FTF$. We modeled the individual mechanical responses of the inner and outer composites using an additive decomposition $\Psi \xaf=\Psi \xafIM+\Psi \xafFN$, with an isotropic (neo-Hookean) matrix $\Psi \xafIM(I\xaf1)=\mu (I\xaf1\u22123)/2,$ where $\mu >0$ is the shear modulus of the underlying matrix in the reference configuration, $I\xaf1=trC\xaf$ is the first invariant of $C\xaf$, and with a contribution from a network of fibers [34,35]

where $D$ is a second-order, symmetric, positive-definite tensor defined in spherical coordinates, $sin\u2009\theta $ accounts for the change from Cartesian to spherical variables, and with $1/4\u222b\Omega \rho (M)d\Omega =1$, where $\Omega =M\u2208\mathbb{R}3:|M|=1$ is the unit sphere.

To model the through-thickness biomechanical heterogeneity at the colonic, intermediate, and rectal segments of the colorectum we leveraged the model parameters (*μ*, *k*_{1}, *k*_{2}, *α*), we previously established for the inner (mucosa and submucosa) and outer (muscular layers and serosa) composite layers [32].

### 2.3 Finite Element Simulations Including Residual Stretches/Stresses.

We established FE models for predicting (a) the opening-angle test of the composite colorectum ring (for validation) and (b) the pressure-diameter test of the composite colorectum under applied intraluminal pressure (to determine the intratissue distributions of stretch and stress in vivo with and without considering residual stretches/stresses). To create our FE models, we specified the reference configurations (inner and outer diameters and layer thicknesses) using data in Table 1, see Fig. 1.

presented in cylindrical coordinates (*r*, *θ*, *z*), where $\lambda p,r,\u2009\lambda p,\theta $, and $\lambda p,z$ are layer-specific, prescribed prestretches. We prescribed $\lambda p,\theta $ for the inner and outer composites as given in Table 4, while the remaining prestretches $\lambda p,r=\lambda p,z=1$.

Segment | Inner composite | Outer composite |
---|---|---|

Colonic | 0.983 | 1.025 |

Intermediate | 0.979 | 1.031 |

Rectal | 0.981 | 1.047 |

Segment | Inner composite | Outer composite |
---|---|---|

Colonic | 0.983 | 1.025 |

Intermediate | 0.979 | 1.031 |

Rectal | 0.981 | 1.047 |

We completed all simulations using febio [36].

### 2.4 Validation: The Opening-Angle Test.

To predict the opening angles at the colonic, intermediate, and rectal segments, we exploited the symmetry of the boundary value problem. We modeled only half model of the composite (intact) colorectum as two-layered, residually stressed cylinder which we meshed using 1200 tri-linear hexahedral elements for each of the two layers. To release the residual stresses, we modeled a longitudinal cut (see Fig. 1(a)) along the length of the colorectum, wherein nodes on the cut surface had no constraints on their displacement degrees-of-freedom. We enforced a fixed boundary condition on remaining symmetry face (without the longitudinal cut), and we directly measured the opening angle from the simulations at equilibrium.

### 2.5 Predictions: The Pressure-Diameter Test.

To predict the intratissue stretches/stresses at the colonic, intermediate, and rectal segments during the pressure-diameter test we modeled the composite (intacted) colorectums as two-layered, tubular segments both with and without accounting for residual stretches/stresses. Exploiting plane-strain conditions, we modeled only a 2 mm longitudinal segments of intact colorectums meshed using 2400 tri-linear hexahedral elements for each of the two layers. We applied symmetry boundary conditions on both faces of the model normal to the longitudinal direction, fixed two radial rows of nodes normal to the radial direction (to prevent rigid-body rotations), and stretched the model axially by 30%, consistent with the experimentally determined axial elongation of the colorectum undergoing intraluminal distension with 60 mmHg internal pressure. We then increased the intraluminal pressure linearly from 0 to 100 mmHg. To assess the effects of residual stretches/stresses on the intratissue stretches/stresses within the colorectum under intraluminal pressure, we performed three related simulations: (1) zero intraluminal pressure including residual stresses, (2) 100 mmHg intraluminal pressure without including residual stresses, and (3) 100 mmHg intraluminal pressure including residual stresses.

## 3 Results

### 3.1 Validation: The Opening-Angle Test.

### 3.2 Predictions: The Pressure-Diameter Test.

We successfully simulated the pressure-diameter test with the intraluminal pressure increasing linearly from 0 to 100 mmHg at (1) zero intraluminal pressure including residual stresses, (2) 100 mmHg intraluminal pressure without including residual stresses, (3) 100 mmHg intraluminal pressure including residual stresses. To compare the results qualitatively, we first plotted the distributions in circumferential stresses for each segment (colonic, intermediate, rectal) in the three conditions detailed above, see Fig. 3. Specifically, Figs. 3(a), 3(d), and 3(g), 3(c), 3(e), and 3(h), and 3(d), 3(f), and 3(i) show the simulation results considering (1) only residual stresses, (2) no residual stresses at *P *=* *100 mmHg, and (3) the two effects combined, respectively.

To compare the results quantitatively, we plotted the (principal) Cauchy stresses ($\sigma \theta \theta $, *σ _{zz}*, and

*σ*) through the normalized thickness (in the radial direction) for the colonic (Fig. 4), intermediate (Fig. 5), and rectal (Fig. 6) segments of the colorectum.

_{rr}## 4 Discussion

We focused this study on predicting residual stresses within colorectums and quantifying the macro- and microscale intratissue mechanics of the colorectum. We modeled the colorectum as a two-layered, residually stressed composite. We included residual stresses in our models as a form of prestretch calibrated from our experimental measurements of separated layers of tissue. As validation of our approach, we successfully predicted the (independent) experimentally determined opening angles. With confidence in our models, we quantified the distributions intratissue stretches/stresses in the colorectal wall during intraluminal distension with and without considering residual stresses. These intratissue mechanical conditions may provide insight both to structure-function relationships and to mechanobiology within the colorectum.

### 4.1 Validation: The Opening-Angle Test.

Independently and accurately predicting independent experimental measurements of the opening angle test validates our method to include residual stresses within our models of the colorectum, cf. Fig. 2. Separating the layers of intact (composite) specimens of tissue leads to significant circumferential elongation of the inner composite and corresponding circumferential shrinkage of the outer composite, cf. Table 1. These deformations relieve residual stresses and indicate that the inner and outer composites are physiologically in compression and tension, respectively. For tubular composite (Fig. 2(a)), the longitudinal cut releases the residual stresses and produces deformed configuration with opening angles (*θ*). In Fig. 2, our simulated results of *θ* agree well with the mean experimental values and fall well within one standard deviation from the mean. Moving proximal to distal along the colorectum (colonic to rectal segments), the colorectum presents increased opening angles, consistent with the measured residual stretches in Table 1, and thus, the internal residual stretches/stresses increase along the length.

### 4.2 Predictions: The Pressure-Diameter Test.

Prior to reviewing the intratissue distributions of stresses and stretches, both qualitatively and quantitatively, we note that an intracolonic pressure of zero mmHg is innocuous, whereas 60 mmHg is in the noxious pressure range for mice [38]. An intracolonic pressure of 100 mmHg recapitulates the magnitude of focal mechanical stresses in experimental studies of punctate probing of colorectal wall with a von Frey-like nylon filament [39]. In the cylindrical colorectum (modeled as a two-layered, residually stressed composite) undergoing internal pressurization, the cylindrical basis is coincident with the principal basis, such that $\sigma 1=\sigma \theta \theta ,\u2009\sigma 2=\sigma zz$, and $\sigma 3=\sigma rr$.

Starting from our qualitative results, Figs. 3(a), 3(d), and 3(g) shows that with only the initial distributions of residual stresses, compressive circumferential stresses appear within the softer inner composite, which may trigger mechanical instabilities with buckling [40,41]. Figures 3(b), 3(e), and 3(h) shows that, in response to pure intraluminal pressure at 100 mmHg, tensile circumferential stresses appear in both the inner and outer composite layers, and the maximum stresses present at the inner radius (submucosal space) with values decreasing toward the outermost radius, where we observed the lowest circumferential stresses.

Finally, Figs. 3(c), 3(f), and 3(i) shows that the residual stretches smooth the distribution of circumferential stresses in the radial direction of the colorectum tissue The heterogeneity of the colorectal wall generates the discontinuity in the circumferential stresses at the interface of the composite layers.

Quantitatively, plots of the distributions of principal stresses (Figs. 4–6) and strains (Figs. 7–9) through the wall thickness of the pressurized colorectum highlight the physiological impact of residual stresses. Overall, under applied internal pressure, we found the composite in tension in the circumferential and axial directions through the thickness, while there was compression in the radial direction. Furthermore, the magnitude of radial stresses (*σ _{rr}*) was much smaller than that of circumferential and axial stresses in the cases both with and without residual stresses. The stress components $\sigma \theta \theta $,

*σ*, and

_{zz}*σ*are all discontinuous at the interface of the inner and outer composite layers. We found that higher internal pressures resulted in larger discontinuities in the stresses, and that residual stresses decreased both the overall magnitude of the stresses and also the interface discontinuity, particularly under noxious internal pressures. Our findings in the colorectum are consistent with prior analyses of arteries where residual stresses create a more uniform distribution of stresses in the homeostatic state [27,42].

_{rr}We observed the maximum circumferential stresses occurred at the inner radius of the cylindrical composite, and affected by the residual stresses, the maximum circumferential stresses at *P *=* *100 mmHg reduced from 337 kPa, 278 kPa, and 269 kPa (without residual stresses) to 236 kPa, 190 kPa, and 142 kPa (with residual stresses) in the colonic, intermediate, and rectal segments, respectively. Similarly, affected by residual stresses, the maximum magnitude of axial stresses (in the inner layer composite) at *P *=* *100 mmHg decreased from 656 kPa, 477 kPa, and 470 kPa (without residual stresses) to 475 kPa, 330 kPa, and 278 kPa (with residual stresses) in the colonic, intermediate, and rectal segments, respectively. Simultaneously, the principal strains (Figs. 7–9) demonstrated similar distributions through the wall thickness and were similarly impacted by including residual stresses, except for the axial strain which we fixed at 0.3 (boundary condition) to mimic the experiment (we designed the experiment itself to mimic the conditions in vivo). Note that the stresses $\sigma \theta \theta $ and *σ _{zz}* show comparable magnitudes when $E\theta \theta \u22480.3$ ($P\u224860$ mmHg) and $Ezz\u22480.3$ ($P\u224810$ mmHg), which is consistent with our observations experimentally.

Our simulations indicated that the most important (physiologically relevant) influence of the residual stresses is the reduction in magnitudes of the maximum circumferential and axial stresses. Residual stretches, for example, in the rectal segment lead to reductions of $\u200947.2%$ and $\u200941.7%$ in $\sigma \theta \theta $ and *σ _{zz}*, respectively. Additionally, we found that larger residual stretches resulted in more pronounced reductions in stresses, cf. Ref. [27].

Our results highlight the importance of modeling colorectum as a residually stressed, multilayered composite where differences in the mechanical properties of the layers lead to discontinuous stress/strain patterns through the wall thickness. Physiologically, the predicted distributions of circumferential stresses within the colorectum strongly indicates that the inner (mucosal and submucosal) layer is the main load bearing structure of the colorectum, consistent with our previous experimental observations [13,14]. Residual stresses within the inner and outer composite layers play a critical role in the physiological, biomechanical performance of the colorectum. Due to the highly nonlinear behavior of the colonic tissue [13,14,32], our results indicate the importance of including residual stretches/stresses in analyses of macro- and microscale intratissue mechanics of the colorectum. Accurate predictions of intratissue mechanical state become especially important when analyzing visceral pain evoked by mechanical stimuli, where thresholds above a homeostatic stress or strain may trigger a response.

Our results indicate the importance of including residual stretches/stresses in analyses of macro- and microscale intratissue mechanics of the colorectum. Sensory nerve endings embedded in the colorectal wall undertake visceral mechanotransduction and nociception, which encode intratissue mechanical stresses and stretches into trains of action potentials to inform the central nervous system. Our recent anatomic study on optically cleared colorectum indicates concentrated sensory nerve endings in the submucosa [12], exactly where our current simulation results indicate that the in-plane mechanical stresses are greatest throughout the thickness of the colorectal wall. In addition, residual stretches/stresses significantly affect the magnitudes of stresses in the submucosa during colorectal distension. These insights in combination suggest that changes in residual stretches/stresses under pathophysiological conditions likely contribute to altered visceral mechanotransduction and nociception. We suggest future studies to establish the causal link between residual stretches/stresses and visceral hypersensitivity in symptoms like irritable bowel syndrome.

### 4.3 Limitations and Outlook.

We acknowledge several limitations of our study. We modeled the colorectum as a two-layered, hyperelastic (cylindrical) composite, where each layer assumes homogeneous mechanical properties. In reality, the colorectum contains at least five distinct (if not load-bearing) layers, each of which is likely heterogeneous both radially and axially. Additionally, the colorectum may present time-dependent, viscoelastic behaviors, e.g., Karimi et al. [43]. We also simplified the morphology of the networked fibers in each layer, and the model could be further refined by including additional micromechanical data acquired via imaging, e.g., confocal microscopy with second harmonic generation.

Overall, our results showed the profound impact of residual stresses on the magnitudes and distributions of local mechanical stresses and strains through the wall thickness of the colorectum. Neglecting residual stresses lead to marked over estimations of circumferential stresses, particular at the interface between the inner and outer composite layers. Our simulation results further confirmed our prior experimental observations that the submucosa is the load-bearing structure of the colorectum. Hence, afferent nerve endings strategically located within the submucosa are likely nociceptors that detect concentrations of mechanical stresses to evoke the perception of pain from the viscera. Our models and modeling approaches allow researchers to predict both organ and intratissue biomechanics in the colorectum, and may facilitate better understanding of the underlying mechanical mechanisms of visceral pain and thus aid in establishing treatment targets.

## Funding Data

Division of Civil, Mechanical and Manufacturing Innovation, NSF (Grant No. 1727185; Funder ID: 10.13039/100000147).

National Institute of Diabetes and Digestive and Kidney Diseases, NIH (Grant No. 1R01DK120824-01; Funder ID: 10.13039/100000062).

## Conflicts of Interest

We have no conflicts of interest to report.

## References

**127**, pp.