# Variation of Passive Biomechanical Properties of the Small Intestine along Its Length: Microstructure-Based Characterization

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Biologic Tissue and Biomechanical Testing

_{2}in O

_{2}) calcium-free Krebs solution (37 °C); EGTA was added to the tissue bath to abolish smooth muscle tone. The pressure range in our experiments was selected to encompass sub-physiologic (0–4 mmHg), physiologic (4–12 mmHg), and supra-physiologic pressures (12–15 mmHg) [2], while allowing for a suitable deformation range. A suitable range of axial stretches encompassing the physiologic condition (~1.1) was also applied to the specimens. Preconditioning was achieved by applying four pressurization cycles at each axial stretch (Figure 1). The inflating portion of a fifth stable cycle over the same pressure range was used for data analysis. The effect of the pressurization rate was previously examined [11] by varying the driving speed of the syringe pump. Little tissue stiffening was observed when increasing the pressurization rate between 0.1 and 1.5 mmHg/s, that is, less than 5% rise in both principal stresses at the maximum strain levels. These findings, together with the slim hysteresis after three inflation/extension cycles, qualified the selected pressurization rate as quasi-static and indicated that pseudo-elasticity was a valid approximation for the small intestinal tissue. A pressure transducer (BLPR; World Precision Instruments, Hertfordshire, UK) measured the lumen pressure, a force transducer (Fort 100; World Precision Instruments, Hertfordshire, UK) the axial force, and a laser micrometer (LS-3100; Keyence Corp, Osaka, Japan) the external diameter. The device and peripheral components were controlled with a computer running a LabView program (v7.1; National Instruments, Austin, TX, USA). After testing, four rings were removed from the midpoint of the small intestinal specimens for the determination of the no-load and zero-stress states.

#### 2.2. Histomorphometric Evaluation

#### 2.3. Microstructure-Based Material Models

**C**. Given the large amount of smooth muscle cells in the small intestinal wall, especially in the thick muscle layer, their passive contribution, along with the ground matrix within which the fibrous elements and smooth muscle cells reside, were assumed to determine the neo-Hookean model. The exponential terms in Equation (1) accounted for the anisotropic characteristics of collagen fibers in the tissue. Although radial fibers, caused by the presence of mucosal foldings, were histologically observed in the no-load state of the tissue, they were not included in the model for simplicity. We assumed that under physiologic pressures, when collagen fibers are engaged, the small intestine had attained an axisymmetric cylindrical geometry internally, without foldings, and collagen fibers of the mucosa no longer occurred in the radial direction.

#### 2.4. Parameter Estimation

**R**for the estimated model parameters, $det\left(R\right)<{10}^{-4}$ being the limit set to determine over-parameterization.

#### 2.5. Statistical Analysis

## 3. Results

#### 3.1. Comprehensive Model

^{2}~0.90, and the low values of root-mean-square error, ε~0.26, for these segments. The correspondence between the model and data for the remaining segments was less good (R

^{2}~0.87 and ε~0.32), due to the inadequate fit to the force data of the 1.2 axial stretch in five out of the twelve examined duodenal specimens (Figure 2a,b), the inadequate fit to the force data of the 1.1 and 1.3 axial stretches in two out of the six middle ileal specimens (Figure 4b), and the inadequate fit to the pressure data of the 1.1 and/or 1.3 axial stretches in two out of the six distal ileal specimens (Figure 4c).

^{−4}in most small intestinal specimens (Tables S1–S3 of the Supplementary Materials). On top of that, the error for one or more parameters $\mu ,\text{}{k}_{1}^{c},$ and ${k}_{2}^{c}$ was relatively large as compared to their very small values, and their dependence was almost unity (data not shown), strongly indicating that the comprehensive model was over-parameterized.

#### 3.2. Parametric Analysis

^{−4}values in Table S7. Still, the computed root-mean-square error and determination coefficient values of $\epsilon $~0.5 and ${R}^{2}$~0.5 were indicative of greatly diminished goodness of fit compared to the comprehensive model and the variants without the neo-Hookean or circumferential-fiber family term.

#### 3.3. Reduced Model

^{−4}was avoided, and the $\epsilon $ and ${R}^{2}$ values of the reduced model resembled those of the comprehensive model; again, being noticeably better for the three jejunal segments and the proximal ileum, in comparison to the $\epsilon $ and ${R}^{2}$ values found for the proximal and distal duodenum and the middle and distal ileum. See the comparable fitting quality in Figure 2, Figure 3 and Figure 4 and Figure 6, Figure 7 and Figure 8.

#### 3.4. Histologic Findings

## 4. Discussion

#### 4.1. General Findings

#### 4.2. Consideration of Microstructure-Based Material Models for the Small Intestine

#### 4.3. Structural Interpretation of Model Parameters: Consideration of Segmental Differences and Physiologic Implications

#### 4.4. Limitations and Future Studies

## 5. Conclusions

## Supplementary Materials

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Online record of preconditioning for a distal duodenal specimen at (

**a**) ${\mathsf{\lambda}}_{\mathrm{z}}=$ 1.1, (

**b**) ${\mathsf{\lambda}}_{\mathrm{z}}=$ 1.2, and (

**c**) ${\mathsf{\lambda}}_{\mathrm{z}}=$ 1.3. The 2nd, 3rd, and 4th pressure-diameter cycles overlapped and were enveloped by the larger 1st cycle. Τhe area enclosed by the 1st and 2nd force-pressure cycles increased with increasing ${\mathsf{\lambda}}_{\mathrm{z}}$, with all cycles overlapping at ${\mathsf{\lambda}}_{\mathrm{z}}=$ 1.1 and the 3rd and 4th cycles being very slim and nearly overlapping at ${\mathsf{\lambda}}_{\mathrm{z}}=$ 1.2 and 1.3.

**Figure 2.**Plot of measured lumen pressure (red color) and axial force (blue color) vs. external diameter data at three different fixed axial stretches ${\lambda}_{z}=$ {1.1, 1.2, 1.3} for a characteristic specimen from the proximal and distal duodenum and fits (solid lines for lumen pressure and dashed lines for axial force) calculated using the neo-Hookean and four-fiber family microstructure-based model of Equation (1) with the following best-fit model parameters for the (

**a**) proximal duodenum: $\mu =$ 3.7 × 10

^{−11}kPa, ${k}_{1}^{d}=$ 0.312 kPa, ${k}_{2}^{d}=$ 16.078, ${k}_{1}^{a}$ = 8.467 kPa, ${k}_{2}^{a}$ = 3.143, ${k}_{1}^{c}$ = 9.5 × 10

^{−9}kPa, ${k}_{2}^{c}$ = 3.922, ${a}_{0}$ = 0.670 rad, ε = 0.321, ${R}^{2}=$ 0.870, $det\left(R\right)=$ 4.7 × 10

^{−5}and (

**b**) distal duodenum: $\mu =$ 5.0 × 10

^{−6}kPa, ${k}_{1}^{d}=$ 0.336 kPa, ${k}_{2}^{d}=$ 6.940, ${k}_{1}^{a}$ = 2.470 kPa, ${k}_{2}^{a}$ = 7.840, ${k}_{1}^{c}$ = 0.127 kPa, ${k}_{2}^{c}$ = 3.4 × 10

^{−8}, ${a}_{0}$ = 0.308 rad, ε = 0.322, ${R}^{2}=$ 0.881, $det\left(R\right)=$ 6.3 × 10

^{−6}. Data are shown every 0.5 mmHg for clarity.

**Figure 3.**Plot of measured lumen pressure (red color) and axial force (blue color) vs. external diameter data at three different fixed axial stretches ${\lambda}_{z}=$ {1.1, 1.2, 1.3} for a characteristic specimen from the proximal, middle, and distal jejunum and fits (solid lines for lumen pressure and dashed lines for axial force) calculated using the neo-Hookean and four-fiber family microstructure-based model of Equation (1) with the following best-fit model parameters for the (

**a**) proximal jejunum: $\mu =$ 7.0 × 10

^{−12}kPa, ${k}_{1}^{d}=$ 3.121 kPa, ${k}_{2}^{d}=$ 9.743, ${k}_{1}^{a}$ = 25.331 kPa, ${k}_{2}^{a}$ = 3.590, ${k}_{1}^{c}$ = 1.7 × 10

^{−10}kPa, ${k}_{2}^{c}$ = 1.662, ${a}_{0}$ = 0.691 rad, ε = 0.239, ${R}^{2}=$ 0.927, $det\left(R\right)=$ 2.9 × 10

^{−5}, (

**b**) middle jejunum: $\mu =$ 1.7 × 10

^{−13}kPa, ${k}_{1}^{d}=$ 0.150 kPa, ${k}_{2}^{d}=$ 7.627, ${k}_{1}^{a}$ = 4.723 kPa, ${k}_{2}^{a}$ = 3.307, ${k}_{1}^{c}$ = 0.115 kPa, ${k}_{2}^{c}$ = 3.1 × 10

^{−7}, ${a}_{0}$ = 0.310 rad, ε = 0.312, ${R}^{2}=$ 0.871, $det\left(R\right)=$ 1.5 × 10

^{−4}, and (

**c**) distal jejunum: $\mu =$ 4.7 × 10

^{−12}kPa, ${k}_{1}^{d}=$ 1.237 kPa, ${k}_{2}^{d}=$ 5.379, ${k}_{1}^{a}$ = 1.264 kPa, ${k}_{2}^{a}$ = 0.944, ${k}_{1}^{c}$ = 2.4 × 10

^{−10}kPa, ${k}_{2}^{c}$ = 0.662, ${a}_{0}$ = 0.692 rad, ε = 0.342, ${R}^{2}=$ 0.824, $det\left(R\right)=$ 7.1 × 10

^{−4}. Data are shown every 0.5 mmHg for clarity.

**Figure 4.**Plot of measured lumen pressure (red color) and axial force (blue color) vs. external diameter data at three different fixed axial stretches ${\lambda}_{z}=$ {1.1, 1.2, 1.3} for a characteristic specimen from the proximal, middle, and distal ileum and fits (solid lines for lumen pressure and dashed lines for axial force) calculated using the neo-Hookean and four-fiber family microstructure-based model of Equation (1) with the following best-fit model parameters for the (

**a**) proximal ileum: $\mu =$ 0.103 kPa, ${k}_{1}^{d}=$ 0.006 kPa, ${k}_{2}^{d}=$ 6.312, ${k}_{1}^{a}$ = 1.966 kPa, ${k}_{2}^{a}$ = 2.102, ${k}_{1}^{c}$ = 2.4 × 10

^{−10}kPa, ${k}_{2}^{c}$ = 0.008, ${a}_{0}$ = 0.513 rad, ε = 0.260, ${R}^{2}=$ 0.905, $det\left(R\right)=$ 3.5 × 10

^{−6}, (

**b**) middle ileum: $\mu =$ 3.2 × 10

^{−11}kPa, ${k}_{1}^{d}=$ 0.694 kPa, ${k}_{2}^{d}=$ 4.890, ${k}_{1}^{a}$ = 3.722 kPa, ${k}_{2}^{a}$ = 7.2 × 10

^{−11}, ${k}_{1}^{c}$ = 4.0 × 10

^{−11}kPa, ${k}_{2}^{c}$ = 1.548, ${a}_{0}$ = 0.852 rad, ε = 0.380, ${R}^{2}=$ 0.788, $det\left(R\right)=$ 9.2×10

^{−6}, and (

**c**) distal ileum: $\mu =$ 7.0 × 10

^{−14}kPa, ${k}_{1}^{d}=$ 3.629 kPa, ${k}_{2}^{d}=$ 24.221, ${k}_{1}^{a}$ = 6.730 kPa, ${k}_{2}^{a}$ = 4.560, ${k}_{1}^{c}$ = 0.724 kPa, ${k}_{2}^{c}$ = 1.023, ${a}_{0}$ = 0.626 rad, ε = 0.333, ${R}^{2}=$ 0.864, $det\left(R\right)=$ 5.3 × 10

^{−5}. Data are shown every 0.5 mmHg for clarity.

**Figure 5.**Plots of measured data from the characteristic proximal ileum specimen shown in Figure 4 and fits by the neo-Hookean and four-fiber family model with vanishing (

**a**) neo-Hookean term: $\mu =$0 kPa, ${k}_{1}^{d}=$ 0.027 kPa, ${k}_{2}^{d}=$ 5.128, ${k}_{1}^{a}$ = 1.544 kPa, ${k}_{2}^{a}$ = 2.524, ${k}_{1}^{c}$ = 8.3 × 10

^{−11}kPa, ${k}_{2}^{c}$ = 0.086, ${a}_{0}$ = 0.512 rad, ε = 0.269, ${R}^{2}=$ 0.899, $det\left(R\right)=$ 1.7 × 10

^{−5}, (

**b**) diagonal-fiber families: $\mu =$ 0.307 kPa, ${k}_{1}^{d}=$ 0 kPa, ${k}_{2}^{d}=$ 0, = 4.488 kPa, ${k}_{2}^{a}$ = 0.812, ${k}_{1}^{c}$ = 0.002 kPa, ${k}_{2}^{c}$ = 0.295, ${a}_{0}$ = 0.785 rad, ε = 0.687, ${R}^{2}=$ 0.336, $det\left(R\right)=$ 0.001, (

**c**) axial-fiber family: $\mu =$ 3.5 × 10

^{−9}kPa, ${k}_{1}^{d}=$ 0.110 kPa, ${k}_{2}^{d}=$ 4.471, ${k}_{1}^{a}$ = 0 kPa, ${k}_{2}^{a}$ = 0, ${k}_{1}^{c}$ = 6.6 × 10

^{−9}kPa, ${k}_{2}^{c}$ = 0.020, ${a}_{0}$ = 0.487 rad, ε = 0.394, ${R}^{2}=$ 0.782, $det\left(R\right)=$ 9.4 × 10

^{−5}, and (

**d**) circumferential-fiber family: $\mu =$ 0.102 kPa, ${k}_{1}^{d}=$ 0.006 kPa, ${k}_{2}^{d}=$ 6.310, ${k}_{1}^{a}$ = 1.965 kPa, ${k}_{2}^{a}$ = 2.102, ${k}_{1}^{c}$ = 0 kPa, ${k}_{2}^{c}$ = 0, ${a}_{0}$ = 0.513 rad, ε = 0.260, ${R}^{2}=$ 0.905, $det\left(R\right)=$ 4.8 × 10

^{−4}.

**Figure 6.**As Figure 2 but using the neo-Hookean and (diagonal and axial)-fiber family model with the following best-fit model parameters for the (

**a**) proximal duodenum: $\mu =$ 3.7 × 10

^{−11}kPa, ${k}_{1}^{d}=$ 0.312 kPa, ${k}_{2}^{d}=$ 16.078, ${k}_{1}^{a}$ = 8.467 kPa, ${k}_{2}^{a}$ = 3.143, ${a}_{0}$ = 0.670 rad, ε = 0.321, ${R}^{2}=$ 0.870, $det\left(R\right)=$ 0.018 and (

**b**) distal duodenum: $\mu =$ 3.1 × 10

^{−6}kPa, ${k}_{1}^{d}=$ 0.707 kPa, ${k}_{2}^{d}=$ 5.332, ${k}_{1}^{a}$ = 0.915 kPa, ${k}_{2}^{a}$ = 9.892, ${a}_{0}$ = 0.319 rad, ε = 0.375, ${R}^{2}=$ 0.838, $det\left(R\right)=$ 3.7 × 10

^{−5}. Data are shown every 0.5 mmHg for clarity.

**Figure 7.**As Figure 3 but using the neo-Hookean and (diagonal and axial)-fiber family model with the following best-fit model parameters for the (

**a**) proximal jejunum: $\mu =$ 7.0 × 10

^{−12}kPa, ${k}_{1}^{d}=$ 3.121 kPa, ${k}_{2}^{d}=$ 9.743, ${k}_{1}^{a}$ = 25.331 kPa, ${k}_{2}^{a}$ = 3.590, ${a}_{0}$ = 0.691 rad, ε = 0.239, ${R}^{2}=$ 0.927, $det\left(R\right)=$ 0.020, (

**b**) middle jejunum: $\mu =$ 1.1 × 10

^{−13}kPa, ${k}_{1}^{d}=$ 0.175 kPa, ${k}_{2}^{d}=$ 7.071, ${k}_{1}^{a}$ = 2.920 kPa, ${k}_{2}^{a}$ = 4.464, ${a}_{0}$ = 0.319 rad, ε = 0.353, ${R}^{2}=$ 0.835, $det\left(R\right)=$ 9.6 × 10

^{−4}, and (

**c**) distal jejunum: $\mu =$ 4.7 × 10

^{−12}kPa, ${k}_{1}^{d}=$ 1.237 kPa, ${k}_{2}^{d}=$ 5.379, ${k}_{1\text{}}^{a}$ = 1.264 kPa, ${k}_{2}^{a}$ = 0.944, ${a}_{0}$ = 0.692 rad, ε = 0.342, ${R}^{2}=$ 0.824, $det\left(R\right)=$ 0.011. Data are shown every 0.5 mmHg for clarity.

**Figure 8.**As Figure 4 but using the neo-Hookean and (diagonal and axial)-fiber family model with the following best-fit model parameters for the (

**a**) proximal ileum: $\mu =$ 0.102 kPa, ${k}_{1}^{d}=$ 0.006 kPa, ${k}_{2}^{d}=$ 6.310, ${k}_{1}^{a}$ = 1.965 kPa, ${k}_{2}^{a}$ = 2.102, ${a}_{0}$ = 0.513 rad, ε = 0.260, ${R}^{2}=$ 0.905, $det\left(R\right)=$ 4.8 × 10

^{−4}, (

**b**) middle ileum: $\mu =$ 3.2 × 10

^{−11}kPa, ${k}_{1}^{d}=$ 0.694 kPa, ${k}_{2}^{d}=$ 4.890, ${k}_{1}^{a}$ = 3.722 kPa, ${k}_{2}^{a}$ = 7.5 × 10

^{−11}, ${a}_{0}$ = 0.852 rad, ε = 0.380, ${R}^{2}=$ 0.788, $det\left(R\right)=$ 0.002, and (

**c**) distal ileum: $\mu =$ 6.3 × 10

^{−14}kPa, ${k}_{1}^{d}=$ 3.281 kPa, ${k}_{2}^{d}=$ 29.283, ${k}_{1}^{a}$ = 7.588 kPa, ${k}_{2}^{a}$ = 4.554, ${a}_{0}$ = 0.678 rad, ε = 0.378, ${R}^{2}=$ 0.825, $det\left(R\right)=$ 0.016. Data are shown every 0.5 mmHg for clarity.

**Figure 9.**Representative adjacent transverse histologic sections for the (

**a**–

**c**) distal duodenum, (

**d**–

**f**) middle jejunum, and (

**g**–

**i**) middle ileum, stained with hematoxylin-eosin, orcein, and Sirius red for the identification of cells, elastin, and collagen, respectively. The scale bar applies to all the images.

**Figure 10.**Thickness of the mucosa, submucosa, muscle, and serosa in the small intestinal segments. Symbols †, #, *, ‡, and ^ denote significant difference vs. proximal duodenum, distal duodenum, proximal jejunum, middle jejunum, and distal jejunum.

**Table 1.**Parameters of the neo-Hookean and four-fiber family model fitted to experimental data of eight small intestinal segments.

μ [kPa] | ${\mathit{k}}_{\mathbf{1}}^{\mathit{d}}\phantom{\rule{0ex}{0ex}}\left[\mathbf{kPa}\right]$ | ${\mathit{k}}_{\mathbf{2}}^{\mathit{d}}\phantom{\rule{0ex}{0ex}}[-]$ | ${\mathit{k}}_{\mathbf{1}}^{\mathit{a}}\phantom{\rule{0ex}{0ex}}\left[\mathbf{kPa}\right]$ | ${\mathit{k}}_{\mathbf{2}}^{\mathit{a}}\phantom{\rule{0ex}{0ex}}[-]$ | ${\mathit{k}}_{\mathbf{1}}^{\mathit{c}}\phantom{\rule{0ex}{0ex}}\left[\mathbf{kPa}\right]$ | ${\mathit{k}}_{\mathbf{2}}^{\mathit{c}}\phantom{\rule{0ex}{0ex}}[-]$ | ${\mathit{a}}_{\mathbf{0}}\text{}\phantom{\rule{0ex}{0ex}}\left[\mathbf{rad}\right]$ | ε [-] | ${\mathit{R}}^{\mathbf{2}}\text{}\phantom{\rule{0ex}{0ex}}[-]$ | $\mathit{d}\mathit{e}\mathit{t}\left(\mathit{R}\right)\text{}\phantom{\rule{0ex}{0ex}}[-]$ | |
---|---|---|---|---|---|---|---|---|---|---|---|

PD | 0.042 ± 0.042 | 1.256 ± 0.527 * | 17.118 ± 4.047 | 16.751 ± 6.176 | 3.788 ± 0.730 | 0.003 ± 0.003 ^{&} | 1.546 ± 0.605 | 0.651 ± 0.029 ^{#} | 0.325 ± 0.014 | 0.865 ± 0.011 | (14.1 ± 9.2) × 10 ^{−5} |

DD | (8.3 ± 8.3) × 10 ^{−7} | 0.955 ± 0.216 * | 5.964 ± 0.641 | 10.592 ± 4.936 * | 5.235 ± 1.351 | 0.167 ± 0.031 | 0.057 ± 0.046 ^{@} | 0.378 ± 0.028 | 0.332 ± 0.026 | 0.857 ± 0.024 | (9.0 ± 4.2) × 10 ^{−5} |

PJ | (1.8 ± 1.2) × 10 ^{−12} | 3.760 ± 0.757 | 10.521 ± 1.926 | 49.491 ± 18.385 | 2.327 ± 0.566 | (1.7 ± 1.4) × 10^{−9 &} | 0.607 ± 0.269 ^{@} | 0.665 ± 0.028 ^{#} | 0.287 ± 0.019 | 0.883 ± 0.018 | 0.002 ± 0.002 |

MJ | (2.8 ± 2.7) × 10 ^{−9} | 0.615 ± 0.131 * | 4.447 ± 1.006 | 19.571 ± 7.680 | 3.252 ± 0.988 | 0.045 ± 0.020 | 0.205 ± 0.186 ^{@} | 0.560 ± 0.056 ^{#} | 0.263 ± 0.015 | 0.908 ± 0.009 | (6.2 ± 2.3) × 10 ^{−5} |

DJ | (13.5 ± 8.9) × 10 ^{−11} | 1.064 ± 0.191 * | 10.974 ± 3.093 | 16.812 ± 6.229 | 2.836 ± 0.570 | 0.090 ± 0.063 | 1.191 ± 0.649 | 0.622 ± 0.031 ^{#} | 0.271 ± 0.019 | 0.898 ± 0.017 | (3.8 ± 1.5) × 10^{−4} |

PI | 0.065 ± 0.048 | 0.642 ± 0.265 * | 9.651 ± 2.983 | 13.258 ± 4.926 * | 3.226 ± 0.420 | (3.4 ± 1.8) × 10^{−8 &} | 0.095 ± 0.092 ^{@} | 0.599 ± 0.050 ^{#} | 0.236 ± 0.021 | 0.919 ± 0.011 | (12.0 ± 5.1) × 10 ^{−5} |

MI | (2.5 ± 1.7) × 10 ^{−7} | 1.261 ± 0.213 * | 18.490 ± 5.010 | 3.713 ± 0.668 * | 2.518 ± 0.966 | (1.6 ± 1.4) × 10^{−4&} | 5.298 ± 2.608 | 0.704 ± 0.048 ^{#} | 0.319 ± 0.018 | 0.859 ± 0.019 | (5.7 ± 3.9) × 10 ^{−5} |

DI | (6.2 ± 3.6) × 10 ^{−9} | 2.102 ± 0.658 | 14.938 ± 2.796 | 9.541 ± 2.091 * | 4.540 ± 0.664 | 0.240 ± 0.108 | 1.751 ± 0.283 | 0.639 ± 0.018 ^{#} | 0.304 ± 0.020 | 0.882 ± 0.017 | (9.8 ± 4.7) × 10 ^{−5} |

^{#}, *,

^{@}, and

^{&}denote significant difference vs. DD, PJ, MI, and DI. Refer to Tables S1–S3 in the Supplementary Materials for the individual parameter values of the duodenum, jejunum, and ileum, respectively.

**Table 2.**Validation of the parameters of the neo-Hookean and four-fiber family terms for pooled data from the duodenum, jejunum, and ileum.

μ [kPa] | ${\mathit{k}}_{\mathbf{1}}^{\mathit{d}}\text{}\phantom{\rule{0ex}{0ex}}\left[\mathbf{kPa}\right]$ | ${\mathit{k}}_{\mathbf{2}}^{\mathit{d}}\text{}\phantom{\rule{0ex}{0ex}}[-]$ | ${\mathit{k}}_{\mathbf{1}}^{\mathit{a}}\text{}\phantom{\rule{0ex}{0ex}}\left[\mathbf{kPa}\right]$ | ${\mathit{k}}_{\mathbf{2}}^{\mathit{a}}\text{}\phantom{\rule{0ex}{0ex}}[-]$ | ${\mathit{k}}_{\mathbf{1}}^{\mathit{c}}\text{}\phantom{\rule{0ex}{0ex}}\left[\mathbf{kPa}\right]$ | ${\mathit{k}}_{\mathbf{2}}^{\mathit{c}}\text{}\phantom{\rule{0ex}{0ex}}[-]$ | ${\mathit{a}}_{\mathbf{0}}\text{}\phantom{\rule{0ex}{0ex}}\left[\mathbf{rad}\right]$ | ε [-] | ${\mathit{R}}^{\mathbf{2}}\text{}\phantom{\rule{0ex}{0ex}}[-]$ | $\mathit{d}\mathit{e}\mathit{t}\left(\mathit{R}\right)\text{}\phantom{\rule{0ex}{0ex}}[-]$ | |
---|---|---|---|---|---|---|---|---|---|---|---|

Zero Neo-Hookean Term | |||||||||||

D | 0 | 1.118 ± 0.330 | 11.320 ± 2.968 | 13.155 ± 4.550 | 4.557 ± 0.920 | 0.088 ± 0.033 | 0.857 ± 0.434 | 0.519 ± 0.051 | 0.357 ± 0.022 | 0.835 ± 0.019 | (3.4 ± 1.0) × 10^{−4} |

J | 0 | 1.817 ± 0.480 | 8.709 ± 1.633 | 28.980 ± 10.048 | 2.857 ± 0.491 | 0.046 ± 0.027 | 0.738 ± 0.292 | 0.613 ± 0.029 | 0.272 ± 0.012 | 0.895 ± 0.012 | (8.4 ± 2.5) × 10^{−4} |

I | 0 | 1.377 ± 0.319 | 14.084 ± 3.669 | 8.338 ± 2.182 | 3.438 ± 0.522 | 0.080 ± 0.051 | 2.542 ± 1.165 | 0.649 ± 0.030 | 0.290 ± 0.016 | 0.883 ± 0.012 | (5.6 ± 2.1) × 10^{−4} |

Zero Diagonal-Fiber Families | |||||||||||

D | 0.103 ± 0.103 | 0 | 0 | 20.853 ± 6.232 | 3.422 ± 0.701 | 0.161 ± 0.053 | 2.483 ± 1.105 | 0 | 0.665 ± 0.031 | 0.442 ± 0.038 | 0.031 ± 0.011 |

J | (1.9 ± 1.8) × 10 ^{−9} | 0 | 0 | 36.421 ± 10.349 | 2.048 ± 0.347 | 0.185 ± 0.051 | 0.964 ± 0.279 | 0 | 0.664 ± 0.024 | 0.392 ± 0.041 | 0.018 ± 0.004 |

I | 0.064 ± 0.047 | 0 | 0 | 14.093 ± 3.330 | 2.507 ± 0.400 | 0.259 ± 0.094 | 0.890 ± 0.244 | 0 | 0.673 ± 0.024 | 0.389 ± 0.037 | 0.013 ± 0.003 |

Zero Axial-Fiber Family | |||||||||||

D | 0.144 ± 0.014 | 1.939 ± 0.623 | 10.332 ± 3.270 | 0 | 0 | 0.110 ± 0.054 | 0.924 ± 0.424 | 0.487 ± 0.054 | 0.572 ± 0.041 | 0.562 ± 0.062 | 0.010 ± 0.004 |

J | (1.3 ± 1.2) × 10 ^{−9} | 2.529 ± 0.563 | 7.478 ± 1.339 | 0 | 0 | 0.093 ± 0.065 | 0.496 ± 0.170 | 0.572 ± 0.029 | 0.577 ± 0.032 | 0.521 ± 0.055 | 0.011 ± 0.007 |

I | (2.7 ± 2.7) × 10 ^{−4} | 2.979 ± 0.817 | 9.869 ± 3.525 | 0 | 0 | 0.265 ± 0.164 | 1.381 ± 0.346 | 0.562 ± 0.028 | 0.511 ± 0.023 | 0.645 ± 0.030 | 0.002 ± 0.001 |

Zero Circumferential-Fiber Family | |||||||||||

D | 0.025 ± 0.025 | 1.185 ± 0.320 | 11.016 ± 3.043 | 12.280 ± 4.562 | 5.127 ± 1.043 | 0 | 0 | 0.528 ± 0.049 | 0.375 ± 0.022 | 0.819 ± 0.020 | 0.007 ± 0.002 |

J | (1.1 ± 1.1) × 10^{−9} | 1.804 ± 0.483 | 8.660 ± 1.630 | 28.639 ± 10.076 | 3.075 ± 0.568 | 0 | 0 | 0.619 ± 0.028 | 0.285 ± 0.014 | 0.886 ± 0.011 | 0.015 ± 0.003 |

I | 0.026 ± 0.020 | 1.298 ± 0.311 | 14.431 ± 3.755 | 8.638 ± 2.228 | 3.329 ± 0.482 | 0 | 0 | 0.671 ± 0.031 | 0.302 ± 0.019 | 0.873 ± 0.015 | 0.008 ± 0.002 |

**Table 3.**Parameters of the neo-Hookean and (diagonal and axial)-fiber family model fitted to experimental data of eight small intestinal segments.

μ [kPa] | ${\mathit{k}}_{\mathbf{1}}^{\mathit{d}}\text{}\phantom{\rule{0ex}{0ex}}\left[\mathbf{kPa}\right]$ | ${\mathit{k}}_{\mathbf{2}}^{\mathit{d}}\text{}\phantom{\rule{0ex}{0ex}}[-]$ | ${\mathit{k}}_{\mathbf{1}}^{\mathit{a}}\text{}\phantom{\rule{0ex}{0ex}}\left[\mathbf{kPa}\right]$ | ${\mathit{k}}_{\mathbf{2}}^{\mathit{a}}\text{}\phantom{\rule{0ex}{0ex}}[-]$ | ${\mathit{a}}_{\mathbf{0}}\text{}\phantom{\rule{0ex}{0ex}}\left[\mathbf{rad}\right]$ | ε [-] | ${\mathit{R}}^{2}\text{}\phantom{\rule{0ex}{0ex}}[-]$ | $\mathit{d}\mathit{e}\mathit{t}\left(\mathit{R}\right)\text{}\phantom{\rule{0ex}{0ex}}[-]$ | |
---|---|---|---|---|---|---|---|---|---|

PD | 0.042 ± 0.042 | 1.237 ± 0.517 * | 17.196 ± 4.024 | 16.724 ± 6.175 | 3.822 ± 0.735 | 0.652 ± 0.028 ^{#} | 0.325 ± 0.014 | 0.865 ± 0.011 | 0.011 ± 0.002 |

DD | (5.2 ± 5.2) × 10^{−7} | 1.127 ± 0.196 * | 5.145 ± 0.522 | 9.183 ± 4.722 * | 6.136 ± 1.497 | 0.397 ± 0.030 | 0.364 ± 0.024 | 0.827 ± 0.024 | 0.004 ± 0.002 |

PJ | (1.8 ± 1.2) × 10^{−12} | 3.760 ± 0.757 | 10.521 ± 1.926 | 49.491 ± 18.385 | 2.327 ± 0.566 | 0.665 ± 0.028 ^{#} | 0.286 ± 0.019 | 0.883 ± 0.018 | 0.028 ± 0.005 |

MJ | (2.9 ± 2.7) × 10^{−9} | 0.565 ± 0.130 * | 4.549 ± 1.012 | 19.244 ± 7.821 | 3.594 ± 1.140 | 0.570 ± 0.056 ^{#} | 0.281 ± 0.024 | 0.894 ± 0.016 | 0.006 ± 0.002 |

DJ | (14.5 ± 9.3) × 10^{−11} | 1.086 ± 0.189 * | 10.816 ± 3.110 | 16.349 ± 6.061 | 3.030 ± 0.714 | 0.628 ± 0.029 ^{#} | 0.284 ± 0.019 | 0.888 ± 0.017 | 0.010 ± 0.002 |

PI | 0.065 ± 0.048 | 0.642 ± 0.265 * | 9.650 ± 2.983 | 13.258 ± 4.927 * | 3.226 ± 0.420 | 0.599 ± 0.050 ^{#} | 0.236 ± 0.021 | 0.919 ± 0.011 | 0.004 ± 0.002 |

MI | (2.6 ± 1.7) × 10 ^{−7} | 1.251 ± 0.214 * | 18.474 ± 5.005 | 3.669 ± 0.671 * | 2.520 ± 0.965 | 0.709 ± 0.045 ^{#} | 0.323 ± 0.020 | 0.855 ± 0.021 | 0.009 ± 0.002 |

DI | (6.5 ± 3.9) × 10 ^{−9} | 1.967 ± 0.648 | 15.262 ± 3.629 | 9.918 ± 1.806 * | 4.396 ± 0.484 | 0.695 ± 0.027 ^{#} | 0.333 ± 0.028 | 0.858 ± 0.022 | 0.011 ± 0.003 |

^{#}and * denote significant difference vs. DD and PJ. Refer to Tables S4–S6 in the Supplementary Materials for the individual parameter values of the duodenum, jejunum, and ileum, respectively.

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**MDPI and ACS Style**

Sokolis, D.P.
Variation of Passive Biomechanical Properties of the Small Intestine along Its Length: Microstructure-Based Characterization. *Bioengineering* **2021**, *8*, 32.
https://doi.org/10.3390/bioengineering8030032

**AMA Style**

Sokolis DP.
Variation of Passive Biomechanical Properties of the Small Intestine along Its Length: Microstructure-Based Characterization. *Bioengineering*. 2021; 8(3):32.
https://doi.org/10.3390/bioengineering8030032

**Chicago/Turabian Style**

Sokolis, Dimitrios P.
2021. "Variation of Passive Biomechanical Properties of the Small Intestine along Its Length: Microstructure-Based Characterization" *Bioengineering* 8, no. 3: 32.
https://doi.org/10.3390/bioengineering8030032