## Abstract

Birthing mechanics are poorly understood, though many injuries during childbirth are mechanical, like fetal and maternal tissue damage. Several biomechanical simulation models of parturition have been proposed to investigate birth, but many do not include the uterus. Additionally, most solid models rely on segmenting anatomical structures from clinical images to generate patient geometry, which can be time-consuming. This work presents two new parametric solid modeling methods for generating patient-specific, at-term uterine three-dimensional geometry. Building from an established method of modeling the sagittal uterine shape, this work improves the uterine coronal shape, especially where the fetal head joins the lower uterine wall. Solid models of the uterus and cervix were built from five at-term patients' magnetic resonance imaging (MRI) sets. Using anatomy measurements from MRI-segmented models, two parametric models were created—one that employs an averaged coronal uterine shape and one with multiple axial measurements of the coronal uterus. Through finite element analysis, the two new parametric methods were compared to the MRI-segmented high-fidelity method and a previously published elliptical low-fidelity method. A clear improvement in the at-term uterine shape was found using the two new parametric methods, and agreement in principal Lagrange strain directions was observed across all modeling methods. These methods provide an effective and efficient way to generate three-dimensional solid models of patient-specific maternal uterine anatomy, advancing possibilities for future research in computational birthing biomechanics.

## 1 Introduction

In 2022, there were 3.66 × 10^{6} births in the United States [1]. Many deliveries in the United States receive clinical intervention, with 32% undergoing cesarean section, 32% undergoing induction of labor (artificially prompting uterine contractions), and 3% having an operative vaginal delivery (use of vacuum or forceps) [1–3]. Recognizing the importance of improving maternal and fetal outcomes and overall maternal childbirth experience, the American College of Obstetricians and Gynecologists recommends limiting interventions during labor and birth for low-risk women in spontaneous labor [4]. However, progress in the understanding of maternal anatomy and soft-tissue mechanical changes that occur with advancing pregnancy, labor, and delivery remains elusive. To balance the push toward reduced medical intervention during childbirth while ensuring maximal patient safety, attention has turned to studying birthing mechanics via computational methods.

Existing computational models of parturition have helped provide an enhanced understanding of maternal pelvic floor injuries, fetal injuries, and the mechanics of clinical labor practices, but few include the uterus [5]. The inclusion of the uterus is important for a complete mechanical understanding of labor, as its contractions dilate the cervix and push the fetus through the birth canal during normal vaginal delivery [6]. Despite its importance, few methods exist to generate at-term uterine geometry for computational simulation. Many works rely on manual segmentation of magnetic resonance images (MRI) to generate at-term uterine models, which is a tedious process due to the inability of contrast-based segmentation methods to distinguish the uterus from surrounding tissue [7–10]. Other studies utilize a uterine surface mesh of a near-term pregnant patient made available by the FEMONUM project, which was also derived via manual MRI segmentation [11–17]. Additionally, certain work omits derivation from clinical images and instead generates at-term uterine geometry by wrapping a fetal model with a shell of uniform thickness [18,19]. These uterine models have provided a necessary starting point for complex computational birth simulations. Moving forward, a goal of our research is to provide fast and efficient modeling methods that can capture key anatomical features from large medical imaging datasets. The key drawbacks of the initial modeling methods are they do not efficiently allow for patient-specific uterine modeling, nor can they explore the potential impacts of anatomic variation. These challenges can be addressed through parametric modeling approaches.

Parametric modeling is a method in which solid model features and constraints are established through boolean geometry, allowing for fast and flexible size and shape changes given a set of dimension values. In previous work, a parametric model of the uterus and cervix was developed based on measurements from two-dimensional (2D) ultrasound to study the effect of cervical geometry and stiffness on cervical loading during gestation [20]. Using 2D ultrasound to obtain patient-specific measurements allows broad use of the parametric model, as 2D ultrasound is part of routine prenatal care. The modeling protocol has since been updated to refine the shape of the uterus in the sagittal plane, accounting for variances in posterior uterine wall shape [21]. Previously published work has quantified how the sagittal and axial uterine dimensions change throughout gestation, but little is known about how the uterine coronal shape evolves. Until now, an elliptical coronal shape was assumed for the uterus before 37 weeks of gestation [21]. Though this assumption may be sufficient in modeling the uterus during these gravid time points, there are significant changes in fetal position during the last few weeks of pregnancy. During this time, the fetal head turns downward and descends into the pelvic area, which may affect the distribution of the load on the uterus and cervix. As a result, further investigation is warranted to establish an at-term (gestational age between 37 and 41 weeks) patient-specific parametric uterine model.

The parametric modeling methods in this study aim to capture the coronal at-term uterine shape observed in MRI imaging data and corresponding MRI segmented solid models of five at-term pregnancies [22,23]. Two new parametric uterine modeling methods are introduced: one relying on the same set of measurements as in previously published work with an updated coronal shape (averaged) and one including several additional measurements of the coronal uterine shape (multimeasure) (Fig. 1) [21]. The new parametric modeling methods are compared to MRI segmented models and the existing parametric uterine modeling method (elliptical) through an initial finite element analysis (FEA) of uterine pressurization. Anatomical dimension measurements from MRI data and the MRI segmented models are also reported and compared.

## 2 Methodology

### 2.1 Uterine and Cervical Anatomical Measurements.

Anatomical dimension measurements were collected from MRI images (Sec. 2.1.1) and corresponding MRI segmented models (Sec. 2.1.2) of five at-term pregnant patients. The MRI images are the original clinical images, and the MRI segmented models are solid computer-aided design models generated via manual segmentation of the MRI images. The same dimension measurements were collected from the MRI images and MRI segmented models to confirm findings on the uterine coronal shape and ensure all dimension measurements could be collected from MRI images. The MRI images used in this research were collected for and used by Joyce et al. in a study approved by the institutional review board at MetroHealth Medical Center/Case Western Reserve University [22]. The pregnant patients (gestational age $38\u221239$ weeks) who underwent MRI were scheduled for repeat cesarean section delivery. There were no reported pregnancy complications or anomalies. The MRI images were later segmented to generate solid models of the uterus and cervix and made publicly available [23]. From both the MRIs and MRI segmented models, two sets of dimension measurements were collected: (1) dimension measurements of the overall uterus and cervix using a previously published parametric method (Fig. 2) and (2) dimension measurements characterizing the at-term uterine coronal shape (Fig. 3) [20,21].

Dimension measurements based on a previously published measurement protocol were collected as follows (Fig. 2) [20,21]

UD1: inferior-superior intrauterine diameter

UD2: anterior intrauterine diameter

UD3: posterior intrauterine diameter

UD3a: posterior intrauterine diameter extrema superior to UD3

UD1a: locator dimension for UD3a from inferior side of UD1

UD3b: posterior intrauterine diameter extrema inferior to UD3

UD1b: locator dimension for UD3b from inferior side of UD1

UD4: left-right intrauterine diameter

UT1: fundal uterine wall thickness

UT2: anterior uterine wall thickness

UT3: left/right uterine wall thickness

UT4: lower uterine segment thickness

PCO: posterior cervical offset

AUCA: anterior uterocervical angle

CL: cervical length

CD1: outer cervical diameter

CD2: cervical canal diameter

To characterize the coronal shape of the at-term uterus, measurements were taken of the left-right (axial) intra-uterine diameter, left uterine wall thickness, and right uterine wall thickness at multiple locations along the inferior-superior axis of the uterus (Fig. 3). The percentage error between the MRI and MRI segmented model measurement sets was calculated. See Online Resource 1, available through Columbia University's academic commons, for further dimension measurement details [24].

#### 2.1.1 Measurements From Magnetic Resonance Imaging Images.

ImageJ was used to acquire measurements from the MRI image stacks, which had sagittal stack intervals of 5.5 mm and axial stack intervals of 5 or 5.5 mm, and the Bioformats plugin was used to load the image stacks [25,26]. The sagittal MRI image with the widest cervical canal was used as the sagittal measurement plane. The axial measurement plane was the axial MRI image with the largest left-right intra-uterine diameter. Additional left-right intra-uterine diameters, left uterine wall thickness, and right uterine wall thicknesses were measured from the axial MRI images. These measurements were taken where the left and right uterine wall thicknesses were adequately visible, not necessarily in every image.

#### 2.1.2 Measurements From Magnetic Resonance Imaging Segmented Models.

Solidworks 2018-19 (Dassault Systémes, Vélizy-Villacoublay, France) was used to obtain measurements from the MRI segmented models (Sec. 2.2.1) [23]. The sagittal measurement plane was chosen as the sagittal plane with the widest cervical canal, and the axial measurement plane as the axial plane having the largest left-right intra-uterine diameter. Fifteen left-right intra-uterine diameters, left uterine wall thicknesses, and right uterine wall thicknesses were collected. Ten of these measurements were taken equidistantly along the inferior-superior uterine axis, and five were concentrated in the inferior-most section of the uterus for a detailed evaluation near the uterocervical junction (Fig. 3). The left-right intra-uterine diameters were measured as the longest in-plane horizontal diameter, and the left and right uterine wall thicknesses were taken at the ends of the longest left-right intra-uterine diameter.

### 2.2 Patient-Specific Solid Models.

Four methods were used for each of the five patients to generate models of their uterus and cervix, one relying on MRI segmentation and the other three based on parametric methods (Fig. 1). All parametric models' sagittal shape was built as described in previous work [21]. The parametric models assumed intra-uterine symmetry over the midsagittal plane and used the measurements taken from the MRI segmented model due to differences in uterine wall thickness measurements found between the MRI images and MRI segmented models (Sec. 3.1). Additionally, all parametric models were established as default configurations in SolidWorks 2018-19 (Dassault Systémes, Vélizy-Villacoublay, France), where the default configuration provided a template for automatic construction of all subsequent patient-specific parametric models. The primary difference between the parametric models lies in their coronal shape approach:

Multi-measured: coronal shape based on 45 axial measurements (15 intrauterine diameters, 15 left uterine wall thicknesses, and 15 right uterine wall thicknesses)

Averaged: coronal shape based on the average coronal shape across the five patients

Elliptical: coronal shape is assumed to be an ellipse

#### 2.2.1 Magnetic Resonance Imaging Segmented Model.

MRI images were obtained from eight at-term pregnant women ($38.4\xb10.4$ weeks gestation) before cesarean section, of which five were ultimately used in this study [22]. The sagittal view of the MRI stacks was used to perform manual segmentation in Seg3D (CIBC, Utah). Manual segmentation was used as traditional thresholding tools were ineffective due to the similar contrast of the uterus and cervix to the surrounding abdominal tissues. Three entities were segmented: the gross uterine body (including the amniotic sac and the cervix), the uterine cavity (the combination of the amniotic sac and the placenta), and the cervical canal (visible as the mucus plug in MRI). A Boolean operation was applied to derive the final volume of the uterus and cervix as the subtraction of the uterine cavity and the cervical canal from the gross uterine body. For efficiency, segmentation was performed on one out of every three sagittal images, except when the images contained critical features such as the cervical canal, the cervical boundary, and the uterine boundary. An interpolation was performed between all segmented images to render a three-dimensional geometry, and the result was exported as a stereolithography (STL) formatted surface. Conservative smoothing was done on the segmented isosurface in 3DCoat (Pilgway, Kiev, Ukraine), using the smoothing and filling tool to eliminate minor surface imperfections while preserving the overall geometric features. After smoothing, three geometries had incomplete uterine walls. Therefore, only five of the original eight patients were used in this research.

#### 2.2.2 Elliptical Parametric Model.

The elliptical model was built using a published parametric modeling protocol, which was developed for the uterus and cervix before 37 weeks gestation [21,27]. As suggested in the S2 Appendix of previous work, the posterior uterine wall was formulated using a method to capture the extrema, which generated more accurate sagittal uterine shapes overall [21]. The coronal shape was modeled as an ellipse, though the method of geometry generation does not enforce elliptical axial cross section due to the loft function executing from left to right.

#### 2.2.3 Averaged Parametric Model.

A detailed workflow for generating the averaged parametric model in SolidWorks 2018-19 (Dassault Systémes, Vélizy-Villacoublay, France) is available through Columbia University Library's Academic Commons (Online Resource 2) [24]. Trends observed in the left-right intra-uterine diameter in four sections along the inferior-superior intra-uterine axis informed the averaged parametric model's coronal profile, with 0% at the inferior-most point and 100% at the superior-most point (see Sec. 3.1):

$0\u20139$%: arc connecting inferior point of the inferior-superior axis to a left-right intrauterine diameter at 9% of the inferior-superior axis with a value that is 44% of the largest left-right intrauterine diameter

$9\u201338$%: line connecting left-right intrauterine diameter at 9% of the inferior-superior axis to a left-right intrauterine diameter at 38% of the inferior-superior axis with a value that is 84% of the largest left-right intrauterine diameter

$38\u201362$%: arc connecting the left-right intrauterine diameter at 38% of the inferior-superior axis to a left-right intrauterine diameter at 62% of the inferior-superior axis with a value that is the largest left-right intrauterine diameter

$62\u2013100$%: semi-ellipse connecting the left-right uterine diameter at 62% of the inferior-superior axis to the superior point of the inferior-superior axis

Uterine wall thicknesses were added to generate the outer uterine coronal shape.

#### 2.2.4 Multi-Measured Parametric Model.

A detailed workflow for generating the multimeasured parametric model in SolidWorks 2018-19 (Dassault Systémes, Vélizy-Villacoublay, France) is available through Columbia University's Academic Commons (Online Resource 3) [24]. The intra-uterine coronal shape was generated using the left-right intra-uterine diameter measurements from the MRI segmented model along the inferior-superior intra-uterine axis. Left and right uterine wall thicknesses measured from the MRI segmented model were added at the corresponding intra-uterine diameter locations to generate the outer uterine coronal shape.

### 2.3 Finite Element Simulation Models.

To compare the effect of the four solid modeling methods on uterine wall strain, FEA models of the five patients with contraction-level pressure were built. Half-geometries generated in SolidWorks 2018-19 (Dassault Systémes, Vélizy-Villacoublay, France) were imported into Hypermesh 2020 (Altair Engineering, Inc., Troy, MI) and discretized into elements. The finite element simulation was setup, and the analysis was performed in FEBio Studio v1.3.0 [28]. The resultant principal Lagrange strain magnitudes and directions were compared between the parametric and MRI segmented models [28].

#### 2.3.1 Mesh Generation.

All geometries were discretized using the three-dimensional volume tetra generation tool in Hypermesh 2020 (Altair Engineering, Inc., Troy, MI). Each geometry was comprised of linear tetrahedral elements, and the element size (Table 1) used was determined through a mesh size convergence study, where visual agreement in principal Lagrange strain direction and magnitude pattern was used as convergence criteria.

#### 2.3.2 Material Properties.

The uterus and cervix were treated as a continuous body, with the same material properties assigned throughout. Since a purely kinematic comparison between modeling methods was desired, rather than a realistic childbirth simulation, a nearly incompressible neo-Hookean material was chosen for computational simplicity. A Young's modulus (*E*) of 500 kPa and Poisson's ratio (*ν*) of 0.48 was used in FEBio [29]. At-term uterine tissue has very nonlinear mechanical behavior, with higher strains exhibiting stiffer responses [30]. A Young's modulus of 500 kPa was chosen to represent the mechanical response of the highly deformed pregnant uterus, though it does not represent the uterus across strain regimes.

#### 2.3.3 Boundary Conditions and Loading.

The plane that separated the superior half of the uterus from the inferior half was fixed in the *x*, *y*, and *z* directions (Fig. 4). A contraction level intra-uterine pressure of 8 kPa was applied to the intra-uterine and cervical canal surfaces and is similar to previously used contraction-level intra-uterine pressures in parametric modeling [20,31]. This approach is a simplification of in vivo contractions, which pulsate in a bell-shaped wave, and it is on the higher end of normal reported first stage of labor peak uterine pressures, which generally range from 1.3 to 6.7 kPa, but can reach 8 to 9.3 kPa [6].

#### 2.3.4 Finite Element Analysis and Interpretation.

Finite element analyses were setup, performed, and visualized in FEBio Studio (v1.3.0) [28]. For each model, heat maps of first, second, and third principal Lagrange strain magnitude and direction were generated and visually compared. The method for FEBio's Lagrange strain computation can be found in the FEBio theory manual [32].

*D*, which is the maximum deviation Kolomogorov statistic (Eq. (1))

*i*th principal Lagrange strain in the

*j*th section of the parametric model for patient

*X*, and

*m*is the number of elements in section

*j*of the parametric model. Similarly, $EiiMRIsegX,j$ is a $1\xd7n$ vector of the

*i*th principal Lagrange strain in the

*j*th section of the parametric model for patient

*X*, and

*n*is the number of elements in section

*j*of the MRI segmented model. $DiModelX,j$ is the resulting value from the KS test between $EiiModelX,j$ and $EiiMRIsegX,j$. $DiModelX,j$ ranges from 0 to 1, where values closer to 0 indicate similarity between the two distributions. The similarity metric in the

*i*th principal Lagrange strain in the

*j*th section of the parametric model for patient

*X*was therefore set to be $1\u2212DiModelX,j$, such that values closer to 1 indicate more similarity. This was then multiplied by $VjMRIsegX$, which is the ratio of the volume of section

*j*over the total volume in all 12 uterine sections. Adding the weighted similarity across sections yielded the similarity metric for the

*i*th principal Lagrange strain in the parametric model of patient

*X*, $K\xafiModelX$ (Eq. (2))

*X*(Eq. (3))

*j*for all principal strains and patients (Eq. (5))

Similarity scores between 0 and 1 are possible, with similarity scores closer to 1 indicating high similarity in strain distribution between the MRI-segmented and parametric model. Therefore, when comparing parametric models to the MRI-segmented models, the parametric model with the highest similarity score was de-emed most similar, and the model with the lowest similarity score was de-emed least similar. Though these similarity scores provide a method of quantitative comparison between the parametric and MRI-segmented models, it is most likely not suitable as a means of quantitative comparison in specific at-term simulations. Future use of these models should evaluate model fitness using quantities of interest in the context of use amongst other credibility assessment methods [34].

## 3 Results

### 3.1 Anatomical Measurements From Magnetic Resonance Imaging and Magnetic Resonance Imaging Segmented Models.

For the dimensions depicted in Fig. 2, as described in Westervelt et al. and Louwagie et al., measurement values differed between the MRI and associated MRI segmented model with percentage difference varying with dimension (Fig. 6, Table 2) [20,21]. Small average errors (less than 15%) in intra-uterine diameter dimension measurements between the MRI segmented models MRI images were found, except for UD3 (39% error). Dimension measurements describing the cervix size and placement also had small average errors, with the exception of CD2 (36%). The largest disagreement in dimension measurement values between the MRI and MRI segmented models was in uterine wall thicknesses, with three out of four having large average errors (greater than 40%). The exception was UT1, with an average error of 4.5%. With these disparities established, the additional 15 coronal dimension measurement sets were examined.

Dimension | Average error (%) | Error range (%) |
---|---|---|

UD1 | 2 | 1–4 |

UD2 | 6 | 0–20 |

UD3 | 39 | 6–76 |

UD3a | 13 | 2–39 |

UD1a | 6 | 2–10 |

UD3b | 8 | 2–14 |

UD1b | 12 | 4–24 |

UD4 | 6 | 4–11 |

PCO | 12 | 4–26 |

UT1 | 4 | 1–8 |

UT2 | 48 | 7–78 |

UT3 | 76 | 1–246 |

UT4 | 90 | 56–174 |

CL | 15 | 3–35 |

CD1 | 11 | 1–31 |

CD2 | 36 | 10–66 |

AUCA | 9 | 2–18 |

Left-right diameter | 7 | 0–25 |

Left wall thickness | 120 | 1–489 |

Right wall thickness | 137 | 3–328 |

Dimension | Average error (%) | Error range (%) |
---|---|---|

UD1 | 2 | 1–4 |

UD2 | 6 | 0–20 |

UD3 | 39 | 6–76 |

UD3a | 13 | 2–39 |

UD1a | 6 | 2–10 |

UD3b | 8 | 2–14 |

UD1b | 12 | 4–24 |

UD4 | 6 | 4–11 |

PCO | 12 | 4–26 |

UT1 | 4 | 1–8 |

UT2 | 48 | 7–78 |

UT3 | 76 | 1–246 |

UT4 | 90 | 56–174 |

CL | 15 | 3–35 |

CD1 | 11 | 1–31 |

CD2 | 36 | 10–66 |

AUCA | 9 | 2–18 |

Left-right diameter | 7 | 0–25 |

Left wall thickness | 120 | 1–489 |

Right wall thickness | 137 | 3–328 |

The average error for the left-right intra-uterine diameters between the MRIs and MRI segmented models was small (7%), and a nonelliptical coronal profile was observed (Fig. 7). To quantify trends in left-right uterine diameter changes along the inferior-superior axis, regions were identified from the inferior-most point (0%) to the superior-most point (100%) of the inferior–superior axis. Because left-right diameter measurements could infrequently be collected in the MRI in the $0\u20139$% range of the inferior-superior axis due to poor uterine wall resolution, the MRI segmented model was used to find trends in coronal uterine shape in four sections.

$0\u20139%$: left-right intrauterine diameter increases at a decreasing rate to an average intrauterine diameter that is 44% (range = $40\u201355$%) of the largest left-right intrauterine diameter

$9\u201338$%: linear increase in left-right intrauterine diameter (R-squared value range = $0.98\u22121$), increasing to an average of 84% (range = $70\u201392$%) of the largest left-right intrauterine diameter

$38\u201362$%: left-right intrauterine diameter increases to the location of largest left-right intrauterine diameter (range = $55\u201364$% of inferior-superior axis)

$62\u2013100$%: left-right intrauterine diameter decreases in a manner resembling a semi-elliptical shape (R-squared value range = $0.92\u20130.98$)

The averaged parametric model was created by leveraging these trends in left-right diameter over the inferior-superior axis.

No trends were identified for left and right uterine wall thicknesses along the inferior-superior axis, and large errors were found between the MRI and MRI segmented model measurement values (Table 2). Looking specifically at the left uterine wall thickness (Fig. 8), measurements taken from MRI are almost universally smaller than those taken from the MRI segmented model. Exceptions are observed in patient 1 (two pairs of measurements are about equal) and patient 3 (MRI wall thickness is greater than the MRI segmented model). Additionally, the measured left uterine wall thickness range is larger in the MRI solid models than in the MRIs themselves, except in patient 3, where the range is approximately equal. Similarly, the right uterine wall thickness is usually smaller in the MRIs than in the MRI segmented models (Fig. 9), except for patients 2 and 5, for which regions of the MRI segmented models have a thinner right wall measurement than the MRIs. Again, the range of right uterine wall thicknesses in each MRI segmented model was greater than that of the corresponding MRI stack. All measurements from MRI and MRI segmented models are available through Columbia University Library's Academic Commons (Online Resource 1) [24].

### 3.2 Qualitative Solid Model Comparison.

Visual comparison of the models across all five patients showed that the elliptical model does not capture the coronal shape of the at-term uterus (Fig. 10). The intra-uterine coronal shape produced by the averaged parametric and multimeasure parametric models closely resembles the intra-uterine coronal shape in the MRI segmented model. However, the multimeasure model better captured the variations in left-right wall thickness seen in the MRI segmented model compared to the averaged parametric model. None of the parametric models captured the asymmetries in the intra-uterine coronal shape seen in the MRI segmented models. All solid models generated are available as STL files through Columbia University Library's Academic Commons [35].

### 3.3 Finite Element Model Comparison.

Though the MRI segmented models were not found to capture the corresponding MRIs precisely, they were adequate for the purposes of this study as they represent the highest possible fidelity for patient-specific modeling and were considered the “ground truth” for parametric model comparison. In comparing the 1st principal Lagrange strain results for all patients and modeling methods for the posterior (Fig. 11) and anterior (Fig. 12) uterus, the FEA results further than 10% of UD1 from the fixed boundary condition were considered. The cervix was excluded from the analyses.

The highest magnitude strains were in the MRI segmented model, except for patient 2, in which it occurs in the multimeasure model, and patient 5, in which it occurs in the averaged model (Fig. 12). The strain distribution in the multimeasured model was the best overall match to the MRI segmented model, and had the best similarity in three of the five patients (Table 3). The averaged model had the second best overall strain distribution similarity to the MRI segmented model, and the best similarity score in patient 1. The elliptical model had the lowest overall strain distribution similarity score to the MRI segmented model, though it marginally had the best strain distribution similarity in patient 3. In the section strain distribution similarity scores, the multimeasure model had the highest score in 8 of the 12 sections (Table 4). The sections in which the multimeasure model did not have the highest strain similarity score are the posterior-superior, posterior-mid-dle, posterior-inferior, and right-inferior sections.

Patient | $K\xafMulti\u2212measure$ | $K\xafAveraged$ | $K\xafElliptical$ |
---|---|---|---|

1 | 0.63 | 0.67 | 0.54 |

2 | 0.60 | 0.56 | 0.54 |

3 | 0.61 | 0.64 | 0.64 |

4 | 0.68 | 0.62 | 0.61 |

5 | 0.75 | 0.66 | 0.64 |

Overall | 0.65 | 0.63 | 0.60 |

Patient | $K\xafMulti\u2212measure$ | $K\xafAveraged$ | $K\xafElliptical$ |
---|---|---|---|

1 | 0.63 | 0.67 | 0.54 |

2 | 0.60 | 0.56 | 0.54 |

3 | 0.61 | 0.64 | 0.64 |

4 | 0.68 | 0.62 | 0.61 |

5 | 0.75 | 0.66 | 0.64 |

Overall | 0.65 | 0.63 | 0.60 |

Section | $K\xafMulti\u2212measure$ | $K\xafAveraged$ | $K\xafElliptical$ |
---|---|---|---|

A-S | 0.50 | 0.49 | 0.48 |

A-M | 0.51 | 0.49 | 0.47 |

A-I | 0.79 | 0.60 | 0.57 |

L-S | 0.77 | 0.71 | 0.73 |

L-M | 0.76 | 0.71 | 0.65 |

L-I | 0.66 | 0.47 | 0.46 |

P-S | 0.53 | 0.58 | 0.56 |

P-M | 0.44 | 0.46 | 0.45 |

P-I | 0.56 | 0.71 | 0.71 |

R-S | 0.80 | 0.75 | 0.66 |

R-M | 0.72 | 0.65 | 0.50 |

R-I | 0.41 | 0.43 | 0.36 |

Section | $K\xafMulti\u2212measure$ | $K\xafAveraged$ | $K\xafElliptical$ |
---|---|---|---|

A-S | 0.50 | 0.49 | 0.48 |

A-M | 0.51 | 0.49 | 0.47 |

A-I | 0.79 | 0.60 | 0.57 |

L-S | 0.77 | 0.71 | 0.73 |

L-M | 0.76 | 0.71 | 0.65 |

L-I | 0.66 | 0.47 | 0.46 |

P-S | 0.53 | 0.58 | 0.56 |

P-M | 0.44 | 0.46 | 0.45 |

P-I | 0.56 | 0.71 | 0.71 |

R-S | 0.80 | 0.75 | 0.66 |

R-M | 0.72 | 0.65 | 0.50 |

R-I | 0.41 | 0.43 | 0.36 |

When comparing principal strain direction across models, a relatively consistent agreement was found between all parametric models. On the anterior side of the uterus, all patient-specific modeling methods showed the first principal Lagrange strain oriented circumferentially, the second principal Lagrange strain oriented longitudinally, and the third principal Lagrange strain oriented radially. This principal strain directionality was the same for the posterior uterus. However, the elliptical parametric model does not show as strong of a preference as the other modeling methods. The multimeasure parametric model for patient 1 displayed first principal Lagrange strains in the longitudinal direction and second principal Lagrange strains in the circumferential direction. All FE result files are available through Columbia University Library's Academic Commons, along with all heat maps showing the FEA results' first, second, and third principal Lagrange strain magnitude and direction (Online Resource 4) [24,35].

## 4 Discussion

This work presents two new methods for building patient-specific, parametric models of at-term uteri based on magnetic resonance images. To create these models, measurements of at-term uteri from MRI and MRI segmented models were collected, and trends in uterine coronal shape were identified. In comparing the measurements taken in accordance with previous work on parametric modeling of the uterus and cervix [20,21], the greatest difference in measured value between MRI and the MRI segmented models occurred in dimension measurements of uterine wall thickness. This was also observed in the additional measurements of uterine coronal shape, where agreement was seen between MRI and MRI segmented models for left-right intra-uterine diameters, but large differences existed in left and right uterine wall thickness. These findings suggest that although segmentation can generate high-fidelity models, steps in the model-generating process may cause the solid model to be an inexact match to its source image set. From these results it was decided to base the averaged parametric model on trends in left-right intra-uterine diameter measurements taken from the MRI segmented model. All parametric models were therefore built using the MRI segmented model measurements rather than those from MRI to ensure differences observed qualitatively and in FEA would arise from the modeling approach rather than measurement values.

The two new parametric and elliptical parametric modeling methods were compared to the MRI segmented model visually and through FEA. The averaged parametric model was constructed using trends found in the left-right intra-uterine diameter along the inferior-superior axis, taking on a widest shape at about 62% of the inferior-superior axis and narrowing substantially toward the inferior uterus. The multimeasure model incorporated all additional left-right measurements collected. Both the averaged and multimeasured parametric models provided suitable visual matches to the MRI segmented model, and it was clear that the elliptical coronal shape in the elliptical model is not appropriate for generating at-term uterine geometry. Additionally, the multimeasure model could better capture the local changes in uterine wall thickness than the averaged model.

Parametric models were also evaluated through FEA by comparing Lagrange strain results to the MRI segmented model in a simple pressurization simulation. Across all models, the highest strain levels were found in the MRI segmented model, with the exception of patient 2. The multimeasured parametric model had the best overall strain distribution similarity score to the MRI segmented model, with the averaged model having a better overall strain distribution similarity score than the elliptical model (Table 3). The multimeasure model also had the highest strain distribution similarity score in 8 of the 12 uterine sections (Table 4). Additionally, the principal strain direction in the at-term uterus was conserved across the MRI segmented, multimeasured, and averaged parametric models. The first principal Lagrange strain was typically oriented circumferentially, the second principal Lagrange strain is oriented longitudinally, and the third principal Lagrange strain is oriented radially.

The circumferential orientation of 1st principal strain has relevance to existing histological and clinical theories. First, it is in accordance with the prevailing theory of the overall orientation of uterine muscular fibers, which has a family of circumferential fibers [36,37]. This fiber orientation suggests that as the fetus grows and stretches the uterus, the hypertrophied uterine muscular fibers are aligned in the direction that will experience the greatest tensile deformation. The circumferential direction of the 1st principal strain may also account for clinical reports on uterine rupture rates, which vary depending on cesarean scar number, location, and type. Patients with prior vertical classical uterine cesarean incisions or myomectomy incisions have a greater risk of uterine rupture than patients with lower transverse incisions [38]. Based on the FEA results, it is hypothesized that vertical uterine scars rupture more frequently than low transverse uterine scars because the scar tissue is perpendicular to the greatest tensile strain rather than parallel, pulling apart rather than along the scar tissue, though changes in myometrial thickness, edema, fiber disarray, and impaired vascularity due to scarring probably also play a role [39]. Existing computational models of cesarean section scars have found the dependence of increased regions of stress on scar positioning, scar defects, and thinning of the uterine walls [40].

A significant dependence on the local uterine wall thickness for strain magnitude was observed throughout the FEA results. The MRI segmented modeling method had the overall highest strain magnitude in four of the five patients because of the localized wall thinning, particularly in the central posterior wall. High levels of posterior wall strain are observed in all but patient 1's MRI segmented model (Fig. 11). The difference in central posterior wall strain magnitude is hypothesized to result from the modeling approach of the sagittal posterior uterus, which assumes the same uterine wall thickness at the anterior and posterior uterine walls. This assumption stems from the difficulties in measuring posterior wall thickness in 2D ultrasound, so additional posterior uterine wall thickness measurements in these MRI-based models may be necessary. The higher levels of strain observed in the posterior patient 1's multimeasure parametric model, when compared to the MRI segmented model, are believed to also result from the assumption of equal anterior and posterior uterine wall thicknesses.

The localized thinning in the left and right uterine walls also caused local strain concentrations to develop. In patient 2, thinning of the lower left and right uterine walls produced a high-strain region in the MRI segmented model. The multimeasure parametric model, which incorporates multiple axial thickness measurements, could reproduce this thinning and, therefore, reports a similar strain magnitude and has high strain distribution similarity scores in the left-middle and right-middle uterine sections (0.91 and 0.74, respectively, see Online Resource 5) [24]. On the other hand, the averaged and elliptical parametric models measure the axial wall thickness in only one location and are thus less able to capture these strain patterns. However, the MRI-segmented model has higher uterine wall thickness variability than observed directly from the MRI stacks. This raises the question of whether the range of strain magnitude observed in the MRI segmented model FEA results is simply due to the segmentation. Thus, it is possible that the averaged parametric model sufficiently captures at-term uteri, but further studies are required. An advantage of the averaged parametric model over the multimeasured is its basis of dimension measurements from a 2D ultrasound-based parametric modeling approach [20,21]. Thus, the averaged parametric model could potentially be built from 2D ultrasound images and not require MRI.

### 4.1 Limitations.

Though the work presented here provides compelling parametric methods for generating patient-specific models from magnetic resonance images, it has several limitations. The small sample size of five patients cannot draw definitive conclusions from the presented results. This is particularly true with regard to the average coronal shape, which is central to the assumptions made in the averaged parametric model. Because the axial MRIs were acquired every 5 or 5.5 mm, finding the MRI stack's inferior-most and superior-most image was unclear. Therefore, it is possible that the locations along the longest inferior-superior axis where intra-uterine diameter and uterine wall thickness measurements were collected from MRI are inconsistent with those taken from the MRI segmented models. In developing the parametric modeling methods, asymmetry across the sagittal plane of the intra-uterine cavity was not accounted for. Including intra-uterine asymmetry in the averaged and multimeasured parametric models may improve FEA results compared to the MRI segmented models. Though a visual qualitative comparison of the parametric models to the MRI segmented model was undertaken, no quantitative shape analysis was conducted. A simple neo-Hookean material was chosen in the FEA rather than a more realistic fibrous material in the uterus for computational simplicity. Future use of these models in FEA should include realistic uterine material properties. Lastly, uterine loading was explored during childbirth with only one FEA configuration, and the modeling methods were not compared under other relevant birthing scenarios or with anatomically correct boundary conditions. The strain similarity metric used for comparison was developed to compare strain distributions in sections of the uterus, and therefore, future use of the model should assess parametric model performance with metrics targeted to the simulation at hand. Despite these limitations, both the averaged and multimeasure parametric models provide a viable method for efficiently generating patient-specific geometry compared to the labor-intensive process of generating MRI segmented solid models.

## 5 Conclusion

This work presents two new patient-specific parametric models of the at-term uterus based on MRI measurements. Both were developed by identifying trends in uterine coronal shape found in at-term MRI and MRI segmented uterus models and were compared to the existing elliptical parametric uterine model qualitatively and through FEA. Both the multimeasured and averaged parametric models visually captured the tapering of the uterus toward the cervix. In the FEA simulations, the multimeasure model had the highest overall strain distribution similarity score. The averaged model had a higher overall strain distribution similarity score than the elliptical model, though they both used the same number of measurements. Further research must assess quantitative shape similarity and confirm the averaged uterine coronal shape through a larger patient dataset. The framework set forth in these parametric methods allows for the efficient building of patient-specific uteri, opening the door for research that could significantly improve childbirth simulation capabilities, provide a better understanding of birthing mechanics, and guide clinical interventions in the future.

## Acknowledgment

The authors thank Dr. John J. Moore (Department of Neonatology, Metrohealth, Cleveland, OH) for providing the MRI images used in this work, and Serena Russell and Nicholas Bible for their counsel on the quantitative strain comparison. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.

## Funding Data

Fu Foundation School of Engineering and Applied Science (Award No. Summer@SEAS Program; Funder ID: 10.13039/100008162).

National Institute of Child Health and Human Development (Award No. R01HD091153; Funder ID: 10.13039/100000071).

## Data Availability Statement

The data and information that support the findings of this article are freely available at: Columbia University Academic Commons Online resources^{2}.