Human airway material characterization via inverse finite element analysis and neural network surrogate.

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Learn more: PMC Disclaimer | PMC Copyright Notice Biomech Model Mechanobiol . 2026 Jun 3;25(3):61. doi: 10.1007/s10237-026-02081-7 Human airway material characterization via inverse finite element analysis and neural network surrogate Arif Badrou 1 , Crystal A Mariano 1 , Talyah M Nelson 1 , Mona Eskandari 1, 2, ✉ Author information Article notes Copyright and License information 1 Department of Mechanical Engineering, University of California Riverside, Riverside, CA USA 2 BREATHE Center, School of Medicine, University of California Riverside, Riverside, CA USA ✉ Corresponding author. Received 2025 Jul 25; Accepted 2026 May 8; Issue date 2026. © The Author(s) 2026 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ . PMC Copyright notice PMCID: PMC13233878  PMID: 42234231 Abstract Central airway obstruction can be caused by lung cancer and may severely diminish respiratory function to necessitate airway stenting. However, the mechanical properties of airway tissues remain poorly characterized, leading to a mismatch between stent mechanical behavior and airway compliance, which can reduce stent biocompatibility and clinical effectiveness. Such mismatches can result in abnormal stress transfer at the stent–airway interface and cause stent migration, local tissue irritation or inflammation, and impaired long-term performance. Predictive models are often used in treatment planning, however, without a comprehensive understanding of airway properties, advancements in modeling remain highly limited. In this study, we develop an experimental to numerical pipeline for identifying the mechanical properties of the human airway branching network, specifically the trachea, right bronchus, and left bronchus tissues. Biaxial planar tensile tests are used to capture the tissues’ unique, anisotropic mechanical response in order to inform the computational routine—where inverse finite element analysis is employed to calibrate a Holzapfel–Gasser–Ogden constitutive model. To overcome the computational burden of traditional IFEA, a neural network (NN) surrogate model is trained to enable rapid material identification. We identify parameter values for the various tissues, such as C 10 = 0.53 ± 0.25 kPa, k 1 = 0.17 ± 0.30 kPa, k 2 = 6.1 ± 2.0, and κ = 0.08 ± 0.01 for the trachea. Notably, this NN-approach reduced computational time from weeks to mere minutes, enabling fast material characterization. Thus, we also provide an efficient modeling framework and user-friendly MATLAB application for future studies. Ultimately, the findings here critically contribute to our current understanding of airway biomechanics and provide essential data for accurate modeling and optimization of airway stenting strategies. Supplementary Information The online version contains supplementary material available at 10.1007/s10237-026-02081-7. Keywords: Human airway, Inverse finite element analysis, Central airway obstruction, Lung cancer, Anisotropy, Biaxial tensile testing Introduction Lung cancer is the leading cause of cancer-related mortality worldwide, with more than two million new cases diagnosed in 2018 (Chaitanya Thandra et al. 2021 ). By 2030, lung cancer is projected to account for 3% of total global deaths (Yousefi et al. 2023 ). Beyond its devastating human toll, lung cancer also imposes a significant and irreversible economic burden. In the USA alone, the total economic impact of lung cancer including medical costs, lost productivity, and premature mortality exceeded $13 billion in 2019 (Kratzer et al. 2024 ). Among those affected, approximately 30% of patients develop central airway obstruction (CAO), primarily involving the trachea, right primary bronchus, and left primary bronchus (Saji et al. 2010 ). Without timely intervention, CAO drastically reduces survival rates (Umar et al. 2023 ). As such, it is of utmost significance to enable solutions to mitigate CAO. Airway stenting is one of the most effective interventions for CAO—where the insertion of a hollow tube restores airflow to improve lung oxygenation (Salguero et al. 2024 ; Saji et al. 2010 ). Despite recent advancements, such as more effective patient-specific stent designs, complications still prevail (Li et al. 2023 ). For instance, long-term issues such as stent migration (i.e., position shift) are common: Dialani et al. (Dialani et al. 2008 ) reported how stent migration away from surgical placement in the proximal right bronchus led to partial collapse of a patient’s right upper lung lobe. Such complications occur frequently, in 5–17% of CAO patients, and are often attributable to design and implementation issues, highlighting the need for improved stent structures and placement strategies (Madan et al. 2024 ). Numerical models play a crucial role in minimizing complications such as stent migration, as they provide preoperative insights into the mechanical response of stented airways (Malvè et al. 2011 ; Saeidi et al. 2023 ). However, the accuracy of these models fundamentally depends on precise characterization of airway tissue mechanical properties. Current finite element (FE) models of airway stenting suffer from a critical limitation: they predominantly assume isotropic material behavior (Farahani et al. 2023 ), despite recent experimental evidence demonstrating the highly anisotropic nature of mammalian airways (Eskandari et al. 2018 ; Sattari et al. 2023 ). This represents a current limitation that can compromise the predictive capability of stent simulations. Previous studies have characterized the mechanical properties of the trachea across different species, including humans, using a range of experimental approaches, most commonly uniaxial tensile testing or compression (Trabelsi et al. 2010 ; Safshekan et al. 2016 ; Huang et al. 2021 ; Jau-Yi Wang et al. 2011 ). While these studies have provided valuable baseline insights into airway mechanics, simplified loading conditions such as uniaxial tension are known to offer a limited representation of the anisotropic and nonlinear behavior of soft tissues, motivating the need for more comprehensive experimental and modeling frameworks (Jiang et al. 2021 ). As a result, accurately translating such experimental measurements into constitutive material parameters suitable for computational modeling remains a nontrivial challenge. The two common approaches for identifying material properties in numerical models are direct stress–strain curve-based parameter fitting and inverse finite element analysis (IFEA). Direct curve-based fitting involves optimizing parameters of a function that describes the material behavior, minimizing the difference between experimental and numerical data. Although this approach is relatively straightforward, it provides only an approximate estimate of tissue properties because it relies on a predefined mathematical model that may not fully capture the complexity of the tissue's mechanical response, particularly in fitting the global response of the data (i.e., how the force measure responds to an imposed displacement). The second approach, IFEA, uses an FE model to simulate the experimental setup and test results (e.g., tissue biaxial testing), where the material parameters are adjusted until the simulation results agree with the experimental data (Bäker and Shrot 2013 ). Consequently, IFEA is known for its precision, as the FE model incorporates the actual physics and boundary conditions of the soft tissues, which are often too complex to describe with a simplified curve-fitting mathematical function (Kauer et al. 2002 ). However, IFEA’s complexity can make the problem ill-posed and is computationally expensive. In particular, for nonlinear constitutive models with multiple parameters, different combinations of material parameters can produce very similar force–displacement responses, leading to non-unique solutions. In addition, strong coupling between parameters and sensitivity to experimental noise and boundary conditions further complicate parameter identification, requiring repeated simulations during optimization (Bäker and Shrot 2013 ; Liu et al. 2014 ). For example, Chawla et al. (Chawla et al. 2009 ) reports that characterizing human passive muscle using IFEA requires approximately 10 h of computational time to obtain optimized material parameters. Despite these methodological advances, a critical gap remains: Existing studies have not yet systematically combined biaxial experimental characterization of human airway tissues with a computationally efficient inverse finite element framework capable of identifying anisotropic material parameters in a manner suitable for airway stenting applications. In this study, we address this gap by introducing a new approach to airway tissue characterization using IFEA, establishing an integrated experimental-to-modeling pipeline aimed at improving physiological relevance while reducing computational cost. To this end, we present the first biaxial characterization of the mechanical properties of human airway tissues, including the trachea, right bronchus, and left bronchus tissue samples, with methods based on recent advancements from our group (Mariano and Eskandari 2024 ; Sattari et al. 2023 ; Mariano et al. 2023 ; Nelson et al. 2024 ). These data are then used in a numerical pipeline that efficiently identifies material parameters using an inverse approach. By integrating a feedforward neural network (NN) as a surrogate model within the optimization loop, our method drastically reduces the computational cost of FE simulations, enabling rapid and accurate calibration. This new application not only enhances current models of human airways but also provides a highly efficient framework for future research, significantly improving simulations related to soft tissue characterization. Materials and methods Experimental to numerical framework An overview of the experimental and numerical pipeline is presented in Fig. 1 A. Biaxial tensile experiments were conducted, and then, a FE model was developed to represent the airway samples during testing, incorporating geometry and boundary conditions. To accelerate material parameter identification, a NN surrogate model was constructed and trained using FE simulations with varying material properties. The NN provided instantaneous force predictions for imposed displacements, which were then compared with experimental force data. To optimize the material parameters, an inverse problem was formulated and solved through an optimization procedure, where an objective function was defined to minimize the discrepancy between the model predictions and the experimental results. Finally, a graphical user interface (GUI) was developed (Fig. 1 B), allowing users to efficiently characterize human airway samples without running costly FE simulations. The following sections describe each of these steps in-depth, including a brief overview of the biaxial experiments performed on human airway samples, the development of the FE model representing the biaxial test, and the construction and application of the NN in an inverse framework to calibrate material parameters. Finally, a MATLAB application ( BiaFit ) was developed to allow users to apply the trained neural network and characterize HGO material parameters. The application is available at: https://www.mathworks.com/matlabcentral/fileexchange/181573-biafit . Fig. 1. Open in a new tab A General workflow for characterizing the material parameters of human airway tissue using an inverse finite element and neural network framework. B A GUI allows the user to input the experimental file from biaxial testing, and the application then provides the calibrated material properties along with the corresponding fit Experimental biaxial tensile testing of human airways Seven transplant-eligible lung donors were obtained (IRB exemption approval HS 20-180; some specimens were also previously utilized (Quiros et al. 2026 ; Nelson and Eskandari 2026 )), having been deemed healthy enough for clinical use but then reallocated for research purposes due to logistical issues or declining lung function. Airway tissue from each lung was dissected and tested within 48 h of surgical retrieval, as provided by the transplant site. The parenchyma was removed, leaving only the trachea, left primary bronchus, and right primary bronchus. The availability and condition of donor lungs limited the regions that could be consistently sampled across all specimens. As a result, tracheal samples were collected from the region immediately above the carina, which was the only tracheal segment reliably accessible and suitable for mechanical testing from all donors. Where feasible, uniform soft smooth muscle tissue square samples (5.2 × 5.2 mm) were cut away from the cartilage of each airway region (N = 1–3), yielding a total of 46 tissue samples from the seven lungs. Typically, approximately two samples were available per subject per the different airway regions. Each sample was mounted on a biaxial tensile testing system (CellScale Biomaterials Testing, Canada) equipped with 5N load cells and secured using metal rakes (Fig. 2 ) (Nelson et al. 2024 ). Samples were extracted from their native tissue immediately before testing. Sample hydration during testing was maintained using 1X phosphate-buffered saline (PBS). Planar equi-biaxial tensile tests were performed on all samples to ensure consistent and controlled loading across various proximal and distal airway regions, while minimizing the risk of tissue damage in smaller bronchial specimens. Samples were stretched to a maximum strain of 50% at a rate of 2.5%/s (Ramirez et al. 2024 ). Although this strain exceeds typical physiological airway strains (8–31%) (Zhao et al. 2021 ), it was chosen to allow characterization of both the low-strain response and the transition to higher-strain behavior, while avoiding tissue failure, providing a more thorough mechanical characterization. During protocol development, strains above 60% consistently led to tissue damage near the attachment points. Prior to data collection, each sample underwent 10 preconditioning cycles to establish a consistent reference state. Force–displacement data were then recorded for analysis. Details regarding experimental protocol development and established methodological pipelines, including specific to human airways, can be found in previous studies (Mariano and Eskandari 2024 ; Nelson et al. 2024 ; Mariano 2025 ; Eskandari et al. 2018 ; Sattari et al. 2023 ; Mariano et al. 2023 ). Fig. 2. Open in a new tab A Airway smooth muscle tissue was extracted from each sample (created with BioRender.com). B The tissue was mounted on a biaxial tensile testing system and secured using metal rakes. C Ten preconditioning cycles (green) were applied to achieve a steady and repeatable the mechanical response before the final test (blue) was recorded FE model construction A FE model was developed in ABAQUS (Dassault Systèmes, France) to simulate the biaxial tensile test under quasi-static conditions. The planar dimensions of the model are shown in Fig. 3 . The rake diameters and inter-rake distances were set based on manufacturer specifications, while the apron distance (space between rakes and sample edge) was measured across multiple samples, averaging 1.0 ± 0.2 mm. To reduce the computational cost, symmetry was leveraged, and only one-quarter of the FE model was represented. The model thickness was adjusted to match each specific sample. The rakes were modeled indirectly as holes with the same diameter. A reference point was assigned at the center of each hole, and a kinematic constraint was applied between this reference point and the nodes along the semicircular edge (shown in blue in Fig. 3 ) to simulate rake movement, in accordance with past studies (Fehervary et al. 2016 ). The experimental displacements prescribed during the biaxial test (Fig. 2 ) were imposed on the center nodes, while symmetric boundary conditions were applied. Fig. 3. Open in a new tab A Dimensions of the FE model of the tissue under tensile testing (holes = rake placement) with the applied boundary conditions. B Mesh visualization, including a close-up of the region expected to undergo significant deformation during the biaxial test. C Schematic representation of an airway, its wall microstructure, and axial and circumferential orientations (created with BioRender.com) The importance of accurately modeling boundary conditions in biaxial testing has been widely emphasized in the literature (Jacobs et al. 2013 ). To validate our approach, a preliminary analysis was conducted comparing two models (see Appendix 1 ): one assuming a simple square sample with uniform displacements applied along its edges and another incorporating the rake representation. The comparison revealed notable differences between the two, highlighting the importance of precise boundary condition representation and supporting the validity of our modeling approach. The model was meshed with 28,920 quadratic membrane elements, assuming plane stress conditions, a standard approach for modeling thin biological tissues (Kroon 2011 ; Fehervary et al. 2016 , 2019 ). A mesh convergence analysis was conducted to ensure accuracy, with details provided in Appendix 2 . The Holzapfel–Gasser–Ogden (HGO) material model was used to describe the mechanical behavior of the tissue, as it effectively captures the stress–strain response of anisotropic biological tissues (Gasser et al. 2006 ; Nolan et al. 2014 ). The HGO model was defined by the strain energy function U as follows: U = C 10 I ¯ 1 - 3 + k 1 2 k 2 e k 2 κ I ¯ 1 - 3 + 1 - 3 κ I ¯ 4 - 1 2 - 1 1 where C 10 , k 1 , k 2 , and κ are material parameters and κ drives fiber dispersion and is zero when fibers are perfectly aligned in the axial direction (Fig. 2 ). I ¯ 1 and I ¯ 4 represent the first and fourth invariants of the deviatoric Cauchy–Green strain tensor, respectively. The invariants are defined as: I ¯ 1 = tr C ¯ I ¯ 4 = a 0 · C ¯ a 0 where C ¯ = J - 2 / 3 C is the deviatoric right Cauchy–Green tensor, C ¯ = F T F , J = det ( F ) , and a 0 is the unit vector defining the mean fiber direction in the reference configuration. In the model, the fibers were assumed to be primarily aligned in the axial direction (Fig. 3 ), following experimental observations (Mariano and Eskandari 2024 ; Eskandari et al. 2019 ). IFEA calibration The FE model was first used in an IFEA to estimate material properties for nine samples serving as references, randomly selected from the 46 available samples (three from each of the three airway regions). This analysis helped define the parameter space of interest for training an NN surrogate model. In the second approach, a NN was trained within this parameter space and then used to identify the material properties of all 46 samples. The IFEA was conducted using lsqnonlin in MATLAB (MathWorks, USA), estimating the material parameters of HGO model (Eq. 1 ): C 10 , k 1 , k 2 , and κ (Maghsoudi-Ganjeh et al. 2010b ). The analysis was performed separately for samples taken from the trachea, left primary bronchus, and right primary bronchus. The initial parameter ranges (Table 1 ) were determined through preliminary trial-and-error analyses exploring different orders of magnitude and assessing their qualitative influence on the force–displacement response and were further informed by existing literature on soft biological tissues (Trabelsi et al. 2010 ; Canales et al. 2023 ; Nolan et al. 2014 ). These ranges were selected to ensure numerical stability and parameter identifiability within the inverse finite element framework. Parameter values or combinations outside these ranges frequently led to poor convergence or divergence of the numerical simulations, preventing reliable inverse identification. Accordingly, the bounds should be interpreted as model- and protocol-specific limits. Table 1. Material parameters associated with the HGO model and used in the IFEA. Initial values correspond to the initial guesses used in the lsqnonlin optimization, and lower and upper values define the parameter bounds Parameters Initial value Lower value Upper value C 10 (kPa) 1.0 0.01 10.0 k 1 (kPa) 0.1 0.0 10.0 k 2 (–) 1.0 0.0 10.0 κ (–) 0.0 0.0 0.33 Open in a new tab The objective of the IFEA was to minimize the difference between experimental and numerical outputs, specifically the force values (Eq. 2 ). The objective function, denoted as φ , was expressed as: φ = ∑ i = 1 n W ( F exp ) F exp i - F num i 2 2 where F exp and F num ​ represent the experimental and numerical force values, respectively. Forces from both loading directions of the biaxial test were included in the objective function. The index i represents each experimental data point used in the fitting procedure. To account for sensor noise and differences in force magnitudes, a weighting factor W was introduced: W = 2 , if F exp ≥ 100 mN 1 , if F exp ≥ 30 mN and F exp < 100 mN 0 , otherwise 3 This weighting scheme prioritizes more reliable force values while reducing the influence of noisy measurements below 30 mN. Additionally, forces exceeding 100 mN are given greater emphasis, as higher-magnitude values tend to be more accurate (Nelson et al. 2024 ). A total of n = 50 values were analyzed (as test force values were recorded every 2 s, resulting in 25 values per circumferential and axial direction), and this number of data points ensured an accurate representation of the features of the force–displacement curve. To improve the robustness of the optimization process, the MultiStart option with three starting points was used to search for a global minimum. The first starting point corresponds to the initial parameter values reported in Table 1 , while the two additional starting points were automatically generated by the MultiStart algorithm within the prescribed parameter bounds (The MathWorks Inc. 2022 ). The convergence criteria were based on those effectively employed in a previous IFEA study (Badrou et al. 2025 ). Neural network and characterization of human airway samples From the nine sets of calibrated material properties obtained through IFEA using the described FE model, the parameter space for training the NN was refined. This refinement involved narrowing the ranges of the four material properties based on the calibrated values and incorporating sample thickness ( t ) as an additional parameter, with its minimum and maximum values determined from measurements of the 46 available samples (Table 2 ). Table 2. Parameter space used to train the neural network (NN) Parameters Lower value Upper value C 10 (kPa) 0.01 2.0 k 1 (kPa) 0.005 1.0 k 2 (–) 1.0 10.0 κ (–) 0.0 0.10 t (mm) 1.24 2.84 Open in a new tab A feedforward NN was constructed using MATLAB’s fitnet function (The MathWorks Inc. 2022 ). The input vector of the neural network consisted of the four HGO material parameters ( C 10 , k 1 , k 2 , κ ) and the sample thickness t . The output vector consisted of the force–displacement curves predicted by the neural network, which was trained using FE simulations under prescribed biaxial loading. The NN architecture consisted of two hidden layers with 128 and 64 neurons, respectively, guided by recommendations in the literature (Stathakis 2009 ; Uzair and Jamil 2020 ). Training was conducted using 2,000 FE simulations generated via the Latin Hypercube Sampling method (Ye 1998 ), ensuring comprehensive coverage of the parameter space. The number of snapshots and neurons in each layer were optimized through iterative experimentation, as detailed in Appendix 3 . To enhance the NN’s ability to model complex relationships, tansig activation functions were used in the hidden layers, while a linear function was applied to the output layer (Vogl et al. 1988 ). Bayesian regularization was implemented to mitigate overfitting, ensuring the robustness and generalizability of the NN (MacKay 1992 ; Burden and Winkler 2008 ). The training of the neural network was stopped when any of the following conditions were met: (i) the mean squared error (MSE) between the predicted and actual values reached 1 e - 7 , (ii) the gradient of the loss function, which indicates the rate of error reduction when adjusting model parameters, fell below 1 e - 4 , or (iii) the maximum number of 1000 epochs was reached. The NN was validated through two approaches: (i) retrieving results for the nine reference samples to verify its effectiveness in inverse problems for characterizing HGO material properties and (ii) comparing NN predictions with a batch of 200 new FE simulations generated using Latin hypercube sampling within the prescribed parameter bounds (Table 2 ), and distinct from the 2,000 FE simulations used for training. The comparison was conducted using the normalized root mean square error (NRMSE in %; Eq. 4 ) to assess accuracy on unseen data from the 200 new samples: NRMSE % = ∑ j = 1 m ( F NN j - F FE j ) 2 ∑ j = 1 m ( F FE j ) 2 × 100 4 where F NN ​ and F FE ​ represent the force predicted by the NN and the FE model, respectively. The index j corresponds to each force value along the force–displacement curves, and m is the total number of force values across the 200 samples, i.e., m = 200 × 50 = 10 , 000 . Once validated, the NN replaced the FE model in the IFEA process, enabling the calibration of all 46 samples with significantly reduced computational cost. During post-processing, metrics were employed to assess the quality of the fit for each sample, including the residual norm (sum of squared differences between predicted and experimental forces (The MathWorks Inc. 2022 )), and the coefficient of determination ( R 2 ), computed as: R 2 = 1 - ∑ k = 1 n F exp k - F num k 2 ∑ k = 1 n F exp k - F exp ¯ 2 5 where F exp ¯ represents the mean of the experimental force values and the index k corresponds to each force value along the force–displacement curves. Additionally, the computational time required for each optimization was recorded, to highlight the efficiency of the NN-based approach. Results Reference samples: FE simulations and material calibration Using two Intel Xeon 2.10 GHz CPUs, the FE model required an average of one hour to compute for each sample. The primary challenge was handling the high strains induced by displacement and the rakes, which caused localized tissue stretching. The model's performance is illustrated in Fig. 4 , where it is qualitatively compared to one of the experimental samples at different time steps. Fig. 4. Open in a new tab Biaxial tensile test of a representative sample. Top row: experimental images with the corresponding finite element model edges overlaid in green, showing good agreement between experiment and model geometry. Bottom row: numerically predicted true strain magnitude maps for the same loading states, illustrating the spatial distribution of strain within the sample Using the FE model, IFEA was performed for the nine reference samples. All three optimization runs converged to an identical solution, supporting the robustness of the inverse procedure. The computational cost for each sample was approximately one day. The optimized HGO material parameters are reported in Table 3 , and representative stress–strain curves fitted to the experimental data are shown in Fig. 5 . Engineering (first Piola–Kirchhoff) stress was computed from the measured force using the specimen thickness and the effective testing area, while strain was defined as the elongation divided by the initial length. The remaining fitted curves are provided in Appendix 4 . It can be observed that the FE model, combined with the integrated HGO material law, successfully captured the complexity of the curve, which is characterized by a bilinear response, where the second portion typically exhibited a high peak value. The parameter κ, which governs fiber dispersion, converged to a value close to 0, indicating that the fibers are primarily aligned in the axial direction. Based on these reference results, the parameter ranges were refined to train the neural network in a subsequent phase (see Table 2 ). Table 3. HGO material resultant parameters for the nine reference samples, obtained from IFEA (blue) and NN predictions (green). The coefficient of determination (R 2 ), defined in Eq. 5 , indicates the quality of fit of both models to the experimental force–displacement data Region Sample C 10 (kPa) k 1 (kPa) k 2 (–) κ (–) R 2 Trachea #1 0.60 0.59 0.031 0.040 4.9 4.7 0.07 0.07 0.96 0.96 #2 0.42 0.48 0.841 0.730 3.4 3.6 0.09 0.09 0.99 0.98 #3 0.12 0.12 0.122 0.121 6.6 6.6 0.07 0.07 0.99 0.99 Left Bronchus #1 0.54 0.55 0.031 0.030 6.1 6.2 0.08 0.08 0.98 0.98 #2 0.47 0.48 0.075 0.068 4.5 4.6 0.02 0.02 0.97 0.97 #3 0.57 0.60 0.036 0.006 3.6 5.1 0.02 0.00 0.95 0.95 Right Bronchus #1 0.62 0.62 0.014 0.013 8.0 8.2 0.08 0.08 0.98 0.98 #2 0.90 0.91 0.131 0.100 2.4 2.7 0.09 0.09 0.97 0.09 #3 0.57 0.67 0.018 0.010 8.1 8.9 0.07 0.07 0.99 0.99 Open in a new tab Fig. 5. Open in a new tab Stress–strain curves for representative samples of the trachea, left bronchus, and right bronchus selected from the nine reference samples (trachea #3, left bronchus #1, and right bronchus #1; see Table 3 ). Results are shown for both axial and circumferential directions after material calibration using the neural network (NN) surrogate and the finite element (FE) model NN validation Using five Intel Xeon 2.10 GHz CPUs, the 2000 FE simulations were completed in approximately 8 days. The NN was then trained for 609 epochs (see Fig. 6 ; performance vs. number of epochs), achieving an error of 0.9% when compared to the FE model results for the 200 new test samples. Fig. 6. Open in a new tab Mean squared error computed between the predicted force values from the neural network and the experimental values during the training phase, plotted against the number of epochs The NN was then used to retrieve the calibrated material parameters via IFEA by replacing the FE model with the NN for the optimization process. The results (blue: FE model; green: NN) are summarized in Table 3 . Similar results were observed, except for discrepancies in sample #3 of the left bronchus: the force values for this sample were small and noisy, leading to multiple possible solutions. It was hypothesized that the difference was due to the variation in parameter ranges between the NN and the FE model. To verify this, an IFEA was performed using the same parameter range as the inverse method with the NN and obtained identical results, mitigating concerns. NN validation The NN was subsequently used to characterize the remaining 37 samples. The comprehensive results, including the metrics defined in section “ Neural network and characterization of human airway samples ”, are provided in Appendix 5 . Table 4 summarizes the calibrated HGO material parameters for the trachea, left primary bronchus, and right primary bronchus, reported as mean ± standard deviation. Figure 7 presents the comparison of these parameters across airway regions. Regional differences were assessed using one-way ANOVA with Bonferroni post hoc correction. No statistically significant differences were observed between regions for any of the fitted parameters (all comparisons: p ≥ 0.05). Although variations in group mean values were observed, substantial overlap between regions was present for all parameters. And while no significant regional differences were observed, modest variations in group mean values were apparent. In particular, the trachea and right bronchus tended to exhibit higher mean values of C 10 , k 1 , and k 2 compared to the left bronchus, while the trachea showed a slightly lower mean C 10 value relative to the right bronchus. These trends should be interpreted cautiously due to substantial overlap between regions. Table 4. Calibrated HGO material parameters for the 46 samples, obtained through optimization. The mean ± standard deviation values are reported across respective tracheal and bronchial regions Region C 10 (kPa) k 1 (kPa) k 2 (–) κ (–) Trachea 0.53 ± 0.25 0.17 ± 0.30 6.1 ± 2.0 0.08 ± 0.01 Left Bronchus 0.60 ± 0.22 0.17 ± 0.32 5.7 ± 2.7 0.07 ± 0.03 Right Bronchus 0.46 ± 0.13 0.13 ± 0.23 5.0 ± 2.0 0.07 ± 0.03 Open in a new tab Fig. 7. Open in a new tab Bar plots showing the calibrated HGO material parameters C 10 , k 1 , k 2 , and κ for the trachea, right primary bronchus, and left primary bronchus. Individual data points are shown together with group means ± standard deviation. Regional differences were assessed and no statistically significant differences were observed between airway region parameters These numerical results correlated well with experimental trends and bilinear behavior (Fig. 5 ). The high values of k 2 (5.0 to 6.1) across all regions were influenced by the bilinearity of the material response, where a higher k 2 value suggests a delayed activation of fiber contribution, which aligns with the observed mechanical behavior. The value of the parameter κ , which governs fiber dispersion, remained relatively low (0.08) indicating strong fiber alignment along the axial direction. This finding supports the hypothesis and past experimental works that collagen fibers are predominantly oriented in the axial direction, playing a key role in the mechanical behavior of airway structures (Mariano et al. 2023 ). The total computational cost for calibrating all 46 samples was approximately 6 min, compared to several weeks using the traditional IFEA process (estimated based on results from the nine reference samples). This highlights the necessity of using the surrogate NN in the inverse problem to significantly reduce computational time. Discussion This study presents a comprehensive pipeline to identify the material parameters of the human trachea, right bronchus, and left bronchus smooth muscle using the HGO material model. While direct comparisons with previously reported HGO parameters are limited by differences in constitutive form and experimental loading protocol, Trabelsi et al. reported a matrix shear parameter C 10 = 0.88 kPa for a for a modified two-fiber-family HGO model fitted to uniaxial human trachea tests (Trabelsi et al. 2010 ). This magnitude is comparable to our fitted matrix response ( C 10 = 0.53 kPa), suggesting a similar order of stiffness for the isotropic smooth muscle matrix. However, Trabelsi et al. reported substantially larger exponential nonlinearity parameters for the fiber families (analogous to k 2 ; e.g., k 2 = 34 and k 4 = 14 ) than those obtained here ( k 2 < 10 ). This discrepancy is expected given (i) the different model structure (two fiber families versus our parameterization) and (ii) different loading conditions and strain ranges (uniaxial tests versus our biaxial experiments), both of which strongly affect identifiability of the exponential stiffening parameters. We therefore restrict cross-study interpretation primarily to the matrix stiffness scale ( C 10 ) and treat the fiber exponential parameters as protocol and model dependent. More broadly, quantitative characterization of airway soft tissue mechanics remains limited in the pulmonary biomechanics literature, with most existing studies focusing either on the whole composite airway behavior or on bulk parenchymal lung tissue. Only a few studies have directly characterized the mechanical properties of the trachea (Safshekan et al. 2016 ; Wang et al. 2011 ; Huang et al. 2021 ). For example, Huang et al. characterized the trachea as a composite whole in porcine, caprine, and canine models and reported stiffness values ranging from 1.2 to 2.6 MPa, while emphasizing that the dominant contribution to overall stiffness arises from the rigid cartilage rings (Huang et al. 2021 ; Safshekan et al. 2016 ). We can derive an effective stiffness from our measured material parameters and compare it with reported values for pulmonary soft tissues. Specifically, from the matrix shear parameter C 10 , an effective Young’s modulus E can be estimated assuming incompressibility, such that E = 6 C 10 , yielding E ≈ 3.2 kPa for the trachea. Although this comparison warrants further investigation, the methods used to quantify stiffness differ from those in the current study. Additionally, Polio et al. reported Young’s moduli for bulk lung tissue ranging from approximately 1 to 6 kPa using multiple characterization techniques, including uniaxial testing (Polio et al. 2018 ). Although these measurements primarily reflect the mechanical behavior of lung parenchyma rather than airway tissue specifically, they support the plausibility of the stiffness scale obtained in this study. While alveolar walls and airway tissues differ anatomically, both are composed of compliant, extracellular-matrix-dominated constituents that govern the low-strain mechanical response prior to fiber recruitment. Consistent with this, multiscale modeling studies of lung parenchyma have estimated alveolar wall elastic moduli on the order of 4–6 kPa, with substantially stiffer fibrous components contributing at higher strains (Bou Jawde et al. 2020 ). Together, these comparisons support the interpretation that the effective small-strain modulus inferred here ( E ≈ 3.2 kPa) represents a physiologically reasonable matrix-level stiffness within pulmonary soft tissues. Considering the experimental biaxial characterization data only, we further compared our results with a previous study on cartilage-free porcine airways (Sattari et al. 2023 ). As in prior work, the initial and ultimate moduli were obtained directly from experimental stress–strain curves using linear fits with R 2 > 0.9 (Smith et al. 2008 ; Sattari et al. 2023 ). At 5%/s, porcine tracheal tissues exhibited axial and circumferential initial moduli of 14.1 ± 3.6 kPa and 38.9 ± 16.2 kPa, respectively, while the large bronchi were less stiff ( 9.1 ± 3.1 kPa axially and 26.6 ± 9.2 kPa circumferentially). Using the same procedure, human airway tissues showed comparable magnitudes, with axial initial moduli of 22.3 ± 7.4 kPa in the trachea and ∼ 20.1 kPa in the bronchi (averaged left/right), and lower circumferential initial moduli of 13.2 ± 6.5 kPa in the trachea and ∼ 10.0 kPa in the bronchi. Ultimate moduli were substantially higher in both species. At 5%/s, porcine tracheal tissues reached 588.4 ± 335.9 kPa axially and 367.6 ± 236.5 kPa circumferentially, compared with 293.4 ± 206.8 kPa axially and 107.0 ± 31.2 kPa circumferentially in the bronchi. Human airway tissues exhibited similar magnitudes, with axial ultimate moduli of 592.3 ± 542.1 kPa in the trachea and ∼ 403.9 kPa in the bronchi (averaged left/right), and circumferential ultimate moduli of 284.1 ± 355.3 kPa in the trachea and ∼ 145.4 kPa in the bronchi. To more efficiently calibrate material parameters compared to computationally-intensive traditional IFEA methods, the NN was implemented as a surrogate model within the optimization framework. This approach, relying on built-in MATLAB functions, circumvents the need for intrusive methods that embed FE equations directly into the NN (Meethal et al. 2023 ; Hu et al. 2024 ). The pipeline was designed with a rigorous methodology, ensuring a smooth transition from biaxial experimental data to parameter identification. By integrating this machine learning approach, the computational burden associated with traditional IFEA was significantly reduced, making large-scale parameter identification feasible (Torun et al. 2022 ; Badrou et al. 2025 ). Ultimately, the improved inverse modeling approach described in this study enhances the accuracy and speed of numerical simulations and will be a useful framework for future computational modeling endeavors. Moreover, this study’s material characterization of tracheal and bronchial tissues offers valuable insights into airway mechanics. These findings have direct implications for optimizing interventions, including stent design and placement, to ultimately reduce complications, such as stent migration, in the case of central airway obstruction due to lung cancer. Limitations and future directions While the results are promising and the proposed framework provides a solid foundation for future developments, several aspects of the study should be considered for refinement in future works. For instance, noisy force measurements at low force levels required the use of a weighting factor in the objective function to improve the robustness of the parameter identification. This weighting scheme was introduced to address experimental and numerical limitations rather than to directly represent physiological loading levels. The force thresholds of 30 mN and 100 mN were selected based on experimental observations, where measurements below 30 mN were strongly affected by sensor noise, while higher force values were more stable and reproducible (Cleveland 1979 ). Importantly, the applied deformation levels were chosen based on physiological considerations, whereas the weighting factor was used only during the fitting process to prioritize reliable portions of the force–displacement response. Although further refinement of this weighting approach could be explored in future studies, this method was necessary to ensure stable and accurate identification of the material parameters. Another limitation of this study is the dispersion observed in the experimental data. This variability is mainly related to the difficulty of handling and testing very soft airway tissues, especially when working with small samples (Nelson et al. 2024 ). Due to their low stiffness and limited dimensions, these samples are highly sensitive to minor variations in handling, gripping, alignment, and boundary conditions, which can introduce uneven pre-strain or small differences in loading and increase variability in the measured response no matter how meticulous the experimental preparation was. In addition, real biological differences between subjects are expected and likely contribute to the observed spread in the calibrated material parameters. From a modeling point of view, this dispersion means that airway tissue properties should not be represented by a single fixed set of parameters. Instead, the reported variability provides useful information on realistic parameter ranges that can be used in computational models. In future work, we aim to reduce this dispersion by increasing the number of samples and by grouping specimens based on subject-specific characteristics. From a modeling perspective, another limitation of this work is the potential non-uniqueness of the identified HGO material parameters. Like other exponential constitutive models, the HGO formulation can lead to different combinations of parameters producing similar force–displacement responses. In this study, the objective was to identify a consistent and stable combination of parameters that reproduces the experimental response within the considered loading range. To limit non-uniqueness, parameter bounds were imposed based on numerical stability and preliminary analyses, the full force–displacement curve was used in the objective function, and a MultiStart optimization strategy was applied. Validation of the FE model remains a critical step to ensure that the NN is trained on accurate predictive simulations. Nevertheless, it is difficult to define appropriate comparison metrics between numerical and experimental results. Ideally, strain measurements obtained using digital image correlation (DIC) could be used to define realistic deformation levels, as demonstrated in previous studies from our group (Mariano et al. 2020 ; Quiros et al. 2022 ; Nelson et al. 2023 ; Shankel et al. 2025a , b ; Quiros et al. 2024 ; Mariano et al. 2022 ). However, in the present study, accurate DIC strain measurements during biaxial testing were difficult to obtain. This was due to the rapid tissue deformation, combined with the small sample dimensions and limitations in speckle pattern quality, which ultimately reduced the reliability of the DIC-based strain measurements. As a result, despite collecting the data, strains from DIC were not used beyond qualitative assessment in the neural network framework. As a result, strain data were not used in the neural network framework, and force–displacement responses were instead selected as the primary outputs. Future work will focus on improved optical setups and imaging strategies to enable reliable strain-based inputs. Geometric variability across samples posed another limitation in the modeling process. Rake dimensions and planar geometry were based on standard documentation, while thickness was measured experimentally and incorporated into the model. However, differences in inter-rake spacing and apron dimensions between samples were not accounted for, potentially affecting the resulting material parameters (Pei et al. 2021 ). Future work should explore sample-specific modeling techniques using imaging data to generate subject-specific geometries (Fehervary et al. 2019 ) and assess the influence of the apron (Di Leonardo et al. 2024 ). Additionally, a plane-stress assumption was adopted when modeling the airway samples. However, the ratio of sample length to thickness is below the commonly accepted threshold of 10:1 and represents a limitation of the present study. Nevertheless, the use of plane stress was motivated by prior studies employing similar approaches (Fehervary et al. 2016 , 2019 ) and its common use in the modeling of biological tissue samples (Holzapfel and Weizsäcker 1998 ; Kroon 2011 ). Future work should investigate the impact of this assumption by developing full 3D models of the biaxial test. In parallel, experimental strategies aimed at reducing sample thickness could help better satisfy plane-stress conditions. The choice of the HGO material model was motivated by the nonlinearity of lung tissue. While the model provided a good fit, it may not fully capture the whole complexity of airway mechanics (Eskandari et al. 2019 ; Maghsoudi-Ganjeh et al. 2021a ). Lung tissue exhibits not only anisotropic but also heterogeneous and viscoelastic behavior (Eskandari et al. 2018 ; Bayliss and Robertson 1939 ; Hurtado et al. 2017 ; Amelon et al. 2011 ; Nelson et al. 2025 ; Sattari et al. 2020 ). Incorporating viscoelastic experimental data into the pipeline would require modifications to the material formulation, likely increasing the number of parameters and, consequently, the computational cost of the FE model—however the tradeoff may well be warranted. Further improvements must also be made on the machine learning approach. The current study relied on a feedforward neural network, but advancements in deep learning, such as recurrent neural networks, could enhance predictive capabilities (Shi et al. 2024 ; Tao et al. 2024 ). Emerging techniques that bypass the computational expense of FE simulations for training surrogate models could also be integrated into this pipeline (Motiwale et al. 2024 ). Finally, the current pipeline was developed using lungs from transplant-grade specimens that were rejected due to logistical reasons or reduced lung function and were therefore designated for research use. Importantly, these lungs were not obtained from patients with severe lung disease and were considered representative of relatively healthy tissue. As a result, this study does not capture the mechanical behavior of diseased airways. Extending the present pipeline to include pathological cases is an important next step, as airway material properties are known to change significantly in disease (Strohl et al. 2012 ; Williamson et al. 2011 ). Such extensions would also strengthen the connection with clinical applications, particularly for patients with respiratory diseases who require airway stenting procedures. In addition, the material properties reported in this study were obtained from cadaveric lung specimens tested within 48 h post mortem and therefore may differ from in vivo physiological conditions. Post-mortem changes, the absence of smooth muscle tone, lack of perfusion, and differences in loading environment may influence the measured mechanical response. Conclusion In this study, we developed a numerical pipeline to identify the mechanical properties of human airway tissue. Biaxial tensile testing is combined with an inverse finite element analysis (IFEA) procedure, where the optimization process is then improved with construction of a neural network (NN) surrogate model to allow highly efficient parameter identification. By replacing the computationally expensive traditional IFEA approach with the trained NN in the inverse optimization process, we significantly reduced the time required for material characterization, while maintaining accuracy. The obtained material parameters provide a refined understanding of airway biomechanics, addressing an important knowledge gap in the literature. The findings not only improve current airway models but also represent a step forward in the development of predictive computational frameworks for respiratory medicine. These models could be integrated into clinical applications, such as patient-specific simulations for airway interventions, stent design optimization, and surgical planning, ultimately contributing to improved treatment strategies for airway-related conditions. Furthermore, the methodology and graphical interface can readily be extended to study airway tissues across different species or adapted for other soft biological tissues, such as brain, skin or arterial structures, broadening its applications in computational biomechanics. Supplementary Information Below is the link to the electronic supplementary material. Supplementary file1 (DOCX 2153 kb) (2.1MB, docx) Supplementary file2 (DOCX 18 kb) (18.6KB, docx) Supplementary file3 (DOCX 19 kb) (20KB, docx) Supplementary file4 (DOCX 907 kb) (907.2KB, docx) Supplementary file5 (XLSX 21 kb) (21.1KB, xlsx) Acknowledgements Supported in part by a Dassault Systèmes U.S. Foundation Grant and as part of the Living Lung Project, the Opportunity to Advance Sustainability Innovation and Social Inclusion (OASIS) UCR and State of California Climate Action Through Resilience Program, and the National Science Foundation Graduate Research Fellowship. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Author contributions A.B. designed the research, developed the numerical implementation, conducted numerical simulations, performed the analysis, prepared the figures and manuscript. C.A.M. designed and performed the experiments and contributed to figure preparation. T.M.N. revised the manuscript. M.E. designed and supervised the research, analyzed the results, and prepared the manuscript. Data availability Considering the restrictions and nature of the human tissues involved in this study, the raw data supporting the conclusions of this article will be made available by the authors, upon reasonable request. Declarations Conflict of interest The authors declare no competing interests. Footnotes Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. References Amelon R, Cao K, Ding K, Christensen GE, Reinhardt JM, Raghavan ML (2011) Three-dimensional characterization of regional lung deformation. J Biomech 44(13):2489–2495. 10.1016/j.jbiomech.2011.06.009 [ DOI ] [ PMC free article ] [ PubMed ] [ Google Scholar ] Badrou A, Mariano CA, Ramirez GO, Shankel M, Rebelo N, Eskandari M (2025) Towards constructing a generalized structural 3D breathing human lung model based on experimental volumes, pressures, and strains. PLoS Comput Biol 21(1):e1012680. 10.1371/journal.pcbi.1012680 [ DOI ] [ PMC free article ] [ PubMed ] [ Google Scholar ] Bäker M, Shrot A (2013) Inverse parameter identification with finite element simulations using knowledge-based descriptors. Comput Mater Sci 69:128–136. 10.1016/j.commatsci.2012.11.059 [ Google Scholar ] Burden F, Winkler D (2008) Bayesian regularization of neural networks. In: Artificial Neural Networks, edited by David J. Livingstone, vol. 458, edited by John M. Walker. Methods in Molecular Biology TM . Humana Press. 10.1007/978-1-60327-101-1_3 [ DOI ] [ PubMed ] Bayliss LE, Robertson GW (1939) The visco‐elastic properties of the lungs. Q J Exp Physiol Cogn Med Sci 29(1):27–47. 10.1113/expphysiol.1939.sp000792 [ Google Scholar ] Bou Jawde S, Takahashi A, Bates JHT, Suki B (2020) An analytical model for estimating alveolar wall elastic moduli from lung tissue uniaxial stress-strain curves. Front Physiol 11(February):121. 10.3389/fphys.2020.00121 [ DOI ] [ PMC free article ] [ PubMed ] [ Google Scholar ] Canales C, García-Herrera C, Rivera E, Macías D, Celentano D (2023) Anisotropic hyperelastic material characterization: stability criterion and inverse calibration with evolutionary strategies. Mathematics 11(4):922. 10.3390/math11040922 [ Google Scholar ] Chaitanya Thandra K, Barsouk A, Saginala K, Aluru JS, Barsouk A (2021) Epidemiology of lung cancer. Wspolczesna Onkol 25(1):45–52. 10.5114/wo.2021.103829 [ DOI ] [ PMC free article ] [ PubMed ] [ Google Scholar ] Chawla A, Mukherjee S, Karthikeyan B (2009) Characterization of human passive muscles for impact loads using genetic algorithm and inverse finite element methods. Biomech Model Mechanobiol 8(1):67–76. 10.1007/s10237-008-0121-6 [ DOI ] [ PubMed ] [ Google Scholar ] Cleveland WS (1979) Robust locally weighted regression and smoothing scatterplots. J Am Stat Assoc 74(368):829–836. 10.1080/01621459.1979.10481038 [ Google Scholar ] Dialani V, Ernst A, Sun M et al (2008) MDCT detection of airway stent complications: comparison with bronchoscopy. Am J Roentgenol 191(5):1576–1580. 10.2214/AJR.07.4031 [ DOI ] [ PubMed ] [ Google Scholar ] Eskandari M, Arvayo AL, Levenston ME (2018) Mechanical properties of the airway tree: heterogeneous and anisotropic pseudoelastic and viscoelastic tissue responses. J Appl Physiol 125(3):878–888. 10.1152/japplphysiol.00090.2018 [ DOI ] [ PubMed ] [ Google Scholar ] Eskandari M, Nordgren TM, O’Connell GD (2019) Mechanics of pulmonary airways: linking structure to function through constitutive modeling, biochemistry, and histology. Acta Biomater 97(October):513–523. 10.1016/j.actbio.2019.07.020 [ DOI ] [ PMC free article ] [ PubMed ] [ Google Scholar ] Farahani MM, Bakhtiyari A, Beshkoofe S et al (2023) Numerical simulation of the effect of geometric parameters on Silicone airway stent migration. Front Mech Eng 9(July):1215895. 10.3389/fmech.2023.1215895 [ Google Scholar ] Fehervary H, Smoljkić M, Vander Sloten J, Famaey N (2016) Planar biaxial testing of soft biological tissue using rakes: a critical analysis of protocol and fitting process. J Mech Behav Biomed Mater 61(August):135–151. 10.1016/j.jmbbm.2016.01.011 [ DOI ] [ PubMed ] [ Google Scholar ] Fehervary H, Vander Sloten J, Famaey N (2019) Development of an improved parameter fitting method for planar biaxial testing using rakes. Int J Numer Methods Biomed Eng 35(4):e3174. 10.1002/cnm.3174 [ DOI ] [ PubMed ] [ Google Scholar ] Gasser TC, Ogden RW, Holzapfel GA (2006) Hyperelastic modelling of arterial layers with distributed collagen fibre orientations. J R Soc Interface 3(6):15–35 [ DOI ] [ PMC free article ] [ PubMed ] Guvenir Torun S, Torun HM, Hansen HHG et al (2022) Multicomponent material property characterization of atherosclerotic human carotid arteries through a Bayesian optimization based inverse finite element approach. J Mech Behav Biomed Mater 126(February):104996. 10.1016/j.jmbbm.2021.104996 [ DOI ] [ PubMed ] [ Google Scholar ] Holzapfel GA, Weizsäcker HW (1998) Biomechanical behavior of the arterial wall and its numerical characterization. Comput Biol Med 28(4):377–392. 10.1016/S0010-4825(98)00022-5 [ DOI ] [ PubMed ] [ Google Scholar ] Hu H, Qi L, Chao X (2024) Physics-informed neural networks (PINN) for computational solid mechanics: numerical frameworks and applications. Thin-Walled Struct 205(December):112495. 10.1016/j.tws.2024.112495 [ Google Scholar ] Huang Z, Wang L, Zhang C-X et al (2021) Biomechanical strength dependence on mammalian airway length. J Thorac Dis 13(2):918–926. 10.21037/jtd-20-2970 [ DOI ] [ PMC free article ] [ PubMed ] [ Google Scholar ] Hurtado DE, Villarroel N, Andrade C, Retamal J, Bugedo G, Bruhn A (2017) Spatial patterns and frequency distributions of regional deformation in the healthy human lung. Biomech Model Mechanobiol 16(4):1413–1423. 10.1007/s10237-017-0895-5 [ DOI ] [ PubMed ] [ Google Scholar ] Jacobs NT, Cortes DH, Vresilovic EJ, Elliott DM (2013) Biaxial tension of fibrous tissue: using finite element methods to address experimental challenges arising from boundary conditions and anisotropy. J Biomech Eng 135(2):021004. 10.1115/1.4023503 [ DOI ] [ PMC free article ] [ PubMed ] [ Google Scholar ] Jiang M, Sridhar RL, Robbins AB, Freed AD, Moreno MR (2021) A versatile biaxial testing platform for soft tissues. J Mech Behav Biomed Mater 114:104144. 10.1016/j.jmbbm.2020.104144 [ DOI ] [ PubMed ] [ Google Scholar ] Kauer M, Vuskovic V, Dual J, Szekely G, Bajka M (2002) Inverse finite element characterization of soft tissues. Med Image Anal 6(3):275–287. 10.1016/S1361-8415(02)00085-3 [ DOI ] [ PubMed ] [ Google Scholar ] Kratzer TB, Bandi P, Freedman ND et al (2024) Lung cancer statistics, 2023. Cancer 130(8):1330–1348. 10.1002/cncr.35128 [ DOI ] [ PubMed ] [ Google Scholar ] Kroon M (2011) A constitutive framework for modelling thin incompressible viscoelastic materials under plane stress in the finite strain regime. Mech Time-Depend Mater 15(4):389–406. 10.1007/s11043-011-9159-4 [ Google Scholar ] Di Leonardo S, Monteleone A, Caruso P, Meecham-Garcia H, Pitarresi G, Burriesci G (2024) Effect of the apron in the mechanical characterisation of hyperelastic materials by means of biaxial testing: a new method to improve accuracy. J Mech Behav Biomed Mater 150:106291. 10.1016/j.jmbbm.2023.106291 [ DOI ] [ PubMed ] [ Google Scholar ] Li L, Zhang X, Shi J et al (2023) Airway stents from now to the future: a narrative review. Respiration 102(6):439–448. 10.1159/000530421 [ DOI ] [ PubMed ] [ Google Scholar ] Liu H, Sangpradit K, Li M, Dasgupta P, Althoefer K, Seneviratne LD (2014) Inverse finite-element modeling for tissue parameter identification using a rolling indentation probe. Med Biol Eng Comput 52(1):17–28. 10.1007/s11517-013-1118-6 [ DOI ] [ PubMed ] [ Google Scholar ] MacKay DJC (1992) Bayesian interpolation. In: Smith CR, Erickson GJ, Neudorfer PO (eds) Maximum entropy and Bayesian methods. Springer Netherlands. 10.1007/978-94-017-2219-3_3 [ Google Scholar ] Madan M, Sharma S, Kumar R, Ish P, Kaushik R, Gupta N (2024) Tracheal stent migration in malignant central airway obstruction—a case report and systematic review of literature. Chest Disease Rep 12(July):12731. 10.4081/cdr.12.12731 [ Google Scholar ] Maghsoudi-Ganjeh M, Sattari S, Eskandari M (2021a) Mechanical behavior of the airway wall in respiratory disease. Curr Opin Physiol 22:100445. 10.1016/j.cophys.2021.05.008 [ Google Scholar ] Maghsoudi-Ganjeh M, Mariano CA, Sattari S, Arora H, Eskandari M (2021b) Developing a lung model in the age of COVID-19: a digital image correlation and inverse finite element analysis framework. Front Bioeng Biotechnol. 10.3389/fbioe.2021.684778 [ DOI ] [ PMC free article ] [ PubMed ] Malvè M, Pérez Del Palomar A, Mena A et al (2011) Numerical modeling of a human stented trachea under different stent designs. Int Commun Heat Mass Transfer 38(7):855–862. 10.1016/j.icheatmasstransfer.2011.04.012 [ Google Scholar ] Mariano CA, Sattari S, Maghsoudi-Ganjeh M, Tartibi M, Lo DD, Eskandari M (2020) Novel mechanical strain characterization of ventilated ex vivo porcine and murine lung using digital image correlation. Front Physiol 11:600492. 10.3389/fphys.2020.600492 [ DOI ] [ PMC free article ] [ PubMed ] [ Google Scholar ] Mariano CA, Sattari S, Ramirez GO, Eskandari M (2023) Effects of tissue degradation by collagenase and elastase on the biaxial mechanics of porcine airways. Respir Res 24(1):105. 10.1186/s12931-023-02376-8 [ DOI ] [ PMC free article ] [ PubMed ] [ Google Scholar ] Mariano CA, Eskandari M (2024) Human Airway Tissue Biaxial Tensile Mechanics. Paper presented at 2024 American Society of Mechanical Engineers Bioengineering Division: Summer of Biomechanics Bioengineering Biotransport Conference. June 11. Mariano CA (2025) The breathing strains of artificially and physiologically ventilated humn cadaveric lungs. American Society of Mechanical Engineers Bioengineering Division: Summer of Biomechanics Bioengineering Biotransport Conference, June 22 Mariano CA, Sattari S, Quiros KAM, Nelson TM, Eskandari M (2022) Examining lung mechanical strains as influenced by breathing volumes and rates using experimental digital image correlation. Resp Res 23(1). 10.1186/s12931-022-01999-7 [ DOI ] [ PMC free article ] [ PubMed ] Meethal RE, Kodakkal A, Khalil M et al (2023) Finite element method-enhanced neural network for forward and inverse problems. Adv Model Simul Eng Sci 10(1):6. 10.1186/s40323-023-00243-1 [ Google Scholar ] Motiwale S, Zhang W, Feldmeier R, Sacks MS (2024) A neural network finite element approach for high speed cardiac mechanics simulations. Comput Methods Appl Mech Eng 427:117060. 10.1016/j.cma.2024.117060 [ Google Scholar ] Nelson TM, Eskandari M (2026) Human lung parenchyma: tensile mechanics and the effects of smoking. J R Soc Interface 23:20250721. 10.1098/rsif.2025.0721 [ DOI ] [ PubMed ] [ Google Scholar ] Nelson TM, Quiros KAM, Dominguez EC, Ulu A, Nordgren TM, Eskandari M (2023) Diseased and healthy murine local lung strains evaluated using digital image correlation. Sci Rep 13(1):4564. 10.1038/s41598-023-31345-w [ DOI ] [ PMC free article ] [ PubMed ] [ Google Scholar ] Nelson TM, Mariano CA, Ramirez GO et al (2024) Lung mechanics: material characterization of pulmonary constituents for an experimentally informed computational pipeline. Curr Protoc 4(9):e70001. 10.1002/cpz1.70001 [ DOI ] [ PubMed ] [ Google Scholar ] Nelson TM, Quiros KA, Dominguez EC, Ulu A, Nordgren TM, Nair MG, Eskandari M (2024) Healthy and diseased tensile mechanics of mouse lung parenchyma. Result Eng 22:102169. 10.1016/j.rineng.2024.102169 Nelson TM, Quiros KAM, Eskandari M (2025) Interactive effects of pulmonary pathologies and ventilation modes driving heterogeneous and anisotropic regional strain mechanics. Sci Rep 15(1). 10.1038/s41598-025-27146-y [ DOI ] [ PMC free article ] [ PubMed ] Nolan DR, Gower AL, Destrade M, Ogden RW, McGarry JP (2014) A robust anisotropic hyperelastic formulation for the modelling of soft tissue. J Mech Behav Biomed Mater 39:48–60. 10.1016/j.jmbbm.2014.06.016 [ DOI ] [ PubMed ] [ Google Scholar ] Pei M, Zou D, Gao Y et al (2021) The influence of sample geometry and size on porcine aortic material properties from uniaxial tensile tests using custom-designed tissue cutters, clamps and molds. PLoS ONE 16(2):e0244390. 10.1371/journal.pone.0244390 [ DOI ] [ PMC free article ] [ PubMed ] [ Google Scholar ] Polio SR, Kundu AN, Dougan CE et al (2018) Cross-platform mechanical characterization of lung tissue. PLoS ONE 13(10):e0204765. 10.1371/journal.pone.0204765 [ DOI ] [ PMC free article ] [ PubMed ] [ Google Scholar ] Quiros KAM, Nelson TM, Sattari S et al (2022) Mouse lung mechanical properties under varying inflation volumes and cycling frequencies. Sci Rep 12(1):7094. 10.1038/s41598-022-10417-3 [ DOI ] [ PMC free article ] [ PubMed ] [ Google Scholar ] Quiros KAM, Mariano CA, Shankel M, Nelson TM, Eskandari M (2026) Human ex-vivo lung mechanics differ under positive- and negative-pressure ventilation. Respir Res 27(1):92. 10.1186/s12931-026-03518-4 [ DOI ] [ PMC free article ] [ PubMed ] [ Google Scholar ] Quiros KAM, Nelson TM, Ulu A, Dominguez EC, Biddle TA, Lo DD, Nordgren TM, Eskandari M (2024) A comparative study of ex-vivo murine pulmonary mechanics under positive- and negative-pressure ventilation. Ann Biomed Eng 52(2):342–354. 10.1007/s10439-023-03380-1 [ DOI ] [ PMC free article ] [ PubMed ] Ramirez GO, Mariano CA, Carter D, Eskandari M (2024) Visceral pleura mechanics: Characterization of human pig and rat lung material properties. Acta Biomaterialia 189:388–398. 10.1016/j.actbio.2024.09.003 [ DOI ] [ PubMed ] Saeidi M, Selahi E, Nemati H (2023) Performance simulations of human trachea during respiratory reflexes after stent implantation. J Braz Soc Mech Sci Eng 45(5):276. 10.1007/s40430-023-04206-w [ Google Scholar ] Safshekan F, Tafazzoli-Shadpour M, Abdouss M, Shadmehr M (2016) Mechanical characterization and constitutive modeling of human trachea: age and gender dependency. Materials 9(6):456. 10.3390/ma9060456 [ DOI ] [ PMC free article ] [ PubMed ] [ Google Scholar ] Saji H, Furukawa K, Tsutsui H et al (2010) Outcomes of airway stenting for advanced lung cancer with central airway obstruction. Interact Cardiovasc Thorac Surg 11(4):425–428. 10.1510/icvts.2010.238196 [ DOI ] [ PubMed ] [ Google Scholar ] Salguero BD, Agrawal A, Kaul V et al (2024) Airway stenting for liberation from positive pressure ventilation in patients with central airway obstruction presenting with acute respiratory failure. Respir Med 225:107599. 10.1016/j.rmed.2024.107599 [ DOI ] [ PubMed ] [ Google Scholar ] Sattari S, Mariano CA, Eskandari M (2023) Biaxial mechanical properties of the bronchial tree: characterization of elasticity, extensibility, and energetics, including the effect of strain rate and preconditioning. Acta Biomater 155:410–422. 10.1016/j.actbio.2022.10.047 [ DOI ] [ PubMed ] [ Google Scholar ] Sattari S, Eskandari M (2020) Characterizing the viscoelasticity of extra- and intra-parenchymal lung bronchi. J Mech Behav Biomed Mater 110:103824. 10.1016/j.jmbbm.2020.103824 [ DOI ] [ PubMed ] Shankel M, Nelson TM, Quiros KAM et al (2025a) Lung recruitment mechanics: coalescing tissue strains with organ expansion. Respir Res 26(1):58. 10.1186/s12931-025-03118-8 [ DOI ] [ PMC free article ] [ PubMed ] [ Google Scholar ] Shankel M, Badrou A, Nelson T, Eskandari M (2025b) Surfactant-depleted rat lungs: a characterization of organ- and tissue-level mechanics under negative and positive pressure ventilation. J Biomech Eng. 10.1115/1.4069142 [ DOI ] [ PubMed ] [ Google Scholar ] Shi Y, Chong W, Zhao W, Li S, Li B, Sun X (2024) A new recurrent neural network based on direct discretization method for solving discrete time-variant matrix inversion with application. Inf Sci 652:119729. 10.1016/j.ins.2023.119729 [ Google Scholar ] Smith LJ, Byers S, Costi JJ, Fazzalari NL (2008) Elastic fibers enhance the mechanical integrity of the human lumbar anulus fibrosus in the radial direction. Ann Biomed Eng 36(2):214–223. 10.1007/s10439-007-9421-8 [ DOI ] [ PubMed ] [ Google Scholar ] Stathakis D (2009) How many hidden layers and nodes? Int J Remote Sens 30(8):2133–2147. 10.1080/01431160802549278 [ Google Scholar ] Strohl KP, Butler JP, Malhotra A (2012) Mechanical properties of the upper airway. In: Prakash YS (ed) Comprehensive physiology, 1st ed. Wiley. 10.1002/cphy.c110053 [ DOI ] [ PMC free article ] [ PubMed ] [ Google Scholar ] Tao Y, Lv Z, Liu W, Qi H, Hu P (2024) Recurrent neural network‐based simultaneous cardiac T1, T2, and T1ρ mapping. NMR Biomed 37(8):e5133. 10.1002/nbm.5133 [ DOI ] [ PubMed ] [ Google Scholar ] The MathWorks Inc. (2022) Optimization Toolbox Version: 9.4 (R2022b). The MathWorks Inc., released. https://www.mathworks.com Trabelsi O, Pérez Del Palomar A, López-villalobos JL, Ginel A, Doblaré M (2010) Experimental characterization and constitutive modeling of the mechanical behavior of the human trachea. Med Eng Phys 32(1):76–82. 10.1016/j.medengphy.2009.10.010 [ DOI ] [ PubMed ] [ Google Scholar ] Umar Z, ul Rasool MH, Hosna AU et al (2023) Malignant airway obstruction and endobronchial stent placement: a systematic review on the efficacy and safety. Cureus. 10.7759/cureus.40912 [ DOI ] [ PMC free article ] [ PubMed ] [ Google Scholar ] Uzair M, Jamil N (2020) Effects of hidden layers on the efficiency of neural networks. In: 2020 IEEE 23rd International Multitopic Conference (INMIC), November 5, 1–6. 10.1109/INMIC50486.2020.9318195 Vogl TP, Mangis JK, Rigler AK, Zink WT, Alkon DL (1988) Accelerating the convergence of the back-propagation method. Biol Cybern 59(4–5):257–263. 10.1007/BF00332914 [ Google Scholar ] Wang J-Y, Mesquida P, Lee T (2011) Young’s modulus measurement on pig trachea and bronchial airways. In: 2011 annual international conference of the IEEE engineering in medicine and biology society, August, 2089–92. 10.1109/IEMBS.2011.6090388 [ DOI ] [ PubMed ] Williamson JP, McLaughlin RA, Noffsinger WJ et al (2011) Elastic properties of the central airways in obstructive lung diseases measured using anatomical optical coherence tomography. Am J Respir Crit Care Med 183(5):612–619. 10.1164/rccm.201002-0178OC [ DOI ] [ PubMed ] [ Google Scholar ] Ye KQ (1998) Orthogonal column Latin Hypercubes and their application in computer experiments. J Am Stat Assoc 93(444):1430–1439. 10.1080/01621459.1998.10473803 [ Google Scholar ] Yousefi M, Jalilian H, Heydari S, Seyednejad F, Mir N (2023) Cost of lung cancer: a systematic review. Value Health Reg Issues 33:17–26. 10.1016/j.vhri.2022.07.007 [ DOI ] [ PubMed ] [ Google Scholar ] Zhao J, Feng Y, Koshiyama K, Wu H (2021) Prediction of airway deformation effect on pulmonary air-particle dynamics: a numerical study. Phys Fluids 33(10):101906. 10.1063/5.0065309 [ Google Scholar ] Associated Data This section collects any data citations, data availability statements, or supplementary materials included in this article. Supplementary Materials Supplementary file1 (DOCX 2153 kb) (2.1MB, docx) Supplementary file2 (DOCX 18 kb) (18.6KB, docx) Supplementary file3 (DOCX 19 kb) (20KB, docx) Supplementary file4 (DOCX 907 kb) (907.2KB, docx) Supplementary file5 (XLSX 21 kb) (21.1KB, xlsx) Data Availability Statement Considering the restrictions and nature of the human tissues involved in this study, the raw data supporting the conclusions of this article will be made available by the authors, upon reasonable request. 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