Automated discovery of interpretable hyperelastic material models for human brain tissue with EUCLID - Flaschel, Moritz; Yu, Huitian; Reiter, Nina; Hinrichsen, Jan; Budday, Silvia; Steinmann, Paul; Kumar, Siddhant; De Lorenzis, Laura (Journal of the Mechanics and Physics of Solids, 2023)
We propose an automated computational algorithm for simultaneous model selection and parameter identification for the hyperelastic mechanical characterization of biological tissue and validate it on experimental data stemming from human brain tissue specimens. Following the motive of the recently proposed computational framework EUCLID (Efficient Unsupervised Constitutive Law Identification and Discovery) and in contrast to conventional parameter calibration methods, we construct an extensive set of candidate hyperelastic models, i.e., a model library including popular models known from the literature, and develop a computational strategy for automatically selecting a model from the library that conforms to the available experimental data while being represented as an interpretable symbolic mathematical expression. This computational strategy comprises sparse regression, i.e., a regression problem that is regularized by a sparsity promoting penalty term that filters out irrelevant models from the model library, and a clustering method for grouping together highly correlated and thus redundant features in the model library. The model selection procedure is driven by data stemming from mechanical tests under different deformation modes, i.e., uniaxial compression/tension and simple torsion. The data is acquired through conventional mechanical tests that deliver labeled one-dimensional data pairs, and thus the method can be interpreted as a supervised counterpart to the originally proposed EUCLID that is informed by full-field displacement data and global reaction forces. The proposed method is verified on synthetical data with artificial noise. In addition, we present for the first time an experimental investigation of the EUCLID framework by validating the proposed method on experimental data acquired through mechanical tests of human brain specimens, proving that the method is capable of discovering hyperelastic models that exhibit both high fitting accuracy to the data as well as concise and thus interpretable mathematical representations.
CRIS-Team Scopus Importer:2023-09-08
Keywords: Brain tissue; Constitutive models; Hyperelasticity; Interpretable models; Sparse regression

Compression‐Tension‐Asymmetry and Stiffness Nonlinearity of Collagen‐Matrigel Composite Hydrogels - Böhringer, David; Hinrichsen, Jan; Gataulin, Radik; Wiedenmann, Sandra; Spörrer, Marina; Sherifova, Selda; Steinmann, Paul; Holzapfel, Gerhard A.; Fabry, Ben; Budday, Silvia (Advanced Healthcare Materials, 2025)
Collagen type I hydrogels, which self-assemble into 3D fiber networks, are commonly used for cell culture and tissue engineering applications. Collagen hydrogels replicate the nonlinear stress–strain relationship of collagenous tissue under extension. However, they buckle and soften under compression, whereas natural tissue exhibits significant stiffening due to the presence of cells and other matrix components. To more closely mimic the mechanical properties of natural tissue, varying concentrations of the basement membrane extract Matrigel are added to collagen. The stress–strain relationship of the resulting composite hydrogels is then analyzed under compression, tension, and shear. It is found that the addition of Matrigel increases the stiffness and reduces the compression-tension asymmetry. This can be explained by a reduced degree of freedom for collagen fiber buckling due to the constraints imposed by the surrounding fine-meshed Matrigel network. Consistent with this explanation, it is found that the collapse of composite hydrogels under uniaxial strain decreases with increasing concentration of Matrigel and other filler materials, such as alginate. Taken together, by adjusting the ratio of Matrigel to collagen, the mechanical compression-tension asymmetry and nonlinearity of composite hydrogels can be tuned to more closely mimic natural tissue and tailor cell behavio

Identifying composition-mechanics relations in human brain tissue based on neural-network-enhanced inverse parameter identification - Hinrichsen, Jan; Feiler, Lea; Reiter, Nina; Bräuer, Lars; Schicht, Martin; Paulsen, Friedrich; Budday, Silvia (Mathematics and Mechanics of Solids, 2024)
The mechanical properties of human brain tissue remain far from being fully understood. One aspect that has gained more attention recently is their regional dependency, as the brain’s microstructure varies significantly from one region to another. Understanding the correlation between tissue components and the mechanical behavior is an important step toward better understanding how human brain tissue properties change in space and time and to develop highly spatially resolved constitutive models for large-scale brain simulations. Here, we analyze the correlation between human brain tissue components quantified through enzyme-linked immunosorbent assays (ELISA) and material parameters obtained through an inverse parameter identification scheme based on a hyperelastic Ogden model and multimodal mechanical testing data for eight regions of the brain. We use neural networks as a metamodel to save computational costs. The networks are trained on finite element simulation outputs and are able to replace the simulations in the initial optimization step. We identified strong dependencies between mechanical properties and Iba1 associated with microglia cells, collagen VI, GFAP associated with astrocytes, and collagen IV. These results advance our understanding of microstructure-mechanics relations in human brain tissue and will contribute to the development of highly spatially resolved microstructure-informed constitutive models.<br

Inverse identification of region-specific hyperelastic material parameters for human brain tissue - Hinrichsen, Jan; Reiter, Nina; Bräuer, Lars; Paulsen, Friedrich; Käßmair, Stefan; Budday, Silvia (Biomechanics and Modeling in Mechanobiology, 2023)
The identification of material parameters accurately describing the region-dependent mechanical behavior of human brain tissue is crucial for computational models used to assist, e.g., the development of safety equipment like helmets or the planning and execution of brain surgery. While the division of the human brain into different anatomical regions is well established, knowledge about regions with distinct mechanical properties remains limited. Here, we establish an inverse parameter identification scheme using a hyperelastic Ogden model and experimental data from multi-modal testing of tissue from 19 anatomical human brain regions to identify mechanically distinct regions and provide the corresponding material parameters. We assign the 19 anatomical regions to nine governing regions based on similar parameters and microstructures. Statistical analyses confirm differences between the regions and indicate that at least the corpus callosum and the corona radiata should be assigned different material parameters in computational models of the human brain. We provide a total of four parameter sets based on the two initial Poisson’s ratios of 0.45 and 0.49 as well as the pre- and unconditioned experimental responses, respectively. Our results highlight the close interrelation between the Poisson’s ratio and the remaining model parameters. The identified parameters will contribute to more precise computational models enabling spatially resolved predictions of the stress and strain states in human brains under complex mechanical loading conditions.<br

Inverse identification of region-specific hyperelastic material parameters for human brain tissue - Hinrichsen, Jan; Reiter, Nina; Bräuer, Lars; Paulsen, Friedrich; Käßmair, Stefan; Budday, Silvia (Biomechanics and modeling in mechanobiology, 2023)
The identification of material parameters accurately describing the region-dependent mechanical behavior of human brain tissue is crucial for computational models used to assist, e.g., the development of safety equipment like helmets or the planning and execution of brain surgery. While the division of the human brain into different anatomical regions is well established, knowledge about regions with distinct mechanical properties remains limited. Here, we establish an inverse parameter identification scheme using a hyperelastic Ogden model and experimental data from multi-modal testing of tissue from 19 anatomical human brain regions to identify mechanically distinct regions and provide the corresponding material parameters. We assign the 19 anatomical regions to nine governing regions based on similar parameters and microstructures. Statistical analyses confirm differences between the regions and indicate that at least the corpus callosum and the corona radiata should be assigned different material parameters in computational models of the human brain. We provide a total of four parameter sets based on the two initial Poisson’s ratios of 0.45 and 0.49 as well as the pre- and unconditioned experimental responses, respectively. Our results highlight the close interrelation between the Poisson’s ratio and the remaining model parameters. The identified parameters will contribute to more precise computational models enabling spatially resolved predictions of the stress and strain states in human brains under complex mechanical loading conditions.
CRIS-Team Scopus Importer:2023-09-15
Keywords: Finite element method; Finite hyperelasticity; Human brain tissue; Ogden model; Parameter identification

Model-driven exploration of poro-viscoelasticity in human brain tissue: be careful with the parameters! - Greiner, Alexander; Reiter, Nina; Hinrichsen, Jan; Kainz, Manuel P.; Sommer, Gerhard; Holzapfel, Gerhard A.; Steinmann, Paul; Comellas, Ester; Budday, Silvia (Interface Focus, 2024)
The brain is arguably the most complex human organ and modelling its mechanical behaviour has challenged researchers for decades. There is still a lack of understanding on how this multiphase tissue responds to mechanical loading and how material parameters can be reliably calibrated. While previous viscoelastic models with two relaxation times have successfully captured the response of brain tissue, the Theory of Porous Media provides a continuum mechanical framework to explore the underlying physical mechanisms, including interactions between solid matrix and free-flowing interstitial fluid. Following our previously published experimental testing protocol, here we perform finite element simulations of cyclic compression–tension loading and compression–relaxation experiments on human brain white and gray matter specimens. The solid volumetric stress proves to be a crucial factor for the overall biphasic tissue behaviour as it strongly interferes with porous effects controlled by the permeability. An inverse parameter identification reveals that poroelasticity alone is insufficient to capture the time-dependent material behaviour, but a poro-viscoelastic formulation captures the response of brain tissue well. We provide valuable insights into the individual contributions of viscous and porous effects. However, due to the strong coupling between porous, viscous, and volumetric effects, additional experiments are required to reliably determine all material parameter

Multilayer 3D bioprinting and complex mechanical properties of alginate-gelatin mesostructures - Ahmadi Soufivand, Anahita; Faber, Jessica; Hinrichsen, Jan; Budday, Silvia (Scientific Reports, 2023)
In the biomedical field, extrusion-based 3D bioprinting has emerged as a promising technique to fabricate tissue replacements. However, a main challenge is to find suitable bioinks and reproducible procedures that ensure good printability and generate final printed constructs with high shape fidelity, similarity to the designed model, and controllable mechanical properties. In this study, our main goal is to 3D print multilayered structures from alginate-gelatin (AG) hydrogels and to quantify their complex mechanical properties with particular focus on the effects of the extrusion process and geometrical parameters, i.e. different mesostructures and macroporosities. We first introduce a procedure including a pre-cooling step and optimized printing parameters to control and improve the printability of AG hydrogels based on rheological tests and printability studies. Through this procedure, we significantly improve the printability and flow stability of AG hydrogels and successfully fabricate well-defined constructs similar to our design models. Our subsequent complex mechanical analyses highlight that the extrusion process and the mesostructure, characterized by pore size, layer height and filament diameter, significantly change the complex mechanical response of printed constructs. The presented approach and the corresponding results have important implications for future 3D bioprinting applications when aiming to produce replacements with good structural integrity and defined mechanical properties similar to the native tissue, especially in soft tissue engineering. The approach is also applicable to the printing of gelatin-based hydrogels with different accompanying materials, concentrations, or cells.
CRIS-Team Scopus Importer:2023-07-21

On the importance of using region-dependent material parameters for full-scale human brain simulations - Griffiths, Emma; Hinrichsen, Jan; Reiter, Nina; Budday, Silvia (European Journal of Mechanics A-Solids, 2023)
An accurate finite element (FE) model of the human brain reliably predicting its response to loading offers great possibilities for brain injury prevention, disease prediction, and surgical guidance. However, the brain is a very complex organ. One important aspect is the structural heterogeneity, both on a macroscopic and microscopic level. This also results in heterogeneous mechanical properties, which highly depend on the brain regions. The extent to which this heterogeneity affects the mechanical response of an FE simulated full brain remains largely unexplored. In this work, we investigate the importance of using region-specific material parameters when modelling the full human brain. Additionally, we assess how a prefixed Poisson's ratio used for the determination of regional parameters affects the response of the full brain under loading. Finally, we compare the simulation results when using parameter sets that have been determined from the unconditioned and preconditioned behaviour of human brain tissue. Our results show that accounting for region-dependent properties leads to significant differences in the predicted strain state compared to simulations assuming homogeneous material properties. Also the Poisson's ratio and using unconditioned or preconditioned data sets significantly affects the results of full-scale brain simulations, highlighting the importance of carefully selecting the material parameters used. The presented analyses have important implications for choosing appropriate region-dependent material parameters for full-scale FE human brain simulations in the future to ensure reliable predictions.
CRIS-Team Scopus Importer:2023-03-08
Keywords: Finite element method; Human brain; Material modelling; Material parameters; Regional heterogeneity

Poro-viscoelastic material parameter identification of brain tissue-mimicking hydrogels - Kainz, Manuel P.; Greiner, Alexander; Hinrichsen, Jan; Kolb, Dagmar; Comellas, Ester; Steinmann, Paul; Budday, Silvia; Terzano, Michele; Holzapfel, Gerhard A. (Frontiers in Bioengineering and Biotechnology, 2023)
Understanding and characterizing the mechanical and structural properties of brain tissue is essential for developing and calibrating reliable material models. Based on the Theory of Porous Media, a novel nonlinear poro-viscoelastic computational model was recently proposed to describe the mechanical response of the tissue under different loading conditions. The model contains parameters related to the time-dependent behavior arising from both the viscoelastic relaxation of the solid matrix and its interaction with the fluid phase. This study focuses on the characterization of these parameters through indentation experiments on a tailor-made polyvinyl alcohol-based hydrogel mimicking brain tissue. The material behavior is adjusted to ex vivo porcine brain tissue. An inverse parameter identification scheme using a trust region reflective algorithm is introduced and applied to match experimental data obtained from the indentation with the proposed computational model. By minimizing the error between experimental values and finite element simulation results, the optimal constitutive model parameters of the brain tissue-mimicking hydrogel are extracted. Finally, the model is validated using the derived material parameters in a finite element simulation.
CRIS-Team Scopus Importer:2023-05-05
Keywords: biomechanical testing; brain tissue; hydrogel; indentation; parameter identification; polyvinyl alcohol; poroelasticity; viscoelasticity

Tissue-Scale Biomechanical Testing of Brain Tissue for the Calibration of Nonlinear Material Models - Faber, Jessica; Hinrichsen, Jan; Greiner, Alexander; Reiter, Nina; Budday, Silvia (Current Protocols, 2022)
Brain tissue is one of the most complex and softest tissues in the human body. Due to its ultrasoft and biphasic nature, it is difficult to control the deformation state during biomechanical testing and to quantify the highly nonlinear, time-dependent tissue response. In numerous experimental studies that have investigated the mechanical properties of brain tissue over the last decades, stiffness values have varied significantly. One reason for the observed discrepancies is the lack of standardized testing protocols and corresponding data analyses. The tissue properties have been tested on different length and time scales depending on the testing technique, and the corresponding data have been analyzed based on simplifying assumptions. In this review, we highlight the advantage of using nonlinear continuum mechanics based modeling and finite element simulations to carefully design experimental setups and protocols as well as to comprehensively analyze the corresponding experimental data. We review testing techniques and protocols that have been used to calibrate material model parameters and discuss artifacts that might falsify the measured properties. The aim of this work is to provide standardized procedures to reliably quantify the mechanical properties of brain tissue and to more accurately calibrate appropriate constitutive models for computational simulations of brain development, injury and disease. Computational models can not only be used to predictively understand brain tissue behavior, but can also serve as valuable tools to assist diagnosis and treatment of diseases or to plan neurosurgical procedures. © 2022 The Authors. Current Protocols published by Wiley Periodicals LLC.
CRIS-Team Scopus Importer:2022-04-15
Keywords: brain tissue; constitutive modeling; indentation; large strains; multi-modal mechanical testing; parameter identification

Tuning the mechanical properties of alginate dialdehyde–gelatin (ADA–GEL) bioinks for bioprinting approaches by varying the degree of oxidation - Faber, Jessica; Hinrichsen, Jan; Ahmadi Soufivand, Anahita; Lu, Hsuan-Heng; Rosenberger, Tanja; Karakaya, Emine; Detsch, Rainer; Boccaccini, Aldo R.; Budday, Silvia (Journal of the Mechanical Behavior of Biomedical Materials, 2025)
Extrusion-based 3D bioprinting is one of the most promising and widely used technologies in bioprinting. However, the development of bioprintable, biocompatible bioinks with tailored mechanical and biological properties remains a major challenge in this field. Alginate dialdehyde–gelatin (ADA–GEL) hydrogels face these difficulties and enable to tune the mechanical properties depending on the degree of oxidation (% DO) of ADA. Here, we present a holistic approach for characterizing the influence of the % DO on the mechanical properties of ADA–GEL hydrogels under multiple loading modes, compression, tension, and torsional shear in the large-strain regime. We evaluate complex mechanical characteristics including nonlinearity, hysteresis, conditioning, and stress relaxation. We calibrate hyperelastic material models to determine the corresponding material parameters inversely. Our results confirm that decreasing the % DO of ionically crosslinked ADA–GEL hydrogels leads to an increase in stiffness, more distinct nonlinearity, more pronounced hysteresis, and minor preconditioning effects, while the relaxation behavior is slightly affected. The fabrication technique – molding or printing – does only slightly affect the complex mechanical properties and stress relaxation behavior. Ionically and enzymatically dual-crosslinked ADA–GEL hydrogels showed higher stresses during cyclic loading and less viscous effects during stress relaxation in all three loading modes. We conclude that the % DO and the crosslinking procedure are crucial parameters to tune the mechanical behavior of ADA–GEL hydrogels. Careful choice of these parameters might facilitate the fabrication of biomaterials that closely mimic the properties of native tissues for advanced tissue engineering applications.
CRIS-Team Scopus Importer:2025-01-31
Keywords: Alginate dialdehyde–gelatin hydrogels; Bioprinting; Degree of oxidation; Large strains; Multimodal mechanical testing

Using dropout based active learning and surrogate models in the inverse viscoelastic parameter identification of human brain tissue - Hinrichsen, Jan; Ferlay, Carl; Reiter, Nina; Budday, Silvia (Frontiers in Physiology, 2024)
Inverse mechanical parameter identification enables the characterization of ultrasoft materials, for which it is difficult to achieve homogeneous deformation states. However, this usually involves high computational costs that are mainly determined by the complexity of the forward model. While simulation methods like finite element models can capture nearly arbitrary geometries and implement involved constitutive equations, they are also computationally expensive. Machine learning models, such as neural networks, can help mitigate this problem when they are used as surrogate models replacing the complex high fidelity models. Thereby, they serve as a reduced order model after an initial training phase, where they learn the relation of in- and outputs of the high fidelity model. The generation of the required training data is computationally expensive due to the necessary simulation runs. Here, active learning techniques enable the selection of the “most rewarding” training points in terms of estimated gained accuracy for the trained model. In this work, we present a recurrent neural network that can well approximate the output of a viscoelastic finite element simulation while significantly speeding up the evaluation times. Additionally, we use Monte-Carlo dropout based active learning to identify highly informative training data. Finally, we showcase the potential of the developed pipeline by identifying viscoelastic material parameters for human brain tissue.
CRIS-Team Scopus Importer:2024-02-16
Keywords: active learning; human brain tissue; neural network; parameter identification; surrogate model