Comparison between constitutive models for the solid phase of biphasic agarose/chondrocytes constructs for knee cartilage engineering

A 3-D constitutive model for finite element analyses of agarose with a range of gel concentrations

Hydrogels have seen widespread application across biomedical sciences and there is considerable interest in using hydrogels, including agarose, for creating in vitro three-dimensional environments to grow cells and study mechanobiology and mechanotransduction. Recent advances in the preparation of agarose gels enable successful encapsulation of viable cells at gel concentrations as high as 5%. Agarose with a range of gel concentrations can thus serve as an experimental model mimicking changes in the 3-D microenvironment of cells during disease progression and can facilitate experiments aimed at probing the corresponding mechanobiology, e.g. the evolving mechanobiology of chondrocytes during the progression of osteoarthritis. Importantly, whether stresses (forces) or strains (displacements) drive mechanobiology and mechanotransduction is currently unknown. We can use experiments to quantify mechanical properties of hydrogels, and imaging to estimate microstructure and even strains; however, only computational models can estimate intra-gel stresses in cell-seeded agarose constructs because the required in vitro experiments are currently impossible. Finite element modeling is well-established for (computational) mechanical analyses, but accurate constitutive models for modeling the 3-D mechanical environments of cells within high-stiffness agarose with varying gel concentrations are currently unavailable. In this study we aimed to establish a 3-D constitutive model of high-stiffness agarose with a range of gel concentrations. We applied a multi-step, physics-based optimization approach to separately fit subsets of model parameters and help achieve robust convergence. Our constitutive model, fitted to experimental data on progressive stress-relaxations, was able to predict reaction forces determined from independent experiments on cyclical loading. Our model has broad applications in finite element modeling aimed at interpreting mechanical experiments on agarose specimens seeded with cells, particularly in predicting distributions of intra-gel stresses. Our model and fitted parameters enable more accurate finite element simulations of high-stiffness agarose constructs, and thus better understanding of experiments aimed at mechanobiology, mechanotransduction, or other applications in tissue engineering.

Comparative assessment of intrinsic mechanical stimuli on knee cartilage and compressed agarose constructs

A well-established cue for improving the properties of tissue-engineered cartilage is mechanical stimulation. However, the explicit ranges of mechanical stimuli that correspond to favorable metabolic outcomes are elusive. Usually, these outcomes have only been associated with the applied strain and frequency, an oversimplification that can hide the fundamental relationship between the intrinsic mechanical stimuli and the metabolic outcomes. This highlights two important key issues: the firstly is related to the evaluation of the intrinsic mechanical stimuli of native cartilage; the second, assuming that the intrinsic mechanical stimuli will be important, deals with the ability to replicate them on the tissue-engineered constructs. This study quantifies and compares the volume of cartilage and agarose subjected to a given magnitude range of each intrinsic mechanical stimulus, through a numerical simulation of a patient-specific knee model coupled with experimental data of contact during the stance phase of gait, and agarose constructs under direct-dynamic compression. The results suggest that direct compression loading needs to be parameterized with time-dependence during the initial culture period in order to better reproduce each one of the intrinsic mechanical stimuli developed in the patient-specific cartilage. A loading regime which combines time periods of low compressive strain (5%) and frequency (0.5Hz), in order to approach the maximal principal strain and fluid velocity stimulus of the patient-specific cartilage, with time periods of high compressive strain (20%) and frequency (3Hz), in order to approach the pore pressure values, may be advantageous relatively to a single loading regime throughout the full culture period.

A mathematical model of tissue-engineered cartilage development under cyclic compressive loading

In this work a coupled model of solute transport and uptake, cell proliferation, extracellular matrix synthesis and remodeling of mechanical properties accounting for the impact of mechanical loading is presented as an advancement of a previously validated coupled model for free-swelling tissue-engineered cartilage cultures. Tissue-engineering constructs were modeled as biphasic with a linear elastic solid, and relevant intrinsic mechanical stimuli in the constructs were determined by numerical simulation for use as inputs of the coupled model. The mechanical dependent formulations were derived from a calibration and parametrization dataset and validated by comparison of normalized ratios of cell counts, total glycosaminoglycans and collagen after 24-h continuous cyclic unconfined compression from another dataset. The model successfully fit the calibration dataset and predicted the results from the validation dataset with good agreement, with average relative errors up to 3.1 and 4.3 %, respectively. Temporal and spatial patterns determined for other model outputs were consistent with reported studies. The results suggest that the model describes the interaction between the simultaneous factors involved in in vitro tissue-engineered cartilage culture under dynamic loading. This approach could also be attractive for optimization of culture protocols, namely through the application to longer culture times and other types of mechanical stimuli.

Influence of the scaffold geometry on the spatial and temporal evolution of the mechanical properties of tissue-engineered cartilage: insights from a mathematical model

The production of tissue-engineered cartilage in vitro with inhomogeneous mechanical properties is a problem yet to be solved. Different geometries have been studied to overcome this caveat; however, the reported measurements are limited to average values of some properties and qualitative measures of spatial distributions. We will apply a coupled model to extend knowledge about the introduction of a macrochannel in a scaffold by calculating spatiotemporal patterns for several interest variables related to the remodeling of the mechanical properties. Model parameters were estimated based on experimental data on the temporal patterns of glycosaminoglycans, collagen and compressive Young's modulus for channel-free constructs. The model reproduced the experimental data trends in both geometries, with experimental-numerical correlations between 0.84 and 0.97. The channel had a higher impact on the reduction in spatial heterogeneities and delay of saturation of core properties than in the improvement of average properties. Despite the possible improvement of cell densities for longer periods than 56 days, it is estimated that it will not cause further significant improvements of the mechanical properties. The degrees of spatial heterogeneity of the Young's modulus and permeability in the channeled geometry are 23 and 27 % of the channel-free values. While the average Young's modulus values are in the range of native cartilage, the permeabilities are one to three degrees of magnitude higher than the native cartilage, suggesting that limiting factors such as scaffold porosity and initial permeability are more relevant than scaffold geometry to effectively decrease the tissue permeability.