A backward pre-stressing algorithm for efficient finite element implementation of in vivo material and geometrical parameters into fibril-reinforced mixture models of articular cartilage.

Classical continuum mechanics has been widely used for implementation of the material models of articular cartilage (AC) mainly with the aid of the finite element (FE) method, which, in many cases, considers the stress-free configuration as the initial configuration. On the contrary, the AC experimental tests typically begin with the pre-stressed state of both material and geometrical properties. Indeed, imposing the initial pre-stress onto AC models with the in vivo values as the initial state would result in nonphysiologically expansion of the FE mesh due to the soft nature of AC. This change in the model configuration can also affect the material behavior kinematically in the mixture models of cartilage due to the intrinsic compressibility of the tissue. Although several different fixed-point backward algorithms, as the most straightforward pre-stressing methods, have already been developed to incorporate these initial conditions into FE models iteratively, such methods focused merely on the geometrical parameters, and they omitted the material variations of the anisotropic mixture models of AC. To address this issue, we propose an efficient algorithm generalizing the backward schemes to restore stress-free conditions by optimizing both the involving variables, and we hypothesize that it can affect the results considerably. To this end, a comparative simulation was implemented on an advanced and validated multiphasic model by the new and conventional algorithms. The results are in support of the hypothesis, as in our illustrative general AC model, the material parameters experienced a maximum error of 16% comparing to the initial in vivo data when the older algorithm was employed, and it led to a maximum variation of 44% in the recorded stresses comparing to the results of the new method. We conclude that our methodology enhanced the model fidelity, and it is applicable in most of the existing FE solvers for future mixture studies with accurate stress distributions

Impacts of water stress, environment and rootstock on the diurnal behaviour of stem water potential and leaf conductance in pistachio (Pistacia vera L.)

Little information is available on the diurnal behaviour of water potential and leaf conductance on pistachio trees despite their relevance to fine tune irrigation strategies. Mature pistachio trees were subject to simultaneous measurements of stem water potential (Ψx) and leaf conductance (gl) during the day, at three important periods of the irrigation season. Trees were grown on three different rootstocks and water regimes. An initial baseline relating Ψx to air vapor pressure deficit (VPD) is presented for irrigation scheduling in pistachio. Ψx was closely correlated with VPD but with a different fit according to the degree of water stress. No evidence of the variation of Ψx in relation to the phenology of the tree was observed. Furthermore, midday Ψx showed more accuracy to indicate a situation of water stress than predawn water potential. Under well irrigated conditions, gl was positively correlated with VPD during stage II of growth reaching its peak when VPD reached its maximum value (around 4 kPa). This behaviour changed during stage III of fruit growth suggesting a reliance of stomatal behaviour to the phenological stage independently to the tree water status. The levels of water stress reached were translated in a slow recovery of tree water status and leaf conductance (more than 40 days). Regarding rootstocks, P. integerrima showed little adaptation to water shortage compared to the two other rootstocks under the studied conditions

Bridging Diverse Physics and Scales of Knee Cartilage With Efficient and Augmented Graph Learning

Articular cartilage (AC) is essential for minimizing friction in the human knee, but its healthy function is highly influenced by biomechanical factors such as weight bearing. Non-invasive biomechanical and numerical simulations are widely used to study AC but often require complex and costly numerical approximations. Machine learning (ML) provides a more efficient alternative and uses data from these numerical methods for training. Hybrid ML models (HML) complemented by reduced-order numerical models can achieve similar outcomes with minimal data input but may have problems with generalizability across different scales. In this study, we present an extended HML framework (EHML) for developing a multiscale surrogate model specifically tailored for knee cartilage simulations. Our approach is based on integrating hybrid graph neural networks (GNNs) with tissue-scale data and aims to achieve remarkable few-shot learning and potential zero-shot generalizability for large-scale analysis. The main proposed idea is a physics-constrained data augmentation (DA) technique coupled with a set of pre-processing and customization algorithms to bridge the scales. Specifically, we integrate feature transformations, resampling, and cost-sensitive functions to manage the observed data imbalances, all within a customized, memory-efficient training framework. Our rigorous testing using an advanced multi-physics cartilage model demonstrates the viability of our approach. Comparative analyses underscore the significant role of pre-processing and DA methods in enhancing generalizability and efficiency. They helped reduce the normalized mean squared errors to 0.1 or less (compared to the ablated model with its error of 2 or higher). Therefore, this work represents an important step towards addressing the challenges of limited generalizability and efficiency of existing ML-based surrogate models and opens new possibilities for their application in more complex simulations

The effect of highly inhomogeneous biphasic properties on mechanical behaviour of articular cartilage

Background and objective Investigating the biomechanics of cartilage could help to understand the unique load-bearing property of the cartilage and optimize the scaffold design in tissue-engineering. It is important to model the cartilage as a highly inhomogeneous fibril-reinforced biphasic material to represent its complex composition and structure. The depth-dependent and strain-dependent properties of the cartilage would also play an important role in its mechanical behaviour. However, the differences in representing the cartilage as a highly inhomogeneous model or as simplified models still remain unclear. Hence, in this study, a highly inhomogeneous fibril-reinforced biphasic cartilage model considering both the depth-dependent and strain-dependent properties was constructed; the effect of highly inhomogeneous properties on the mechanical behaviour of articular cartilage was investigated. Methods A finite element model of the cartilage was developed based on a flat-ended indentation test. Compressive forces were applied to four various inhomogeneous layered models through a porous indenter (Model 1: nine layers with strain-dependent permeability; Model 2: three layers with strain-dependent permeability; Model 3: single layer with strain-dependent permeability; Model 4: nine layers with constant permeability). Results Models 1 and 2 provided similar results with less than 3% difference in the peak effective stress, contact pressure, fluid pressure as well as fluid support ratio. However, Model 1 to Model 3 differed in stress and strain distribution patterns along depth over prolonged loads, which may provide an important insight into the highly inhomogeneous depth-dependent properties of cartilage. In addition, Model 1 with strain-dependent permeability demonstrated an enhanced capability on fluid pressurisation as compared with Model 4 which had constant permeability. Conclusions A highly inhomogeneous fibril-reinforced biphasic model considering both depth-dependent and strain-dependent properties was developed in this study, in order to illustrate the effect of highly inhomogeneous properties on the mechanical behaviour of the articular cartilage. The number of layers in the models with depth-dependent properties should be selected according to the research questions and clinical demands. The model with strain-dependent permeability offers an enhanced capability on fluid pressurisation. In future studies, the proposed model could be adopted in cell-models to provide more in-depth information or in tissue-engineering to optimize the depth-dependent scaffold structure