A phenomenological mathematical model of the articular cartilage damage

A chemo-mechano-biological modeling framework for cartilage evolving in health, disease, injury, and treatment

BACKGROUND AND OBJECTIVE

Osteoarthritis (OA) is a pervasive and debilitating disease, wherein degeneration of cartilage features prominently. Despite extensive research, we do not yet understand the cause or progression of OA. Studies show biochemical, mechanical, and biological factors affect cartilage health. Mechanical loads influence synthesis of biochemical constituents which build and/or break down cartilage, and which in turn affect mechanical loads. OA-associated biochemical profiles activate cellular activity that disrupts homeostasis. To understand the complex interplay among mechanical stimuli, biochemical signaling, and cartilage function requires integrating vast research on experimental mechanics and mechanobiology-a task approachable only with computational models. At present, mechanical models of cartilage generally lack chemo-biological effects, and biochemical models lack coupled mechanics, let alone interactions over time.

METHODS

We establish a first-of-its kind virtual cartilage: a modeling framework that considers time-dependent, chemo-mechano-biologically induced turnover of key constituents resulting from biochemical, mechanical, and/or biological activity. We include the "minimally essential" yet complex chemical and mechanobiological mechanisms. Our 3-D framework integrates a constitutive model for the mechanics of cartilage with a novel model of homeostatic adaptation by chondrocytes to pathological mechanical stimuli, and a new application of anisotropic growth (loss) to simulate degradation clinically observed as cartilage thinning.

RESULTS

Using a single set of representative parameters, our simulations of immobilizing and overloading successfully captured loss of cartilage quantified experimentally. Simulations of immobilizing, overloading, and injuring cartilage predicted dose-dependent recovery of cartilage when treated with suramin, a proposed therapeutic for OA. The modeling framework prompted us to add growth factors to the suramin treatment, which predicted even better recovery.

CONCLUSIONS

Our flexible framework is a first step toward computational investigations of how cartilage and chondrocytes mechanically and biochemically evolve in degeneration of OA and respond to pharmacological therapies. Our framework will enable future studies to link physical activity and resulting mechanical stimuli to progression of OA and loss of cartilage function, facilitating new fundamental understanding of the complex progression of OA and elucidating new perspectives on causes, treatments, and possible preventions.

Medial knee cartilage is unlikely to withstand a lifetime of running without positive adaptation: a theoretical biomechanical model of failure phenomena

Runners on average do not have a high risk of developing knee osteoarthritis, even though running places very high loads on the knee joint. Here we used gait analysis, musculoskeletal modeling, and a discrete-element model of knee contact mechanics to estimate strains of the medial knee cartilage in walking and running in 22 young adults (age 23 ± 3 years). A phenomenological model of cartilage damage, repair, and adaptation in response to these strains then estimated the failure probability of the medial knee cartilage over an adult lifespan (age 23-83 years) for 6 km/day of walking vs. walking and running 3 km/day each. With no running, by age 55 the cumulative probability of medial knee cartilage failure averaged 36% without repair and 13% with repair, similar to reports on incidence of knee osteoarthritis in non-obese adults with no knee injuries, but the probability for running was very high without repair or adaptation (98%) and remained high after including repair (95%). Adaptation of the cartilage compressive modulus, cartilage thickness, and the tibiofemoral bone congruence in response to running (+1.15 standard deviations of their baseline values) was necessary for the failure probability of walking and running 3 km/day each to equal the failure probability of walking 6 km/day. The model results suggest two conclusions for further testing: (i) unlike previous findings on the load per unit distance, damage per unit distance on the medial knee cartilage is greater in running vs. walking, refuting the "cumulative load" hypothesis for long-term joint health; (ii) medial knee cartilage is unlikely to withstand a lifetime of mechanical loading from running without a natural adaptation process, supporting the "cartilage conditioning" hypothesis for long-term joint health.

Single-leg forward hopping exposures adversely affect knee joint health among persons with unilateral lower limb loss: A predictive model

Single-leg hopping is an atypical, yet convenient, method of ambulation for individuals who have sustained unilateral lower limb-loss. Hopping is generally discouraged by therapists but many patients report hopping, and the potential deleterious effects of frequent hopping on knee joint health remains unclear. Mechanical fatigue due to repeated exposures to increased or abnormal loading on the intact limb is thought to be a primary contributor to the high prevalence of knee osteoarthritis among individuals with unilateral lower limb amputation. We aimed to compare knee joint mechanics between single-leg hopping and walking at self-selected paces among individuals with unilateral lower limb-loss, and estimated the associated probability of knee cartilage failure. Thirty-two males with traumatic unilateral lower limb-loss (22 transtibial, 10 transfemoral) hopped and walked at a self-selected pace along a 15-m walkway. Peak knee moments were input to a phenomenological model of cartilage fatigue to estimate the damage and long-term failure probability of the medial knee cartilage when hopping vs. walking. We estimate that each hop accumulates as much damage as at least 8 strides of walking (p < 0.001), and each meter of hopping accumulates as much damage as at least 12 m of walking (p < 0.001). The 30-year failure probability of the medial knee cartilage exceeded a "coin-flip" chance (50%) when performing more than 197 hops per day. Although a convenient mode of ambulation for persons with unilateral lower limb-loss, to mitigate risk for knee osteoarthritis it is advisable to minimize exposure to single-leg forward hopping.

Use of Computational Modeling to Study Joint Degeneration: A Review

Osteoarthritis (OA), a degenerative joint disease, is the most common chronic condition of the joints, which cannot be prevented effectively. Computational modeling of joint degradation allows to estimate the patient-specific progression of OA, which can aid clinicians to estimate the most suitable time window for surgical intervention in osteoarthritic patients. This paper gives an overview of the different approaches used to model different aspects of joint degeneration, thereby focusing mostly on the knee joint. The paper starts by discussing how OA affects the different components of the joint and how these are accounted for in the models. Subsequently, it discusses the different modeling approaches that can be used to answer questions related to OA etiology, progression and treatment. These models are ordered based on their underlying assumptions and technologies: musculoskeletal models, Finite Element models, (gene) regulatory models, multiscale models and data-driven models (artificial intelligence/machine learning). Finally, it is concluded that in the future, efforts should be made to integrate the different modeling techniques into a more robust computational framework that should not only be efficient to predict OA progression but also easily allow a patient’s individualized risk assessment as screening tool for use in clinical practice.

Obtaining the sGAG distribution profile in articular cartilage color images

The articular cartilage tissue is an essential component of joints as it reduces the friction between the two bones. Its load-bearing properties depend mostly on proteoglycan distribution, which can be analyzed through the study of the presence of sulfated glycosaminoglycan (sGAG). Currently, sGAG distribution in articular cartilage is not completely known; it is calculated by means of laboratory tests that imply the inherent inaccuracy of a manual procedure. This paper presents an easy-to-use desktop software application for obtaining the sGAG distribution profile in tissue. This app uses color images of stained cartilage tissues taken under a microscope, so researchers at the Trinity Centre for Bioengineering (Dublin, Ireland) can understand the qualitative distribution of sGAG with depth in the studied tissues.

Simulation of Subject-Specific Progression of Knee Osteoarthritis and Comparison to Experimental Follow-up Data: Data from the Osteoarthritis Initiative

Economic costs of osteoarthritis (OA) are considerable. However, there are no clinical tools to predict the progression of OA or guide patients to a correct treatment for preventing OA. We tested the ability of our cartilage degeneration algorithm to predict the subject-specific development of OA and separate groups with different OA levels. The algorithm was able to predict OA progression similarly with the experimental follow-up data and separate subjects with radiographical OA (Kellgren-Lawrence (KL) grade 2 and 3) from healthy subjects (KL0). Maximum degeneration and degenerated volumes within cartilage were significantly higher (p < 0.05) in OA compared to healthy subjects, KL3 group showing the highest degeneration values. Presented algorithm shows a great potential to predict subject-specific progression of knee OA and has a clinical potential by simulating the effect of interventions on the progression of OA, thus helping decision making in an attempt to delay or prevent further OA symptoms.

On mechanical mechanism of damage evolution in articular cartilage

Superficial lesions of cartilage are the direct indication of osteoarthritis. To investigate the mechanical mechanism of cartilage with micro-defect under external loading, a new plain strain numerical model with micro-defect was proposed and damage evolution progression in cartilage over time has been simulated, the parameter were studied including load style, velocity of load and degree of damage. The new model consists of the hierarchical structure of cartilage and depth-dependent arched fibers. The numerical results have shown that not only damage of the cartilage altered the distribution of the stress but also matrix and fiber had distinct roles in affecting cartilage damage, and damage in either matrix or fiber could promote each other. It has been found that the superficial cracks in cartilage spread preferentially along the tangent direction of the fibers. It is the arched distribution form of fibers that affects the crack spread of cartilage, which has been verified by experiment. During the process of damage evolution, its extension direction and velocity varied constantly with the damage degree. The rolling load could cause larger stress and strain than sliding load. Strain values of the matrix initially increased and then decreased gradually with the increase of velocity, and velocity had a greater effect on matrix than fibers. Damage increased steadily before reaching 50%, sharply within 50 to 85%, and smoothly and slowly after 85%. The finding of the paper may help to understand the mechanical mechanism why the cracks in cartilage spread preferentially along the tangent direction of the fibers.

Parameter-dependent behavior of articular cartilage: 3D mechano-electrochemical computational model

BACKGROUND AND OBJECTIVE

Changes in mechano-electrochemical properties of articular cartilage play an essential role in the majority of cartilage diseases. Despite of this importance, the specific effect of each parameter into tissue behavior remains still obscure. Parametric computational modeling of cartilage can provide some insights into this matter, specifically the study of mechano-electrochemical properties variation and their correlation with tissue swelling, water and ion fluxes. Thus, the aim of this study is to evaluate the influence of the main mechanical and electrochemical parameters on the determination of articular cartilage behavior by a parametric analysis through a 3D finite element model.

METHODS

For this purpose, a previous 3D mechano-electrochemical model, developed by the same authors, of articular cartilage behavior has been used. Young's modulus, Poisson coefficient, ion diffusivities and ion activity coefficients variations have been analyzed and quantified through monitoring tissue simulated response.

RESULTS

Simulation results show how Young's modulus and Poisson coefficient control tissue behavior rather than electrochemical properties. Meanwhile, ion diffusivity and ion activity coefficients appear to be vital in controlling velocity of incoming and outgoing fluxes.

CONCLUSIONS

This parametric study establishes a basic guide when defining the main properties that are essential to be included into computational modeling of articular cartilage providing a helpful tool in tissue simulations.

Computational analysis of cartilage implants based on an interpenetrated polymer network for tissue repairing

Interpenetrated polymer networks (IPNs), composed by two independent polymeric networks that spatially interpenetrate, are considered as valuable systems to control permeability and mechanical properties of hydrogels for biomedical applications. Specifically, poly(ethyl acrylate) (PEA)-poly(2-hydroxyethyl acrylate) (PHEA) IPNs have been explored as good hydrogels for mimicking articular cartilage. These lattices are proposed as matrix implants in cartilage damaged areas to avoid the discontinuity in flow uptake preventing its deterioration. The permeability of these implants is a key parameter that influences their success, by affecting oxygen and nutrient transport and removing cellular waste products to healthy cartilage. Experimental try-and-error approaches are mostly used to optimize the composition of such structures. However, computational simulation may offer a more exhaustive tool to test and screen out biomaterials mimicking cartilage, avoiding expensive and time-consuming experimental tests. An accurate and efficient prediction of material's permeability and internal directionality and magnitude of the fluid flow could be highly useful when optimizing biomaterials design processes. Here we present a 3D computational model based on Sussman-Bathe hyperelastic material behaviour. A fluid structure analysis is performed with ADINA software, considering these materials as two phases composites where the solid part is saturated by the fluid. The model is able to simulate the behaviour of three non-biodegradable hydrogel compositions, where percentages of PEA and PHEA are varied. Specifically, the aim of this study is (i) to verify the validity of the Sussman-Bathe material model to simulate the response of the PEA-PHEA biomaterials; (ii) to predict the fluid flux and the permeability of the proposed IPN hydrogels and (iii) to study the material domains where the passage of nutrients and cellular waste products is reduced leading to an inadequate flux distribution in healthy cartilage tissue. The obtained results show how the model predicts the permeability of the PEA-PHEA hydrogels and simulates the internal behaviour of the samples and shows the distribution and quantification of fluid flux.

Cartilage dysfunction in ALS patients as side effect of motion loss: 3D mechano-electrochemical computational model

Amyotrophic lateral sclerosis (ALS) is a debilitating motor neuron disease characterized by progressive weakness, muscle atrophy, and fasciculation. This fact results in a continuous degeneration and dysfunction of articular soft tissues. Specifically, cartilage is an avascular and nonneural connective tissue that allows smooth motion in diarthrodial joints. Due to the avascular nature of cartilage tissue, cells nutrition and by-product exchange are intermittently occurring during joint motions. Reduced mobility results in a change of proteoglycan density, osmotic pressure, and permeability of the tissue. This work aims to demonstrate the abnormal cartilage deformation in progressive immobilized articular cartilage for ALS patients. For this aim a novel 3D mechano-electrochemical model based on the triphasic theory for charged hydrated soft tissues is developed. ALS patient parameters such as tissue porosity, osmotic coefficient, and fixed anions were incorporated. Considering different mobility reduction of each phase of the disease, results predicted the degree of tissue degeneration and the reduction of its capacity for deformation. The present model can be a useful tool to predict the evolution of joints in ALS patients and the necessity of including specific cartilage protectors, drugs, or maintenance physical activities as part of the symptomatic treatment in amyotrophic lateral sclerosis.

Finite element modeling of soft tissues: material models, tissue interaction and challenges

BACKGROUND

Musculoskeletal soft tissues, such as articular cartilage, ligaments, knee meniscus and intervertebral disk, have a complex structure, which provides elasticity and capability to support and distribute the body loads. Soft tissues describe an inhomogeneous and multiphasic structure, and exhibit a nonlinear, time-dependent behavior. Their mechanical response is governed by a substance composed of protein fiber-rich and proteoglycan-rich extracellular matrix and interstitial fluid. Protein fibers (e.g. collagen) give the tissue direction dependent stiffness and strength. To investigate these complex biological systems, the use of mathematical tools is well established, alone or in combination with experimental in vitro and in vivo tests. However, the development of these models poses many challenges due to the complex structure and mechanical response of soft tissues.

METHODS

Non-systematic literature review.

FINDINGS

This paper provides a summary of different modeling strategies with associated material properties, contact interactions between articulating tissues, validation and sensitivity of soft tissues with special focus on knee joint soft tissues and intervertebral disk. Furthermore, it reviews and discusses some salient clinical findings of reported finite element simulations.

INTERPRETATION

Model studies extensively contributed to the understanding of functional biomechanics of soft tissues. Models can be effectively used to elucidate clinically relevant questions. However, users should be aware of the complexity of such tissues and of the capabilities and limitations of these approaches to adequately simulate a specific in vivo or in vitro phenomenon.