Development of a computational-experimental framework for enhanced mechanical characterization and cross-species comparison of the articular cartilage superficial zone

Abstract

To provide a better understanding of the contribution of specific constituents (i.e., proteoglycan, collagen, fluid) to the mechanical behavior of the superficial zone of articular cartilage, a complex biological tissue with several time-dependent properties, a finite element model was developed. Optimization was then used to fit the model to microindentation experiments. We used this model to compare superficial zone material properties of mature human vs. immature bovine articular cartilage. Non-linearity and stiffness of the fiber-reinforced component of the model differed between human and bovine tissue. This may be due to the more complex collagen architecture in mature tissue and is of interest to investigate in future work.

INTRODUCTION:

Articular cartilage is a heterogeneous tissue with biphasic and dissipative properties that contribute to its mechanical behavior. Recently, efforts have grown to elucidate material properties and structure-function relationships in cartilage. Advancements in high-resolution mechanical testing methods, such as microindentation, have made it possible to investigate small-scale tissue properties. However, obtaining material parameters requires the application of empirical or numerical models on the resulting indentation curves, classically Oliver-Pharr (Oliver and Pharr 1992) or Hertzian contact models (Garcia et al. 2017). These models are used to obtain the reduced modulus (E*) which, while useful for gross comparisons, neglects the individual contributions of the different constituents of articular cartilage to material behavior. Furthermore, these models do not consider time-dependent properties, which are crucial to cartilage function (Han et al. 2018).

Finite element analysis (FEA) has been used to model cartilage behavior and offers a way to examine material parameters that are challenging to measure. In a well-designed model, parameters can be attributed to specific structures within the tissue, including assessments of pericellular matrix strain during applied load (Khoshgoftar et al. 2018) and changes in extracellular matrix and fixed charge density following injury (Orozco et al. 2018) or surgery (Nagel and Kelly 2013). To enhance our understanding of the mechanical behavior of the superficial zone (SFZ) of articular cartilage, we developed an FEA model of specifically the SFZ using a published cartilage material model (Görke et al. 2010). Material parameters were then optimized to load-displacement curves obtained from microindentation of immature bovine tissue samples.

We also investigated the SFZ material properties of mature human articular cartilage. While the literature contains many studies using bovine tissue for understanding of cartilage tribological properties, including our own (Yuh et al. 2021a; Yuh et al. 2021b), the extent by which these studies can be generalized to human cartilage remains unclear. It is important to distinguish between the mechanical, structural, and biochemical properties of animal cartilage compared to that of human, which can vary in content and structure of constituents such as chondrocytes, proteoglycans, and type II collagen. Therefore, it was of interest to determine how immature bovine cartilage and mature human cartilage differ in surface properties. To do this, the combined computational/experimental workflow was applied on adult human ankle cartilage, and the resulting material parameters were compared to juvenile bovine tissue.

METHODS:

A published (Görke et al. 2010) biphasic material model (Figure 1) was implemented in an FEA model of cartilage indentation within Abaqus/2019 (Dassault Systèmes). Model geometry and boundary conditions are described in Table 1. A biased mesh was used in the FEA model, with an element edge length of 1μm near the indenter and 25,005 C3D8RP elements total (Figure 2). A convergence study was used to determine the element size near the indenter, the total number of elements, and the domain size, with outputs of von Mises stress, reaction force on the indenter, and pore pressure used to determine convergence. Fiber orientation within the cartilage was set as parallel to the surface.

Figure 1:

The “overlay” concept used in the material model of the developed FEA model. The 2^nd Piola-Kirchoff Stress, T, was separated into three separate contributions (fluid, fibers, isotropic contributions), each with their own constitutive equation.

Table 1:

Geometry and boundary condition definitions for our FEA model of cartilage.

Feature	Definition
Cartilage SFZ Geometry	150×150×150 μm cube quarter symmetry model
Indenter Tip Geometry	20 μm radius 90° conospherical tip.
Boundary Condition	Bottom surface is fixed Outer surfaces allowed to move freely All surfaces have free draining boundary conditions except for those along the symmetry of the quarter model and areas in contact with the indenter

Figure 2:

A) Final Mesh of our cartilage FEA model. B) Lengths and positions of each mesh region. C) Number and types of elements used in each mesh region.

The model replicated a published (Yuh et al. 2021a; Yuh et al. 2021b) microindentation protocol. For the indentation experiments, we procured cartilage explants from two human tali (Figure 3, Gift of Hope Tissue & Organ Donor Network of Illinois, IRB-exempt, male, age:42, K/L Grade 0), and twenty bovine cartilage explants from the trochlear groove of fourteen juvenile bovine stifle joints. A 3×1 array of indents (Hysitron TI 950, Bruker Inc.) was performed, 100 μm apart, submerged in 1× PBS. Stress-relaxation was applied experimentally and simulated in the FEA model: 1) “setpoint” step: 7.5 μN load applied, 2) lift off step: tip is retracted by 750 nm, 3) measurement step: probe displaced by 8 μm into the tissue (rate: 8 μm/s), 4) 60 seconds “hold” phase, 5) retraction of the tip (8 μm/s). Force-time curves were obtained for human (N=9) and bovine (N=180) samples and averaged to generate target functions for each species. E* for experimental and simulated curves was determined using a Hertzian approach (Garcia et al. 2017; Yuh et al. 2021). The Isight Execution Engine was used to optimize four material parameters relating to structures that contribute to cartilage material behavior: collagen (fiber stiffness: C4, fiber nonlinearity: β), permeability (κ), and proteoglycan (fixed charge density: FCD).

Figure 3:

14-mm Ø, 1-mm thick human articular cartilage explants were obtained from a pair of healthy tali.

RESULTS:

Optimized FEA models for both human and bovine cartilage represented the trend of the experimental data (Figure 4). All portions of the simulated load-time curves fell within the 95% confidence intervals of the experimental data. E* were similar between the simulations (118.3 kPa human, 170.8 kPa bovine) and experiments (117.8 kPa human, 174.1 kPa bovine). Multiple parameters differed between human and bovine (Table 2), with C4, FCD, β, and κ increased in human compared to bovine.

Figure 4:

Top Row- Force vs. Time curves for bovine (blue) and human (red) cartilage. Bottom Row- Force vs. Displacement curves for bovine and human cartilage.

Table 2:

Constitutive equation and coefficient descriptions for the surface of bovine and human articular cartilage. Resulting optimized parameters (shaded) from the optimization are depicted in the far-right column of the table.

Parameter	Represents	Bovine Cartilage	Human Cartilage
α	Non-linearity of the solid matrix	0.0	0.0
C1 (MPa)	Stiffness of the solid matrix	0.0020	0.0020
D2 (MPa)	Bulk modulus of the solid matrix	0.0022	0.0022
C4	Stiffness of the fibers	0.06	0.08
β	Non-linearity of the fibers	2.11	3.50
κ (μm4 μN−1 s−1)	Permeability	430	468
FCD (meq μm−3)	Fixed Charge Density	2.97 × 10−14	3.74 × 10−14
Cext (mmol μm−3)	External Ion Concentration	1.5 × 10−13	1.5 × 10−13
ϕ	Void Ratio	3.6	3.6

DICUSSION:

In this study, we compared the material properties of the SFZ of immature bovine and mature human cartilage using microindentation and FEA, with the goal of determining whether immature bovine tissue is an appropriate experimental model for drawing conclusions relevant to mature human tissue. All optimized parameters were in the range of those reported for the SFZ (Chen et al. 2001; Fujie and Imade 2015). Overall, the shape of the indentation curve was observed to be similar between the surfaces of immature bovine and mature human cartilage. Interestingly, the non-linearity (ß) and stiffness (C4) of the fiber-reinforced component of the model differed between human and bovine cartilage. This finding may be due to the increased complexity of the collagen architecture in mature tissues and is of interest to investigate in future work.

A limitation of the presented model is the lack of viscoelasticity in the constitutive equation. Viscoelastic effects in cartilage due to the rearrangement of the collagen fibril structure is well-documented (McGann et al. 2014; Han et al. 2018). It is likely that viscoelasticity may explain the slight mismatch between simulation and experiment during initial relaxation in the force-time curve. It is important to note that increasing the number of human cartilage samples will be necessary to draw more substantial conclusions. Nevertheless, this study presents a first step in describing mechanical variations between human and bovine tissue samples at the cartilage surface.