A review of the combination of experimental measurements and fibril-reinforced modeling for investigation of articular cartilage and chondrocyte response to loading

Abstract

The function of articular cartilage depends on its structure and composition, sensitively impaired in disease (e.g. osteoarthritis, OA). Responses of chondrocytes to tissue loading are modulated by the structure. Altered cell responses as an effect of OA may regulate cartilage mechanotransduction and cell biosynthesis. To be able to evaluate cell responses and factors affecting the onset and progression of OA, local tissue and cell stresses and strains in cartilage need to be characterized. This is extremely challenging with the presently available experimental techniques and therefore computational modeling is required. Modern models of articular cartilage are inhomogeneous and anisotropic, and they include many aspects of the real tissue structure and composition. In this paper, we provide an overview of the computational applications that have been developed for modeling the mechanics of articular cartilage at the tissue and cellular level. We concentrate on the use of fibril-reinforced models of cartilage. Furthermore, we introduce practical considerations for modeling applications, including also experimental tests that can be combined with the modeling approach. At the end, we discuss the prospects for patient-specific models when aiming to use finite element modeling analysis and evaluation of articular cartilage function, cellular responses, failure points, OA progression, and rehabilitation.

Figure 1

Presentation of the collagen network organization in maturing articular cartilage based on articular cartilage from rabbit, pig, and sheep [–4]. When maturing, a nonorganized collagen fibril network slowly forms into a traditional Benninghoff-type arcade structure. At the same time, cartilage thickness is decreased [1, 2, 4, 15].

Figure 2

Fibril orientation in the vector-based fibril reinforced models can be implemented with any given structure. The principle is that the fibril vector is given a direction at each point in the model (a). Two examples of the implemented structure are presented [16]. On the left, (b) an articular cartilage sample is modeled in unconfined compression geometry with a typical Benninghoff-type arcade structure including the superficial, middle, and deep layer. On the right, (c) a submodel of the global model presented on the left is implemented with pericellular matrix fibrils tangential to the cell surface. The extracellular matrix fibril vectors are not presented in (c) for clarity. The axis of symmetry has been indicated.

Figure 3

Multiscale modeling allows for simulation of macroscale effects on microscale. In this example, a macromodel (global model, on the right) was used to simulate knee joint function and the effects on a single chondrocyte in femoral cartilage were simulated in microscale using submodeling (on the left). The transient boundary conditions were driven by the global model.

Figure 4

Convergence of an objective function (mean squared error, MSE) during the optimization of a 2-step stress-relaxation experiment [17]. (a) With initial parameters MSE was 17.73% and after optimization it was 0.96%. Five model parameters were optimized using a multidimensional minimization routine (the Nelder-Mead simplex method). One local minimum is indicated with a grey circle. (b) Normalized parameter values after each iteration with respect to the initial values. Less than 0.5% change between iterations in any of the optimized parameters was observed at the end of optimization, after the solution had converged.

Figure 5

Convergence test for finite element mesh. As the number of elements is increased in an inhomogeneous finite element model, the simulation outcome begins to converge towards true value (i.e., case with infinite number of elements). In this case, the finite element mesh was homogenous and the number of elements was systematically increased to observe convergence in the simulated peak reaction force during unconfined compression experiment (a). To optimize the performance of the model, an appropriate amount of elements is required in the model to obtain reliable results (within 5% error from an excessive amount of elements). If an excessive number of elements are used, the computational cost (CPU time) increases, and the model performance suffers. In this demonstration, the model used was a fibril reinforced poroviscoelastic model [17, 18]. In the model, collagen fibrils are implemented with a Benninghoff-type arcade structure. Therefore, to observe the effect of the bending of the fibrils, it is essential to have a sufficient amount of elements. Two finite element meshes with the fibril orientations are shown (b).

Figure 6

To demonstrate the role of experimental testing in optimization, we present a stress-relaxation experiment with 10 steps (2%-strain/step). We optimized 2 models (elastic and inhomogeneous fibril reinforced poroelastic, FRPE) to that test. The elastic model was fitted to all steps simultaneously by minimizing the mean squared error while the FRPE was fitted to 2 steps and 10 steps. The elastic model was unable to predict the data from the stress-relaxation curve, while the FRPE model agreed better with the experimental data. However, when the FRPE model parameters were optimized for 2-step data, the predicted data in the following 8 steps did not agree well with the experimental data. Instead, when the model was fitted to all 10 steps, a good agreement was achieved.

Figure 7

(a) The measurement setup of osmotic loading experiment of cells within intact cartilage tissue, a representative confocal microscopic image and a 3D presentation of a cell used for cell volume and morphology analysis. (b) Chondrocyte volume change in osmotically or mechanically loaded intact cartilage, cartilage explant, collagenase-degraded cartilage, and osteoarthritic cartilage (as a result of anterior cruciate transection). Experimentally determined cell volume change in osmotically challenged intact cartilage tissue is compared with a computational, microscale, fibril reinforced model (<1 ~ volume decrease, >1 ~ volume increase).

Figure 8

Subject-specific joint geometry can be imaged using, for example, MRI, from which using contrast agents, imaging protocols and image-analysis techniques structural and compositional details of articular cartilage can be measured and implemented into a finite element model. Using human movement analysis, the loading conditions can be determined realistically for individual subjects. Combining imaging, movement analysis, and finite element methods, realistic joint stresses and strains can be evaluated and effects on cells and matrix adaptation predicted using validated theories. Such analysis will aid in understanding and predicting advancement and onset of joint injuries and osteoarthritis.