Edited by: Ridha Hambli, Polytechnique Orleans, France
Reviewed by: Massimiliano Zingales, Università degli Studi di Palermo, Italy; Salah Ramtani, Université Paris 13, France
This article was submitted to Biomechanics, a section of the journal Frontiers in Bioengineering and Biotechnology.
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Traumatic injuries of articular cartilage result in the formation of a cartilage lesion and contribute to cartilage degeneration and the risk of osteoarthritis (OA). A better understanding of the framework for the formation of a cartilage lesion formation would be helpful in therapy development. Toward this end, we present an age and space-structured model of articular cartilage lesion formation after a single blunt impact. This model modifies the reaction-diffusion-delay models in Graham et al. (
The degenerative joint disease known as osteoarthritis (OA) is among the most common causes of disability worldwide. While OA involves multiple joint tissues including bone, tendons, ligaments and synovium, articular cartilage degeneration, and erosion is the proximal cause of loss of joint function. Articular cartilage is a thin layer of connective tissue that covers the ends of long bones in synovial joints such as the shoulder, hip, knee, and ankle, where it distribute mechanical loads and allows for smooth joint motion. These functions are attributable to the unique composition and structure of cartilage extracellular matrix (ECM), which consists of water (>70%), proteoglycan (15%), and collagen (15%) (Martin et al.,
Even though articular cartilage is only 1–3 mm thick, it has four distinct zones (superficial, transitional, radial, calcified). Different zones have different cell morphologies, matrix composition, and collagen fibril properties. Compared to other zones, the superficial zone is specialized to resist tensile stresses and minimize surface friction. In this paper, we focus on the properties of the superficial zone. We assume the solid cartilage matrix to be a homogeneous system, so that the chemical and cell properties remain the same inside the space. Cartilage cells, known as chondrocytes, are distributed sparsely within the tissue (104 cells/mm3). They are largely trapped inside the ECM, so there is no appreciable cell motility. Chondrocytes are solely responsible for the maintenance of the cartilage matrix, and engage in complex biochemical signaling/regulation via the synthesis and release or recognition of signaling molecules such as cytokines and oxidants, among others. We classify chondrocytes to be in different states in our model with respect to the chemical signaling processes. In addition to synthesizing ECM components, chondrocytes also release matrix proteases that cause matrix degradation.
There are various ways to cause traumatic articular cartilage injury, but they all share high loading rates and a high peak stress amplitude, which initiates the damage (lesion). The damage done by injuries seldom heals spontaneously and often leads to the progressive cartilage degeneration characteristic of post-traumatic osteoarthritis (PTOA). The strain in the superficial zone under a single blunt impact can easily exceed 40%, and when combined with the high loading rate and excessive stress, is lethal to chondrocytes and detrimental to the ECM. Chondrocyte death inside the superficial zone can be assumed to be directly caused by this impact. The main focus of most therapies for PTOA has been to prevent chondrocyte death and dysregulation. The chondrocyte depletion and ECM degradation process is illustrated in Figure
Pro-inflammatory cytokines are the main reason for cell apoptosis. Moreover, the pro-inflammatory cytokines can cause severe aggrecan depletion, which leads to loss of strength and elevated strain in affected cartilage. Even though it is quite limited, chondrocytes still have some self-repair ability. Anti-inflammatory cytokines such as erythropoietin (EPO) can antagonize the effect of pro-inflammatory cytokines and result in reduced cell apoptosis and ECM degradation. The “balancing act” (Graham et al.,
This paper is organized as follows. We describe the mathematical models and numerical methods used to solve the model equations. We then describe the materials and methods used for the experimental validation of our computational results. We then present the computational results and the experimental validation. We finish with a discussion of these results.
In this section, we describe the age- and space-structured model developed for the inflammatory response after a single blunt impact injury. The cartilage lesion caused by a single severe traumatic event was described in a reaction-diffusion-delay model by Graham et al. (
We assume circular symmetry so that the system can be reduced to a one-dimensional model with respect to space. The components of the system depend on radius (
There are two main categories of components in our mathematical model, cells and chemicals. A schematic of the system is presented in Figure
Let
We assume negligible motility on the part of chondrocytes, so there are no diffusion terms for the cells equations. The different cell states correspond to the different chemical signals. The injury kills cells inside the impact area by rendering them necrotic (
The chemical components are
signals healthy cells ( signal catabolic cells ( cause both catabolic and EPOR-active cells to become apoptotic, degrade the extracellular matrix, which in turn increases the level of DAMPs, resulting in further damage of the system. The degradation of ECM is a slow and complex process. However, we assume for mathematical convenience that inflammatory cytokines directly damage ECM, limit the production of EPO.
We assume that the chemicals diffuse throughout the whole region. The diffusion coefficients were estimated by Graham et al. (
In addition to the chemicals, we track the extracellular matrix density:
The equations for the chemical components of our system are
The initial and boundary conditions are
Our chemical equations are similar to those in Graham et al. (
The ECM is assumed to be degraded by inflammatory cytokines such as IL-6, measured in terms of decreased proteoglycan concentration in the matrix. When ECM is intact, the sulfate groups are kept inside the ECM. The release of sulfate groups is an indication of ECM degradation, which can be estimated by the decrease in concentration of SO4. The average concentration of SO4 in normal undamaged cartilage is 30 g/L (Farndale et al.,
We define the Heaviside function,
The equation for the ECM dynamics is
A substantial difference between our model and Graham et al. (
The equation for the population density of healthy cells not yet signaled by ROS is
The equation for the population density of healthy cells signaled by ROS and in the process of becoming catabolic is
The equation for the population density of healthy cells signaled by ROS and producing EPO is
The equations for the population density of sick cells in the catabolic state is
The equation for the population density of EPOR-active sick cells is
The equation for the necrotic cell population is
with initial condition
The computational methods used to solve the age- and space-structured differential equations were developed and analyzed in Ayati (
The relative errors obtained in our computational convergence studies of the numerical results are given in Table
Variable number of age intervals | 100 | 200 | 400 | 800 |
---|---|---|---|---|
Healthy normal ( |
4.1060E-04 | 2.2420E-05 | 2.8220E-06 | 9.5317E-07 |
Healthy pre-catabolic ( |
1.0342E-03 | 1.1883E-04 | 8.5328E-05 | 1.0001E-04 |
Healthy EPO producing ( |
1.4318E-03 | 7.7614E-05 | 1.0574E-05 | 3.1481E-06 |
Catabolic ( |
1.6966E-03 | 1.8716E-04 | 4.1704E-05 | 6.8092E-05 |
EPOR-active ( |
1.7993E-03 | 1.0926E-04 | 1.2838E-05 | 4.6493E-06 |
ECM ( |
5.5031E-07 | 6.9499E-08 | 2.8295E-08 | 1.5726E-08 |
IL-6 ( |
8.5298E-04 | 7.1301E-05 | 1.5445E-05 | 5.9705E-06 |
EPO ( |
1.8318E-03 | 1.1066E-04 | 1.9243E-05 | 4.6840E-06 |
DAMPs ( |
1.8736E-03 | 8.5046E-05 | 2.8458E-05 | 6.3479E-06 |
ROS ( |
2.1849E-03 | 5.2076E-04 | 1.2122E-04 | 3.0536E-05 |
Osteochondral explants were surgically excised from bovine lateral tibial plateaus (25 mm × 25 mm × 10 mm). All specimens were allowed to equilibrate for two days in culture media in a low oxygen environment (5% O2, 5% CO2). After equilibration (day 0), explants either underwent a sham impact procedure or were impacted with a 5.5 mm rounded edge indenter from a drop tower, imparting an energy of 2.18 J/cm2. Unimpacted explants were removed on day 0, preserved and embedded in paraffin. Impacted explants were placed back in fresh culture media, and removed at day 1, 7, or 14 for preservation and paraffin embedding. Culture media was changed every two days for explants cultured out to day 7 or 14.
Paraffin embedded cartilage explants were cut into 5 μm thick sections on glass slides to be processed for immunohistochemical detection of interleukin-6 (IL-6), erythropoietin (EPO), or the erythropoietin receptor (EPOR). All cartilage sections for each stain were processed in batch at the same time under identical conditions. The sections were deparaffinized, quenched of endogenous peroxidase activity, and then underwent antigen retrieval using a 0.01 M citric acid solution for 4 h at 65°C. Non-specific antigen binding was then blocked for 1 h with PBS containing 10% goat serum, 1% bovine serum albumin (w/v), and 0.1% tween 20. This blocking solution was then used to dilute the primary antibodies against IL-6 (Abcam ab193799), EPO (Abcam ab65394), and the EPOR (Abcam ab83696) at a ratio of 1:100 from their starting concentration. The solution containing the primary antibody was incubated on the slides overnight at 4°C. The slides were then rinsed and blocking solution applied for 30 min at room temperature. A biotinylated secondary antibody against the primary antibody was then incubated on the slides at a 1:250 dilution (Vector Labs). The slides were then rinsed and incubated with the VECTASTAIN® ABC Reagent (Vector Labs) for 30 min at room temperature. After another rinsing, DAB (3, 3-diaminobenzidine) HRP substrate solution (Vector Labs DAB substrate kit) was incubated on the sections for 2–10 min (depending on primary antibody). The sections were then rinsed for a final time, counter stained with eosin, dehydrated, and a glass coverslip was attached and sealed with permount (Fisher).
Slides were then digitally scanned on an Olympus VS110. The Olympus software controlled the slide stage in all axes and automatically focused/captured high resolution images (322 nm/pixel) which were then tiled together, resulting in full explant immunohistochemically labeled images. The resolution of the resulting images was then reduced to 20% of their original size and exported as tiff images for analysis.
To obtain the validation data, we analyzed 12 cartilage slices. The slices were separated into six categories with two slices per category: EPO at days 1, 7, 14; and IL-6 at days 1, 7, 14. The length of each slice is around 1″ (2.5 cm approximately). The sham impact was placed in the middle of each cartilage slice and formed a dent (see Figure
For image processing, we split each slice into two half-slices, divided at the point of impact. From these we could obtain four measurements of each of EPO and IL-6 at each radial value. We further subdivided each half-slice into pieces approximately 0.15 cm long; the pieces correspond roughly to the radius intervals [0, 0.15] cm, [0.15, 0.3] cm, [0.3, 0.45] cm, [0.45, 0.6] cm, [0.6, 0.75] cm, and [0.75, 0.9] cm.
We cropped an 800 × 800 pixel sample image from each piece and estimated the cell numbers in each sample. Figure
We simulate the change of the chondrocyte population densities over a 14-day period after an initial cartilage injury in the center of a disk with a radius of 2.5 cm. The radius of the area of impact is 0.25 cm. We assume the initial cells density is 100,000 cells/cm2.
The parameters used for the simulations are shown in Table
Parameter | Value | Units | Source |
---|---|---|---|
0.1 | Graham et al. ( |
||
0.05 | Graham et al. ( |
||
0.005 | Graham et al. ( |
||
0.05 | Graham et al. ( |
||
δ |
60 | Wang et al. ( |
|
δ |
0.5545 | Ito et al. ( |
|
δ |
0.5545 | Wedlock et al. ( |
|
δ |
3.326 | Brines and Cerami ( |
|
δ |
0.0193 | Lu et al. ( |
|
σ |
0.0024 | Zhou et al. ( |
|
σ |
5.17 × 10−7 | Terada et al. ( |
|
σ |
2.35 × 10−7 | Terada et al. ( |
|
σ |
4.2 × 10−5 | Brines and Cerami ( |
|
σ |
0.0154 | Lu et al. ( |
|
Λ | 0.5 | nanomolar | Approximated |
λ |
5 | nanomolar | Approximated |
λ |
0.5 | nanomolar | Approximated |
λ |
0.5 | nanomolar | Approximated |
λ |
0.5 | nanomolar | Approximated |
α1 | 1 | Approximated | |
α2 | 1 | Approximated | |
β11 | 100 | Approximated | |
β12 | 50 | Approximated | |
β13 | 10 | Approximated | |
κ1 | 10 | Approximated | |
κ2 | 10 | Approximated | |
1 | Brines and Cerami ( |
||
0.5 | Approximated | ||
0.1 | Approximated | ||
0.05 | Approximated | ||
0.5 | days | Graham et al. ( |
|
1 | days | Graham et al. ( |
When we initiate the simulation at
Figure
Simulation results show that after the initial injury, the cells adjacent to the impact area quickly sense the release of ROS and begin to switch states. At the same time, DAMPs released by necrotic cells start to trigger the pre-catabolic cell population (
In the simulation period of 14 days, the decrease of ECM density is quite small. This is because ECM degradation is a much slower process than the apoptosis of cells.
Figures
The main competition between pro- and anti-inflammatory cytokines results in a steady increase of IL-6 across the cartilage, whereas EPO is more heavily concentrated just outside the penumbra of the inflamed region. The net result is a slowing, but not full cessation, of the spread of the inflammation.
We remark that in the simulations presented, we never reach the threshold
We examined the approximated parameters in Table for λ for λ for λ for λ for α1, all variables, except for EPOR-active cells ( for β11, all healthy cells ( for β12, all healthy cells ( for β13, normal healthy cells ( for κ1, healthy pre-catabolic cells ( for κ2, healthy pre-catabolic cells ( for for for
Parameter | Base value | Perturbed values | Interval |
---|---|---|---|
λ |
5 | {1, 3, 7, 9} | [3, 7] |
λ |
0.5 | {0.1, 0.3, 0.7, 0.9} | [0.3, 0.7] |
λ |
0.5 | {0.1, 0.3, 0.7, 0.9} | [0.3, 0.7] |
λ |
0.5 | {0.1, 0.3, 0.7, 0.9} | [0.3, 0.9] |
α1 | 1 | {0.1, 0.5, 1.5, 2} | [0.5, 2] |
β11 | 100 | {50, 75, 125, 150} | [60, 70] |
β12 | 50 | {40, 45, 55, 60} | [40, 60] |
β13 | 10 | {1, 5, 15, 20} | [5, 15] |
κ1 | 10 | {1, 5, 15, 20} | [5, 20] |
κ2 | 10 | {1, 5, 15, 20} | [5, 20] |
0.5 | {0.1, 0.3, 0.7, 0.9} | [0.1, 0.9] | |
0.1 | {0.01, 0.05, 0.15, 0.2} | [0.01, 0.2] | |
0.05 | {0.01, 0.03, 0.07, 0.09} | [0.01, 0.09] |
The most sensitive parameters in the system are λ
The sequence and distributions of the cell states match what is expected from past and current understanding of the “penumbra” around a lesion, in which an initial burst of pro-inflammatory cytokine expression is eventually attenuated by an increase in anti-inflammatory cytokine expression. Our major simulation result predicting this progression (Figure
We bring attention to some specific features of the validation result. Figure
We clarify that our current experimental techniques cannot extract precise concentration values for cytokines throughout the cartilage matrix. Instead, they measure the numbers of chondrocytes expressing each cytokine at a level detectable by immunohistochemistry (“positive cells”). Figure
The immunohistochemistry analysis indicates relative rises and declines in the amounts of cytokine present in and around individual cells where they may act in autocrine or paracrine fashion. Thus, the immunohistochemical approach gives a rough estimation of the distribution of bioavailable cytokines. In this sense, we see agreement with our predictions: the broad, nearly uniform, distributions of IL-6, peaking near day 7; and the more unimodal distributions of EPO, increasing steadily.
We present an age- and space-structured model to simulate the development of an articular cartilage lesion after a single blunt impact. We simplified the model to a radially symmetric geometry. Based on previous experimental findings, we hypothesized that the consequences of mechanical trauma to cartilage depend on the balance between competing chondrolytic and chondroprotective responses of local chondrocytes. Whether damaged cartilage is stabilized, or begins down the path to progressive degeneration, has been regarded as a matter of great biologic complexity. However, our mathematical simulation confirms that the complex behavior of this system can be modeled using just two cytokines with opposing pro- and anti-inflammatory activities. The post-injury pattern of expression for IL-6 and EPO indicated by immunohistology suggests they may play roles in this binary system.
Our model did not predict runaway degeneration after a simulated impact injury. This result is appropriate and matches bench experiments showing that explanted cartilage recovers from impacts of similar magnitude, which cause minimal structural damage. Structurally damaging impacts deserve attention, but will require consideration of spatial heterogeneity in the impact site, which is beyond the scope of the current work. In addition, extrinsic factors that are sure to influence cartilage recovery
In summary, although our simulation results were obtained using a relatively small number of approximated parameters, our calculations provide useful predictions of the formation of cartilage lesions after a single blunt impact. Although we limited simulations to only 2 weeks in this work, it is possible to predict outcomes over the much longer time frames. Thus, our
XW and BA developed the mathematical model and software used to solve the model equations. XW conducted the numerical simulations. MB conducted the immunohistochemistry, and slide preparation and scanning for the validation experiments. MB and JM designed the validation experiments. XW, MB, BA, and JM contributed to the writing and editing of the manuscript.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Anneliese D. Heiner conducted the tissue culture experiments that generated the sample set used for model validation.