In this section we describe the mathematical model developed to represent the biological interactions of chondrocytes and cytokines described in [5
]. We aim for a minimal model based on known mechanisms considered to be the dominant factors in articular cartilage lesion abatement. By a minimal model we mean one in which the removal of any component results in behavior that is inconsistent with the typical injury responses in cartilage as discussed in [5
]. We note that the model described below can be expanded or modified to include further interactions as new results from experiment and observation dictate. In particular, further chemical pathways such as interleukin-1 β
) can easily be incorporated if necessary
. The chemical species included below are chosen for their functionality that is their action on chondrocytes during injury response. Hence they can be replaced with any other chemical species whose effects on chondrocytes are functionally the same. The chondrocyte cell states described below represent the biological actions of the cells in response to signaling by a particular cytokine during the typical injury response. These actions are assumed to be analogous to those of other cell types, for which aspects of innate immune response, such as local inflammation considered in this paper, are well established.
We consider a population of chondrocytes fixed in matrix, which is typical of articular cartilage. Sub-populations of chondrocytes are assumed to exist in different states corresponding to the chemical signals being received by the cells during injury response. Figure shows the states in which subpopulations of chondrocytes may exist. We refer to the normal state of a subpopulation of chondrocytes as the healthy state. We denote by C
the population density (cells per unit area) of healthy chondrocytes at a given time and location. As a result of inflammation and injury healthy chondrocytes can enter into a "sick" state. Cells in this state are at risk death (via apoptosis) unless their signaling by TNF-α
is somehow limited. We consider two subpopulations of cells in the sick state. We denote by ST
the population density of cells in the "catabolic" state. Catabolic cells are chondrocytes that have been signaled by alarmins and are capable of synthesizing TNF-α
and other cytokines associated with inflammation. Healthy cells signaled by DAMPs or TNF-α
enter into the catabolic state and begin to synthesize TNF-α
and produce reactive oxygen species (ROS). Catabolic cells that are signaled by TNF-α
express a receptor (EPOR) for EPO and make up the subpopulation of sick cells we refer to as EPOR active. It should be noted that there is a time delay of 8-12 hours before a cell expresses the EPO receptor after being signaled to become EPOR active [5
]. We denote the population density of EPOR active cells by SA
. Since EPOR active cells express a receptor for EPO, they may switch back to the healthy state if signaled by EPO. However, as discussed in [5
limits production of EPO. Thus there is a balance between EPO and TNF-α
that determines the spreading behavior of cartilage lesions. The catabolic and EPOR active cells together make up the population of cells forming the penumbra as illustrated in figure . We also consider a "dead" state for subpopulations of chondrocytes. This includes necrotic cells DN
and apoptotic cells DA
. We note that for the purposes considered herein that apoptotic cells do not feed back into the system and thus are not explicitly represented in the mathematical model. Due to the abrupt nature of the injury, we assume that the initial injury results in necrosis of cells at the injury site. Furthermore, we assume that cell death due to secondary cytokine-induced injury is strictly through apoptosis. The reasoning here is that necrosis is a nonspecific event that occurs in cases of severe pathological cell and tissue damage, whereas secondary cytokine-induced injury corresponds with a physiologic form of cell death used to remove cells in a more orderly and regulated fashion and there is evidence that often, this is via apoptosis [8
]. While these may be simplifying assumption, it is not entirely clear what type of cell death dominates in osteoarthritis, and there is some evidence to support our assumptions [8
]. Necrotic cells release alarmins such as DAMPs that initiate the injury response. Either catabolic or EPOR active cells become apoptotic if signaled by inflammatory cytokines.
Figure 3 States of chondrocytes during stereotypical injury response. Healthy cells (C) become sick. Typically a healthy cell will first become catabolic (ST) and either enter apoptosis (DA), or become EPOR active (SA). EPOR active cells may be saved by EPO to (more ...)
Figure details the chemical signaling and the switching of chondrocyte cell states represented in the mathematical model presented in the following. An initial injury creates a population of necrotic cells which release alarmins (such as DAMPs) [6
]. We denote by M
the concentration of DAMPs at a given time and location. The DAMPs signal healthy cells near the injury to enter the catabolic state resulting in the production of TNF-α
. We denote the concentration of TNF-α
at a given time and location by F
. The inflammatory cytokine TNF-α
has several effects on the system: It
Figure 4 Signaling involved in cartilage injury response. The occurrence of an injury begins a sequence of chemical productions that promote the inflammatory response. Lysing (necrotic) cells give off DAMPs (M) resulting in a population of catabolic cells that (more ...)
1. feeds back to continue to switch healthy cells into the catabolic state,
2. causes catabolic cells to enter the EPOR active state [5
3. influences apoptosis of catabolic and EPOR active cells,
4. degrades extracellular matrix (denoted by U) which results in increased concentrations of DAMPs,
5. has a limiting effect on production of EPO [5
Catabolic cells also produce reactive oxygen species (ROS) which influences the production of EPO by healthy cells. We denote the concentration of ROS at a given time and location by R. There is a time delay of 20-24 hours before a healthy cell signaled by ROS will begin to produce EPO.
In developing a mathematical model to represent the scenario described above we assume that the chemicals diffuse throughout a domain. This diffusion is essential in determining the spatial behavior, i.e. the expansion or abatement of the lesion. Since the chondrocytes are fixed in the matrix we do not consider cell motility. First the mathematical model contains four equations (one for each chemical species) describing the dynamics of the chemical concentrations. These equations are each of the form:
Next the model consists of four equations for the population densities of cells in the healthy, catabolic, EPOR active, and necrotic states. Each of these equations are of the form:
Finally there is an equation corresponding to the degradation of extracellular matrix by TNF-α which feeds back into the production of DAMPs.
The model equations for the concentrations of the chemical species are:
The equation describing matrix degradation by TNF-α is:
We note that the right hand side of this equation appears in (4) as part of the production of DAMPs as we have assumed that degraded matrix releases alarmins.
The equations for the switching of cell states are:
In (8) and (9) the function H(·) is given by
The constant Pc represents a critical level of EPO above which the effects of DAMPs and TNF-α on healthy cells is limited.
Based on the diagram in figure and the assumptions underlying that diagram, one could assume that the function H
(·) is identically one, i.e. H
1. However, if we take H
1 then figure implies, and computational results confirm, that there is infinite feedback into the system by alarmins, TNF-α
, and catabolic cells. As noted in [5
] "the pro-inflammatory arm of the injury response is inherently self-amplifying". This does not allow for lesion abatement or limitation of secondary pro-inflammatory cytokine induced injury described in [5
]. This suggests that signaling by anti-inflammatory cytokines such as EPO not only promote the switch from the EPOR active state to the healthy state but also influences the response of healthy cells to alarmins and pro-inflammatory cytokines. Thus the mathematical model suggests that in some way the anti-inflammatory cytokines limit the switch from the healthy state to the catabolic state so that there is not an infinite feedback into the system by alarmins, TNF-α
, and catabolic cells. This is consistent with the observation of Brines and Cerami on the antagonistic relationship between TNF-α
and EPO, that "each is capable of suppressing the biological activity of the other" [5
]. In the model equations (3-11) we take the function H
(·) as in (12) for convenience. However, there may be a more appropriate form for the function H
(·) that must be determined through experiment to discover the nature of the effects that sufficient concentrations of EPO or other anti-inflammatory cytokines have on healthy chondrocytes.
We note some features of the mathematical model (equations (3-11)):
1. we have incorporated the time delays for activation of the EPO receptor and synthesis of EPO (6),(9),(10),
2. it requires a concentration of TNF-α and DAMPs together for apoptosis of catabolic cells (see (9)),
3. after some time necrotic cells decay to an inert state (see (11)) so that there is not a continuous production of DAMPs for all time from the initial injury. This also corresponds to the loss of cartilage such as is sometimes associated with osteoarthritis.
The specific functional forms appearing in the model system (equations (3-11)) have been chosen to capture the critical thresholds and represent the function shapes qualitatively.