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Mathematics is used a lot in the world that average folk are blissfully unaware of. Everyone goes through the education system learning mathematics in the early years as counting from 1 through to 100, and then later on, in their teens, as trigonometry and random equations. From the education we gain as a youngster, it's hard to imagine mathematics to be useful in many areas of employment. This is untrue. Mathematics is everywhere, working behind the scenes. One of these unidentified areas is Medicine.
Mathematic is used extensively to help gain advances in medical research; from the Pharmacology of new drug concoctions to the understanding of the mechanics of complex diseases. Mathematicians are working with medical researchers every day to understand how different triggers affect the human body, creating some form of mathematical equation or system to show this, and then to create a mathematical system to predict how adverse effects can be counteracted.
Immunology is a complex area of Medicine and Biology. Rather than the molecular and cellular study, it deals with complex non-linear biological systems. Although there are an increasing number of attempts to use and develop computational software, these often don't consider the nature of the interactions between the various cellular and molecular components nor do they usually give much biological insight into how the system works. Mathematical modelling is used to predict tumour growth and cancer spread, where as another branch of mathematics, statistics are used to interpret data collected from clinical trials.
As the title suggests, this paper will be concentrating on the mathematical modelling used to aid research into Autoimmune Diseases.
So, what is an Autoimmune Disease? An Autoimmune disease is when the body's immune system becomes overactive or 'confused' and starts to produce a response against these cells. In essence, attacking and damaging them. There are two different responses the immune system can take; it can produce antigens to cause direct damage to a single organ or tissue causing a 'localised' autoimmune disease, or produces a response against multiple tissues and organs resulting in a 'systemic' autoimmune disease.
Autoimmune diseases generally don't just have singular symptoms, or necessarily a specific test for diagnosis, therefore it can be hard for medical practioners to provide a diagnosis. Also many symptoms can overlap from one disease to another; it can be hard for them to establish which disease the patient is suffering from.
There can be many potential triggers to activate the disease activity, and also to determine the severity. Many average life components; like the weather, stress etc., can cause a completely non-linear reaction to a sufferers immune response.
Therefore, because of the many varying symptoms and triggers, there ranges a large number of different drug and physical therapies. Through medical literature there exists plenty of papers on the subject of autoimmune disease and it's complications, yet there is limited theoretical work in mathematical literature. However, over the years, some have proposed mathematical models in relation to T cell vaccination, the behaviour of the immune network and the activation of immune cell activation, for example. For the latter, there is a paper by Wodarz and Jansen (1993) investigating the ratio of cross-presentation to direct-presentation of antigen presenting cells assuming their number is variable. However these models are fairly specific to certain autoimmune trends.
Current understanding of Autoimmunity is that tissue injury causes T cell reactions and the activation of some immune cells. Consequently, this leads us to assume that patient's autoimmune symptoms are based on the population size of healthy (target) cells. Hence, the smaller the population size, the more severe the symptoms. In the following model we will assume that the number of antigen presenting cells (henceforth known as APC's) is constant. More specifically, the Dendritic Cells (DC's), a type of APC, are known to carry out almost all antigen presentations to T cells and the number of these are said to be constant in vivo.
Throughout the remainder of this paper, we will investigate a simple model for general autoimmune disease. In the following model we will be showing the effects of;
The immune system cells, C , which naturally die at the rate Î³
The healthy (target) cells, T , which naturally die at the rate Î¼ and are
produced at a rate of Î»
The damaged cells, D , which naturally die at the rate Î± ;
where Î± Ëƒ Î¼
The immune system cells, C, damage the target cells, T, at a rate proportional to their population, Î²TC. Here, Î² represents the efficacy of the process.
Vicious Cycle of Autoimmunity
Before continuing onto the basic model, the cycle of autoimmunity needs to be explained, in brief.
Fig 1. A Cycle of Autoimmunity
Dynamical Properties of autoimmune disease models: Tolerance, flare-up, dormancy
When Autoimmune disease is initiated by some event, an existing cell in vivo becomes a key effector cell. Then, this key effector cell attacks and damages a healthy cell. Here, the protein of the damaged cell (Antigen) is captured by an APC (as in our model a DC) and the protein is shown as a self-antigen at the lymph vessel. Then the immune cells which are specific to the protein are induced, and these specific immune cells again attack and damage target cells, starting the cycle again.
From the above background, a basic dynamical model is obtained, which is a system of non-linear differential equations. In order to simplify the model we assume that damaged cells already exist and disregard the dynamics of the key effector cells and APC's. We combine the dynamics of immune cells and target cells and obtain a model as follows:
The immune system is a lot more complicated than our simple mathematical model above based on the target cell growth function (g) and the personal immune response function (f), but this will be adequate for the moment. includes the rates that immune cells find and succeed in attacking target cells. Later, we show that target cell growth can induce symptom flare-ups and the personal immune response can induce dormancy of the disease. We can use this model to help us understand the mechanisms of drug therapies.
Target Cell Growth Function
The above mathematical model is based on a common model used to understand HIV infection. In two papers by Martin A. Nowak in 2000 and 1996 and a paper by Alan S. Perelson in 1998, reasonable functions to represent target cell growth in humans have been investigated.
The first is a simple equation which is just Î», the rate at which the new target cells are produced, minus Î¼T (rate of death of target cells x population size of target cells) which was investigated by Martin A. Nowak et al.
The second equation is slightly more complex, investigated by Alan S. Perelson and Patrick W. Nelson, with the addition of an extra term to take into account natural target cell proliferation. Here, Ï represents the maximum proliferation rate and L; the target cell population density at which proliferation shuts off.
However, it is noted that the above equation is not density independent. But an alternative equation has been investigated by Liancheng Wang and Micheal Y. Li in 2006 and is as follows:
For mathematical simplicity without changing the qualitative behaviour of the system, we will assume that the first equation of is the functional form including density dependence.
Personal Immune Response Function
If we define the personal immune response function f(D), the relationship between the immune cell inducement and the symptoms of autoimmune disease can be investigated. Immune response functions can vary from person to person or they may depend on a patient's condition or the kind of immune cells. Immune cell inducement is considered to be a reasonable function shown as:
Fig 2. Personal Immune Response
Dynamical Properties of autoimmune disease models: Tolerance, flare-up, dormancy
The above diagram shows APCs do not produce immune cells if only a few antigens exist, but then immune cells are gradually induced when relatively many antigens exist.
Henceforth, the paper will investigate the behaviour of two personal immune response functions;
Linear Personal Response Function
In this section, the linear personal response function and it's effect on autoimmune disease symptoms will be investigated, i.e. , where represents the average magnitude of activation of Immune Response by APCs per damaged cell.
Here, the number of immune cells induced by APC's is proportional to the number of damaged cells, so D can be considered as the proliferation rate of immune cells by APCs.
Linear Target Cell Growth Function
To start, we select the simplest equation for the population dynamics of target cells; . This gives the following simple model of autoimmune disease dynamics:
The above model is the same as a basic HIV model studied by several researchers, but only the implications for autoimmune disease will be explored in this paper. In a paper by Iwami et al., numerical solutions were found for this system of differential equations with parameters
Now, if is small then the damaged cells, D, vanish and the immune cells, C, are not activated hence the target cells, T, do not deplete. However, if is large then the number of damaged cells, D, increases activating the immune cells, C, and depleting the number of target cells, T. Hence, the larger is, the more likely a patient is to develop an autoimmune disease.
This system has two equilibria:
The tolerance equilibrium, , where ,
The chronic infection equilibrium,
Next, the basic reproductive number is defined as, which shows how many newly damaged cells produced from one damaged cell in an individual who does not have an autoimmune disease currently. It has been shown that is globally asymptotically stable, if , by De Leenheer and Smith (2003). If , under certain conditions is globally asymptotically stable.
The argument of continues here. The chronic infection equilibrium coordinates, from direct calculation, are given by:
Taking the derivative of and , we get;
As increases there is a transferal from tolerance to autoimmune disease activity from the result of . From acute symptoms the disease transfers to a chronic phase. Here, increasing deteriorates the symptoms shown by the results of , and . Namely, increasing activates the immune cells, and reduces the number of target cells. Hence, , which represents the average magnitude of activation of the immune response, affects the severity of the autoimmune disease attack and it's symptoms.
Density-Dependant Target Cell growth Function
In the following section the density-dependant function is used to obtain a similar model, which is also a HIV model that has been previously researched, once more only the autoimmune disease implications are considered.
Again, from Iwami et al., numerical solutions of this system where found with parameters .
As , target cells reproduce themselves. The different values of determine the state of the disease;
gives a tolerance of the immune response resulting in no autoimmune disease,
gives a slow progression of the disease resulting in mild symptoms for the patient: The target cells decrease gradually, however, in the chronic phase, there is still a relatively high level of target cells.
gives repeated flare-ups of an autoimmune disease.
gives a more severe reaction than (ii). There is a quick progression of the disease resulting in the rapid decrease of target cells, and subsequently, a low level of target cells in the chronic phase.
These results show that the values of in (i) and (ii) are similar to the previous results with the linear target cell growth function ; hence, the higher the value of the more severe the disease symptoms for the patient. The result for (iii) behaves interestingly. It shows a periodic pattern for the disease symptoms, which relate to repeated flare-ups of the autoimmune diseases.
The reason for this flare-up pattern in (iii) can be explained as the following; if is relatively small and the number of target cells is small, then the number of target cells increases because of the logistic nature of . Then there is an associated increase in damaged cells and immune cells. The immune cells attack target cells can cause a decrease in the number of target cells and the cycle repeats.
In (iv), target cells cannot be increased any more due to the logistic nature of , if is large. Therefore, the effect of density-dependant target cell growth dramatically changes the symptoms of autoimmune disease.
Mathematically, this system is more interesting than the previous one, as a positive equilibrium point could be unstable. This system has been investigated by De Leenheer and Smith (2003) and also by Iwami et al. (2007). It has two equilibria:
The tolerance equilibrium, ,
The chronic infection equilibrium,
The basic reproductive number is defined as
De Leenheer and Smith (2003) show that if , then is globally asymptotically stable and if then is only globally asymptotically stable under certain conditions but can also be unstable under certain conditions.
Non-Linear Personal Response
In the subsequent section, the non-linear personal immune response function, , and it's relationship with the symptoms of autoimmune disease is to be investigated. In ecological population dynamics, this function is called the "functional response", or more accurately a Holling type III or sigmoid functional response.
Here, I define the parameters of the Functional Response;
= maximum proliferation rate of immune cells caused by APCs
= number of damaged cells at which the proliferation of immune cells is hald
of the maximum
Thus, the function can be regarded as the proliferation rate of immune cells by APCs.
This non-linear response function vividly changes the structure of the equation systems. Namely, both the tolerance and chronic infection equilibriums can be simultaneously stable under certain conditions. Specifically, is always stable.
Linear Target Cell Growth Function
As with the linear personal response function, we start with the linear target cell growth function . Using this, the model changes to the following;
Here is given a stability analysis for this system. This model has three equilibria:
Henceforth, a detailed explaination of the stability of these equilibria is given. To find the stability of these equilibria, the eigenvalues of a jacobian matrix for each of these equilibria coordinates are investigated.
The formula for the jacobian matrix for this differential equation system is;
Hence, the Jacobian matrix of the first equilibrium point is as follows;
Consequently, the eigenvalues of this matrix are and . And so, is always stable.
Moving onto the Jacobian matrix for , the coordinates of which are found to be;
The characteristic equation of this matrix is found as;
In order to make this equation simpler, the coefficients for the indeterminate of the polynomial, s, are denoted by,
All eigenvalues have negative real parts if;
is clearing true, so the final two conditions need to be proven.
By using the coordinates for , is the same as
Then if and only if
which implies that,
which is interpreted as for . However, if it exists, is always unstable.
Finally, the last condition is investigated.
So, if , then and is always stable when it exists.
Now, with , the implications for autoimmune disease suggested by the model in this section are considered. Numerical solutions are investigated with parameters from a paper by Iwami et al.: , with the initial conditions:
(i) gives a representation of dormancy of an autoimmune disease. In the beginning, the target cells (T) remain at a fairly steady large level, while the immune cells (C) remain steady at a low level. At some point, without some evolutionary event (i.e. without changing parameters), the number of target cells suddenly falls to a fairly small level while the amount immune cells increases.