Mathematicss is used a batch in the universe that mean common people are blissfully incognizant of. Everyone goes through the instruction system larning mathematics in the early old ages as numeration from 1 through to 100, and so subsequently on, in their teens, as trigonometry and random equations. From the instruction we gain as a child, it ‘s difficult to conceive of mathematics to be utile in many countries of employment. This is untrue. Mathematics is everyplace, working behind the scenes. One of these unidentified countries is Medicine.
Mathematic is used extensively to assist derive progresss in medical research ; from the Pharmacology of new drug mixtures to the apprehension of the mechanics of complex diseases. Mathematicians are working with medical research workers every twenty-four hours to understand how different triggers affect the human organic structure, making some signifier of mathematical equation or system to demo this, and so to make a mathematical system to foretell how inauspicious effects can be counteracted.
Immunology is a complex country of Medicine and Biology. Rather than the molecular and cellular survey, it deals with complex non-linear biological systems. Although there are an increasing figure of efforts to utilize and develop computational package, these frequently do n’t see the nature of the interactions between the assorted cellular and molecular constituents nor do they normally give much biological penetration into how the system works. Mathematical modeling is used to foretell tumour growing and malignant neoplastic disease spread, where as another subdivision of mathematics, statistics are used to construe informations collected from clinical tests.
As the rubric suggests, this paper will be concentrating on the mathematical modeling used to help research into Autoimmune Diseases.
So, what is an Autoimmune Disease? An Autoimmune disease is when the organic structure ‘s immune system becomes hyperactive or ‘confused ‘ and starts to bring forth a response against these cells. In kernel, assailing and damaging them. There are two different responses the immune system can take ; it can bring forth antigens to do direct harm to a individual organ or tissue doing a ‘localised ‘ autoimmune disease, or produces a response against multiple tissues and variety meats ensuing in a ‘systemic ‘ autoimmune disease.
Autoimmune diseases by and large do n’t merely hold remarkable symptoms, or needfully a specific trial for diagnosing, therefore it can be difficult for medical practioners to supply a diagnosing. Besides many symptoms can overlap from one disease to another ; it can be difficult for them to set up which disease the patient is enduring from.
There can be many possible triggers to trip the disease activity, and besides to find the badness. Many mean life constituents ; like the conditions, emphasis etc. , can do a wholly non-linear reaction to a sick persons immune response.
Therefore, because of the many varying symptoms and triggers, there ranges a big figure of different drug and physical therapies. Through medical literature there exists plentifulness of documents on the topic of autoimmune disease and it ‘s complications, yet there is limited theoretical work in mathematical literature. However, over the old ages, some have proposed mathematical theoretical accounts in relation to T cell inoculation, the behavior of the immune web and the activation of immune cell activation, for illustration. For the latter, there is a paper by Wodarz and Jansen ( 1993 ) look intoing the ratio of cross-presentation to direct-presentation of antigen showing cells presuming their figure is variable. However these theoretical accounts are reasonably specific to certain autoimmune tendencies.
Current apprehension of Autoimmunity is that tissue hurt causes T cell reactions and the activation of some immune cells. Consequently, this leads us to presume that patient ‘s autoimmune symptoms are based on the population size of healthy ( mark ) cells. Hence, the smaller the population size, the more terrible the symptoms. In the undermentioned theoretical account we will presume that the figure of antigen showing cells ( henceforth known as APC ‘s ) is changeless. More specifically, the Dendritic Cells ( DC ‘s ) , a type of APC, are known to transport out about all antigen presentations to T cells and the figure of these are said to be changeless in vivo.
Throughout the balance of this paper, we will look into a simple theoretical account for general autoimmune disease. In the undermentioned theoretical account we will be demoing the effects of ;
The immune system cells, C, which of course die at the rate I?
The healthy ( mark ) cells, T, which of course die at the rate I? and are
produced at a rate of I»
The damaged cells, D, which of course die at the rate I± ;
where I± E? I?
The immune system cells, C, damage the mark cells, T, at a rate proportional to their population, I?TC. Here, I? represents the efficaciousness of the procedure.
Barbarous Cycle of Autoimmunity
Before go oning onto the basic theoretical account, the rhythm of autoimmunity demands to be explained, in brief.
Fig 1. A Cycle of Autoimmunity
Dynamic Properties of autoimmune disease theoretical accounts: Tolerance, flare-up, quiescence
When Autoimmune disease is initiated by some event, an bing cell in vivo becomes a cardinal effecter cell. Then, this cardinal effecter cell onslaughts and amendss a healthy cell. Here, the protein of the damaged cell ( Antigen ) is captured by an APC ( as in our theoretical account a DC ) and the protein is shown as a self-antigen at the lymph vas. Then the immune cells which are specific to the protein are induced, and these specific immune cells once more attack and damage mark cells, get downing the rhythm once more.
Basic theoretical account
From the above background, a basic dynamical theoretical account is obtained, which is a system of non-linear differential equations. In order to simplify the theoretical account we assume that damaged cells already exist and disregard the kineticss of the cardinal effecter cells and APC ‘s. We combine the kineticss of immune cells and mark cells and obtain a theoretical account as follows:
The immune system is a batch more complicated than our simple mathematical theoretical account above based on the mark cell growing map ( g ) and the personal immune response map ( degree Fahrenheit ) , but this will be equal for the minute. includes the rates that immune cells find and win in assailing mark cells. Subsequently, we show that mark cell growing can bring on symptom outbursts and the personal immune response can bring on quiescence of the disease. We can utilize this theoretical account to assist us understand the mechanisms of drug therapies.
Target Cell Growth Function
The above mathematical theoretical account is based on a common theoretical account used to understand HIV infection. In two documents by Martin A. Nowak in 2000 and 1996 and a paper by Alan S. Perelson in 1998, sensible maps to stand for mark cell growing in worlds have been investigated.
The first is a simple equation which is merely I» , the rate at which the new mark cells are produced, minus I?T ( rate of decease of mark cells ten population size of mark cells ) which was investigated by Martin A. Nowak et Al.
The 2nd equation is somewhat more complex, investigated by Alan S. Perelson and Patrick W. Nelson, with the add-on of an excess term to take into history natural mark cell proliferation. Here, I? represents the maximal proliferation rate and L ; the mark cell population denseness at which proliferation shuts off.
However, it is noted that the above equation is non density independent. But an alternate equation has been investigated by Liancheng Wang and Micheal Y. Li in 2006 and is every bit follows:
For mathematical simpleness without altering the qualitative behavior of the system, we will presume that the first equation of is the functional signifier including denseness dependance.
Personal Immune Response Function
If we define the personal immune response map degree Fahrenheit ( D ) , the relationship between the immune cell incentive and the symptoms of autoimmune disease can be investigated. Immune response maps can change from individual to individual or they may depend on a patient ‘s status or the sort of immune cells. Immune cell incentive is considered to be a sensible map shown as:
Fig 2. Personal Immune Response
Dynamic Properties of autoimmune disease theoretical accounts: Tolerance, flare-up, quiescence
The above diagram shows APCs do non bring forth immune cells if merely a few antigens exist, but so immune cells are bit by bit induced when comparatively many antigens exist.
Henceforth, the paper will look into the behavior of two personal immune response maps ;
Linear Personal Response Function
In this subdivision, the additive personal response map and it ‘s consequence on autoimmune disease symptoms will be investigated, i.e. , where represents the mean magnitude of activation of Immune Response by APCs per damaged cell.
Here, the figure of immune cells induced by APC ‘s is relative to the figure of damaged cells, so D can be considered as the proliferation rate of immune cells by APCs.
Linear Target Cell Growth Function
To get down, we select the simplest equation for the population kineticss of mark cells ; . This gives the undermentioned simple theoretical account of autoimmune disease kineticss:
The above theoretical account is the same as a basic HIV theoretical account studied by several research workers, but merely the deductions 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 parametric quantities
Now, if is little so the damaged cells, D, vanish and the immune cells, C, are non activated therefore the mark cells, T, do non consume. However, if is big so the figure of damaged cells, D, increases triping the immune cells, C, and consuming the figure of mark 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 generative figure is defined as, which shows how many freshly damaged cells produced from one damaged cell in an person who does non hold an autoimmune disease presently. 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 statement of continues here. The chronic infection equilibrium co-ordinates, from direct computation, are given by:
Taking the derived function of and, we get ;
As additions there is a transferal from tolerance to autoimmune disease activity from the consequence of. From ague symptoms the disease transportations to a chronic stage. Here, increasing deteriorates the symptoms shown by the consequences of, and. Namely, increasing activates the immune cells, and reduces the figure of mark cells. Hence, , which represents the mean magnitude of activation of the immune response, affects the badness of the autoimmune disease onslaught and it ‘s symptoms.
Density-Dependant Target Cell growing Function
In the undermentioned subdivision the density-dependant map is used to obtain a similar theoretical account, which is besides a HIV theoretical account that has been antecedently researched, one time more merely the autoimmune disease deductions are considered.
Again, from Iwami et al. , numerical solutions of this system where found with parametric quantities.
As, mark cells reproduce themselves. The different values of determine the province of the disease ;
gives a tolerance of the immune response ensuing in no autoimmune disease,
gives a slow patterned advance of the disease ensuing in mild symptoms for the patient: The mark cells lessening bit by bit, nevertheless, in the chronic stage, there is still a comparatively high degree of mark cells.
gives repeated outbursts of an autoimmune disease.
gives a more terrible reaction than ( two ) . There is a speedy patterned advance of the disease ensuing in the rapid lessening of mark cells, and later, a low degree of mark cells in the chronic stage.
These consequences show that the values of in ( I ) and ( two ) are similar to the old consequences with the additive mark cell growing map ; hence, the higher the value of the more terrible the disease symptoms for the patient. The consequence for ( three ) behaves interestingly. It shows a periodic form for the disease symptoms, which relate to repeated outbursts of the autoimmune diseases.
The ground for this flare-up form in ( three ) can be explained as the followers ; if is comparatively little and the figure of mark cells is little, so the figure of mark cells increases because of the logistic nature of. Then there is an associated addition in damaged cells and immune cells. The immune cells attack mark cells can do a lessening in the figure of mark cells and the rhythm repetitions.
In ( four ) , mark cells can non be increased any more due to the logistic nature of, if is big. Therefore, the consequence of density-dependant mark cell growing dramatically changes the symptoms of autoimmune disease.
Mathematically, this system is more interesting than the old one, as a positive equilibrium point could be unstable. This system has been investigated by De Leenheer and Smith ( 2003 ) and besides by Iwami et Al. ( 2007 ) . It has two equilibria:
The tolerance equilibrium, ,
The chronic infection equilibrium,
The basic generative figure is defined as
De Leenheer and Smith ( 2003 ) show that if, so is globally asymptotically stable and if so is merely globally asymptotically stable under certain conditions but can besides be unstable under certain conditions.
Non-Linear Personal Response
In the subsequent subdivision, the non-linear personal immune response map, , and it ‘s relationship with the symptoms of autoimmune disease is to be investigated. In ecological population kineticss, this map is called the “ functional response ” , or more accurately a Holling type III or sigmoid functional response.
Here, I define the parametric quantities of the Functional Response ;
= maximal proliferation rate of immune cells caused by Armored personnel carriers
= figure of damaged cells at which the proliferation of immune cells is hald
of the upper limit
Therefore, the map can be regarded as the proliferation rate of immune cells by APCs.
This non-linear response map vividly changes the construction of the equation systems. Namely, both the tolerance and chronic infection equilibriums can be at the same time stable under certain conditions. Specifically, is ever stable.
Linear Target Cell Growth Function
As with the additive personal response map, we start with the additive mark cell growing map. Using this, the theoretical account alterations to the followers ;
Here is given a stableness analysis for this system. This theoretical account has three equilibria:
Henceforth, a elaborate explaination of the stableness of these equilibria is given. To happen the stableness of these equilibria, the characteristic root of a square matrixs of a jacobian matrix for each of these equilibria co-ordinates are investigated.
The expression for the jacobian matrix for this differential equation system is ;
Hence, the Jacobian matrix of the first equilibrium point is as follows ;
Consequently, the characteristic root of a square matrixs of this matrix are and. And so, is ever stable.
Traveling onto the Jacobian matrix for, the co-ordinates of which are found to be ;
The characteristic equation of this matrix is found as ;
In order to do this equation simpler, the coefficients for the indeterminate of the multinomial, s, are denoted by,
All characteristic root of a square matrixs have negative existent parts if ;
is uncluttering true, so the concluding two conditions need to be proven.
By utilizing the co-ordinates for, is the same as
Then if and merely if
which implies that,
which is interpreted as for. However, if it exists, is ever unstable.
Finally, the last status is investigated.
So, if, so and is ever stable when it exists.
Now, with, the deductions for autoimmune disease suggested by the theoretical account in this subdivision are considered. Numeric solutions are investigated with parametric quantities from a paper by Iwami et Al. : , with the initial conditions:
( I ) gives a representation of quiescence of an autoimmune disease. In the beginning, the mark cells ( T ) remain at a reasonably steady big degree, while the immune cells ( C ) remain steady at a low degree. At some point, without some evolutionary event ( i.e. without altering parametric quantities ) , the figure of mark cells all of a sudden falls to a reasonably little degree while the sum immune cells additions.