Stochastic modelling of lymphocyte dynamics and interactions

The immune system protects the body against invading pathogens. For an immune response to occur, a T cell must encounter a rare antigen-presenting-cell (APC) presenting its cognate antigen. The time it takes for this encounter depends upon how quickly the T cell is moving, as well as how many APCs carrying the T cells cognate antigen are present.
First passage processes are used to derive an equation for the encounter time of a T cell with one of N APCs. Using this time, a rate of encounter is established, and used throughout this thesis. The encounter rate is dependent upon the radius of the lymph node, the effective radius of the APC, and the diffusivity of the T cell. However, the diffusivity of T cells has not been clearly established.
In vivo imaging data is used to develop a systematic method for determining the diffusivity of a population of T cells. Due to in vivo imaging experiments having a limited sized imaging volume, a confinement effect is observed. The expected squared displacement of imaged cells is calculated,
and the level at which a confinement plateau should be observed is determined.
T cell activation, in lymph nodes, relies upon encounters with APCs, but the number of APCs required to initiate a T cell response is currently unknown. Using mathematical models, in combination with experimental work, the probability of T cell-APC encounters can be quantified.
The probability of a T cell, residing in the lymph node for twenty four hours, to interact with APCs is calculated. Extrapolating the developed models to later times and lower cell numbers than can be achieved experimentally, a minimum number of APCs required to initiate a T cell response, for
typical human T cell precursor frequencies, is estimated.
It has been proposed that regulatory T cells suppress effector T cells via a three way interaction with APCs, as a method of preventing autoimmunity. A stochastic model of these interactions is developed and explored. The steady state of the system is found to depend upon the rate of
encounter of T cells and APCs, as well as the number of APCs. Stochastic effects are observed in the model, which affect the state of the system, and are not observed in a deterministic approach.
Interactions between T cells and APCs, in lymph nodes, are crucial for initiating cell-mediated adaptive immune responses. However, how these interactions cause activation of the T cells is not yet fully understood. Three hypotheses have been proposed for the method of T cell activation. These hypotheses are investigated, and models developed, in an attempt to quantify the observed stages of the activation process. It is found that experimental results can, in part, be explained by a probabilistic approach.