The selected major involved biological, physical, and chemical mechanisms in acupunc-ture have led to the conception of a novel mathematical model of mastocyte response to external stimuli.
The first main contribution of this work is the study of the needle motion in the in-terstitial flow approximation through fibrous matrix using a convective Brinkman model.
This approach has shed light on the physical nature of the mechanical stress field induced by pistoning the needle that affects the interstitial flow and in turn the cell activity. The nu-merical results presented have also helped to define a stress function to model the external stimuli of acupuncture.
The second main contribution is the development of the model for the mastocyte re-sponse to external stimuli. The extension of the Keller-Segel model with a forcing term allows us to take into account the stress field previously mentioned. A theoretical study on the chemotaxis model suggests that the recruitment of mastocytes through chemotaxis plays an essential role in the sustainability of the biochemical and cellular responses to acupuncture. A theorem gives the conditions for the effectiveness of the response. The numerical simulations carried out confirm the behavior of the system.
The main limitation of this study is the lack of experimental data. However, our goal was not to establish quantitative models but rather to establish multiscale models that incorporate the major biochemical processes, the cell responses, and mass transfer.
The model developed in this work hopes to be a first step toward the set up of a larger acupuncture framework. The external stimuli trigger a biochemical reaction cascade that leads to the effects of mastocytes. The acupuncture framework includes the external stim-ulation, the signal development at the acupoint region, the signal transmission to the cen-tral nervous system, and the signal processing targeting peripheral organs.
The work presented above focuses on the external stimulation and the local signal development. The correlated works in progress include the transport of calcium ions Ca2+
in ion channels (Poisson-Nernst-Planck/Navier-Stokes model), mastocyte degranulation (granule exocytosis model). The signal transmission has yet to be studied. Future work in this matter should focus on the modeling of mass transport across the blood-brain barrier. The blood-brain barrier is a highly selective permeable barrier that separates the circulating blood from the brain’s extracellular fluid in the central nervous system. It
CONCLUSION AND FUTURE WORK
provides a suitable environment for neuronal signaling. Endocrine messengers liberated at the acupoint conveyed in the blood can modulate neuron activity.
The preliminary results can be expanded with the extension of the mathematical mod-els, the further development of numerical methods, and the inclusion of experimental measurements.
On the theoretical point of view, additional work will be required to extend the anal-ysis to the full parabolic chemotaxis model. Also, it would be interesting to extend the fluid/chemotaxis model presented in this work to the acupuncture study. Fluid/chemotaxis interaction may prevent blow-up of the solution of the chemotaxis model.
The study of a model that takes into account the fiber mechanics could allow extension of the needle pistoning case to the needle rotation case. A first step towards the study of the effects of the needle rotation worth considering is to extend the present numerical study to 3D. The 3D model may lead to a deeper understanding of the stress field.
Last but not least, it is crucial to validate the theoretical findings with experimental results. To our knowledge, the evolution of the spatial distribution and motion of masto-cytes has not yet been fully studied experimentally. Future experimental work could help in the development of the proposed mathematical models.
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