A Predictive Mathematical Model of Acupuncture Based on an Explanation Biological Model

摘要

Acupuncture requires a long-term training to handle acu-points. Four techniques exist: (1) development of a local mechanical stress field by needle motions (lifting - thrusting cycle or rotation) at acupoints; (2) development of a local temperature field by directly applying a heating moxa (mugwort herb) stick on the skin or indirectly by applying this stick on the acupuncture needle (moxibustion) at acupoints; (3) development of a local electrical field by applying a small electric current between a pair of acupuncture needles (electroacupunc-ture, or percutaneous electrical nerve stimulation [PENS]) at acupoints; and (4) laser light excitation independently of heating and other physical means probably via proper G-protein-coupled receptors on the surface of mastocytes. Acupoints are enriched of mastocytes, among other biological structures and cells. Mastocytes are activation by a mechanical stress field (mechanotransduction), heating (thermotransduction), or a electrical field (electrotransduction). Whatever the operation mode, calcium entry in the mastocyte triggers degranulation and release of chemoattractants, neural stimulants, and endocrine substances. The process is sustained by recruitment of mastocytes (chemotaxis). Acupuncture effects result from a set of signals sent from activated mastocytes at given acupoints to local nerve endings and capillaries that are transmitted to the brain and heart for processing and augmenting the flow rate, especially in the vasodilated acupoint. Released substances targets their cognate receptors on nerves and lymph and blood vessels. These two types of conduits deliver fast cues (electrochemical waves) and delayed information (blood transport) to the central nervous system, where they are processed for a desired output. The mathematical model is a system of 5 partial differential equations. Its simplest form describes the evolution of the density of mastocytes and the chemoattractant concentration. A mathematical analysis of a simplified version of the equation set leads to a theorem for blow-up condition (the expected solution) as well as an analytical solution useful for validation. Numerical simulations are also carried out using a finite element method with mesh adaptivity. The computational model based on the home-made FreeFEM++ software demonstrated the occurrence of a stress field that excite mastocytes. It also shows that only adequate pools of mastocyte, that is, acupoints, must be targeted to have marked effects.