MODELING IN ACUPUNCTURE

mechanical stress is large enough; (5) local elevation of vascular permeability and vasodi-lation, induced by released compounds, together with increased local blood flow, induced by targeted cardiotonic messengers, enhanced endocrine signaling, i.e., delayed, perma-nent stimulation of the target brain region, as endocrine messengers are preferentially distributed in active brain regions.

1.3 Modeling in acupuncture

While acupuncture and moxibustion have been safely used for centuries, the reason why moxibustion can strengthen the flow of blood, stimulate the flow of qì, and maintain health remains barely understood by scientists. Model development, analysis and numer-ical simulation in acupuncture and moxibustion can be studied from the macroscopic and microscopic viewpoints.

1.3.1 Electroosmotic meridian model

As does acupuncture via mechanotransduction, that is activation by the stretch of mechanosensitive ion channels at the mastocyte surface, moxibustion may trigger opening of a thermosensitive ion channel. A previous study of the transfer of heat from burning moxa in an indirect moxibustion setting describes the construction of the electroosmotic meridian model for the modeling of a qì-blood interaction [60–64]. The electroosmotic model reads

∂u

∂t + u · ∇u + ∇p − ν∇²u = f (1.1)

The electroosmotic force f on the right hand side of the momentum equation (1.1) for the fluid flow velocity is as follows

f = 

ρλ²_Dψ ∇ (ϕ + ψ) (1.2)

This distinguished driving force stems from the fact that tissue fluid near acupoints is full of ions with the permittivity . In the presence of an externally applied electric field ϕ or the electric potential between the patient and acupuncture practitioner, zeta potential φ will be formed within the electric double layer of thickness λ_D, thereby resulting in the total electrical potential ϕ + ψ. Hydrodynamic equation (1.1) will be solved subject to the divergence free constraint equation

∇ · u = 0. (1.3)

Flow equations used in the continuous porous capillaries with the porosity  and the per-meability κ are as follows

∂

∂t(ρu) + ∇ · (ρuu) = −∇p + ∇ · (τ ) − ²µ

κ u −³C_Fρ

κ^1/2 |u|u, (1.4)

1.3. MODELING IN ACUPUNCTURE

∂

∂t(ρ) + ∇ · (ρu) = 0. (1.5)

In the above, ρ is the fluid density, µ the dynamic viscosity, τ the shear stress tensor and C_Fthe drag factor.

Equations (1.1 - 1.5) have been solved in the three-dimensional meridian model schemat-ically shown in [62] by finite volume method. The results detailed in [60, 62] have re-vealed (1) if the blood is weak, then the qì is weak as well; (2) if qì is blocked, then the blood is in stasis (or if qì is flowing, then blood is flowing as well).

1.3.2 Interstitial flow in acupuncture

Interstitial flow plays important roles in the biological functions of tissues. Shear stress induced by fluid flow can trigger mechanotransduction pathways leading to cell activities such as degranulation of mastocytes. The fluid flows in the interstitial space (collagen-elastin matrix) between blood and lymphatic capillaries and carries nutrients, wastes, and proteins. The flow is driven by the difference in hydrostatic and osmotic pressures between the capillaries and the interstitial space [65]. The interstitial fluid is composed of water, ions, and other small molecules partially coming from the plasma. It interacts with the ground substance, containing proteoglycans and other macromolecules networks, to form a gel-like medium [12].

The interstitial tissue is considered as a fluid-filled porous medium. Low speed flow through rigid porous media is described by the Darcy law:

∇p_f = −µ

Pu,¯ (1.6)

where −_P^µ¯u denotes the Darcy drag, µ the dynamic viscosity of the fluid, P the Darcy permeability, ¯u the averaged velocity, p_f the pressure, and α_f the fluid volume fraction.

This volume fraction corresponds to the effective porosity of the medium.

The incompressible Brinkman equations [66], which permit the computation of flow profile around a solid body in porous media, take the following form:

−µ∇²¯u + ∇p_f = −µ

P¯u in Ω, (1.7)

∇ · ¯u = 0 in Ω. (1.8)

Brinkman equations (1.7-1.8) and the Darcy equation (1.6) constitute a continuum model for the current simulation of microscopic flow in porous media.

In the context of acupuncture, the interstitial flow has been modeled by the Brinkman equations to investigate the effects of shear stress on interstitial cells [67, 68]. A numerical study of the effect of shear stress induced by interstitial fluid flow on the interstitial cells was carried in 2D [67] and 3D [68]. Computed shear stress is shown to be large enough to activate cell functions. Simulations show that the interstitial fluid flow varies with capillary size and density, capillary permeability, blood pressure, interstitial pressure, and interstitial porosity.

1.3. MODELING IN ACUPUNCTURE

1.3.3 Mastocyte dynamics of degranulation

Mastocytes are one of the key components of the physiological response in acupunc-ture treatment. Mastocytes can be activated by fluid shear stress generated, for example, by the needle manipulation in connective tissues. Chemical mediators are released via mechanotransduction process that mainly relies on calcium entry in the mastocyte cy-tosol. This calcium entry enables exocytosis of granules containing chemoattractants, nerve messengers and cardiovascular messengers. The concentration of cytosolic calcium entry can be computed under the applied shear stress. Wiesner et al. [69] developed a mathematical model to reproduce the change in mastocyte cytosolic Ca²⁺ concentration under fluid shear stress in the interstitial flow. The Ca²⁺dynamics model reads as

dc_c where c_c is the cytosolic Ca²⁺ concentration and c_e is the endoplasmic reticulum Ca²⁺

concentration. τ denotes the external applied shear stress and W (τ ) is the strain energy density that corresponds to the function of applied shear stress. The first term of the first equation in (1.9) describes the flux of Ca²⁺ entering the cytoplasm. The second term describes the flux of Ca²⁺ stored in the endoplasmic reticulum. The third term depicts the flux of Ca²⁺ leaving the cytoplasm. The fourth term describes the Na⁺ and Ca²⁺

regulation. Both model (1.9) and observations [53] confirmed that mastocyte cytosolic Ca²⁺concentration increases with shear stress.

In turn, a mathematical model can be used to compute the dynamics of mastocyte degranulation provided that the concentration c_c of cytosolic Ca²⁺ is known [55]. The degranulation dynamics model reads

where the first equation describes the evolution of the activated protein kinase C con-centration, the second equation describes the evolution of phosphorylated synaptosomal-associated protein 23 (SNAP23) concentration, and the third equation depicts the degran-ulation rate.

Both model (1.10) and observations [53] show that the mastocyte degranulation rate increases with the increase of cytosolic Ca²⁺ concentration, that corresponds to a higher