a mathematical model for the instigation and transmission ...w. yao et al. / commun. comput. phys.,...

Commun. Comput. Phys.doi: 10.4208/cicp.091114.170415s

Vol. 18, No. 4, pp. 868-880October 2015

A Mathematical Model for the Instigation and

Transmission of Biological and Neural Signals in

Response to Acupuncture

Wei Yao1, Na Yin1, Hongwei Yang1 and Guanghong Ding1,2,∗

1 Shanghai Key Laboratory of Acupuncture Mechanism and Acupoint Function,Department of Mechanics and Engineering Science, Fudan University, 220 HandanRoad, Shanghai, 200433, P.R. China.2 Shanghai Research Center for Acupuncture and Meridian, 199 Guoshoujing Road,Pudong, Shanghai, 201203, P.R. China.

Received 9 November 2014; Accepted (in revised version) 17 April 2015

Abstract. Acupuncture has been in clinical practice in China for thousands of yearsand its analgesia effect is worldwide accepted. However, the mechanism of acupunc-ture effect is not well understood. The study focus on signaling pathways inducedby acupuncture, analyzes the cooperative action of the acupoints’ structure and theassociated chemical mediators during acupuncture, establishes a mathematical modelclarifying the roadmap of electroneurographic signal startup and transmission mech-anism induced by acupuncture, quantitatively analyzing the response in acupoints toacupuncture. These work contribute to reveal the activation and transmission mech-anism of neural signals induced by acupuncture from systems biology perspective,lay the foundation for the integration of acupuncture theory and modern science andfurther guide the clinical treatment and experimental research of acupuncture.

AMS subject classifications: 92C10, 92C05, 92C42, 92B05

Key words: Mast cell, nerve cell, Ca2+ signaling, biological mediators release, acupuncture.

1 Introduction

Although Traditional Chinese Medicine (TCM) is treated with considerable skepticism,the analgesic effect of acupuncture is well accepted. In 1997, NIH panel issued consensusstatement on acupuncture, the 12-member panel concluded that there are a number ofpain-related conditions for which acupuncture may be effective as an adjunct therapy,an acceptable alternative, or as part of a comprehensive treatment program [1]. We have

∗Corresponding author. Email addresses: [email protected] (W. Yao), [email protected]

(N. Yin), [email protected] (H. W. Yang), [email protected] (G. H. Ding)

http://www.global-sci.com/ 868 c©2015 Global-Science Press

W. Yao et al. / Commun. Comput. Phys., 18 (2015), pp. 868-880 869

retrieved over 1,000 international papers investigating acupuncture analgesia in recentyears, which encouraged us to believe that the mechanism of acupuncture analgesia maybe revealed by scientific methods [2, 3].

Acupuncture is a penetration and manipulation of specific anatomic locations onthe skin, called acupuncture points (acupoints), by thin, solid, generally metallic nee-dles. Recent studies on acupoint morphology using various approaches, such as MRI,anatomy, and XCT, have shown that the structural basis of the acupoints is complex andcomposed of connective tissue with numerous intertwining blood vessels, nerves, mastcells, and lymphatic vessels [4]. The dermic dense connective tissue and subcutaneousloose connective tissue at acupoint form a three dimensional collagen fiber network, con-necting the surface of the body to internal organs. Under normal conditions, collagenfibers wind together and arrange interlaced [5]. By observing collagen and elastic fiberswinding and tightening around the needle during acupuncture, Langevin proposed thatneedle manipulation transmits a mechanical signal into connective tissue via the nee-dle/tissue coupling [6, 7]. We devised a mechanical sensor to detect the real-time forceon the needle during acupuncture manipulation and found if the structure of the collagenfibers at Zusanli (ST36) was destroyed by injection of type I collagenase, the force causedby rotation or lift-thrusting manipulations of the acupuncture needle declined and theanalgesic effects was attenuated accompanying the restraint of the degranulation of mastcells [8]. The results indicated that the collagen fibers participated in the initiation of man-ual acupuncture signal in the acupoints by increasing mast cells’ degranulation. Furtherexperiments showed acupuncture resulted in a remarkable increase in degranulation ofthe mast cells, pretreatment of the acupoints with disodium chromoglycate (DSCG, mastcell stabilizer) not only counteracted the phenomenon of degranulation but also reducedanalgesic effect of acupuncture [9]. Acupuncture analgesia depends on the neural sys-tem, and there’s no acupuncture effect when the acupoint is narcotized [10]. Meanwhile,we found that acupuncture at ST36 can trigger peripheral nerve cell discharge [11], andHan et al. found that acupuncture at ST36 can cause the corresponding neural electricalactivity of spinal cord dorsal root, which suggest that there exists the electroneurographicsignal startup mechanism at acupoint and transmission mechanism in the nerve cell net-works [12].

These studies analyzed the mechanism of acupuncture analgesia effect from collagenfibers, mast cells, nerve cells and neurotransmitters respectively, there was no researchcombining all the aspects, clearly explaining the mechanotransduction pathway inducedby acupuncture and revealing the mechanism of acupuncture analgesia effect. Aboveresearches suggest acupuncture (mechanical stimuli) can activate mast cells to release bi-ological mediators through collagen fiber deformation, biological mediators accumulatein extracellular space (ECS) around acupoint and activate nearby nerve cells, then mod-ulate the multiple pain-processing pathways in response to acupuncture. To explain thisprocess clearly, we suppose to set up a mathematical model and qualitatively describecellular signaling and biological mediators’ dynamics and draw a roadmap of the painmodulating pathways [5].

870 W. Yao et al. / Commun. Comput. Phys., 18 (2015), pp. 868-880

Some mathematical models have been developed for investigating biological medi-ators’ dynamics and electrical signals which may be correlated to signal transmissionpathways modulated by acupuncture. For example, Xu et al. established a nociceptormodel to study the underlying mechanisms in the process of skin thermal pain whichmay correlated to moxibustion (warm acupoints by burning herbs)’s effect [13]. Shi etal. set up a mathematical model to simulate intracellular Ca2+ signals and degranula-tion induced by laser acupuncture [14]. We have set up a mathematical model to exploresignaling pathways in mast cells and clarified mast cells’ participating in acupunctureeffect [15]. Furthermore, we construct a mathematical model for describing mast cell andnerve cell interaction modulated by acupuncture [16].

The object of this paper is to combine these previous mathematical models and setup a comprehensive model describing the signal transmission process that occurs in re-sponse to acupuncture (mechanical stimuli at acupoints). The rest of the paper is orga-nized as follows. In Section 2, we present a mathematical model based on the acupointsstructure. In Section 3, we present numerical simulations under a variety of conditions.In Section 4, we conclude with a brief summary of the paper and a discussion.

2 Methods

Fig. 1 showed the dynamic process of signal pathways induced by manual acupunc-ture at acupoint. Mechanical stimuli induced by collagen fibers’ deformation activatemechano-sensitive (MS) ion channels on mast cells’ membrane and allow Ca2+ entry [17],local intracellular Ca2+ rise induces a cascade of intracellular signaling events and leadsto biological mediators’ release, therefore, increase interstitial biological mediators’ con-centrations at acupoint [18–20]. These biological mediators move (diffuse or flow) inECS and activate adjacent mast cells to form positive cytosol Ca2+ wave propagationin mast cell networks and release more biological mediators, some biological mediatorsmay trigger action potential in the primary sensory neurons and induce passive electricalflow spreading along branches of nerve fibers in nerve cell networks and activate dorsalspinal nerve root, then transmit signals to the brain.

The dynamics’ processes in a mast cell are detailed described in one of our paper [15].MS Ca2+ current (ICa,MS) is described by GHK equation,

ICa,type=Psti

gCa,typeFEm

(

[Ca2+]i−exp(

− Emφ

)

[Ca2+]e

)

φ(

1−exp(

− Emφ

)) , (2.1)

where Psti =1

1+βexp(−τ)is the proportion of MS channels in open state; and β is a mea-

sure of the probability that a channel is in the open state in the no-load case, τ is themechanical stimuli intensity; our mechanical sensor can recoded the stress and torque onthe needle inserting in acupoints, therefore, we simplified the amplitude of the recodingas the mechanical stimuli intensity on mast cells transmit from fiber deformation. The


Figure 1: Dynamic process of signal pathways induced by manual acupuncture at acupoint.

parameter φ=RT/zF where R=8.31 mV coulomb/mM K is the universal gas constant,T is the absolute temperature, z is the valence of ion, and F = 96.485 coulomb/mM isthe Faraday constant. Em is the membrane potential, [Ca2+]i and [Ca2+]e are intracellularand extracellular Ca2+ concentrations respectively.

Local Ca2+ rise activates protein kinase C (PKC) (Eq. (2.2)) and induces biologicalmediators’ release (Eq. (2.3))

d[PKCA]

dt= kaP([PKCT]−[PKCA])[Ca2+ ]i−kdP[PKCA], (2.2)

where [PKCA] is the activated PKC concentration, [PKCT] is the total PKC concentration,kaP and kdP are the PKC activation and deactivation rate parameters, respectively.

[BioM]production=VBioξmax[PKCA]−[PKCA]min,0, (2.3)

where [BioM]production is biological mediators’ production by mast cell (release from mastcell), VBio is the biological mediators’ release rate, [PKCA]min is the threshold concentra-tion that is necessary to prevent small amounts of [PKCA] from being amplified and thusleading to biological mediators release. ξ is a parameter that accounts for the depletionof granules inside the cell. ECS is considered as a continuum medium. To make the prob-lem computationally tractable, the specific geometries of mast cell and nerve cell are notconsidered. Since this is a continuum model, the concept of individual cell is not mean-ingful. Cellular properties refer to the average values at any location within the tissue.In the mast cell networks, each mast cell is arranged in x direction. The distance betweennearby cells Dcell = 3×10−5m for tissues abundant mast cells and Dcell = 1.5×10−4m for


tissues scarce of mast cells. The biological mediators’ move (diffuse and flow) inside ECSis described by standard convective-diffusion equations

∂[BioM]e∂t

=DBio∂2[BioM]e

∂x2−v f low

∂[BioM]e∂x

+[BioM]production, (2.4)

where [BioM]e is the biological mediators’ concentration in ECS, DBio is the diffusion co-efficient and v f low is the interstitial flow speed. Nerve cells locate some distance (r=Dis)from mast cell networks, they can be activated by biological mediators released frommast cells and release biological mediators as a result of local action of biological media-tors (Eq. (2.5)) or the excited membrane potentials (Eq. (2.6)). Here, we didn’t distinguishbetween the biological mediators release from mast cells and those release from nervecells.

∂[BioM]e∂t

=VA[BioM]e7+[BioM]e

−λA[BioM]e, (2.5)

∂[BioM]e∂t

=VNe−0.0044(Em−8.66)2

−λN [BioM]e. (2.6)

The first term on the right-hand side represents the biological mediators’ production innever cells, the second term represents the loss of free biological mediators due to ab-sorption. where VA =600 mM/s, λA =10 s−1, VN =600 mM/s and λN =10 s−1.

All neurons’ membrane potentials have been found to behave qualitatively similarto that described by the Hodgkin-Huxley (H-H) model of nerve excitation [21]. Here,we revised the H-H model to describe the sensory neuron (Eq. (2.7)) and the spinal cordneuron (Eq. (2.9)).

Cm∂Em

∂t=−gNam3h(Em−ENa)−gKn4(Em−EK)−gL(Em−EL)+IBio− Icable, (2.7)

where Cm is the specific capacitance of the membrane, gNam3h(Em−ENa), gKn4(Em−EK)and gL(Em−EL) represent Na+, K+ and leak currents in the H-H model, respectively. IBio

(ion=Na+, K+ and Ca2+) is the current activated by biological mediators which can bedescribed by GHK equation

IBio,ion= gBioPBio

FEm

(

[ion]i−exp(

− Emφ

)

[ion]e

)

φ(

1−exp(

− Emφ

)) , (2.8)

where PBio = 1−exp(∫ t

0−[Bio]edt) is the proportion of receptor channels in open state.

Icable is the coupling cable currents in the nerve fiber that connects the sensory neuron

and the spinal cord neuron: Icable =Em,p−Em,c

Ra[23], where Ra is the equivalent electric con-

ductance; Em,p and Em,c are the Em of sensory neuron and the spinal cord neuron, respec-tively,

Cm∂Em

∂t=−

[

gNam3h(Em−ENa)+gKn4(Em−EK)+gL(Em−EL)]

+nλIcable, (2.9)


where n is the number of activated sensory neurons that connect to the spinal cord neu-ron, and λ is the area ratio of sensory neurons to spinal cord neurons.

The parameter values related to our model come from a number of sources, which aregiven in Table 1.

Table 1: Initial resting values and other relevant parameter values for the computations.

Parameter Value Source

gCa,MS 3.8×10−10Ω

−1 [15]

β 99 [15]

kap 0.06 M−1s−1 [14]

kdp 0.02s−1 [14]

[PKCT] 5×10−7M [14]

VBio 0.1s−1 [15]

[PKCA]min 1.2×10−7M [15]

DBio 3×10−10m2s−1 [22]

VA 600mMs−1 [22]

λA 10 s−1 [22]

VN 600mMs−1 [22]

λN 10 s−1 [22]

gBio 5×10−10Ω

−1 evaluate from [16]

Ra 100 Ω cm [16]

Cm in neuron 0.75F m−2 [16]

Em in mast cell 0 mM [24]

Em in neuron -70 mM [22]

[Ca]e 2 mM [15]

[Ca]i 0.0001 mM [15]

[Ca]ER 0.5 mM [15]

λ 0.5 evaluate

3 Results

3.1 Mast cell networks response to mechanical stimuli at acupoint

We have carried our simulations by applying mechanical stimuli at one mast cell (MC0)at t= 0 s. Fig. 2 shows the response of cytosol Ca2+ and extracellular biological medi-ators in mast cell networks due to biological mediators’ diffusion in ECS. Cytosol Ca2+

responses immediately (t=0 s) to mechanical stimulate [25], [Ca2+]i increase leads to bi-ological mediators release and increase in ECS several seconds later (Fig. 2b, t≈50 s, redarrow mark). Biomediators diffuse in ECS, then activate adjacent mast cells to form the


0

200

400

600 −3−2

−10

12

30.1

0.15

0.2

0.25

0.3

0.35

x (mm)t (s)

[Ca2+

] i (µ

M)

(a) (b)

Figure 2: The response of mast cells network of Dcell =3×10−5m to mechanical stimulate at MC0 (the mast

cell at x=0 position) due to diffusion. (a) Cytosol Ca2+ propagation; (b) extracellular biological mediatorsconcentration.

0

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Figure 3: The response of mast cells network of Dcell = 3×10−5m to mechanical stimulate at MC0 due todiffusion and convection (v f low = 5×10−6ms−1). (a) Cytosol Ca2+ propagation; (b) extracellular biologicalmediators concentration.

cytosol Ca2+ wave propagation (Fig. 2(a)) and extracellular biological mediators concen-tration rise (Fig. 2(b)). Without considering interstitial flow (v f low = 0), biological me-diators diffuses equally to downstream (x direction) and upstream (-x direction), there-fore, Ca2+ and biological mediators signals propagate symmetrically about x= 0. Whentaking into account interstitial flow (v f low > 0), Ca2+ and biological mediators signalspropagate quickly to downstream, while slowly to upstream [25], when the flow is great(v f low > 5×10−6ms−1), there was no Ca2+ and biological mediators propagation in theupstream direction (Fig. 3).


3.2 Nerve cells response to the biological mediators released from theactivated mast cells

Nerve cells share perivascular localization with mast cells and can be activated by biolog-ical mediators release from mast cells. Fig. 4 shows the nerve cells’ (located at differentDists, marked by arrow) responses to the biological mediators released from the activatedmast cells (located at x=0, marked by eclipse). Fig. 4(a)-(c) shows mechanical stimuli ac-tivate mast cells (x= 0) then increase [Bio]e (marked by eclipse). biological mediatorsdiffuse in ECS and activate the nerve cell at Dists (Dist = 200µm, 400 µm and 600 µmrespectively) then induce more biological mediators release from nerve cell and furtherincrease [Bio]e (marked by arrow). Fig. 4(d) shows when the distance between the mastcell and nerve cell is great, the nerve cell is not longer activated by the mast cell acti-vation, therefore there’s no biological mediators release from nerve cell at Dist =800µm.

(a) (b)

(c) (d)

Figure 4: [Bio]e contour in the coupled mast cell (at x= 0) and nerve cell (at x= Dist) model, as function oftime (t) and distance (x), after its exposure to an initiating mechanical stimuli at x=0 during time t=0−60s.(a) Dist =200µm; (b) Dist=400µm; (c) Dist=600µm; (d) Dist =800µm.


Fig. 4 also shows with Dist increase, the response in the nerve cell becomes slower; andwhen Dist≥800µm, the nerve cell is not activated.

3.3 Electrical signal propagate in nerve cells

Application of biological mediators to nerve cells activates membrane currents IBio,ion

(Eq. (2.8)) and induces action potentials in the terminal sensory neuron. The potentialdifference between the sensory neuron (Em,p) and the spinal cord neuron (Em,c) leads to

a passive electrical flow in the nerve fiber (Icable =Em,p−Em,c

Ra), then change Em,c in the end.

Fig. 5(a) showed Em,p and Em,c remain stable in rest state. Fig. 5(b)-(f) showed the oscil-lations of Em,c according to different number of activated sensory neurons connected tothe spinal cord neuron. When there is only one sensory neuron connected to the spinalcord neuron activated (weak electrical current flow from the sensory neuron in the nervefiber), the amplitude of Em,c oscillations becomes weak (Fig. 5(b), several mV). With moresensory neurons being activated, the amplitude of Em,c oscillations is greater. When sixsensory neurons being activated, the amplitude of Em,c reaches the greatest, but the great-est amplitude appears 70ms later than that in the sensory neuron, Fig. 5(f) showed wheneven more sensory neurons being activated, the greatest amplitude appears quickly, onlya few ms later than that in the sensory neuron.

4 Conclusion and discussion

4.1 Local biological responses to acupuncture and the correlation withmeridian

Acupuncture is a method of applying mechanical stimuli into acupoints. We have re-coded the stress and torque on the needle inserting at ST36 [24] (Fig. 6). It showed themean amplitudes of the lift-thrusting and twisting force during acupuncture notable de-creased following pretreatment with type I collagenase at ST36. The data indicate thatcollagen fibers wind and tighten around the needle and transport mechanical stimuli inthe tissue. Our results showed mechanical stimuli can activate mast cells and lead tothe release of biological mediators, these mediators can further activate adjacent cellsand induce biomedical messengers’ signal propagation in the mast cells network (Fig. 2).Further simulation results showed the signals’ propagation speed depends on the dis-tance between the adjacent mast cells (Dcell), the speed decreases with Dcell increasingand there’s no signals’ propagation when Dcell is very large (Fig. 7(a), Dcell =3×10−4m).But when there’s interstitial fluid flow, the biomedical signals propagate faster in down-stream direction and slower (even none) in upstream direction (Fig. 3), the flow can carrythe biomedical messengers to downstream even when mast cells are scarce (Fig. 7(b),Dcell=3×10−4m). Mast cells are found abundant at the areas contain acupoints or merid-ians when compared with areas that do not contain acupoints or meridians [24, 26]. Themechanical force transmitted by collagen fibers from the acupuncture needle stimulates


0 50 100 150 200−74

−73

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V)

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(m

V)

t (ms)

(e) (f)

Figure 5: Simulation results of the Em in the spinal cord neuron and the terminal sensory neurons. Black solidlines represent the terminal sensory neuron, red dashed lines represent the spinal cord neuron. (a) Em in reststate (control); (b) one sensory neuron connected to the spinal cord neuron is activated by [Bio]e (n= 1); (c)two sensory neurons are activated (n=2); (d) four sensory neurons are activated (n=4); (e) six sensory neuronsare activated (n=6); (f) eight sensory neurons are activated (n=8).

mast cell degranulation and releasing biological mediators, the biological mediators dif-fuse and flow along the meridian through the interstitial space and induce downstreammast cell degranulation, thereby generating the propagated sensation along the meridi-ans. Once mast cell degranulation is inhibited or the collagen fibers are destroyed, mast


(a) (b)

Figure 6: Wave form of needle manipulation at ST36 detected by acupuncture-needle real-time force monitor.(a) Lift-thrusting force change. (b) Twisting force change. The dotted and solid lines indicate the force beforeand after the collagenase pretreatment, respectively [8].

(a) (b)

Figure 7: [Bio]e in mast cells network (Dcell=3×10−4m) to mechanical stimulate at MC0. (a) Due to diffusion;

(b) due to diffusion and convection (v f low=5×10−6ms−1).

cells at the acupoints cannot be further activated by manipulation. The transduction ofsignals at the cellular level and the acupuncture analgesia effect are blocked.

4.2 Electroneurographic signal startup and transmission in the nerve cellnetworks induced by acupuncture

We have shown with our model that biological mediators released from mast cells ac-tivate peripheral sensory neurons and trigger action potentials in nerve cells networkwhich responsible for the electroneurographic signal propagate from the peripheral tothe dorsal root ganglia (spinal center). These nerve impulses then transmit into thalamusthough central tegmental tract to activate the central nervous system-Brain. Brain and


spinal cord can expertly integrate the nociceptive information to generate correspondingnerve impulses and modulate the target organ through the nerve system. In this process,bi-directional signal transduction between the peripheral and central nervous system ex-tends continuously and then forms an interactive process of orderly acupunctural signaltransduction. Our simulation results indicate that different number of activated sensoryneurons lead to different electrical signals in the spinal cord neuron (Fig. 5), which mayexplain the acupuncture effect depends on the needle manipulation. Han et al. designedan experiment to obtain action potentials on dorsal spinal nerve root according four dif-ferent acupuncture manipulations at ST36 [12]. By extracting the nonlinear characteristicparameters, they found that the electrical signals have distinguished chaotic features, thesignals induced by four different acupuncture manipulations display different charac-teristics. Our model may reveal the activation and transmission mechanism of neuralsignals induced by acupuncture from systems biology perspective, it may help to ana-lyze electrical signals in the spinal cord neuron, and further guide the clinical treatmentand experimental research of acupuncture.

Acknowledgments

This work was supported by National natural science foundation of China (81473750 and11202053), Shanghai Key Laboratory of acupuncture mechanism and acupoint function(14DZ2260500), National Basic Research Program of China (2012CB518502).

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