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Airway hyperresponsiveness (AHR) is a characteristic feature of asthma, and generally correlates with severity of asthma. Understanding the protection mechanism against excessive airway narrowing and how it breaks down is fundamental to solving the problem of asthma. In this paper we have proposed a stochastic modeling the airway smooth muscle bundle for reproducing AHR such as an increased sensitivity of the airways to an inhaled constrictor agonist, a steeper slope of the dose-response curve, and a greater maximal response to agonist. A large number
N of contractile muscle cells was assumed to repeat themselves in between contraction and relaxation asynchronously. Dynamic equilibrium of statistic physics was applied to the system of ASM bundle. Thus, the relation of dose to response of a piece of ASM bundle was described by
Φ=tanh(
βH) , where
β was Boltzman factor and
H represented energy of contraction induced by constrictor agents. Each of adjacent pair contractile cells was assumed to have Ising-type of antimagnetic interactions of preference energy
J (for the condition of contraction-relaxation) between them. A motion equation for a piece of ASM bundle was described by
Φ=N(H-zJΦ , which explained existence of combined tonic and phasic contractions. Based on observations of Venegas
et al. [4], airway responsiveness was assumed to be assessable by total volume of the ventilation defects (TVD) of 13NN PET-CT images. Interactions via propagation of Ca ion waves between ASM bundles would cause percolation probability by
P_{Φ}=(1+tanh(
βH))
^{2}/4 along the tree, then the relation of dose
βH to TVD was described by TVD=
P_{Φ}[1-(1-
P_{Φ})
^{3}/
P_{Φ}
^{3}]-TVD
_{0}. TVD
_{0} represented the protection mechanism against excessive airway narrowing, which was determined by the ratio of amplitudes between tonic and phasic contractions, thus the balance of amplitudes between tonic and phasic contractions of peripheral lobular smooth muscles would be the determinant of AHR.

The initial observation that bronchoconstriction occurs more rapidly in asthmatics when compared to non-athmathics after exposure to a constrictor agonist was made in 1921 [

AHR can be demonstrated in almost all patients with current symptomatic asthma. Using the standardized method, asthmatic subjects generally have a provocation concentration of methacholine or histamine causing a 20% fall in the forced expiratory volume in one second (FEV_{1.0}) PC_{20} of <8 mg/mL. Most non-asthmatic subjects will have a PC_{20} of >16 mg/mL. The severity of AHR generally correlates with the severity of asthma. Many different factors have been suggested to be involved in causing AHR.

AHR is complex in its mechanisms. One component of AHR is the acute transient and reversible AHR, which is closely associated with effects of acute airway inflammation. Although the link to airway inflammation is strong, the precise mechanism between airway inflamemation and AHR are not entirely clear. The second, more persistent component of AHR appears related to the chronicity of the disease and likely the chronic effects of airway inflammation leading to multitude of structural airway changes known as airway remodeling.

There is currently a major controversy among the molecular and cellular biologists and experts in the lung mechanics. The experts in lung mechanics feel that the disease will be not controlled until we understand the mechanics of excessive airway narrowing that characterizes asthma. From the point of view of a scientist trying to understand the reasons of abnormal airway narrowing in asthma, it is striking that airway smooth muscle (ASM) is capable of shortening to the extent that all airways within the lung could be completely closed. However, in the normal lungs the degree of narrowing cannot be provoked. Normal lungs are protected against excessive airway narrowing. A plateau develops on the bronchial dose response curve so that even an increase in dose of smooth muscle agonists of several orders of magnitude fails to produce an increased response. This protective mechanism is lost in asthma.

Venegas and his colleagues demonstrated the functional images of airflow obstruction as the ventilation defects (VD) during the test of AHR, which would reflect uneven constrictions of lobular bronchioles [

A large number of ASM cells compose the bronchial bundle (the ASM bundle) as a major contractile component of the bronchial tree. A piece of the ASM bundle with the degree of contraction f was assumed to be composed of N cells which consist of m cells in contraction and n cells in relaxation (N = n + m) , then two relations between the degree of contraction and the number of smooth muscles in contraction or relaxation were described as follows,

Each ASM cell was assumed to repeat itself asynchronously in between relaxation (σ = –1) and contraction (σ = +1). According to statistic physics of equilibrium [

where H was the field energy inducing contraction of f, and β was Boltzman’s thermal factor. Thus, for the ASM bundle with the degree of contraction f and the field energy H, we obtained the equation as follows (

or

Adjacent pair cells were assumed to have an energetic preference J to be in the different conditions (σ_{i}, σ_{j}) = (+1, –1) or = (–1, +1) (we called this Ising-type of antimagnetism interactions). The internal energy U of the piece of ASM bundle was able to be defined as follows,

where S of indicated summation of adjacent pair cells across the whole piece of ASM bundle. If each ASM cell repeated itself asynchronously in between relaxation and contraction even in the piece of ASM bundle with the degree of contraction f, we would be able to describe the internal energy consisting of adjacent pair cells i and j in the probability of as

follows,

,

and

Then,

If each cell was surrounded by z adjacent cells, the internal energy of the piece of ASM bundle with the degree of contraction f was able to be described by the following U(f),

Thus, the total energy E of this piece of ASM bundle with the degree of contraction f under the contraction energy H was defined by the following equation,

Inhaled constricting agonists such as methacholine or histamine would induce local tonic contractions of ASM, which would proceed to bronchoconstriction as a whole. Through asynchronous calcium ion waves among ASM cells local tonic contractions of ASM bundle would propagate to both peripheral and central parts of bundle along the binary branching tree in a stochastic manner (

The differential equation for dynamics of ASM was obtained by use of Eq.8,

or

If the field potential H is zero, motions of ASM are oscillations (the phasic contractions with frequency of ω =). In general, dynamics of ASM bundles should be described by the combination of phasic and tonic contractions as follows,

Tonic contractions are cancelled when the amplitude of A is larger than, and phasic ones are visible. Tonic contractions will be observed when A is less than (

If ASM bundles in adjacent pair branches appeared in the same contracted condition by the probability of P(+1, +1), the site probability of p was assumed to be P(+1, +1)

as follows (

Thus, based on Eq.9 and Eq.11 tonic contractions of the ASM bundle in the whole bronchial tree was described by the percolation probability as follows,

Test of airway hyperresponsiveness (AHR) is designed for assessing the degree of tonic bronchoconstriction induced by inhaled methacholine or histamine. Bronchoconstriction is assessed by decrease in forced expiratory volume in one second (FEV_{1.0}). Recent functional images of asthma by Venegas and his colleagues showed the ventilation defects (VD) as peripheral bronchiolar obstructions [_{10}, therefore the degree of bronchoconstriction was proportional to TVD. If TVD was caused by peripheral bronchiolar obstructions induced by inhaled constricting agonists, TVD would be expressed by the percolation probability as follows,

The site probability was described by the equation from Eq.4 as follows,

If the maximal degree of bronchoconstriction was measured by during the provocation test, the parameter TVD_{0} = 1 – TVD_{max} indicated the protection mechanisms of excessive airway narrowing. Thus, the test of provocation was described by Eqs.15a, b and c as follows,

We computed f, p_{f}, and TVD (at TVD_{max} of 0.2, 0.4, 0.7, 1.0) in the domain of between βH = 0 and βH = 3.0 according to Eqs.15a, b and c, and demonstrated the relationships between βH and TVD in _{1.0} vs baseline induced by increasing dose of a bronchoconstrictor stimulus (methacholine) in patients with mild, moderate and severe asthmatics vs healthy individuals [

Previously reported studies on modeling dynamics of the airway smooth muscles (ASM) have been concentrated on changes in dynamic properties of the sub-cellular

molecular elements such as the acto-myosin cross links or the cytoskeleton [10,12]. Many of the biophysical phenomena were observed from the strip of ASM tissue which is composed of a large number of cells. All of the biophysical properties of ASM, such as active force generation and shortening velocity, are based on data from experiments on various strips of ASM tissues. However, no discussion has thought of the multicellularity of the experimentally prepared materials until today. What we proposed in this study was the modeling based on the multicellularity of airway smooth muscle bundle, thus we were able to take account of using the concept of statistic physics on this ASM system (see Eq.4).

There is a well-known statistic physical model of Ising for describing ferromagnetism of materials [6,7]. Ising model in this study was assumed to have antiferromagnetic type of interactions among contractile cells (i.e., the energy preference J was assumed in adjacent pair cells in different conditions such as contraction-relaxation). Then, the model for the ASM bundle we proposed revealed the existence of oscillations or phasic contractions. Recent studies on fetal lung explants have showed that airway peristalsis (AP) during morphogenesis, which appears in the first trimester of pregnancy and then increases in frequency towards birth (~1.0 cycle/min), is the spontaneous phasic propagation of waves of ASM contractility [12-15]. The existence of AP in fetal lungs would support the assumption of Ising-type of antiferromagnetic interactions among muscle cells.

In the healthy intact dog, ASM possesses sufficient force generating capacity to close all airways. But healthy animals or humans are challenged with inhaled contractile agonists in concentrations thought to be sufficient to activate the muscle maximally, resulting airway narrowing limited in extent. Brown and Mitzner [

As described above, healthy animals or humans are challenged with inhaled contractile agonists in concentrations thought to be sufficient to activate the muscle maximally, resulting airway narrowing limited in extent. This is called the protection mechanism against excessive airway narrowing. Professor Peter Macklem [

AHR has been known to consist of the sensitivity and the reactivity [_{0} which was determined by the ratio of H to A. Therefore, the conclusion of this study is that AHR are dependent upon the protection mechanism against the excessive airway narrowing, which is determined by balance of the phasic and tonic contraction-relaxation properties of ASM bundle under the level of potential energy consisting of many extramascluar factors including airway inflammation.