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Carbon nanomaterials, CNMs, are used (or can be formed as by-products) in many applications, which can be associated with an unintentional release of nanometric dust to the air^{1,2,3,4,5,6,7}. The aerosol generated in this way is respirable (inhalable) and easily penetrates to the alveolar region of the respiratory system^{8,9}. A substantial number of inhaled nanoparticles that are deposited in this region can promote direct interactions with the organism^{10,11,12}. The first surface met by inhaled particles in the pulmonary region is a thin layer of alveolar liquid on the top of lung epithelium. This layer contains the mixture of specific compounds of the lung surfactant, LS – the structure, which plays a vital role in the physiological functions of the respiratory system^{13}. It has been recognized that LS is sensitive to inhaled materials^{9,14,15,16}, and the resulting impairment of LS composition and/or properties may contribute to serious health problems, including the acute lung injury and respiratory distress. Accordingly, the analysis of LS properties in the presence of external factors such as micro/nanoparticles or chemicals, can give a preliminary information regarding the possible respiratory health problems that may follow inhalation of these agents^{17,18,19,20}.

Because of the high surface-to-volume ratio, inhaled nanoparticles present a particular threat for health even when their deposited mass is not high^{21}. As shown by the recent studies^{22,23}, effect of different nanomaterials on LS system may be highly specific and depend on several particle properties, such as the specific surface area, SSA or degree of hydrophobicity. Particle dose (concentration) is another essential factor in predicting lung toxicity^{24}. Therefore, the current study is focused on the identification of minimal doses of CNMs with different properties that may cause direct disturbance of dynamic surface properties of LS.

The most important biophysical feature of LS is the ability to modulate surface tension during oscillatory variations of the alveolar interfacial area during breathing^{25,26}. In the recent paper^{22}, the oscillating pendant drop technique was applied to analyze interactions of nanocarbon particles with the model LS at 0.25 Hz as the typical rate of respiration. However, the question arises if the effect of nanoparticle inhalation will remain similar also at variable breathing pattern, which is a quite common situation in the real-life (people working or doing sport exercises, etc.). To answer that, in this paper we propose to investigate CNMs interactions with the model LS at dynamic conditions that correspond to different breathing rates (2–10 s per breath). We demonstrate the usefulness of the concepts of interfacial rheology in the determination of such effects, and we propose new parameters to assess them quantitatively.

### Quantitative analysis and the physiological role of LS dynamics

Several experimental systems have been used to investigate the dynamic surface-active properties of LS in vitro, including Langmuir trough, oscillating bubble tensiometer, constrained bubble/drop tensiometers^{22,27,28}. However, not always appropriate conditions have been assured for such studies (e.g., physiological temperature and the surface area variations relevant to breathing cycle). Due to specific experimental conditions needed to determine dynamic properties of LS at the air/liquid interface, the quantitative analysis of the results also requires special measures. Up to now, the following numerical parameters have been most commonly used:

(a) the minimum value of the surface tension, σ_{min} (mN/m), recorded during periodical expansion–contraction cycles of simulated breathing.

(b) the amplitude of surface tension variations measured at such conditions (σ_{max} − σ_{min}).

Clements et al.^{29} introduced the stability index, *SI,* that can be derived from the above-mentioned surface tension values:

$$SI= frac{{upsigma }_{mathrm{max}}-{upsigma }_{mathrm{min}}}{0.5 left({upsigma }_{mathrm{max}}+{upsigma }_{mathrm{min}}right)}$$

(1)

Periodic variations of the surface tension during oscillations of the air/liquid interfacial area show a time-shift which result in the surface tension hysteresis. The normalized area of the hysteresis loop, *HA*_{n}, was proposed by Notter et al.^{30} as the parameter to quantify this feature:

$${HA}_{n} ~ = ~frac{{left[ {mathop smallint nolimits_{A} upsigma {text{d}}A~} right]_{{{text{expansion}}}} – left[ {mathop smallint nolimits_{A} upsigma {text{d}}A~} right]_{{{text{compression}}}} }}{{{A}_{{max }} – {A}_{{min }} }}$$

(2)

The hysteresis arises due to the time-dependent phenomena (relaxation) that may be attributed both to the intrinsic mechanical properties of the air/liquid interface and to the mass exchange of surface-active molecules between the surface layer and the underlying liquid. These dynamics may be described using 2D rheological formalism, i.e. by determining the apparent surface elasticity and the apparent surface viscosity^{31,32}. Several authors studied the surface rheology of air/liquid interfaces with different LS models^{33,34,35}, however they did not focus on exact relations to the real breathing conditions (in terms of temperature and surface deformation frequency). In the rheological formalism, the departure from the initial value of the surface tension: Δσ = σ − σ_{0} is connected with extensional deformation of the interface, γ:

$$Deltaupsigma ={Deltaupsigma }_{mathrm{E}}+{Deltaupsigma }_{mathrm{V}}={mathrm{ varepsilon gamma }}+upmu {dot{upgamma}}$$

(3)

where (dot{upgamma }) denotes the surface deformation rate. Equation (3) is the Kelvin–Voigt model of a visco-elastic air/liquid interface. The temporary surface dilatational deformation (extension) is defined as:

$$upgamma =frac{A-{A}_{0} }{{A}_{0}}$$

(4)

where *A* denotes the area of the interface at the given time instant *t*, and *A*_{0}—the initial interfacial area (at *t* = 0). Rheological parameters of the interface are denoted as ε (N/m—dilatational surface elasticity) and μ (s N/m—dilatational surface viscosity), while Δσ_{E} and Δσ_{L} represent the elastic and viscous contribution of the surface tension deviation, respectively. Small-amplitude harmonic deformation of the interface with the angular frequency ω (rad/s) is described as:

$$upgamma ={upgamma }_{mathrm{m}} ; mathrm{cos;omega }t$$

(5)

and the corresponding visco-elastic response takes the form:

$$Deltaupsigma ={Deltaupsigma }_{mathrm{m}} ; mathrm{cos} ; left(upomega t+mathrm{varphi }right)$$

(6)

where φ (rad) denotes the phase-shift (loss angle) that appears due to the viscous properties of the interfacial region. If φ > 0, the system shows the hysteresis. γ_{m} and Δσ_{m} in Eqs. (5) and (6) denote the amplitudes of surface deformation and surface tension, respectively. For purely elastic response of the interface, the loss angle φ is zero, and, if the viscosity predominates, φ approaches π/2. In general, the loss angle is equal:

$$mathrm{varphi }=mathrm{arc;tan}frac{mathrm{omega mu }}{upvarepsilon }$$

(7)

which indicates that surface tension hysteresis depends both on the viscosity-to-elasticity ratio and on the frequency of surface deformation. The hysteresis is larger when viscosity dominates over elasticity, and when oscillations are slower (i.e., ω is larger).

For surfactants that undergo mass exchange (dynamic adsorption and desorption) with the air/liquid interface (as in LS system), the situation becomes more complicated since both ε and μ may be not constant. In such a case, these parameters depend on the current surface deformation γ(*t*) because the composition the interface is changing during surface dilation or contraction due to the mass exchange, i.e. surfactant diffusion and adsorption/desorption^{31}. This process act towards the relaxation of Δσ. Obviously, surface deformation rate (dot{upgamma })(*t*) also affects the amount of the surfactant transferred between the liquid and the interface at a given time period. Therefore, in the system considered in this work, both surface elasticity and surface viscosity should be considered as not the real (intrinsic) rheological characteristics of the interface but rather as the apparent parameters of the interfacial region. A more comprehensive discussion of these issues may be found elsewhere^{31,33,34,35,36,37}.

The phenomenon of surface tension hysteresis has been analyzed and discussed in the relation to the physiological functions of LS^{13,26,30,38,39,40}. It has been postulated that the hysteresis is related both to mechanical aspects of breathing (for instance, to the pressure–volume hysteresis in the lungs) and to the mass transfer phenomena on the pulmonary surface, including the hydrodynamic clearance of inhaled deposits from alveoli^{26,38}. The rheological analysis of the interfacial dynamics discussed in the current work facilitates the quantitative assessment of these important processes.

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