Modelling Fluid Secretion
In this chapter a mathematical model of ﬂuid secretion from a single airway epithelial cell is developed. The model incorporates the electrophysiological components understood to be dominant in the ﬂuid secretion process. Particular attention is paid to those ion channels which are known to be controlled through intracellular calcium signalling.
2.1.1 Ion transport to regulate PCL depth
To maintain the optimal PCL depth of around 7 µm , water movement into and out of the airway lumen is regulated through active ion transport processes [410, 230, 55], with many of these processes known to be calcium dependent . Furthermore, it is intracellular calcium which is one of the main messengers initiated by released autocrine messengers after a number of stress states such as hypo-osmotic, compressive, and shear stress [50, 136, 155, 406, 458]. Normal airway epithelium has the capacity to absorb and secrete ions, predominantly Na+ and Cl−, thereby generating osmotic gradients which induce water ﬂow across the epithelium. The epithelial Na+ channel (ENaC) serves as the major conduit for Na+ through the apical membrane, and the Na+-K+-ATPase is the mechanism of basolateral extrusion. Transport of Cl− through the apical membrane is predominantly through the cystic ﬁbrosis transmembrane conductance regulator (CFTR) and the calcium-activated Cl− channel (CaCC). It has been suggested that the CFTR is responsible for resting Cl− currents while the CaCC is activated in response to stress . Under a number of stress conditions, airway epithelial cells release the nucleotides ATP and UTP into the lumen which trigger the release of Ca2+ from internal stores [99, 194, 469, 171, 7, 253]. A rise in [Ca2+]i activates CaCCs and calcium-activated potassium channels (CaKCs), resulting in increased ion currents and a ﬂux of water into the lumen.
The published mathematical models pertaining to airway epithelial function can be separated into two broad categories: those which have focused on the mucociliary transport system, and those which have focused upon epithelial cell electrophysiology.
188.8.131.52 Mucociliary transport models
Mucociliary transport models have in general concentrated upon the hydrodynamic interactions of cilia with the PCL and mucus [29, 42, 43, 72, 134, 128, 204, 251], with the most advanced of these considering ﬂuid interaction with the internal structure of a cilium . Although these models extend our knowledge of the mucociliary system, they address only the physics, and all but ignore the relationship of the system to its biology.
The attempts which have been made to model the mucociliary transport system as a whole have remained highly simpliﬁed, using simplistic geometry, and considering mucus as a onedimensional sheet travelling up the airways [465, 128, 167, 396]. The development of this type of mucociliary transport model has been motivated by the study of particle clearance, and as a consequence has led to the development of simpliﬁed descriptions of mucociliary clearance coupled to complex computational ﬂuid dynamic (CFD) particulate deposition simulations. In general, these models have been parameterised using data for healthy subjects breathing under “normal” conditions. This parameterisation immediately suggests considerable limitations for their use, especially since it is these models which have been used to study drug delivery in the unhealthy subject. More importantly however, none of these models have incorporated information about function at the cellular level which is immediately and directly relevant as it is the ciliary beat frequency, PCL depth, mucous viscosity, and intracellular calcium which inﬂuence the mucus clearance rate [252, 355, 350, 440, 112].
184.108.40.206 Airway epithelial cell models
The lack of models incorporating cellular information could in part be due to the limited number of attempts which have been made to model airway epithelial cell physiology. This lack of models is surprising due to the overwhelming evidence of epithelial cell dysfunction in diseases such as cystic ﬁbrosis. Models for the ciliated airway epithelial cell are few in number and have achieved varying levels of success. The ﬁrst of these were based on models of tight epithelia such as Latta et al. , Lew et al. , or the Verkman & Alpern  renal proximal tubule model. However, airway epithelium is leaky, involving paracellular as well as transcellular ion transport, and this requires additional complexity to be introduced into the model in order to maintain basic physiological consistency.
Of the ﬁve previously published models of airway epithelial cell electrophysiology, those of Hartmann & Verkman  , Duszyk & French , and Horisberger  concentrated speciﬁcally on ion transport rates within the epithelium. However, the inﬂuence of water movement was not incorporated in these models, which would aﬀect any estimation of ion transport rates if the cell were disrupted from homeostasis. Miller  proposed a model which does take into account water movement and cell swelling, but details of the model were not published. Their model attempted to explain regulation of PCL depth through the opening and closing of paracellular ion pathways, suggesting that upon a change in osmolarity the swelling or shrinking of the cell would force the paracellular pathways to open or close. The proposed mechanism has some merit, but immediately falls short due to its inability to explain how cells return to homeostasis after an isotonic challenge. Novotny & Jakobsson  developed a model which takes account of lumenal, intracellular and serosal compartments and more accurately described osmotic water transport. They proposed that PCL depth regulation is controlled through volume-sensing Cl− channels, although the mechanism through which volume is sensed was not identiﬁed. It does however serve to illustrate the eﬀect on PCL depth levels due to modiﬁcations to the apical membrane chloride conductance. There have been some attempts to determine the molecular identity of cell and PCL volume sensitive channels [301, 45], but these are not well, if at all, characterised in airway epithelia.
The most recent of previously published airway epithelial cell models is the model of Horisberger . This model was used to investigate the electrical coupling between the Na+ and Cl− channels. It has been suggested that epithelial cells control ion movement through a blended operation of the Na+ and Cl− channels [36, 412] with regulation of the epithelial sodium channel (ENaC) operating through the cAMP dependent CFTR [397, 399, 336, 334, 335]. This hypothesis has led to the further postulation that it is the absence of Na+ channel inhibition by the CFTR that is responsible for abnormal electrolyte transport in CF airway epithelia. The Horisberger  model however, suggested that the Cl− conductance has a signiﬁcant eﬀect on the rate of Na+ transport through electrical coupling in the parallel and apical pathways. This result suggested that despite no coupling between the two channels – other than through electrical potential – electrolyte transport was strongly coupled. Indeed, this has also been suggested from experimental data by Nagel et al.  who found no evidence that activated CFTR channels regulate Na+ transport through the ENaC channel.
Of these published airway epithelial cell models, only Novotny & Jakobsson  provide a rigourous attempt to consider the important aspect of transcellular water movement. Furthermore, none of the models made any signiﬁcant attempt at including any cellular signalling. It is of course intracellular signalling that is responsible for changes in a cell’s ion and ﬂuid secretion rates. In this chapter no substantial detail of second messengers or autocrine signalling is considered, but instead a model is developed of an epithelial cell that includes the major ion pathways that contribute to the control of water ﬂuxes. Volume sensitive channels have not been included. This allows the response of the model to be investigated despite a lack of direct volume sensing. Intracellular calcium signalling is considered in more detail in Chapter 3.
In this section the mathematical description of the electrophysiology of a ciliated airway epithelial cell is constructed. Fig. 4.5 shows a schematic diagram of the mechanism by which ion channels, pumps, and cotransporters establish an osmotic gradient to drive water ﬂow. Particular attention is paid to the components that are thought to be dominant in the ﬂuid secretion process. The model is divided into three compartments: PCL, intracellular, and serosal, with the constituents of each of these compartments denoted by the subscripts p, i, and, s, respectively. All apical and basolateral membrane components are denoted with the subscripts a and b, respectively.
Volume regulation and water ﬂux
The underlying principle used to describe the change in cell and PCL volume is the conservation of water volume. This principle inherently assumes that the density of water does not change under the operating conditions of the model. Thus in the conditions considered, water must not be compressible or undergo signiﬁcant thermal expansion. Incompressibility is easily justiﬁed given the bulk modulus of water is 2.2 GPa  and therefore would require a pressure of 22 MPa (equivalent to the pressure exerted by water at a depth of 2200 m) to produce a 1% decrease in volume. The volumetric coeﬃcient of thermal expansion for water is a miniscule 207×10−06 K−1 , therefore a 1% increase in volume requires a temperature rise of 48 ◦C. Furthermore, it is assumed here that the solute partial molar volumes are negligible. The diﬀerential equations describing the rate of change of the PCL (wp) and epithelial cell volume (wi) can be written as,
dwp dt = AaJw a − AaJevap, (2.1) dwi dt= AbJw b − AaJw a , (2.2)
where Aa and Ab are the apical and basolateral membrane areas respectively and are assumed to be independent of the cell volume. Therefore, changes in cell and PCL volume are considered to be directly proportional to a change in height. The water ﬂuxes Jw a and Jw b are across the apical and basolateral membranes, respectively, while Jevap is the rate of evaporation (or condensation) from the PCL.
Water ﬂux, like any ﬂuid transport across a semi-permeable membrane, is governed by pressure diﬀerence. Within biological systems it is often assumed that cell membranes cannot support hydrostatic pressure gradients , and therefore the pressure driving force results solely from osmotic pressure. The equations of Kedem & Katchalsky1 are used to describe water movement here, where positive ﬂow is deﬁned in the direction towards the PCL compartment:
Jw a = LaVwψp wp−ψi wi+ ([Cl−]p + [Na+]p + [K+]p) − ([Cl−]i + [Na+]i + [K+]i), (2.3)
Jw b = LbVwψi wi−ψs ws+ ([Cl−]i + [Na+]i + [K+]i) − ([Cl−]s + [Na+]s + [K+]s), (2.4) where Vw is the partial molar volume of water and La and Lb are the hydraulic conductance of the apical and basolateral membrane, respectively. From the formulation in Eqn. 2.3 – 2.4 for water ﬂux, it can be seen that zero water transport will only be achieved when the osmotic gradient across the cell’s membrane is zero. Due to the model only keeping track of the Na+, Cl−, and K+ concentrations, it is necessary to account for other ions, proteins, or environmental conditions which may contribute to osmotic pressures. These additional osmolytes are accounted for by deﬁning a molar value for non-permeable osmolytes in each compartment, denoted by ψp , ψi, and ψs for the PCL, intracellular and serosal spaces, respectively.
Within the PCL, the parameter ψp contains all extra factors capable of generating osmotic forces, such as the capillary surface tension forces acting around cilia. An epithelial cell surface has a density of approximately 8 cilia per µm−2 . Each cilium is approximately 200 nm in diameter and 7 µm in length , and thus cilia increase the apical membrane surface area by approximately 30-fold. It has been calculated that the surface tension generated by the closely spaced cilia generates a pressure corresponding to approximately 40 mOsM . Mucins however are too large (molecular weight of several thousand kDa [431, 432, 436]) to be osmotically active in the airway surface liquid with a concentration of only about 2%
In the experiments of Tarran et al.  the osmolarity of the PCL layer was measured at 320 ±11 mOsM after the cells were left to equilibrate for 36 hours. The osmolarity of the PCL was also found to be comparable to that of the serosal medium. The estimated total ion concentration in the PCL was 16 mOsM greater than the sum of the osmotic contributions from Na+, K+ and Cl−, and 25 mOsM less than the total osmolarity of the PCL. These values suggest that the PCL contains not only additional ionic osmolytes, but also molecules such as glucose or amino acids. Using these data, values were chosen for ψp , ψi, and ψs such that the sum of the osmotic contribution from all species in each compartment produced an osmolarity of 320 mOsM when the ion concentrations (Na+, Cl−, and [K+]s) were at their experimentally measured resting value.
Due to the exact ionic composition of airway surface liquid still being debated, the concentrations used here have been taken from Tarran et al.  in order to maintain consistency between the model and experimental results which will later be simulated. The implications of selecting these ion concentration values will be discussed in Section 2.4.
The high hydraulic permeability within the respiratory epithelium is consistent with transcellular water ﬂow through the aquaporin class of molecular water channels [203, 202, 435, 434]. Aquaporins (AQPs) are integral membrane proteins which function as molecular water channels in a variety of ﬂuid transporting tissues. To date, ten mammalian aquaporin proteins (AQP0 – AQP9) have been identiﬁed [235, 435]. Aquaporin isoforms have a distinctive tissue distribution pattern (e.g., apical versus basolateral membrane in epithelia) and permit rapid and speciﬁc water transport in response to osmotic gradients . Four AQPs have been identiﬁed in the respiratory tract: AQP1, AQP3, AQP4, and AQP5, each with a unique distribution suggesting distinct physiologic roles . In bronchial epithelium, signiﬁcant expression levels of AQP5 in apical membrane and AQP4 in the basolateral membrane have been detected . Fig. 2.2 shows the AQP4 expression in a human bronchus using immunoﬂuorescence.
The apical membrane water permeability La was obtained from cultures of well diﬀerentiated normal human bronchial epithelia . Despite La being measured at 240 µms−1, Matsui et al.  noted that when accounting for unstirred layers in the PCL, the La may exceed the value of 360-545 µms−1 estimated for gallbladder [74, 312]. In the model the initial value of 360 µms−1 is used for the apical membrane permeability which is the lower limit of these estimated values. Furthermore, it has been documented that in airway epithelium the basolateral water permeability is considerably lower than the apical membrane [263, 455]. It is likely that this is due to the expression of diﬀerent AQPs. To account for this reduced basolateral membrane permeability Lb is initially taken to be 1/4 of La.
List of Figures
List of Tables
Acronyms and Abbreviations
List of Symbols
1.1 Physiology and anatomy of the human pathway for ventilation
1.2 Ciliated airway epithelial cells
1.3 The mucociliary transport system
1.4 Multi-scale modelling for multi-scale physiology
1.5 Thesis objectives
2. Modelling Fluid Secretion
2.2 Model development
3. Modelling Cellular Signalling
3.2 Model Development
4. Intercellular Signalling and Multi-scale Modelling
4.1 Multi-scale modelling techniques
4.2 Intercellular calcium signalling
5. Heat and water vapour transport
B Mathematical preliminaries
C The ﬁnite element method
D Computational software
GET THE COMPLETE PROJECT
A Multi-Scale Computational Model of Fluid Transport in the Human Bronchial Airways