Novel Computational Model of the Brain Water Metabolism: Introducing an Interdisciplinary Approach

The brain interstitial and the cerebrospinal fluids are involved in the nutrient and gas transport, non-synaptic intercellular communication (volume transmission), signal transduction, transport and targeted delivery of drugs and metabolites, ionic homeostasis, formation and resolution of the brain β-amyloid deposits, the migration of cells (malignant cells, stem cells), transfer of heat generated by neuractivity [1,2]. Introduction Abstract

The width of the interstitial space that typically occupies 20% of the brain volume is in the range 20-64 nm [12]. From this fact, an opinion has been born and commonly accepted in medical community that the narrowness of the spaces between the cells within the neuropil and the tortuosity of the interstitial space will not permit any significant fluid flow there [13]. Diffusion has been considered a dominant governing mechanism there with the conventional wisdom maintaining that the interstitial space presents a diffusion barrier to fluid movement [13][14][15][16][17].
An interdisciplinary approach makes it possible to view the issue in a different light. The nanodimentionality of the interstitial space suggests a nanofluidic approach to the interstitial space hydrodynamics. Nanofluidics deals with the behaviour of fluids confined to nanoslits, nanochannels, nanopores, etc., where at least one geometrical dimension is in the nanometre scale, usually in the range of 1-100 nm [18]. Surface effects in nanoconfined fluids dominate over the bulk-water properties while the fluids exhibit physical behaviour not observed in larger structures, such as those of micrometre dimensions and above [19] .
A special feature of water movement in nanodimentional spaces manifests itself in significant enhancement of the fluid flux due to the surface hydrodynamic slip [20,21]. Thus aquaporins, the water-conducting nanopores, exhibit water permeability typically three orders of magnitude higher than expected from the classical no-slip framework for the same pore size [22]. From the conventional point of view, this seems counterintuitive and even unsupported when using the no-slip Darcy or Hagen-Poiseuille's equations, as has been done so far to model fluid movement in the brain [23,24].
Recently, there appeared a research paper on monitoring the state of water in the brain extracellular space. On injecting singlewalled carbon nanotubes into the rat brain and using super-resolution imaging, the researchers observed the rheology in live brain interstitial space down to 40 nm. It is the very first direct visualization of the state of nanowater in the interstitial space of the live brain [25]. An interdisciplinary approach to the rheology in the brain interstitial space makes it possible to conceptualize the brain interstitial space as a functional nanofluidic domain where water movement is governed by the slip-flow mechanism.
Pulsatility is a feature intrinsic to the functioning of the brain. Within its rigid confinement, there are generated waves of pulsating pressure related to the heart activity. The intracranial pressure pulses propagate through the brain at the speed of sound with their dynamics being identical over the brain parenchyma [35][36][37]. The osmotic pressure in the blood, the interstitial fluid (ISF) and the cerebrospinal fluid (CSF) is identical and, under physiological conditions, there is isosmotic movement of water from blood into brain tissue. This has been demonstrated in animal experiments with the injection of the tritiated water into the systemic blood [26,38]. The pulsatile intracranial pressure gradients present a driving force behind the isosmotic flow of the brain fluids.
The aim of the ongoing paper has been to develop a computational model for brain water exchange at the capillary level based on the new knowledge and research information. The model should be able to demonstrate good predictability and account for some pertinent experimental and clinical observations on water metabolism in health and disease.
Theoretical and experiment-based assumptions of the model:

Methods
-the brain nanodimentional interstitial space presents a nanofluidic domain with the fluid movement there governed by the slipflow mechanism [18,25]; -aquaporin AQP4 ensures kinetic control over water movement between the blood and the brain interstitial space [26,29,34,39]; -the overall fluid transfer between the capillary blood and the interstitial fluid is isosmotic [26,38]; -the pulsatory intracranial pressure presents a driving force behind the isosmotic fluid exchange between the capillaries and the interstitial space [35,36,40].
A capillary presents a rigid-walled cylinder with Poiseuille's viscous laminar flow of Newtonian fluid. A modified phenomenological fluid flux equation, based on the Kedem-Katchalsky formalism of irreversible thermodynamics [41], has been developed to describe water movement between the capillary blood and the brain interstitial space: where Jv (cm 3 /s) is the volumetric flow rate per unit transfer area (cm 2 ). L p (cm/s/ mmHg per sm 2 ) is a compound hydraulic conductivity coefficient: where is 4 AQP p L the hydraulic conductivity coefficient associated with AQP4 and p L * is the hydraulic conductivity coefficient of possible other parallel water transfer pathways. Within the context of the ongoing work, we assume that The hydraulic conductivity coefficient for the astrocyte endfeet membrane expressing AQP4 has been calculated using the data on the water permeability coefficient (P f ) and AQP4 specific density in the astrocyte endfeet membranes by conversion P f to The capillary longitudinal hydrostatic pressure drops linearly from the arterial to the venous end. The axial hydrostatic pressure gradient, Δp (mmHg/cm), is: where p a (mmHg) and p v (mmHg) is the hydrostatic pressure at the arterial and the venous ends of the capillary, respectively, and L(cm) is the length of the capillary. The longitudinal distance at any point of the capillary axis, starting from the arterial end, is x (cm). The intracranial pressure as a function of time (the pulse waveform) is f(t); π c (mmHg) is the capillary plasma oncotic pressure and π ISF (mmHg) is the interstitial fluid oncotic pressure.
The volume of water, J hc , transferred over a specified period of the heart cycle and a cross-section area is found through numerical integration: The model has been developed, using Wolfram Mathematica language, in Wolfram Mathematica 10 program. Parameters and variables for the basic model are given in Online Resource, 3. Figure 1 present simulation results of water movement between the capillary blood and the ISF over a complete cardiac cycle. The heart rate as well as the parameters of intracranial pressure pulse wave stay within physiological limits. The water filtration rate (water movement out of the capillary) is positive while the water reabsorption rate is negative.

Description of the model
The simulation results in Figure 1, obtained with the intracranial pressure pulse wave W1 (Figure 1b), demonstrate a complex character of the transcapillary water exchange over the length of the capillary. The water flux is positive in the CFA, negative in the CRA and oscillatory in the COA. The overall water flow is pulsatory and synchronized with the phases of the heart cycle. (2) The water flow in the COA presents a special interest. An exploded view of the COA ( Figure 2) demonstrates the details of water exchange there. The infinite plane p, passing through the trough in the COA, separates the systolic part of the COA, on the left side of the divide, from the diastolic part, situated on the right side of the infinite plane. On the time axis, the systolic and the diastolic parts of the COA have coordinates 0 ≤ t ≤ 0.12s and 0.12 ≤ t ≤ 0.83 s, respectively. SFS is the systolic filtration segment; DFS is the diastolic filtration segment; SRS is the systolic reabsorption segment and DRS is the diastolic reabsorption segment. The bar graph shows integral volumes of water, J hc , exchanged over one heart cycle. The filtration volume -1; the reabsorption volume-2; the net volume-3 Figure 3: Schematic presentation of the nanofluidic mechanism for the brain water metabolism The red arrow indicate direction of the blood flow in the capillary. The dashed blue line with AQP4 present this aquaporin in the astrocyte endfeet membrane enveloping the capillary. The functional segments of the capillary, similar to the 3D presentation of the model in Figure 1, are indicated with p 1 , p 2 , p 3 , and p 4 . The blue arrows, with the volumetric flow rate symbols "Jv", "±Jv", and -Jv", indicate the direction of the water flows in the CFA, the COA, and the CRA, respectively. PVS indicates the perivascular spaces. The oncotic pressure in the capillary and the interstitial fluid is π C and π ISF , respectively. Hydrostatic pressure is p a at the arterial end of the capillary and p v at the venular end. The external pulsatory pressure is p(t)= f(t). The square brackets indicate that the water exchange between the capillary and the ISF takes place within the AQP4-controlled nanofluidic domain. Note that the schematic is not drawn to scale and does not represent the true ratio between capillary diameter and the capillary length Figure 2 demonstrates the way the ICP pulse wave, on passing over the COA segment, causes the radial flows of water to oscillate between the filtration and the reabsorption functional modes. The volumes of water in each segment have been obtained on numerical integration of the corresponding 3D graphs (Online resource, 4.2-4.5). The bar graph in Figure 2 presents the lumped quantitative data on water movement in the COA. This graph demonstrates an oscillatory pattern of water exchange between the capillary blood and the ISF. Under physiological conditions, the net fluid flux is positive (Bar 3). This oscillating pattern might ensure a feedback exchange mechanism between the blood and the brain, a possibility discussed later in this paper.
The simulation results, combined with the new knowledge make it possible present an outline of the nanofluidic mechanism for the brain water metabolism, Figure 3.

Simulation results and predictability of the model
It is interesting to see how the model would account for some features of brain water metabolism and the effects of various factors influencing it. Among those are the pulsatile nature of the cerebral fluid flow, capillary length and the transcapillary water exchange, the effect of elevated intracranial and venous pressure on the transcapillary water movement, AQP4 lateral polarization in the astrocyte endfeet membrane and the net water flux, to list a few clinically relevant ones. Figure 4 presents the results of the model-computed predictions on the brain water metabolism. Figure 4a,b present the volumetric flow rate profiles in each capillary segment over a few cardiac cycles. A common feature of the flux pattern in each segment is the pulsatile nature of fluid flow synchronized with the phases of the heart cycle. This might seem obvious from the mathematical description of the model. Indeed, the phenomenological fluid flux equation (Eqn.1) includes the intracranial pressure pulse waveform function, f(t), that accounts for the pulsatile pattern of fluid flow. The intracranial pressure pulse waves present a transformation of the heart-driven pulsations of the larger cerebral arteries [37,42]. Pulsatile flow of the CSF may have a more complex nature due to various factors contributing to its particular pattern like a respiratory component, head position etc. In this paper, we consider the cardiac-induced pulsatility only.

Pulsatile pattern of fluid flow
The length of brain capillaries lies within a wide range [43,44]. Averaged from various geometrically different zones, the mean Capillary length and the net transcapillary fluid exchange   capillary length for the human brain is estimated at about 53 μm [45]. In the present model, the length of a capillary L= 500 µm. Taken against the mean capillary dimension, it gives a possibility to estimate transcapillary fluid exchange with the capillaries of length L * with L > L * . Figure 4c presents water filtration and reabsorption patterns on stepwise reduction of the capillary length starting from either the venular or the arterial end of the capillary. The upper set of the bars demonstrates an increased filtration volumes on the capillary length reduction from the venular end. Reabsorption increases on the capillary length reduction from the arterial end (the set of bars on the negative scale). Variations in the length of a capillary results in establishing different axial hydrostatic pressure profiles. The axial shift of the hydrostatic pressure profiles effects the net driving force, direction, and magnitude of the capillary radial flow rates.

AQP4 polarization in the astrocyte endfeet membrane and the net radial flow rates
AQP4 presents a rate-limiting step in water movement across the BBB. Lateral distribution of AQP4 in the astrocyte endfeet membrane is region-specific and tends to vary under both normal and pathological conditions [34,46]. Polarization of AQP4 in the astrocyte endfeet membrane sheathing a capillary finds its reflection in the region-specific values for the hydraulic conductivity coefficients. This is the way a polarized AQP4 expression may effect fluid movement between blood and the interstitial space.
We modelled the effects of AQP4 polarization by assigning different hydraulic conductivity coefficients (

Elevated intracranial pressure and the net fluid filtration rate
Animal experiments demonstrate that increasing intracranial pressure slows down CSF production [47]. The simulations carried out with the two pulse waveforms differing by their peak intensities, shown in Figure 1a, demonstrate that elevation of the intracranial pressure by 5 mmHg peak intensity results in the overall fluid filtration decrease by 84% (Figure 4e). This might serve for an explanation for the reduced CSF production.

Elevated venous hydrostatic pressure and the transcapillary fluid exchange
Elevation of the venous hydrostatic pressure results in an increased filtration (decreased reabsorption) of the water as shown in Figure 4f. It is interesting to observe that elevation of the venous hydrostatic pressure by mere 0.5 mmHg (the first two bars in Figure 4f) brings about a complete reversal of the capillary fluid exchange pattern: the fluid-absorbing mode of the capillary at 15 mmHg turns into a fluid-filtrating one at 15.5 mmHg. The dependence of the transferred fluid volumes on the hydrostatic venous pressure is linear.
The water metabolism disorders manifest themselves in a broad spectrum of cases and present the centre of attention to many research groups focused on finding effective therapeutic means to correct them. The development of predictive mathematical models of brain water metabolism, that would help to explain the fundamental science behind brain physiology and various clinical conditions, becomes very important.

Discussion
The brain fluids present important inner mobile medium closely involved in physiological functions of the brain [14,48]. It is clearly understood at present that the knowledge of fluid/water dynamics in the brain interstitial space together with that of neural circuits is the way to clarify the complex physiology of the brain [1].
The researchers dissatisfied with the diffusion barrier theory suggest fluid convection in the interstitial space [4,5,7,14,26,50,52]. However computer simulations, using a conventional no-slip approach, demonstrate that any significant convection there would demand unrealistically high hydrostatic pressure gradients [15]. The situation seems to have reached a stalemate.
There is a standing controversy at present about the mechanism of brain water metabolism. Emerging new theories challenge the classical theory and put stress on fluid formation and reabsorption at the microvascular level all over brain parenchyma [8,9,49,50]. The adherents of the diffusion theory maintain that the laws of diffusion are sufficient to explain the events taking place there [15,16]. Within their concept, the interstitial space is looked upon as "a diffusion barrier" [17]. From the characteristic nanodimentional scale of the interstitial space they have come to a conclusion that the spaces between the cells, within the neuropil, are too narrow to permit any significant bulk flow [51].
An interdisciplinary approach makes it possible to see the issue of fluid movements in the brain interstitial space in a different light. The nanodimentional scale of this space has long been recognized [16,51]. The next logical step would have been to consider fluid movement there as governed by the nanofluidics principles. Indeed, in the nanoconfined spaces the surface hydrodynamic slippage of water brings about an increase of the flow rates that become orders of magnitude higher than those predicted from the no-slip Hagen-Poiseuille's or Darcy's equations [18,21]. The nanofluidic approach has the promise to resolve the current stalemate situation and to open new perspectives in understanding brain physiology and diseases.
We first introduced a conceptual nanofluidic approach to water movement in the brain interstitial while conducting an MRI clinical research on cerebral water metabolism [53]. In the present model, we view the brain interstitial space as a distinctive nanofluidic domain whose very dimensionality dictates that the slip-flow phenomena should govern fluid dynamics there. This assumption means that water movement in the interstitial space is fast enough to ensure a considerable convectional flow and to shift the overall kinetic control to AQP4.
Experimental data demonstrate that AQP4 is a key molecule controlling water exchange between the blood and the brain parenchyma. The experiments on the AQP4 knockout animals and those with the AQP4 inhibitors strongly support this AQP4 function [33]. In translational medicine, AQP4 is considered a target molecule in the drug therapy of various pathologies, arising from water metabolism disorders, a guide for developing new therapeutic approaches [54][55][56][57][58][59].
Confronted with the AQP4 water-flow-controlling function, the conventional diffusion-barrier concept becomes a kinetic controversy. The diffusion barrier would be the slowest step in the overall water movement sequence, thus making AQP4 control function redundant. However, this would contradict the vast experimental and clinical data on the AQP4 control function.
The nanofluidic domain concept removes this controversy. It also resolves the problems of the convectional no-slip flow theories that demand high-pressure gradients to ensure convectional water movement through the nanodimentional interstitial space. The nanofluidic approach implies very fast fluid movement in the brain interstitial space, which makes AQP4 the slowest step and puts it in control of the overall fluid flow rate.
The nanofluidic model demonstrates that within brain microvascular network there are found the filtrating and reabsorbing capillaries responsible for water metabolism and whose function depends on the phase of the heart cycle, hydrostatic pressure gradients, capillary length, and AQP4 polarization. These regulatory factors are important for brain fluid homeostasis and physiology.
The simulation results (Figure 4d) demonstrate that altered lateral polarisation of AQP4 in the astrocyte endfeet membrane may have far-reaching consequences for the brain water metabolism. AQP4 content and polarization depend on the brain cortex local physiological activity. Changes in AQP4 subcellular polarization occur in response to various conditions including stroke, Alzheimer's disease, brain tumours and many others pathologies [60,61]. The nanofluidic model describes the ways AQP4 lateral polarization effects brain water metabolism and quantifies its impact. The sensitivity of the brain water metabolism to the changes in the AQP4 lateral polarization underscores the importance of detailed knowledge of AQP4 topography at nano-and microscale in the astrocyte endfeet membrane enveloping a capillary.
The nanofluidic model makes visible the details of the transcapillary water exchange process in relation to the phases of the heart cycle. A distinct oscillatory pattern of fluid movement in the COA deserves a special attention. The oscillatory fluid flux here may serve as a means of communication between the periphery and the central nervous system. The fluid crossing the BBB carries information to the systemic blood about the state of brain metabolism and brings a feedback information from the periphery. The frequency of this exchange is synchronized with the heart cycles.
This feedback exchange within the COA may have important implications for the local neurovascular coupling that requires a close temporal and regional linkage between neural activity and cerebral blood [62]. The shifts in the local functional and metabolic demands of the brain are accompanied by corresponding blood flow responses [45]. Increased capillary flow would necessary result in establishing a new longitudinal pressure profile in a capillary. This may be a way of regulating capillary filtration capacity and switching capillary function from fluid-filtrating to fluid-reabsorbing one. This approach makes it possible to have a glimpse at complex interplay of the capillary fluid exchange function. It might be viewed as a demonstration of how capillarylevel response, in terms of magnitude and direction of radial fluid fluxes, can be attuned to brain homeostatic and physiological demands.
The nanofluidic model may have implications for the translational medicine. The ICP pulsations and intracranial pressure waveforms present much interest in clinical diagnosis of neurological diseases. The model makes it possible to observe the details of the transcapillary fluid exchange in relation the phases of the ICP pulse waveforms generated by the heart activity. It demonstrates the ways the intracranial pulsatility may affect ISF/CSF hydrodynamics and might be useful in solving some experimental and clinical problems in neurology and neurosurgery. The developed nanofluidic model demonstrates the ways the capillary-level water-exchange response, in terms of the magnitude and direction of radial fluid fluxes, is affected by the AQP4 activity and polarization, the intracranial pressure, the capillary length. It might find its use in elucidation of the pathogenic mechanisms of cerebrovascular disease, brain edema, hydrocephalus, Alzheimer, dementia and others.
The model might be used for development of the AQP4-targeted drug therapy, optimization of the intrathecal drug delivery in the brain tumour therapy, and in the therapy of a broad spectrum of water-metabolism-related disorders. An adopted interdisciplinary approach has made it possible to consider the brain interstitial space as a nanofluidic domain where water movement is governed by the slip-flow principles of nanofluidics. The model allows for a new look at the brain fluid physiology.
In the developed computational model, AQP4 kinetically controls the movement of the interstitial and the cerebrospinal fluid in the brain nanofluidic domain. AQP4 presents a molecular target for a directed action of the drugs that might be used for the therapy of a number of clinical conditions. The model demonstrates good predictability in respect to some clinically relevant cases of water metabolism disorders. It underscores the importance of the cardiocerebral relationship for the brain water metabolism. The model describes a molecular-level approach to a possible control and therapy of the water-metabolism-disorder-related conditions.
The author would like to express his gratitude to the Head of the Biophysics Department of the Belarussian State University Professor S. Cherenkevich, Ph.D. for useful discussions. Submit your next manuscript to Annex Publishers and benefit from: Submit your manuscript at http://www.annexpublishers.com/paper-submission.php → Easy online submission process → Rapid peer review process → Open access: articles available free online → Online article availability soon after acceptance for Publication → Better discount on subsequent article submission → More accessibility of the articles to the readers/researchers within the field