Can Electrical Effects Slow the Circulation?
Julie Kim
In collaboration with: Michael Zhang
Supervised by: Dr. Richard Clarke
University of Auckland | Engineering Science
What is the EGL?
Lining our blood vessels is a structure named as a Endothelial Glycocalyx Layer (EGL) which is in a form of hairy brush-like structure sitting on top of its own gel-like structure. The EGL is composed of a range of membrane-bound molecules including glycoproteins imparting a net negative charge to the EGL (1). The EGL is involved in a number of significant regulatory roles regarding the functioning of the blood vessels, such as mechanotransduction, modulation of trans-capillary exchange of water, and regulation of red and white blood cell interactions (2).
Why do we care about them?
Experimental studies (4) and computational modelling (5) of the EGL has revealed that the EGL is involved in an additional functional role of resisting a blood flow which explains the higher resistance values measured in vivo, compared to the values derived from the glass tube assumption. While these findings focused on a single vessel, our project aims to investigate whether the EGL can play a significant role on the overall resistance of the microvascular network. If a contribution of charged effects of the EGL on the overall resistance of the network is significant, then this will open an opportunity to develop a therapeutic strategy concerning the cardiovascular health where the regulation of the flow behaviour in the network can be done via controlling the EGL.
Method
To calculate network resistance values, we employ a lumped parameter model approach. That is, we model a vascular network as an electrical circuit with the following analogies between the variables of each system.
Vascular network
Blood flow
Pressure gradient
Hydraulic resistance
Electrical circuit
Electrical current
Voltage gradient
Electrical resistance
In the lumped parameter model, each vessel segment is characterised with an individual resistance. To use this approach, we need to know two things:
network topology
&
individual resistance values for each vessel
Method
Mathematical modelling
To model a microvessel lined with the electrically charged EGL layer, we need to make a few assumptions first. This will help us answer the question in an efficient and accurate manner. We consider steady-state blood flow without any cellular elements, so effectively modelling plasma flow. Since the vessel deforms only slightly when there are no blood cells, the vessel is modelled as a rigid structure. We assume that the vessel does not exchange fluid with the surroundings, so there is no leakage.
Reality
Mathematical model
Dynamics inside 
the microvessel
Lumen
EGL
Two phases
The core region of the vessel or the lumen consists of two phases: fluid and ions. The fluid experiences two external body forces from the ions: osmotic pressure force and the force due to the electric field.
Osmotic pressure
Osmotic pressure force arises from a local difference in the ion concentrations.Electric field
The electric field is established from a local difference in the net charge, which results in a body force.
Ion transport
The ions are transported throughout the vessel by three different mechanisms: diffusion, advection due to the background flow, and migration due to the electric field.
Diffusion
A local difference in ion concentration causes the ions to diffuse from a region with a lower concentration to a region with a higher concentration.
Advection due to the background flow
The ions are carried by the background flow.
Migration due to the electric field
Due to the charged nature of ions, they migrate following the directionality directed by the electric field established.
Three phases
In the EGL, there is an additional phase of solid. This leads to a momentum loss of the fluid when it comes to interacting with the solid phase.
Dynamics outside 
the microvessel
Even though we model an individual vessel, it needs to be representative of a part of a bigger network. Hence, what’s happening outside of the vessel is reflected in the model through the conditions imposed on the boundaries of the vessel.
Inlet
Mass flow rate
The cardiovascular system works together to transport blood around the body to maintain a given cardiac output. This is reflected in the model by prescribing a mass flow rate of the fluid at the inlet of the vessel.
No net current
A blood vessel is like an open circuit, that is not subject to an externally applied electric field. Hence, a condition to ensure that there is no net current entering the vessel is imposed on the inlet.
Ion concentrations
Ion concentration distribution from the upstream of the vessel is specified on the inlet.
Outlet
Pressure
Reference pressure value that is characteristic to microcirculation is specified on the outlet of the vessel.
No net current
Since there is no net current entering the vessel, there cannot be a net current leaving the vessel.
Ion concentrations / fully developed
On the outlet, ion concentration distribution of the downstream of the vessel is specified, or the ion concentration distribution is assumed to be fully developed and no longer changing in downstream.
Wall
No trans-capillary exchange of fluid allowed in the model imposes the following conditions on the vessel wall.
No slip
A layer of fluid immediately adjacent to the vessel wall sticks to the wall because of viscosity.
Zero ion flux
Ions are not allowed to escape through the vessel wall which is modelled impermeable.
Zero voltage gradient
Zero voltage gradient is specified to reflect the fact that there cannot be a surface charge on the wall while the EGL is charged volumetrically.
Method
Computational modelling
Results
Surface plot of pressure and arrow surface of velocity
Fixed charge concentration = 10%
Fixed charge concentration = 100%
Varicose
Pressure drop = 115.6
Pressure drop = 133.8
Velocity distribution at the centre of the vessel
Fixed charge concentration = 10%
Fixed charge concentration = 100%
To investigate the electrical effect of the EGL on the hydraulic resistance, two different values of fixed charge concentration of the EGL has been simulated for the varicose model: when the fixed charge concentration of the EGL is reduced to be 10% of its normal value, and when it’s at the normal value, which is the same as the neutral salt concentration in blood plasma. The most noticeable difference between the two cases is the appearance of the reversed flow in the EGL region when the fixed charge concentration of the EGL is not reduced from its normal value. This is a consequence of the system trying to counter-balance the current generated by the flow to ensure that there is no net current flowing in the vessel*. Even both cases required to deliver the same flow rate of the fluid, a larger driving force, which is the pressure drop across the inlet and the outlet in a case of the circulation system, is required to required to push the flow when the fixed charge concentration is increased to its normal value. When the fixed charge concentration is reduced, the flow is still retarded by the EGL through its porous structure, but the effect of the resisting mechanism by the EGL is enhanced further by the electrical charge. The results tell us that the charge effect does increase the resistance values of a single vessel.
Note that the quantities stated without the units are the values which have been scaled to the characteristic values of the parameters pertaining to microvessels. This is because we are more interested in qualitative behaviour.
*Why do we observe a reversed flow? →
Without the EGL, the blood flows through the vessel satisfies the electroneutrality condition. That is, positive charges are balanced by the negative charges. 
However, when the negatively charged EGL is present, the electroneutrality condition of the flow is broken, and a streaming current is generated due to a difference in the total flux of positively charged ions and negatively charged ions.
However, when the negatively charged EGL is present, the electroneutrality condition of the flow is broken, and a streaming current is generated due to a difference in the total flux of positively charged ions and negatively charged ions.
As a result, an equal and opposite streaming potential is established to prevent a net current from forming. This can cause a reversed flow in the EGL region.
Resistivity values for individual varicose vessels
The resistivity values for a range of vessel radiuses have been calculated and compared for different geometries: straight-walled channel versus varicose channel. The results show that the resistivities calculated using a simplified geometry of a straight-walled shape are different from the values obtained using a complex shape; a higher resistivity is observed when the shape is varicose. However, the trend of an amount of the resistivity value increased by with increased fixed charge concentration being greater in smaller vessels are still observed in both of the geometries. Now we have individual resistance values, we can use these in our network resistance calculations.
*How is resistivty value calculated? →
Recall that there is a natural similarity between the blood flow in the circulatory system and electric conduction in a circuit. Hence, hydraulic resistance can be calculated by dividing the blood flow (equivalent to electrical current) by the driving force of the circulatory system (which is the voltage gradient in the electrical circuit). To calculate the “driving force” that is a suitable measure to our problem, we carry out the calculations in the following order:
- Evaluate the pressure integral at the inlet and at the outlet, and subtract the latter from the former.
- Divide this quantity, which measures a pressure drop across the inlet and the outlet, by the length of the vessel to obtain the “driving force” that is comparable across different vessel geometries.
When the blood flow rate is divided by this “driving force” we obtain resistivity value which is a resistance per unit length.
Bifurcation
The graph on the left shows resistivity calculations made for the lumped parameter model and the bifurcation model. We see that the resistance value calculation is similar between the two approaches in the parent vessel, but is quite different in the daughter vessels where the resistance value increases by about 30% in the bifurcation model. This implies that the flow cares about the larger scale geometry features.
Conclusions
The magnitude of the fixed charge concentration attributes to the resistance in a similar behaviour regardless of the geometry type.
Controlling the vascular resistance by controlling the fixed charge concentration of the EGL is possible while this will have a greater influence on the smaller vessels.
Bifurcation node, which is a large scale geometry, does not influence the dynamics of the flow in upstream (parent vessel), but it influences the dynamics of the flow in downstream (daughter vessel).
While the lumped parameter model allowed us to solve a big network problem in an efficient manner, a more complex model might be needed to get a better representation of a network.
So, Can electrical effects slow the microcirculation?
Yes
Future work
Add more physics that is characteristic to microvessels such as leakage
Investigate the effect of the charged EGL on larger vessels
Model 3D geometries
References
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- van den Berg BM, Vink H, Spaan JA. The endothelial glycocalyx protects against myocardial edema. Circulation research. 2003 Apr 4;92(6):592-4.
- Smith ML, Long DS, Damiano ER, Ley K. Near-wall μ-PIV reveals a hydrodynamically relevant endothelial surface layer in venules in vivo. Biophysical journal. 2003 Jul 1;85(1):637-45.
- Sumets PP, Cater JE, Long DS, Clarke RJ. Electro-poroelastohydrodynamics of the endothelial glycocalyx layer. Journal of Fluid Mechanics. 2018 Mar;838:284-319.
- Chappell D, Jacob M, Hofmann-Kiefer K, Bruegger D, Rehm M, Conzen P, Welsch U, Becker BF. Hydrocortisone preserves the vascular barrier by protecting the endothelial glycocalyx. Anesthesiology: The Journal of the American Society of Anesthesiologists. 2007 Nov 1;107(5):776-84.
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