blood is a non - newtonian fluid or more precisely , a pseudo - plastic fluid .
this type of fluid is less like water and more like toothpaste or ketchup , flowing more smoothly as the velocity of flow increases .
this smooth flowing phenomenon at higher velocity is related with decreased viscosity and increased shear rate of the blood .
such phenomenon is called as the shear thinning behavior of the blood as a non - newtonian fluid ( fig .
blood flow is pulsatile ; therefore , properties such as pressure , velocity , flow rate , and viscosity change as a function of time .
2a , b ) , indicating that if wall shear rate changes , viscosity might also change .
however , the carreau - yasuda model , one of the non - newtonian models , demonstrates the change in viscosity according to time ( cardiac cycle ) .
this is an important reason why non - newtonian analysis must be used in the pulsatile cardiac conditions .
several studies have emphasized the importance of this non - newtonian behavior of blood by comparing newtonian and non - newtonian modeling research .
gijsen et al .   showed that applying non - newtonian fluid flow at arteries with bifurcations greatly affects the flow features compared to newtonain fluid flow .
molla and paul   also compared newtonian and non - newtonian models in their research of arterial stenosis .
wall shear stress , which is one of the most important factors for diagnosis of aneurysm , is the function of viscosity and shear rate . assuming blood as newtonian fluid means that the effect of viscosity term is ignored .
fisher and rossmann   compared analysis of wall shear stress ( wss ) distribution at aneurysm with both non - newtonian ( casson , carreau , power law ) and newtonian models .
the result showed that wss of non - newtonian models is more influential in aneurysms than that of newtonian model .
applied non - newtonian properties to the analysis of stents in aneurysms , proposing that viscosity and velocity profiles in the aneurysm are largely determined by non - newtonian properties of blood flow .
boundary conditions are the given conditions at the domain boundaries and which account for outer domain properties , are one of the most important factors for a modeling process . as the simulation domain
is restricted to only a small proportion of the total circulatory system , the conditions outside of the domain can not be directly applied to the simulation .
directly measured properties from a specimen may be applied at the boundaries ; however , it is difficult to fully demonstrate the entire vascular system due to the problems of universality , simplicity , and technical deficiency . simplified numerical boundary conditions are instead applied in most circulatory system modeling .
the ' zero pressure outlet ' or ' traction - free'boundary condition is used in various models due to its simplicity .
however , when the purpose of modeling is to analyze blood flow , the zero pressure outlet should not be used due to its inaccurate simulation results .
the zero pressure boundary is as akin to assuming that the vessel is cut and exposed to atmospheric air conditions .
therefore , it neglects the change in pressure and flow rate by ' wave reflection ' from downstream vessels , resulting in considerable difference from actual blood flow .
wave reflection is an influx of blood due to the reflected wave from downstream arteries .
this is mainly due to the pulsatile flow of blood and elasticity of blood vessels .
as pulsatile antegrade flow passes through the elastic vessel , this vessel repeatedly shrinks and expands , causing retrograde flow through the whole vascular system .
this wave reflection results in considerable changes in blood pressure and flow rate due to overlap with the forward flow of blood .
this wave reflection can be used in diagnosis of cardiovascular disease . as wave reflections and arterial stiffness
are associated with an increased risk of cardiovascular disease , the measurement of central arterial pressure can provide a valid aid in cardiovascular risk assessment [ 19 , 20 ] . to demonstrate this wave reflection ,
the well - known outlet conditions in theoretical hemodynamics include resistance , windkessel , and impedance boundary models .
resistance model substitutes the outer domain vessels into one single variable , while windkessel model uses three variables .
these two models can develop wave reflection phenomenon , yet still inappropriate because it can not show the arrival time of wave reflection correctly . however , impedance boundary model reflects the entire vessel geometry and property . by constructing the 1d vessel system that represent the real vessels , the variable called ' impedance '
unlike the other models , this impedance variable includes both the data of magnitude and arrival time of wave reflection .
because it considers wave reflection from bifurcations and geometry of vessels , the impedance model successfully demonstrates the pulsatile aspects of real blood flow .
when there is 75% area reduction stenosis in a branch of the bifurcation vessel model , mean flow of constant pressure boundary condition is divided into 70% and 30% each , because the fluid resistance by the geometry of computational domain is only considered . for the impedance boundary condition ,
on the other hand , the flow is also affected by entire circulatory system so that the difference between those two vessels is smaller than that of the constant pressure boundary condition .
these results agree with representative clinical observations for iliac artery stenosis with less than an 85% area reduction . for these reasons
, the wave reflection should be considered to perform an accurate simulation for the purpose of analysis .
there are various ways to interpret simulation result data , such as plotting pressure and flow rate or calculating wall shear stress .
as the loss of energy is equal to the amount of energy applied to the vessel wall , it is directly related the risk of conditions such as the rupture of aneurysm [ 12 , 22 - 24 ] . however , there are many factors , which are mostly related to the geometry of vessel , to be considered when using the energy loss equation
. it might cause a considerable error on result if those factors are not reflected properly . to predict clinical treatment effects , bernoulli 's energy loss concept is used .
if pi , vi are pressure and velocity at the initial cross sectional area domain respectively and pf , vf the pressure and velocity at the final cross sectional area domain , then the energy loss equation can be written as : ( 1 ) this bernoulli 's energy loss is purely due to mechanical energy loss .
the energy loss consists of a limited number of ' loss factors ' , so some papers have already claimed that it can not be applied in all clinical pathologies such as aneurysms or stenosis [ 25 - 27 ] .
as energy loss can change by depending on its geometrical conditions , the following issues should be considered when using the loss term .
the main properties of energy loss by the pathway of blood are the change in cross sectional area and the direction of flow axis , based on bernoulli 's equation and momentum conservation [ 26 , 28 ] . from fig .
5a , if egeometry is the energy loss from a change in geometry , such as cross - sectional area or directional change , then the mechanical energy balance ( bernoulli 's equation ) is ( 2 ) also , the force - momentum relationship ( momentum conservation ) is ( 3 ) ( pi - pf ) = q ( vf - vi cos)/af = q ( ai - af cos)/aiaf where  is density of blood ,  is angle change in direction , q is flow rate ( m / s ) and a is cross sectional area . using equations 2 and 3 , we can derive the following equation : ( 4 ) from equation 4 , energy loss can occur due to change in cross - sectional area or flow angle .
5b shows this relationship between energy loss and change of cross sectional area or angle .
the velocity difference at initial and final points is related to the momentum conservation equation , (mv)=ft , where m is mass , v is velocity , f is force and t is time .
a change in ' mv ' means that force is applied and the velocity and direction of blood flow are changed by the force of collisions with the vessel wall . the altered velocity and direction result in a change in fluid properties , which are directly related to energy loss .
5b shows that energy loss is also affected by changes in cross sectional area and flow angle .
the larger the flow angle changes , the larger the energy loss becomes . in the case of vessel contraction ,
the larger the change in cross - sectional area , the larger energy loss is .
the dots presented at 0 - 60 degrees represent the cross - sectional area where energy loss is at a minimum .
energy loss tends to decrease until its minimum value , then increase again . as , energy loss approaches  
the cross - sectional area change is larger at each change of angle . in the case of a change in 90
the level of change in cross - sectional area and direction of flow also affects energy loss .
this means that if the rate of change in cross - sectional area or the angle of path is larger , then the value of energy loss also increases [ 26 , 29 ] .
the following equations 6 and 7 show that variation in vessel geometry affects energy loss : ( 6 ) where p0=pi and pn = pf , a0 = ai and an = af 
 ( 7 ) we can rewrite equation 7 , egeometry ( =0 ) = function of bi , f , bi , f = , which is term of bernoulli 's equation . in fig .
6a , it is assumed that the vessel having some gradient of cross sectional area is divided by ' n ' number of steps .
then the momentum conservation between steps ( n-1 ) and ( n ) can be written as equation 8.when =0 , from equations 2 and 6 , equation 7 can be derived to find the value of egeometry by using '  = = ' which represent cross - sectional area ratio .  
is from the bernoulli 's equation and   is the loss factor based on ratio of change in cross sectional area . as observed in fig .
6a ,   gets smaller with increasing n , meaning that the greater the cross - sectional area changes , the less the energy loss is .
when the angle changes by ' n ' number of steps , the energy loss between n-1 and n steps is demonstrated by equation 8 which is derived from equation 4 : ( 8) the sum of the ei , f (  ) values in all n steps can be expressed by equation 9 : ( 9 ) ei , f (  ) is   multiplied by ' energy loss factor ' .
, the energy loss decreases as the angle change becomes more gradual equation 9 can be rewritten as : ( 10 ) it has been shown that , even without an aneurysm there is energy loss due to the geometry of the vessel itself .
hence , energy loss near an aneurysm consists of loss from the aneurysm itself and the loss from the geometry of the vessel , expressed as e=eaneurysm+egeometry .
these equations show that the geometry of vessel must be considered when using the energy loss in diagnosis .
the following equations 6 and 7 show that variation in vessel geometry affects energy loss : ( 6 ) where p0=pi and pn = pf , a0 = ai and an = af 
 ( 7 ) we can rewrite equation 7 , egeometry ( =0 ) = function of bi , f , bi , f = , which is term of bernoulli 's equation . in fig .
6a , it is assumed that the vessel having some gradient of cross sectional area is divided by ' n ' number of steps . then the momentum conservation between steps ( n-1 ) and ( n )
can be written as equation 8.when =0 , from equations 2 and 6 , equation 7 can be derived to find the value of egeometry by using '  = = ' which represent cross - sectional area ratio .  
is from the bernoulli 's equation and   is the loss factor based on ratio of change in cross sectional area . as observed in fig .
6a ,   gets smaller with increasing n , meaning that the greater the cross - sectional area changes , the less the energy loss is .
when the angle changes by ' n ' number of steps , the energy loss between n-1 and n steps is demonstrated by equation 8 which is derived from equation 4 : ( 8) the sum of the ei , f (  ) values in all n steps can be expressed by equation 9 : ( 9 ) ei , f (  ) is   multiplied by ' energy loss factor ' .
, the energy loss decreases as the angle change becomes more gradual equation 9 can be rewritten as : ( 10 ) it has been shown that , even without an aneurysm there is energy loss due to the geometry of the vessel itself .
hence , energy loss near an aneurysm consists of loss from the aneurysm itself and the loss from the geometry of the vessel , expressed as e=eaneurysm+egeometry .
these equations show that the geometry of vessel must be considered when using the energy loss in diagnosis .
three important modeling considerations that are rarely used in circulatory system modeling are explained in this paper .
the non - newtonian properties of blood are found to be critical for the accuracy of modeling .
choosing the appropriate outlet boundary condition also makes an important difference in the results of a simulation .
finally , the energy loss related with blood vessel geometry must be included when the energy term is used in numerical calculations .
there are many researchers developing better modeling techniques and numerical methods , yet these new methods are not routinely applied in circulatory system modeling . establishing accurate and trustworthy modeling of the circulatory system requires the newest techniques and implementation .
 despite recent development of computational fluid dynamics ( cfd ) research , analysis of computational fluid dynamics of cerebral vessels has several limitations . 
 although blood is a non - newtonian fluid , velocity and pressure fields were computed under the assumptions of incompressible , laminar , steady - state flows and newtonian fluid dynamics . 
 the pulsatile nature of blood flow is not properly applied in inlet and outlet boundaries . 
 therefore , we present these technical limitations and discuss the possible solution by comparing the theoretical and computational studies . 
Non-Newtonian Properties
Outlet Boundary Condition
Energy Loss Due to the Vessel's Pathway
A. Change in cross-sectional area but no alteration in direction of flow
B. No change in cross-sectional area but an alteration in direction of flow
SUMMARY