University of Illinois at Chicago



Modeling blood flow in the Human Cerebral VasculatureNakib MansuriUniversity of Illinois at Chicago – Depart of BioengineeringBiological Systems Analysis – Professor Linninger12/1/2013AbstractThe human cerebral circulation has been modeled using linear equations of flow which has been used to simulate the cerebral vasculature of the human brain. This model includes the Circle of Willis which is a network of arteries that are responsible for supplying blood to the brain. A key aspect of the Circle of Willis is its innate symmetry which can be helpful in providing blood flow to the brain even when some arteries have been blocked. The goal of this study is to assess this redundancy and to understand its effectiveness in the event of blockage. Arterial stenosis was simulated in four different arteries within this model and the resultant flow rates within that artery and a symmetrical artery were calculated. The data was examined to reveal a linear relationship between the decreases in flow of a constricted artery and the increase in flow of the symmetric artery. A critical communicating artery within the Circle of Willis was removed completely and the consequential flow rates through the cerebral arteries was calculated. These values were compared to baseline values to show only a small decrease in overall flow. IntroductionBlood delivery to the brain is of critical importance to the functioning of the human body. Blood supplies the necessary oxygen and nutrients to the brain. This allows the brain to function properly and, in turn, regulate all of the other major body organs. Lack of blood in the brain can lead to tissue damage and stroke. For example, the common carotid arteries fork into the internal and external carotid arteries at the throat. This fork is a common site for atherosclerosis which is an inflammatory buildup of plaque that can lead to the narrowing of the arteries. This can have severe effects upon the brain leading to permanent damage [2].An important part of the cerebral vasculature of the brain is called the Circle of Willis. This system of arteries ensures that the brain is able to get an adequate supply of blood even in cases of artery blockage. The goal of this study is to analyze just how well the Circle of Willis can perform under stress. Four different arteries from the Circle of Willis were chosen to study. Each of these arteries also had a corresponding symmetric artery. In each examination, varying degrees of stenosis were simulated by decreasing of the diameter of the chosen artery. The Circle of Willis is also purported to be efficient in blood flow even in the absence of critical arteries. This was also tested by removing a communicating artery from the Circle and analyzing how this change affected overall blood flow within the model.Figure 1 – Network This figure shows the network and the physiological structures that correspond to each face of the network. MethodsThis network allows for the examination of the blood flow through the Circle of Willis and other surrounding arteries and veins. As stated before, the Circle of Willis has an innate symmetry that allows it deliver blood to the brain even when some arteries have been blocked. Baseline flows through each arc and pressure at each node were calculated using boundary conditions found in the literature. Once these baseline values had been established, the experiment could proceed. The aim of the experiment is to model a stenosis in one of the arteries in the Circle of Willis and see how it affects the blood flow in said artery. This would be done by running the simulation multiple times with the diameter of the artery decreasing each time. After the program has ran multiple times, a graph is generated showing the change in blood flow as a function of the diameter.In this network, there are 36 nodes and 42 faces. Of these 36 faces, 7 are terminal. Of those 7, 4 are inlets and 3 are outlets. This means that there are a total of 71 unknowns, 29 nodes and 42 faces. The 7 terminal nodes are boundary conditions and their values have been found in the literature. The inlet nodes are all located in the Circle of Willis. The pressure there was found to be 16 mmHg[3]. Since there are 4 inlet nodes, however, the decision was made to split the pressure 16 mmHg into 4 parts. This resulted in each inlet node having a pressure of 4 mmHg. Of the three outlet nodes, 1 was in the Circle of Willis and the other two were at the ends of the jugular veins. The pressure at the outlet node located in the Circle of Willis was found to be 0 mmHg. The pressure at the jugular veins was found to be 3.6 mmHg[4]. However, since there are two outlet nodes at the jugular veins, the pressure of 3.6 mmHg was split into two parts. This resulted in the two outlet nodes at the jugular veins having pressures of 1.8 mmHg.While the pressures here have been listed in mmHg, the values were all converted to Pascal’s before solving the system.After all of the boundary conditions had been found, balance equations for each node were written. The equations were written similarly to the way that one would analyze the flow of current in an electrical circuit. The basic premise is that the flow in must equal the flow out. If the direction of flow leads into a node, it is deemed positive. If the direction of flow leads away from a node, it is deemed negative. This basic relationship was used to write the equations for the network. MATLAB had been programmed to read in relevant data from text files and generate the balance equations. There were a total of 29 balance equations.Next, the constitutive equations were written for each face. These equations were written using the Hagen-Poiseuille law. The relationship is defined to be that the flow multiplied by the resistance is equal to the pressure drop across 2 nodes. In this network there were 42 arcs so 42 equations were generated. The pressure at the node from which the flow was leaving was deemed negative and the pressure at the node towards which the flow as heading was deemed positive. All of the constitutive equations were written in a similar fashion.After the equations have all been written, a matrix representing them, A, was created. A vector,b, which contained the boundary conditions was also created. The target vector, x, could thus be calculated.ValidationOnce the target vector had been calculated, the validity of the model can be tested. X is a 71 x 1 vector where the first 42 terms represent the flows in the 42 faces and the last 29 values represent the pressures at the 29 nodes. The inlet pressures were 533.29 Pa and the outlet pressures were 239.96 Pa. If the model has worked correctly, the 29 calculated pressures must be between the inlet and outlet pressures and this is indeed the case. Another way to test the model would be to see if the flow into the network equals the flow out of the network. This can be done by adding up the flows in and adding up the flows out. There are 4 inlets so there are 4 inlet flows. These 4 inlet nodes are nodes 1,2,3, and 4 and they correspond to flows 1,2,4 and 6 respectively. The sum of the flows from the x vector is equal to 24.714 mL/min. The same must be done for the outlets. There are 3 outlet nodes which means that there are 3 outlet flows. These 3 outlet nodes are nodes 9, 35, and 36 and they correspond to flows 8, 41, and 42. The sum of the flows from the x vector is equal to 24.449 mL/min. This is very close to the inlet flow which makes sense because the network is supposed to be closed. The small difference can be accounted due to the fact that the computer may have made rounding errors and that the model is not entirely flawless. These two tests indicate that the model and the calculations have been performed correctly. Artery/VeinLiterature[5]Calculated ValueResidualInternal Carotid 341.00342.700.5%Anterior Cerebral 87.00114.0031%Vertebral320.00381.0019%Posterior Circulation165.00165.400.2%Figure 2 – Comparing blood flowThis table compares the calculated blood flow of various groups of arteries and veins against the measured values found in the literature. It can also be noted that the average flow rate in the Circle of Willis was measured to be 11 mL/s and the calculated flow rates do not exceed this value[1].ResultsTo study the effects of arterial stenosis in the Circle of Willis, a pair of arteries were chosen and the diameter of artery from the pair was decreased. Three different diseases were simulated using this process. Atherosclerosis was simulated with a 10% decrease in diameter, mild stenosis was simulated with a 30% decrease in diameter, and severe stenosis was simulated with 70% in diameter. With each decrease in diameter, the flow through that artery and the symmetric artery was calculated. The percent difference in the flow of each artery at a 10% decrease in diameter was also calculated and recorded.Another study conducted dealt with examining blood under the absence of an artery within the Circle of Willis. The Posterior Communicating Artery was eliminated from the model and the blood flow through the cerebral arteries was calculated and compared with the baseline fows.307657540005000Figure 3 – Flow rates under diseases in different arteries Figure 4 – Comparing overall blood flow DiseasePosterior Communicating ArteriesSymmetrical ArteryBaseline 31.9534.34Atherosclerosis22.0235.09Mild Stenosis10.5635.35Severe Stenosis 0.435.87DiseaseMiddle Cerebral ArteriesSymmetrical ArteryBaseline 240.50247.14Atherosclerosis154.40260.11Mild Stenosis 96.41267.95Severe Stenosis 4.96274.24DiseaseInternal Carotid ArteriesSymmetrical ArteryBaseline 350.09369.83Atherosclerosis218.98395.60Mild Stenosis137.04411.45Severe Stenosis 5.69423.31DiseaseVertebral ArteriesSymmetrical ArteryBaseline 655.09700.06Atherosclerosis395.01798.68Mild Stenosis253.46858.36Severe Stenosis 10.44872.26 This figure compares the blood flow in the cerebral arteries under normal and diseased circumstances. The disease refers to the absence of the posterior communicating artery. ArteryDifference in constricted arteryDifference in symmetrical arteryPosterior Comm. 31.05%102.19%Middle Cerebral25.80%105.25%Internal Carotid37.45%106.97%Vertebral Arteries39.70% 114.09% Figure 5 – Flow differentials Figure 5 compares the difference in flows between the baseline flow and the flow when the artery is suffering from atherosclerosis. The difference for the constricted artery and the symmetrical artery is recorded. Figure 4 shows the flow rates through different constricted arteries and the corresponding symmetrical arteries in the Circle of Willis. Baseline flows were calculated for each artery and tests were run with different levels of constriction (10%, 30%, 70%) where each corresponds to a certain disease. Flows were also calculated at each disease. DiscussionFrom the data in Figure 3, it can be seen that as the diameter of an artery decreases to a small fraction of its original value, the flow through that it declines sharply. This is to be expected. However, the interesting phenomenon to note is the increase in flow rate through the corresponding symmetrical artery. This can be attributed to the fact that the Circle of Willis has an innate symmetry that allows it to redirect blood flow in the event of blockage. It is also interesting to note that the change in blood flow in either artery is not constant. It seems that some arteries are very susceptible to blockage and will lose most of their ability to transport blood relatively quickly. This is the case with the Internal Carotid and Vertebral Arteries. To quantify this early loss in blood flow, the difference in flow at 10% constriction was calculated for each vessel and the corresponding symmetric vessel and is available in Figure 5. For the Internal Carotid and Vertebral Arteries, the difference in blood flow is relatively large, which means that these arteries are not allowing much blood to pass. Thus, the symmetrical arteries increase their flow rates quickly to compensate. This effect is not really present in the other tests. When the Posterior Communicating and Middle Cerebral Arteries were constricted, the flow rate did not decrease as sharply. The difference in flow rates is smaller which implies that these arteries are not as susceptible to stenosis. Thus, the flow rates through the symmetrical arteries do not increase as quickly as in the other two tests. Figure 4 shows the blood flow through the major cerebral arterial groups under two different circumstances. One scenario represents blood flow under normal conditions and the other represents blood flow when the Posterior Communicating Artery is absent in the Circle of Willis. As can be noted, the absence of the Posterior Communicating Artery does not impede blood flow in the cerebral arteries to a great degree. This is again due to the fact that there are symmetrical arteries that can compensate for a lack of blood flow. The results from the Figures 3 and 5 have shown that these symmetrical arteries are very efficient at making up for the decrease in blood flow in a blocked artery and the data from Figure 4 shows that the overall blood flow does not decrease by a great factor. This allows people who are born with anatomical anomaly to live normal lives and many people even go years before they become aware that their cerebral vasculature is incomplete.A general trend that can be noticed is that as the difference in flow of the constricted artery graph increases, so does the difference in flow of the symmetric artery. Thus, it can be seen that the control artery’s flow rate is dependent on the constricted artery’s behavior. Also, it can be noticed that overall blood flow does not decrease greatly even if a certain artery is missing completely from the Circle of Willis. This leads to the conclusion that not only does the Circle of Willis have an innate symmetry to allow for continuous blood flow in the event of a constriction, it can also regulate the flow rates to compensate at a reasonable level. This means that if one artery is restricted in the Circle of Willis, the network will not immediately reroute all blood flow away from that artery and into surrounding ones. Instead, blood is rerouted in proportion to the severity of the blockage. This ensures that undue strain will not be caused upon the system and the flow of blood is properly maintained. References[1] Alnaes, M, J Isaksen, et al. "Computation of Hemodynamics in the Circle of Willis."?American Heart Association/American Stroke Association. (2008). Web.[2] Bartlett, ES, TD Walter, et al. "Quantification of carotid stenosis on CT Angiography."?American Journal of Neuroradiology. 27.1 (2006): 13-19. Web. [3] Faraci, F M, and D D Heistad. "Regulation of large cerebral arteries and cerebral microvascular pressure."Circulation Research. 66. (1990): 8-17. Web.[4] Parker, J. L., and C. J. R. Flucker. "Comparison of external jugular and central venous pressures in mechanically ventilated patients."?Anaesthesia. 57.6 (2002): 596-600. Web.[5] Zhao, M., S. Amin-Hanjani, et al. "Regional Cerebral Blood Flow Using Quantitative MR Angiography."American Journal of Neuroradiology. 28. (2007): 1470-1473. Web.AppendixFigure 6 – Network with faces labeledFigure 7 – Network with points labeledFigure 8 – Arc dataNumberVesselsArea (cm^2)Diameter (m)1,6 Internal Carotid Arteries (below)0.140.00422,3,4,5Vertebral arteries0.090.00337,9Basilar artery: first part0.190.00499Basilar artery: second part0.090.003311,13Posterior cerebral arteries0.070.002910,12,21,22Posterior Communicating Arteries0.020.001618,23Internal Carotid Arteries (above)0.140.004219,24Anterior Cerebral Arteries (below)0.070.002914,15,16,17Posterior Cerebral Arteries0.070.002925,26Middle Cerebral Arteries0.120.003920Anterior Cerebral arteries (above)0.120.003927,28Main branches of cerebral arteries0.550.008429Pial Network1.750.014930Intracerebral Arteries0.550.008331Microcirculation380.069532Intracerebral Veins9.490.034733Pial Veins3.860.022134Cerebral Veins0.920.010835Cerebral Veins0.330.006436Longitudinal sinuses: first part0.150.004337Veins0.490.007838Veins0.290.006139Longitudinal sinuses: second part0.470.007740Transverse Sinuses1.650.014441,42Jugular Veins0.430.0073This table lists all of the faces in the network and the veins/arteries that they correspond to along with their lengths and diameters.Equations written by MATLABConservation Balance Equations+F2-F3+F0+F0+F0+F0=0+F4-F5+F0+F0+F0+F0=0+F3+F5-F7+F0+F0+F0=0+F7-F8-F9+F0+F0+F0=0+F9-F10-F12+F0+F0+F0=0+F12-F13-F21+F0+F0+F0=0+F13-F16+F0+F0+F0+F0=0+F16-F17+F0+F0+F0+F0=0+F15+F17+F20+F25+F26-F27=0+F1-F18+F21+F0+F0+F0=0+F18-F19-F26+F0+F0+F0=0+F19-F20+F24+F0+F0+F0=0+F23-F24-F25+F0+F0+F0=0+F6+F22-F23+F0+F0+F0=0+F10-F11-F22+F0+F0+F0=0+F11-F14+F0+F0+F0+F0=0+F14-F15+F0+F0+F0+F0=0+F27-F28+F0+F0+F0+F0=0+F28-F29+F0+F0+F0+F0=0+F29-F30+F0+F0+F0+F0=0+F30-F31+F0+F0+F0+F0=0+F31-F32+F0+F0+F0+F0=0+F32-F33+F0+F0+F0+F0=0+F33-F34-F38+F0+F0+F0=0+F34-F35-F37+F0+F0+F0=0 +F35-F36+F0+F0+F0+F0=0+F36+F37-F39+F0+F0+F0=0+F38+F39-F40+F0+F0+F0=0+F40-F41-F42+F0+F0+F0=0Constitutive Equations(F1*R1)-P1+P15=0(F2*R2)-P2+P5=0(F3*R3)-P5+P7=0(F4*R4)-P3+P6=0(F5*R5)-P6+P7=0(F6*R6)-P4+P19=0(F7*R7)-P7+P8=0(F8*R8)-P8+P9=0(F9*R9)-P8+P10=0(F10*R10)-P10+P20=0(F11*R11)-P20+P21=0(F12*R12)-P10+P11=0(F13*R13)-P11+P12=0(F14*R14)-P21+P22=0(F15*R15)-P22+P14=0(F16*R16)-P12+P13=0(F17*R17)-P13+P14=0(F18*R18)-P15+P16=0(F19*R19)-P16+P17=0(F20*R20)-P17+P14=0(F21*R21)-P11+P15=0(F22*R22)-P20+P19=0 (F23*R23)-P19+P18=0(F24*R24)-P18+P17=0(F25*R25)-P18+P14=0(F26*R26)-P16+P14=0(F27*R27)-P14+P23=0(F28*R28)-P23+P24=0(F29*R29)-P24+P25=0(F30*R30)-P25+P26=0(F31*R31)-P26+P27=0(F32*R32)-P27+P28=0(F33*R33)-P28+P29=0(F34*R34)-P29+P30=0(F35*R35)-P30+P31=0(F36*R36)-P31+P32=0(F37*R37)-P30+P32=0(F38*R38)-P29+P33=0(F39*R39)-P32+P33=0(F40*R40)-P33+P34=0(F41*R41)-P34+P35=0(F42*R42)-P34+P36=0 ................
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