Introduction: Dr. Fady Charbel MD, Chief of UIC Neurosurgery & Staff) have innovated the use of a new Doppler ultrasonic probe (Transonics, Ithaca, NY) which may be used in the operating room to determine flow velocities of intracerebral vessels during cerebrovascular surgery. We have evaluated the flow in the middle cerebral artery recipient vessel and bypass graft in 33 patients who had been treated with extracranial to intracranial bypass (STA-MCA) surgery. We measured the cut-flow, which is the flow in the STA branch prior to anastomosis in EC-IC Bypass. We compared this to the postoperative blood flow in the graft during surgery, but following conduit anastomosis using the new probe to the blood flow values determined with phase contrast magnetic resonance angiography (using the 3D localizer for local cerebral blood flow) CANVASS Study. These values are listed in Table 1 for all of the patients. Wall diameter and wall shear stress were also measured as a part of this study.

Objective: To measure the cut flow of the STA in EC-IC Bypass to evaluate the effectiveness of the bypass procedure. The bypass procedure is evaluated with reference to the clinical prognosis and the PCMRA measurements of the graft postoperatively.We met this objective.

Background: Cerebrovascular surgeons have used a variety of rather time consuming, expensive and laborious techniques to evaluate the patency of EC-IC Bypass grafts during EC-IC Surgery or in the early postoperative period. Intraoperative cerebral angiography, 3D CT angiography, or magnetic resonance angiography are very expensive tools to evaluate bypass patency. We are using a cut flow measurement and a flow probe measurement to make evaluation much quicker, easier, more accurate, and less expensive. We will try to prove foremost that the technique is more accurate.

Our imaging modality of choice for the postoperative period following revascularization is PCMRA or phase-contrast magnetic resonance angiography. PCMRA suppresses nonmobile tissues lacking flow while basing its detection on velocity induced phase shifts in order to image the local cerebral blood flow. It is not, therefore, dependent on T1 values like TOF MRA (time of flight MRA). Because of this fact it has an enhanced capacity to find slowly flowing blood. It is sensitive to pulsatile and turbulent blood flow. This means that it can delineate the average flow when combined with the cardiac gating. The fallback in PCMRA is the aliasing. This is also called the wrap-around and refers to misrepresentation of a high signal frequency, or in any particular case a high flow velocity, as a low signal frequency (velocity). All of this can be utilized for an advantage in PCMRA for the evaluation of areas of slow flow as may occur in a bypass graft. Both TOF and PC use short flip angles, reduced TE/TRs and thin slices to limit the phase dispersion. Furthermore, the technique is capable of 3D and 2 D acquisition.

Cerebral blood flow at the bifurcation of the STA and MCA anastomosis is turbulent flow when modeled using mathematical techniques. The Navier-Stokes and Bernouilli equations describe the flow characteristics quantitatively. Laminar flow is disrupted at the junction of the two vessels. It is highly useful therefore, to know the flow velocity in the vessels intraoperatively to evaluate the quality of the bypass. Empirical observations may be made of the STA donor before anastomosis. The STA donor has a certain diameter and basal flow prior to grafting. We think this is very

important in surgical outcome.

One should think of cerebral blood flow or any blood flow for that matter as a series of moving particles traveling at different speeds depending on proximity to the arterial vessel wall. A fluid element in a flow volume is characteristically positioned between a faster moving element on one side and a slower moving element on the other, similar to a vehicle traveling in a multilane highway. Consequently, the result of this “velocity gradient” is such that the element is in a state of continuous deformation, and there are therefore, elements of the body. The body is in a state of such flow. Flow can therefore, be considered a state of continuous deformation. Blood is considered to be a Newtonian fluid. Several fluids are considered to be Newtonian, including air, oil, and water. Others flow in a Newtonian fashion when the rates of deformation and therefore, velocity gradients within a flow field are thus small, and as non-Newtonian fluids when the gradients are greater. Because blood is corpuscular in nature, this raises the question of whether it should be treated as a continuum, and therefore, the peculiar makeup of plasma, makes it unusual from other common fluids. In the brain blood flow can be turbulent when negotiating the turns and tortuous vessels in the Circle of Willis or in a bypass graft feeding into the middle cerebral artery.

An essential consequence of the viscous property of fluids is that there is fluid flow with ‘step’ changes in velocity at any point within the blood flow field. This is because the velocity gradient at a particular point, which is opposed by the viscosity of the blood. A force is thus essential to maintain the velocity gradient, and the greater this gradient, the higher the force is. A step difference implies that the velocity gradient is infinite at a certain point, which is impossible since the force required to maintain it is therefore, considered to be infinite. Specifically the interface between the solid and fluid boundary at the inner wall of the artery (elastic vessel), the velocity of the blood close to the wall is almost zero at that point. This is well know as the “no-slip" boundary condition which a fundamental property that must be considered in any analysis of viscous flow. Several years ago, it was though the skidding of red blood cells along the artery wall took place, and was therefore thought to mean there is a slip, but it later became apparent that skidding only meant the layer of plasma above it acted as a lubricant.

In the late 1800s, Osborne Reynolds made one of the more significant findings concerning flow in tubes. In several experiments he studied the basic flow qualities. He did this by injecting a dye at the tube’s orifice, then switching the flow rate to find if this affected what happened. Reynolds discovered that at low flow rates the marked elements produced streaklines that were fairly distinct and ran parallel to the tube axis. At higher flow rates the streaklines became turbulent, eventually braking down and leading the dye to diffuse over the entire cross section of the tube, streaklines being no longer distinct or visible. Reynold found two different types of flow: laminar and turbulent. He found that the appearance of turbulence is predicated upon the nondimensional combination R= ρ vd/μ where ρ is viscosity, v velocity, d vessel diameter, and μ elasticity. The Reynold’s number for laminar flow is thus defined. R is 2000 is the lower limit at which turbulent flow is known to occur. In the human aorta which is about 2.5 cm in dimater with an average cardiac output of 5 L/min, the average velocity is v = 5000/ {π(2.5/2)2 X 60}~ 17 cm/sec. Under these assumptions, the Reynolds number is below 2000. With

pulsatile flow which occurs in the cerebral circulation, at the peak of the oscillatory cycle, and under conditions of higher cardiac outputs, the value of R is exceeded greatly, therefore, the chance of turbulent flow occurs at that level of the arterial tree and in that aspect of the oscillatory cycle.

When velocity gradients are not large and the equations of flow are founded on the following equations (See Diagram 1):

σxx = -p + 2μ (δu/δx)σrr = -p + 2μ(δv/δr)σθθ = -p +2μ(1/r δw/δθ + v/r)τxr = τrx = μ (δu/δr + δv/δx)τxθ = τθx = μ(δw/δx +1/r δu/δθ)τrθ = τθr = μ(δw/δr – w/r + 1/r δv/δθ)These equations can be solved for particular flow situations, such as the cerebral

circulation; it is the solutions to them that are compared to experimentally determined values from Doppler flow measurements. The major assumption of these equations is that the linearity between shear stress and velocity gradient occurs. It is well recognized that there is an approximation for small gradients, but there is a range of validity. The value of pressure p is really a thermodynamic property which is also related to laws of physics applied to bodies of fluid at rest. In this case, the pressure is a force which acts perpendicular or normal to the direction of flow. The assumptions inherent in this are mechanical definitions of pressure first derived and defined by Stokes, which is the basis of the continuity equations and the equations of motion.

Below are the two equations described above for reference:ρDV/Dt = -Δp + B + μΔ2 V Stoke’s Equation

σij = -pδij +σij = -pδif +2μeij + δij λ Φ stress tensor & strain equation

Blood flow towards the center of the vessel is laminar in theory and turbulent flow is more likely towards the side of the vessel walls where wall shear stress is higher. Small eddy currents occur at the vessel periphery which is one hemodynamic factor for shear stress on the wall and for promotion of antherosclerosis. Platelets and leukocytes are more likely to adhere or bind to vessel wall interstices and endothelial cell gaps when eddy currents are prominent. Eddy currents contribute in the long term to antherosclerotic buildup inside the graft and the proximal recipient. Asymmetry of the stress tensor can cause forces which have spinning effects on fluid elements. The assumption of symmetry is founded on the fact that the forces are absent under normal blood flow conditions.V = ks/4μ X (r2 –a2) The latter is compatible with Poisseuille Flow in a tube. The volumetric flow q through the tube is obtained by integrating over the cross section of the tube Q = ∫a

0/ (V 2π r )dr = -ks π a4/8μ The resistance to flow at the vessel wall is

τs = 4μ qs/πa3. The rate of the energy produced and necessary to override the resistance to flow at the vessel wall is called the pumping power Hs necessary to maintain the blood

flow, is given by the product of the total force of resistance and the average flow velocity. The total force in turn is the product of the shear stress and the surface area upon which it is acting which is the surface area of the artery. Therefore, Hs = -Δ ps X πa2 x vs

The pressure differential may be defined in terms of the flow rate, by:

–Δ Ps = (8μl/πa4) Qs This is comparable to flow in an electric conductor E =RI The balance of energy is further defined by μ(d2us/dr2 + 1/r dus/dr) = ks. This equation specified the forces of driving pressure and the impeding viscous force on the left side. The equation describes forces per unit volume. The equation must be multiplied by some volume of fluid such that it will represent forces, then determine the rates by which these forces operate. The balance of energy expressed in the vessel is related by:2πμl ∫a

0 r (dus/dr)2 dr = -kslqs.

There is a cube law which represents pulsatile flow in the cardiovascular and cerebrovascular systems. There is a pumping power Hs that is necessary to maintain flows q through the vessel of radius a which is inversely proportional to the fourth power of the radius, that suggests the vessel diameter is as large as it can possibly be. H = A/a4

+ Ba2 . The B is a positive constant representing the metabolic rate of energy expenditure essential for maintaining the volume of the blood, thus: A = 8μlq2

s/π where H is a minimum dH/da = -4A/a5 + 2Ba = 0, where d2H/da2 = 20A/a6 + 2B > 0.Thus, a6 = 2A/B = (2/B) 8μl/μqs

2 where qs is proportional to a3, known as Murray’s Cube Law.

The blood flow dynamics at an arterial bifurcation is described by slightly different equations. A bifurcation occurs at the STA-MCA junction following revascularization most importantly. Another bifurcation occurs at the external and internal carotid artery junctions, as well as at all the branches in the Circle of Willis. One caveat which is essential states that the degree of asymmetry in arterial bifurcations exists, such that wide variations in the degree to which two daughter vessels are of unequal caliber, makes it possible for arterial trees to be highly nonuniform and therefore very nonsymmetrical, as necessitated by the peculiar territory of brain which they supply.

If the diameters of the parent and daughter arterial segments at the bifurcation are defined by a0, a1, a2, and their diameters are described by d0, d1, d2, and using the convention of always using the a1 > a2, then a bifurcation index is defined by α = a2/a1. A very asymmetrical bifurcation is one where the value of α is almost zero, where a symmetrical arterial bifurcation is one where α is 1.0. Another important value is the area ratio of an arterial bifurcation, which is the number value of the combined cross-sectional area of the two daughter vessels divided by the number for the parent artery. Values of β more than 1.0 induce expansion in the entire cross-sectional region available for flow such as it advances from one level of the tree to the next. β = (a2

1 + a22)/ a

20

Conservation of mass necessitates that an arterial bifurcation flow rate in the parent is equal to the two recipient vessels, described by Q: Q0 = Q1 + Q2. The cube law holds true in this case and therefore, a3

0 = a31 +a3

2. In relation to the bifurcation index α, this shows a1/a0 = 1/ (1 + α3)1/3. a2/a0 = α/(1 + α3)1/3. A simple algebraic substitution leads to the

expression for the area ratio β, where β = 1 + α2/ (1 +α3)2/3.Rotational motion is more likely to occur at vessel peripheries where eddy

currents are more prominent and prevalent. The Navier-Stokes equations are complete equations for motion for a viscous Newtonian fluid. For an incompressible fluid, the divergence is 0 and the above equation applies. It is vital to remember that D/Dt and the Laplacian are vector operators and are not applicable to velocity components except in Cartesian coordinates. Though the equation looks simple, its solution is hard by the nonlinear terms on the left-hand side and in fact, points out one of the major challenges in fluid dynamics. The Navier-Stokes equations can be compared with the Euler equations for frictionless flow. The differences are the additional terms involving the viscosity. The values of Boundary Layer Flow and Potential Flow will be discussed later.

Results: Figure 1 demonstrates that graft diameter in mm. Correlates in a linear fashion with the wall shear stress, which is what is predicted by theory. The postoperative flow is linearly related with PCMRA as well, as is seen in Figure 2. In Figure 3, graft diameter is plotted vs. cut flow, which is linear in its correlations. In summary, all of the data corroborates the assertion that the cut flow is a useful index and that the flow probe is a practical and efficient intraoperative device.

The Cut flow index has a high sensitivity and specificity prediction of graft patency acutely based upon the t test we performed. It has an 86% sensitivity and a 73% specificity. This cut flow index is therefore, a highly useful and accurate parameter of graft patency in the early postoperative period.

References:1. Schaum’s Outline W.F. Hughes and John A. Brighton Theory & Problems of

Fluid Dynamics 3rd edition McGraw-Hill Publishers, Inc. (2001)2. Hademenos & Massoud The Physics of Cerebrovascular Diseases AIP Press,

Springer-Verlag, New York (1998).3. Dumoulin C.L., Souza S.P., and Walker M.F. et al. “Three dimensional phase

contrast angiography” Magnetic Resonance Medicine 1989; 9: 139-149. 4. Johnson B.A., and Kelly W.M. “Common MRI artifacts: an overview

(concluded)” MRI Decisions 1989;3:36.5. Okudaira Y., Nakanishi H., Arai H., and K. Sato (2000) “The threshold of

acetazolamide vasoreactivity as an indicator of maximum cerebral blood volume” Neurosurgical Review 23: 132-135.

6. Kirkham S.K., Craine R.E., and A.A. Birch (2001) “A new mathematical model of dynamic cerebral autoregulation based on a flow dependent feedback mechanism” Physiological Measurements 22: 451-473.

7. Rossitti S., and P. Svendsen (1995) “Shear stress in cerebral arteries supplying arteriovenous malformations” Acta Neurochirurgica (Wien) 137: 138-145.

8. Ursino M., and E. Magosso (2001) “Role of Tissue Hypoxia in cerebrovascular regulation: a mathematical modeling study” Annals of Biomedical Engineering 29 563-574.

9. Thorin-Trescases N., and J.A. Bevan (1998) “ High levels of myogenic tone antgonize the dilator response to flow of small rabbit cerebral arteries” Stroke 29: 1194-1201.

10. Masaryk A.M. Frayne R., Unal O., Krupinski E., and C.M. Strother (1999) “In vitro and in vivo comparison of three MR measurement methods for calculating vascular shear stress in the internal carotid artery” Am. Journal of Neuroradiology 20 (2): 237-245.

11. Lorthois S., Lagree P.Y., Marc-Vergnes J.P., and F. Cassot (2000) “Maximal wall shear stress in arterial stenoses: application to the internal carotid arteries” J. of Biomechanical Engineering 122 (6): 661-666.

12. Rossitti S. (1995) “Energetic and spatial constraints of arterial networks” Arq Neuropsiquiatr 53 (2): 333-341.

13. Rossitti S., and J. Lofgren (1993) “Optimality principles and flow orderliness at the branching points of cerebral arteries” Stroke 24 (7): 1029-1032.

14. Rossitti S., and J. Lofgren (1993) “Vascular dimensions of the cerebral arteries follow the principle of minimum work” Stroke 24 (3): 371-377.

15. Rossitti S., Stephensen H. (1994) “Temporal heterogeneity of the blood flow velocity at the middle cerebral artery in the normal human characterized by fractal analysis” Acta Physiologica Scandinavica 151 (2): 191-198.

16. Sakata N., Joshita T., and G. Ooneda (1985) “Topographical study on arteriosclerotic lesions at the bifurcations of human cerebral arteries.” Heart Vessels 1 (2): 70-73.

63 L.STA-MCA 186 8 0.04

53 L.STA-MCA 120 16.9 3.5 100 88.8 0.8

854 R. STA-MCA 58 6 3.8 131.4 23.8 0.1

839 L. STA-MCA 80 38.3 0.4

759 L. STA-MCA 69 20 0.2

855 L.STA-MCA 95 47.1 46.4 0.9

852 R.STA-MCA 76 70 50 0.7

115 L.STA-MCA 78 11 3.4 25.9 18.8 0.7

222 L. STA-MCA 242 12.9 4.8 104 105 0.9

944 B. STA-MCA 9 1 0.1

144 R. STA-MCA 30 34.4 1.1

454 B. STA-MCA 154 37.454 R. STA-MCA 37 35 31 0.8

818 R. STA-MCA 89 14 3 71.2 48 0.6

780 R. STA-MCA 73 57 0.7

877 R. STA-MCA 86 14 3.231 L. STA-MCA 24 12 2 46 56 1.2

146 B. STA-MCA 158 8.8 3 100 26 0.2

621 B. STA-MCA 71 11 3.3 10.6 36.4 3.455 L. STA-MCA 45 4.3 3.9 25.8 16.6 0.6

446 R. STA-MCA 40 7.4 3.2 78.9 62 0.7

869 L. STA-MCA 9.3 3.5 2.5 51.8 28.8 0.5

547 L. STA-MCA 18 11 2.1 32 2.4 0.0

7OA-MCA

70 R. STA-MCA45 L. STA-MCA 42.7 50 1.1

763 L.CCA-MCA vein graft 10046 R. STA-MCA 107 16.4 3.3 68 59 0.8

649 R. STA-MCA vein graft 78 48 0.6

171 R. STA-MCA 17 12.4 0.7

564 L. STA-MCA 88 50 0.5

677 L. STA-MCA 56 120 2.1

475 L. STA-MCA 79 15.6 3 67.8 59 0.8

658 L. STA-MCA 79 22 2.8 99 69 0.6

9