Investigation of brain arterial circle malformations using electrical modelling and simulation

Advances in Electrical and Electronic Engineering 212 INVESTIGATION OF BRAIN ARTERIAL CIRCLE MALFORMATIONS USING ELECTRICAL MODELLING AND SIMULATION K. ápová, V. Blazek, I. áp University of Zilina, Faculty of Electrical Engineering, Velký diel, 010 26 Zilina, Slovakia Tel.: +421-41-513 2100, Fax: +421-41-513 1519, E-mail: [email protected] Summary: The paper deals with the cerebral arterial system investigation by means of electrical modelling and simulations. The main attention is paid to the brain arterial circle malformations (stenoses and aneurysms) and their determination and evaluation by computer-aided methods as tools of a non-invasive diagnostics. The compensation possibilities of brain arterial circle in case of presence of concrete arterial malformations are modelled and simulated. The simulation results of brain arteries blood pressures and volume flow velocities time dependences are presented and discussed under various health conditions. Keywords: brain arterial circle, arterial malformation, electromechanical analogy, electrical modelling and simulations, noninvasive diagnostics, computer-aided methods, state variables. 1. INTRODUCTION During the medical treatment of cerebro-vascular diseases it is often necessary to occlude one of the brain supplying arteries. Due to the arterial circle of Willis (CAW – Circulus Arteriosus Willisii) this intervention contains no further consequences for the majority of patients. But because of an unfavourable vessel anatomy or vessel disease this intervention causes an ischemia in corresponding parts of the brain for some patients who then may suffer from strokes. Until now, this situation can only be estimated by invasive and inaccurate diagnostics, which themselves contain the risk of cerebro-vascular accidents. For this reason profound insights into the complex haemodynamic interactions within the brain were tried to enable by modelling the arterial circle of Willis. Replacing invasive diagnostics by computational simulations, a reliable and noninvasive diagnostic tool estimating the haemodynamic effects of endovascular vessel occlusion is obtained. The model is based upon the analogy of hydrodynamic and electromagnetic state variables called as electromechanical analogy, [1], [2], [3]. The vessel properties are described by coupled first order linear differential equations, which are subsequently nonlinearly extended. The anatomical parameters and boundary conditions of the differential equations are defined by non-invasive, clinically established diagnostics. By these means any given anatomical structure of vessels can be designed. Time continuous simulations of different states of all calculated state variables, such as blood pressure and flow in any given vessel of the model, become feasible and online observable. Critical occlusions can be simulated and corresponding changes in flow or pressure can be observed without exposing the patient to a risk of health, [4] to [7]. On the first stage of development the individual model of each patient must be designed and validated by a comparison with measured data. Although an agreement between the modelled data and those recorded from observation of the subjects is not reached in each case, because the state variables generated by the model behave physiologically. The reaction of individual cerebro-vascular systems in critical situations as occlusions of the main brain arteries is investigated under special conditions. Some deviations between modelled and measured data are based on systematic inaccuracies concerning the determination of the vessels geometry. Using the mentioned electromechanical analogy the equivalent electromagnetic systems have been derived and consequently the computer modelling and simulation of CAW under various conditions were performed. The results of the CAW equivalent electromagnetic systems derivation and their modelling and simulations were published in the previous works [6], [7], [8]. Following from these works the CAW significant malformations have been investigated and evaluated by means of the electrical modelling and simulations in this paper. 2. THEORY The standard model of CAW is the main basis for the individual modelling of vessels networks of individual patients CAW. The principal topology of CAW is illustrated at Fig.1. Investigation of brain arterial circle malformations… 213 Fig. 1. Topology of brain arterial system The basic equations of one vessel segment modelled by the equivalent electric two-port element according to the electromechanical analogy are expressed in terms of the haemodynamic state values in differential equations, [4], [5] [ ] )0()()()()( )( 1 )( 1 0iiiii t i i IdttRtItPtP tL tZ +−−= −  (1) [ ] )( ),( )( )()( )( )( 3 2 )( .1 2 2 tP tC QtQ tItIR tQ tI ltP ex i iui iidi i i i + − +−+         = + ω ρ (2) [ ]  +−= + )0()()()( 1iiii QdttItItQ , (3) where I i is the blood flow, P i blood pressure and Q i blood volume in the i-th vessel segment, R i is a viscose friction resistance of flowing blood and R di resistance corresponding to the friction losses inside the vessel wall of segment i. The vessels parameters in i-th vessel segment, resistance R i and inductance L i in longitudinal direction and the capacitance C i and crossconductance G i in transversal direction, depend on the vessel geometry and the elasticity of vessel walls. The formulas enabling their calculations are given, by following expressions, [2] and [3] )( 8 )( 4 tr l tR i i i π η =, )( )()(2 3 ),( 2 5 0 tr r thE l tC i i idi i i ω π ω = )( )( 2 tr l tL i i i π ρ =, ii ii di lr tht tR 3 0 0 )()( 3 2 )( π η =, (4) where η is the blood viscosity, l i is the length of a vessel segment i, r i vessel internal radius, r 0 the vessel internal radius without drawing out, ρ is the blood density, E di is the relative dynamic part of Young elasticity modulus, h i is the vessel wall thickness of segment i. From the preoperative CAW risk estimation point of view there is the realistic simulation of the arterial malformations, especially stenoses, of the main importance in this area. In order to investigate the influences of an arterial stenosis in terms of the state values as blood pressure or blood velocity, the basic mathematical model, (e.g. in [6]), should be aided by the equation expressing the mutual continuity between the pressure drop caused by the stenosis and the blood flow through the vessel section under stenosis influence, td tId tr Kl tItI r r r K tI tr K tP ust st tv stenose )( )( )()(1 2 )( )(2 )( 2 2 2 2 0 4 0 23 π ρ π ρ π η +         −+=∆ , (5) Advances in Electrical and Electronic Engineering 214 in which the r(t) is the internal radius of the vessel (pipe) run through by blood, r 0 the internal radius in the healthy vessel and r st the internal radius in the place of the highest stenosis (which means maximum narrowing of the vessel). The length of the stenosis is described by l st , η and ρ design the blood viscosity and density correspondingly. The coefficients K t , K u and K v were stated empirically, [5]. The shape (profile) of the stenose has been described by the cosinus function )()( 0 xfrxr −− , where                 += st l x xf πδ 2 cos1 2 )( and      −= 2 , 2 stst ll x (6) The δ is the maximum height of the stenosis, Fig. 2. The stenosis parameters for the simulation are also given at the Fig. 2. The nonlinearities caused by blood whirl, eq. (1), create the main reason for the intensified pressure drop in the vessel under stenosis influence. Fig. 2. Vessel segment with stenosis. According to the previous works the following stenosis parameters of the equivalent electric system of the vessel segment can be defined by the relations )( )( 2 tr Kl tL ust st π ρ =, )(2 )( 3 tr K tR v st π η =, 2 2 2 0 4 0 2 1 2 )( )(         −= st t dp r r r tIK tR π ρ (7) Then the equation (5) can be expressed by relations [ ] td tId tLtItRtRtP stdpststenose )( )()()()()( ++=∆ ( ) [ ]  ++−∆= )0()()()()( )( 1 )( ItdtRtRtItP tL tI dpststenose st (8) The resistance R st (t) is similar to the stationary longitudinal resistance in Hagen - Poisseuille law, [1], and it expresses the fact that with the increasing a vessel narrowing the viscose friction also grows up. Its value is strongly influenced by the stenosis geometry. The coefficient K v is no more constant, but it is a function dependent on the stenosis shape. The term R dp describes the resistance which arises under the influence of the poststenotic turbulences and it depends only a little on the stenosis geometry. The equation (8) is the equivalent one to the differential equation (5), which describes the blood state values of a vessel segment without stenosis. In the case of a vessel segment with the stenosis i the following differential equation is used for the blood flows calculations ( ) [ ] )0()()()()()()( )( 1 )( 1idpstiiii st i ItdtRtRtRtItPtP tL tI +++−−=  − , with )( 8 )( 4 tr l tR i i i π η = (9) By the value L st the effect of the blood inertial mass has been involved into the calculations. In order to respect better the stenosis influence on the CAW haemodynamics the stenosis degree SD has been considered. The formula for the SD determination is 10011001 2 0 2 0 ×         −=×        −= r r A A SG stst (10) where A 0 is the healthy vessel cross-section and A st x [cm] 1 2 3 0,04 f (x) δ r 0 r st l st Investigation of brain arterial circle malformations… 215 is the cross-section of the vessel under the stenosis. For the modeling reasons the stenosis has been placed in the Arteria carotis interna , concretely 5 cm above the splitting of Arteria carotis communis in the Arteria carotis interna and externa, Fig.1. The input signal has been created by blood flow velocities in the right Arteria vertebralis and the left Arteria carotis interna. The stenosis model is given at Fig. 3, where the modeled prestenotic vessel shunt is shown. Fig.3. Topology of the modelled prestenotic shunt. P 1 P 2 I 1 P 1 ’= P 1 P 3 < P 2 I 1 I 3 I 2 I 1 = I 2 + I 3 90 70 50 30 10 -10 Degree of stenosis 100 90 70 80 60 50 A. cerebri media dex. 100 90 70 80 60 50 A. commun. post. dex. 100 90 70 80 60 50 0 1 2 3 4 5 6 A. basilaris A. cerebri media sin. A. carotis int. sin. A. pericallosa sin. (A2) A.cerebri post.sin. (P2) 100 90 70 80 60 50 100 90 70 80 60 50 100 90 70 80 60 50 100 90 70 80 60 50 0 1 2 3 4 5 6 Pressure [kPa x 130] Time [s] Time [s] Pressure [kPa x 130] % Fig. 4. The blood pressure time dependence in CAW in the case of stenosis in Arteria Carotis Interna Advances in Electrical and Electronic Engineering 216 Because of stenosis the reduced blood flow I leaves the prestenotic vessel part (I 3 < I 1 ). In order to keep the continuity principle the aided blood flow I 2 leaves the modeled vessel segment through the shunt. The blood pressure in the prestenotic part hardly increases while the considerable pressure drop P in the poststenotic vessel part is remarkable (P 3 < P 2 ). In the principle the flow I 2 corresponds to a prestenotic blood roundabout way into other vessels, in this case benefit to Arteria carotis externa or Arterie subclavia and it leads to a less congestion of the poststenotic areas. 3. RESULTS The following simulation results according to the CAW topology, Fig. 1, and with a slowly continuous increase of stenosis degree are presented in Fig. 4 and Fig. 5. The graphs represent the blood pressure and blood flow time dependences in various intracranial arteries (of CAW). The vessel segment (Arteria carotis interna sinistra) parameters, [4], [5], used for the simulation are given by following way, radius r 0 = 0,192cm, length l = 5cm, elasticity E l = 1,1652x10 6 g/cm.s 2 and pulse wave velocity PWV = 925cm/s. The maximum stenosis degree was SG m = 83%, the stenosis length was l st = l = 5cm. The mean arterial pressure corresponds in all vessels before the stenosis introduction to the mean pressure values given in the literature, [4], [5], Fig. 1. In the case of blood pressure the significant changes occur as far as the stenosis degree value is about 70%. The small blood pressure drop is by the reduced congestion in the CAW evident and in accordance with the principle of continuous pipes the pressure drop in all other vessels is also possible to explain. Also in the case of blood flow, a nonlinear part of the whole resistance R dp becomes more important than the linear one R st mainly with the growth of the stenosis degree. The most distinct is the reducing of the blood flow in the right Arteria carotis interna behind the stenosis, while the flow in the left Arteria 90 70 50 30 10 -10 Degree of stenosis A. pericallosa dex. (A2) 100 110 70 50 160 110 60 10 500 400 300 200 100 0 1 2 3 4 5 6 A. cerebri ant. dex. (A1) A. carotis int. dex. A. carotis int. sin. A. pericallosa si n. (A2) A. commun. anterior A. cerebri ant. sin. (A1) 110 90 70 50 80 60 40 20 0 -20 160 110 60 10 500 400 300 200 100 0 1 2 3 4 5 6 Volume flow [ml/min) [%] Volume flow [ml/min) Time [s] Time [s] Fig. 5. The blood flow time dependence in CAW in the case of stenosis in Arteria Carotis Interna . Investigation of brain arterial circle malformations… 217 carotis interna is nearly the same in spite of the stenosis, Fig. 2. From the Fig. 2 there is possible to show the CAW as the responsible and reliable one for the vascularization of the brain. The simulation results show how because of stenosis in one vessel segment the flow distribution in CAW develops in order to ensure a sufficient blood supply of the brain. In the left Arteria cerebri anterior (A1) the blood flow increases so that not only the left one but also the right (A2) through the Arteria communicans anterior is supplied by blood. Simultaneously the blood flow in the A1 decreases and so significantly more blood can flow from the stenotic Arteria carotis interna into the right Arteria cerebri media. 4. CONCLUSION The obtained simulation results for the blood pressure and mainly for the blood flow show the compensation possibilities of CAW in case of one brain arterial occlusion. The critical occlusions of an individual cerebral model were simulated and corresponding changes in flow and pressure were observed without exposing the patient to a risk of health. The reactions of the individual arterial system in critical situations were investigated and compared with the recorded data from observations of the subject. The elaborated method of a non/invasive preoperative risk/estimation can be assessed to be efficient. There is hope that this flexible, time saving and reproducible method makes a valuable contribution to avoid complications of induced vessel occlusion during medical treatment by an improved operation planning. REFERENCES [1] Jager, G. N., Westerhof, N., Noordergraaf, A.: Oscillatory Flow Impedance in Electrical Analogy of Arterial System, Circulation Research 16 (1965), 121-133. [2] Gaelings, E.W.: Numerische Simulation Haemodynamischer Prozesse in Vascularen Netzen, Shaker Verlag Aachen, 1996, ISBN-3-8265-1509-9. 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