Subclavian Steal Syndrome – A Computer Model Approach

Subclavian Steal Syndrome – A Computer Model Approach C. Manopoulos and S. Tsangaris National Technical University of Athens/School of Mechanical Engineering, Laboratory of Biofluidmechanics & Biomedical Engineering, Greece Abstract— Subclavian steal syndrome is a constellation of signs and symptoms that arise from retrograde (reversed) flow of blood in the vertebral artery or the internal thoracic artery, due to a proximal stenosis (narrowing) and/or occlusion of the subclavian artery. The present study aim to describe this syndrome in patients with subclavian steal steno-occlusive disease based on a computer lumped parameter model. It is studied the cerebral circulation starting from the aortic arch till the arterial branches leaving the circle of Willis. The governing equations of the model are based on the conservation of mass on each node of arteries and on conservation of energy in every loop consisted of artery branches. In addition, the energy loss equation is employed for every artery branch. The system of the equations describing the model is solved with MATLAB. The results show that when the short low resistance path (along the subclavian artery) becomes a high resistance path (due to narrowing) blood flows around the narrowing via the arteries that supply the brain (left and right vertebral artery, left and right internal carotid artery). Keywords— Subclavian Steal Syndrome, Subclavian Steal Steno-Occlusive Disease, Arterial Stenoses, Cerebral Circulation, Circle of Willis.

I. INTRODUCTION The subclavian steal syndrome (SSS) occurs when there is stenosis or occlusion of the subclavian artery proximal to the origin of the vertebral artery. This may cause flow reversal in the ipsilateral vertebral artery as blood is 'stolen' from the circular vertebro-basilar system, to supply the distal territory of the occluded or stenosed artery. Retrograde flow in the vertebral artery, associated with a subclavian or innominate (brachiocephalic) artery stenosis, can be an incidental finding during Doppler ultrasound examination of the cerebral supply. Contorni first described retrograde flow in the vertebral artery in 1960 [1]. Reivich in 1961 first recognized the association between this phenomenon and neurologic symptoms [2]. The same year Fisher dubbed this combination of retrograde vertebral flow and neurologic symptoms subclavian steal syndrome, suggesting that blood is stolen by the ipsilateral vertebral artery from the contra lateral vertebral artery [3]. It was later suggested that such "steal" may cause brainstem ischemia and stroke, either continuously or secondary to arm exercise.

Lord et al. evaluated statistically in vivo the arteriograms and other findings of 42 patients with the subclavian steal syndrome to determine which factors predisposed to vertebrobasilar ischemia [4]. In vitro studies on the syndrome were done by Rodkiewicz et al. in 1992-93 [5] [6]. They used an experimental model of the arterial system in order to show that, for some specific large occlusions magnitude in the left or right subclavian, or in the brachiocephalic artery the blood flow reverses its direction in the left or the right vertebral or right common carotid, or the right internal carotid arteries. It is shown that besides the known single steal syndrome there exists also a double steal syndrome, i.e., blood reverses its flow direction simultaneously in two arteries, both on the right side of the arterial system. This blood is taken from the circle of Willis, which at the same time is significantly supplemented by the increased blood flow through the other arteries leading into the circle of Willis.

II. MATERIALS AND METHODS A. Cerebral Circulation The blood supply of the brain is achieved through a network of blood vessels (arteries and veins) consisting the cerebral circulation. The arteries deliver oxygenated blood, glucose and other nutrients to the brain and the veins carry deoxygenated blood back to the heart, removing carbon dioxide, lactic acid, and other metabolic products. Since the brain is very vulnerable to compromises in its blood supply, the cerebral circulatory system has many safeguards. Failure of these safeguards results in cerebrovascular accidents, commonly known as strokes. Blood is supplied to the human brain by two pairs of arteries: the internal carotid arteries (right & left) and the vertebral arteries (right & left). They have called the supra-aortic arteries. The arteries of the brain originate from the two internal carotid arteries and from the basilar artery, which is formed by the union of the two vertebral arteries (Fig. 1). The intracranial circulation assumes the form of a polygon, called the circle of Willis [7]. The posterior cerebral artery is the anatomical and functional junction between the anterior circulation (carotid system) and the posterior one (vertebro-basilar system) of the circle of Willis.

P.D. Bamidis and N. Pallikarakis (Eds.): MEDICON 2010, IFMBE Proceedings 29, pp. 764–767, 2010. www.springerlink.com

Subclavian Steal Syndrome – A Computer Model Approach

765

B. The Theoretical Model Figure 2 shows a lumped parameter model focused on cerebral circulation. This model assumes symmetric circulation routes via the brain vessels (with non-flexible walls) from the aorta to the right heart atrium. The model shown in Figure 2 is equivalent to the circuit shown in Figure 1. For simplicity reasons, the local resistances of the individual vessels are expressed in lumped parameters.

Fig. 1 Heart to brain main artery circuit. Are depicted: (1) Circle of WillisCoW, (2) arch of aorta, (3) ascending aorta, (4) descending aorta, (5) right & left main coronary arteries, (6) brachiocephalic trunk or innominate artery-IA, (7) left subclavian artery-SA, (8) right subclavian artery-SA, (9) left [right] common carotid artery-CCA, (10) left [right] external carotid artery-ECA, (11) left [right] internal carotid artery- ICA, (12) left [right] vertebral arteryVA, (13) basilar artery-BA, (14) left [right] posterior cerebral artery-PCA, (15) left [right] posterior communicating artery-PCoA, (16) left [right] middle cerebral artery-MCA, (17) left [right] anterior cerebral artery-ACA, (18) anterior communicating artery-ACoA, (a) right [left] posterior inferior cerebellar artery, (b) anterior spinal artery, (c) right [left] anterior inferior cerebellar artery, (d) pontine arteries, (e) right [left] superior cerebellar artery, (f) right [left] anterior choroidal artery, (g) right [left] ophthalmic artery, (h) anteromedial central (perforating) arteries, (i) Heubner's recurrent artery, (j) right [left] labyrinthine (internal acoustic) artery

The blood supply to the brain in a given time defined as cerebral blood flow (CBF). In an adult, CBF is typically 750 ml/min or 15% of the cardiac output. This equates to 50 to 54 ml of blood per 100 grams of brain tissue per minute [8]. CBF is tightly regulated to meet the brain's metabolic demands. The arrangement of the brain's arteries into the circle of Willis (Fig. 1) creates redundancies in the cerebral circulation. If one part of the circle becomes blocked or narrowed (stenosed) or one of the arteries supplying the circle is blocked or narrowed, blood flow from the other blood vessels can often preserve the cerebral perfusion well enough to avoid the symptoms of ischemia [9].

Fig. 2 Lumped-parameter

model of the cerebral circulation shows extra stenosis resistance (Rs) for the right subclavian artery

Assuming the pathological condition where the right subclavian artery demonstrates stenosis equal to a percentage α %≥50 % which is defined as follows: α =ˆ

A0 − As × 100 % A0

(1)

where Α0 is the internal cross-section of the healthy proximal part of the right subclavian artery and Αs the minimum cross-section of the narrowed part of right subclavian artery due to the development of atheromatous plaque. In this circuit, for each side, R1 is the total resistances due to the heart cardiovascular system, R0 is the total resistance after the aortic arch up to the right atrium, R2 is the total of resistances from the end of subclavian artery up to the right atrium, Rs is the resistance that causes stenosis to the right subclavian artery, R3 is the total of resistances from the end of common carotid artery up to the right atrium and R4, R5 and R6 are the total of resistances from

IFMBE Proceedings Vol. 29

766

C. Manopoulos and S. Tsangaris

the end of posterior, middle and anterior cerebral artery, respectively, up to the right atrium. Considering steady blood flow and assuming that the pressure gradient between aorta and right atrium is 80 mmHg with mean blood supply equal to 6.1 l/min (corresponding to the normal total cardiac output) is supposed that a normal reference blood flow distribution is achieved as described by reference [6] from the aortic valve up to the circle of Willis. By this assumption are determined the unknown resistances and the normal distribution of cerebral blood flow of the model taking into account the blood flow symmetry in the brain and the ratio R6/R5=2 according to reference [10]. The resistances of the normal (unoccluded) arteries are calculated by the equation resulting from the Poiseuille formula: Ri−j =

8μL i − j πri4− j

(2)

where μ=0.0365 poise is the dynamic viscosity of the blood, Li-j and ri-j is the length and the radius of the artery, respectively starting from the i node and ending to the j one. The length and diameter of each arterial segment have been taken from the references [6] and [10]. The solution of the circuit of Figure 2 for the non pathological condition when α=0% is achieved by way of calculating all the blood flow-rates (Qi-j)n of the branches and the unknown resistances. The equations used are based on the Continuity of mass on each node of arteries and on Conservation of energy in every loop consisted of artery branches. In addition, the energy loss equation is employed for every artery branch. The non-linear system of the equations describing the model is solved with MATLAB. After the normal distribution blood flow assessment of the model, the solution of the circuit of Figure 2 is achieved calculating all flow-rates of the branches when the stenosis percentages equal to α=50, 60, 70, 80, 90 and 100 %. The resistance caused by the stenosis in the right subclavian artery is calculated according to the following equation: Rs =

8πμLs 4.8 πμ ρ + + 2 Qs A s2 As A s3

(3)

which represents the stenosis as a sudden contraction followed by a sudden expansion [11]. In equation (3), Ls=10 mm is the length of the stenosis in the right subclavian artery showing the development area size of atheromatous plaque, ρ=1.055 gr/cm3 is the blood density and Qs is the blood flow-rate in the right subclavian artery. As the stenosis percentage increases the Reynolds number becomes higher due to the turbulence and inertia phenomena

occur especially in the blood outlet from the stenosis point. The Reynolds number is calculated as follows: Re =

4A 0 ρQs

(4)

πμA 0

III. RESULTS AND DISCUSSION The quantitative analysis of the results concerning the blood flow-rates for different degrees of stenosis is shown in the Table 1. The numerical order of the nodes in the first column of the Table 1 shows the direction of normal blood flow when α=0%. Table 1 Dimensionless blood flow-rates Qi-j/(Qi-j)n for several stenosis percentages α,[(Qi-j)n is the normal blood flow of each branch when α=0%] Branch (i-j) 2-6 6-17 2-5 2-3 3-4 4-9 5-9 3-7 6-0 7-0 4-0 5-0 9-11 7-14 11-12 11-13 13-10 12-8 17-13 14-12 17-18 14-21 14-15 17-16 15-20 16-19

α 50%

60%

70%

80%

90%

100%

1.0953

1.1376

1.2139

1.3651

1.6870

2.1290

1.1333

1.1926

1.2993

1.5108

1.9612

2.5796

1.0786

1.1151

1.1807

1.3109

1.5882

1.9688

0.9116

0.8716

0.7996

0.6570

0.3532

-0.0638

0.9194

0.8817

0.8139

0.6795

0.3931

0.0000

0.8122

0.7243

0.5663

0.2529

-0.4146

-1.3308

1.1975

1.2893

1.4544

1.7817

2.4789

3.4359

0.9059

0.8643

0.7894

0.6408

0.3245

-0.1097

0.9997

0.9996

0.9994

0.9990

0.9981

0.9969

1.0003

1.0004

1.0007

1.0012

1.0022

1.0037

0.9914

0.9874

0.9801

0.9658

0.9353

0.8934

0.9999

0.9999

0.9998

0.9997

0.9994

0.9990

1.0033

1.0046

1.0068

1.0113

1.0207

1.0336

0.8680

0.8096

0.7045

0.4961

0.0524

-0.5567

1.0144

1.0206

1.0316

1.0535

1.1003

1.1644

0.9923

0.9886

0.9820

0.9689

0.9411

0.9028

0.9954

0.9933

0.9895

0.9820

0.9659

0.9439

0.9953

0.9931

0.9892

0.9814

0.9648

0.9420

1.1085

1.1621

1.2584

1.4495

1.8564

2.4150

0.2916

-0.0173

-0.5729

-1.6747

-4.0211

-7.2423

0.9973

0.9961

0.9939

0.9895

0.9803

0.967

1.0027

1.0039

1.0061

1.0104

1.0196

1.0323

0.6141

0.4433

0.1358

-0.4738

-1.7720

-3.5542

1.4028

1.5818

1.9037

2.5423

3.9020

5.7687

1.0075

1.0108

1.0167

1.0286

1.0537

1.0883

0.9922

0.9887

0.9825

0.9701

0.9437

0.9075

Observing the blood flow-rate values (dimensionless form) of the Table 1 the following behavior of the blood

IFMBE Proceedings Vol. 29

Subclavian Steal Syndrome – A Computer Model Approach

flow distribution is noticed for stenosis percentages α>50%. As the degree of stenosis in the right subclavian artery increases, the flow-rates to the left side increase too in comparison with the ones to the right side, where the flow-rates decrease. Namely, flow-rates from the aortic arch up to the CoW of LSA, LCCA, LICA and LVA increase significantly, while the corresponding ones of RSA, RCCA, RICA and RVA decrease significantly, with the last three reversing their directions for high degrees of stenosis. Figure 3 presents the dimensionless flow-rates of some vessels, along with the increment stenosis in the RSA. 3.5

LVA

3

LICA

2.5 2 1.5

Q/Qn

LCCA LSA

increment area

1

reduction area

0.5 0

RSA RCCA

reversed area

-0.5

767

significantly, while Q11-12 rises a little. Moreover, flow-rate of ACoA is directed from the left to the right (16 to 15) and increases, as the degree of stenosis does the same.

IV. CONCLUSIONS A simple model has been developed in order to describe the subclavian steal syndrome, when the heart to brain main artery circuit is occluded at one location on the right side (e.g. beginning of RSA). It is considered steady blood flow in a circuit with non flexible vessels. It has been shown that when the short low resistance path (along the subclavian artery) becomes a high resistance path (due to narrowing) blood flows around the narrowing via the arteries that supply the brain (left and right vertebral artery, left and right internal carotid artery). Three areas of blood flow are distinguished as the RSA is narrowed. The area of blood flow increment that is on the left side vessels, the reduction and the reversed areas of blood flow that are on the right side. However, further investigation is needed to show the blood flow influence due to more occluded locations (e.g. in the IA and/or LSA), and an unsteady blood flow consideration should be taken into account, in order to achieve more realistic results.

RICA

-1

REFERENCES

RVA

-1.5 0

10

20

30

40

50

α (%)

60

70

80

90

100

Fig. 3 Dimensionless blood flow-rates Q /Qn along with stenosis percentages α, Qn is the normal blood flow of each branch when α=0% The reversed flow is mentioned to the IA too, since it lies on the right side. Α slight blood flow-rate increment is observed in BA. This increment helps the blood to move (together with the rise of the LICA flow-rate) through CoW around stenosis. In this way the blood circulation is maintained. Due to the high values of resistances R2 and R3 the blood flow-rates to the branches 4-0, 5-0, 6-0 and 7-0 are kept almost constant. Slight variations of blood flow are mentioned in the brain, as the percentage of the stenosis increases. Namely, a small blood flow reduction happens in the LPCA, RPCA, LMCA and LACA, while a small increment takes place in the RMCA and RACA. Finally, in the CoW the blood flow reverses in the RPCoA, as soon as α=60%. More stenosis increment, up to 75%, also makes the blood flow-rate Q14-15 to reverse. Flow-rates Q17-13 of LPCoA and Q17-16 rise

1. Contorni L (1960) The vertebro-vertebral collateral circulation in obliteration of the subclavian artery at its origin. Minerva Chir 15:268-271. 2. Reivich M, Holling H E, Roberts B, Toole J F (1961) Reversal of blood flow through the vertebral artery and its effect on cerebral circulation. N Engl J Med 265:878-885. 3. Fisher C M (1961) A new vascular syndrome: "The subclavian steal". N Engl J Med 265:912-913. 4. Lord R S A, Adar R, Stein R L (1969) Contribution of the Circle of Willis to the Subelavian Steal Syndrome. Circulation XL:871-878. 5. Rodkiewicz C M, Centkowski J, Zajac S (1992) On the subclavian steal syndrome. In vitro studies. J Biomech Eng 114:527-532. 6. Rodkiewicz C M, Centkowski J, Zajac S (1993) On the subclaviancarotid transportation for the subclavian steal syndrome. In vitro studies. J Biomech Eng 115:205-206. 7. Feneis H, Dauber W (2000) Pocket Atlas of Human Anatomy. Thieme, Stuttgart - New York. 8. Nuffield Department of Anaesthetics at: http://www.nda.ox.ac.uk/wfsa/html/u08/u08_013.htm 9. De Boorder M J, Van der Grond J, Van Dongen A J, Klijn C J M, Kappelle J L, Van Rijk P P, Hendrikse J (2006) SPECT measurements of regional cerebral perfusion and carbondioxide reactivity: Correlation with cerebral collaterals in internal carotid artery occlusive disease. J Neurol 253:1285–1291. 10. Hillen B, Hoogstraten H W, Post L (1986) A mathematical model of the flow in the circle of Willis. J Biomech 19:187-194. 11. May A G, De Weese J A, Rob C G (1963) Hemodynamic effects of arterial stenosis. Surgery 53:513-524.

IFMBE Proceedings Vol. 29