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Stroke. 1996;27:1072-1083

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(Stroke. 1996;27:1072-1083.)
© 1996 American Heart Association, Inc.

Articles

Risk of Intracranial Arteriovenous Malformation Rupture Due to Venous Drainage Impairment

A Theoretical Analysis

George J. Hademenos, PhD Tarik F. Massoud, MD

From the Endovascular Therapy Service, Department of Radiological Sciences, University of California at Los Angeles School of Medicine.


*    Abstract
up arrowTop
 *Abstract
down arrowIntroduction
 down arrowMaterials and Methods
 down arrowResults
 down arrowDiscussion
 down arrowReferences
 
Background and Purpose Increased resistance in the venous drainage of intracranial arteriovenous malformations (AVMs) may contribute to their increased risk of hemorrhage. Venous drainage impairment may result from naturally occurring stenoses/occlusions, or if draining veins (DVs) undergo occlusion before feeding arteries during surgical removal, or after surgery in the presence of "occlusive hyperemia." We employed a detailed biomathematical AVM model using electrical network analysis to investigate theoretically the hemodynamic consequences and the risk of AVM rupture due to venous drainage impairment.

Methods The AVM model consisted of a noncompartmentalized nidus with 28 vessels (24 plexiform components and 4 fistulous components), 4 arterial feeders, and 2 DVs. An expression for the risk of AVM nidus rupture was derived on the basis of functional distribution of the critical radii of component vessels. Risk was calculated from biomathematical simulations of volumetric flow rate with both DVs patent and for four stages of venous drainage obstruction: (1) 25%, (2) 50%, (3) 75%, and (4) 100%. Each stage of occlusion was applied to each DV while the other DV was patent and then to the patent DV while the other DV was totally occluded.

Results For flow through the AVM when both DVs were unobstructed, the baseline risk of AVM nidus rupture ranged from 4.4% to 91.2%. Theoretical rupture occurred in nidus components proximal to the DVs when the risk exceeded 100%, as was observed with the obstruction of DV1 and a patent DV2. The ranges for risk of rupture across the nidus for the four stages were (1) 4.7% to 90.5%, (2) 5.9% to 86.9%, (3) 0% to 98.4%, and (4) 0% to 106.3%, respectively. Rupture was observed for an 86% occlusion of DV1 (ie, the DV fed by the intranidal fistula) and DV2 patent, primarily because of the dramatic shift in the hemodynamic burden toward the weaker plexiform nidus vessels.

Conclusions On theoretical grounds, venous drainage impairment was predictive of AVM nidus rupture and was strongly dependent on AVM morphology (presence of intranidal fistulas and their spatial relation to DVs) and hemodynamics. Specifically, stenosis/occlusion of a high-flow DV induces a rapid redistribution of blood into the weak plexiform vessels of the opposing region of the nidus, causing a hemodynamic overload and an increased risk of rupture. These findings should be carefully considered among all factors affecting the natural history of intracranial AVMs and the mechanisms implicated in their spontaneous rupture. They may also provide a theoretical rationale for some of the hemorrhagic complications that occur during and after surgical treatment.


Key Words: cerebral arteriovenous malformations • cerebral hemorrhage • hemodynamics • models, theoretical


*    Introduction
up arrowTop
 up arrowAbstract
 *Introduction
down arrowMaterials and Methods
 down arrowResults
 down arrowDiscussion
 down arrowReferences
 
Preoperative evaluation and management of intracranial AVMs are based primarily on the natural history, diagnostic presentation and interpretation, and clinical status of the patient.1 Because the majority of AVMs are symptomatic lesions,2 an aggressive therapeutic approach should be implemented on diagnosis. Attempts to accurately assess nidus stability and risk of impending rupture have been proposed on the basis of a number of factors such as perfusion arterial pressure,3 4 nidus size,5 intranidal aneurysms,6 7 8 angioarchitecture,9 10 11 12 and other structural characteristics, including morphological and hemodynamic features of their DVs.13 14 15 16

The influence of venous drainage impairment as seen in clinical observations, ie, the presence of stenosis or occlusion, on the risk of spontaneous AVM rupture has been controversial.17 18 19 Although factors such as type of venous drainage (central, peripheral, or mixed) and number and IVP of the DVs have been studied, the influence of venous drainage impairment and the resultant altered hemodynamics on the risk of nidus hemorrhage have not been investigated adequately. Venous drainage impairment has been described as the "most critical determinant of what happens within and surrounding an AVM nidus."20 It thus becomes important to qualitatively and quantitatively assess intranidal hemodynamics in relation to venous drainage impairment.

One method of theoretical investigation of intranidal hemodynamics of an AVM involves the use of biomathematical models.21 22 23 24 This technique may be extended to assess the intranidal stability and risk of AVM rupture as a direct result of venous drainage impairment. In this article, a novel biomathematical model of a normal intracranial AVM based on network analysis was used to quantify and visualize hemodynamic processes within the nidus after simulated venous drainage impairment. The observed hemodynamic parameters, combined with biomechanical, anatomic, and histopathological information of the involved vessels, were then used to theoretically determine the intranidal stability and predict risk of hemorrhage of the AVM, as may occur either spontaneously or during/after surgical treatment of human AVMs.


*    Materials and Methods
up arrowTop
 up arrowAbstract
 up arrowIntroduction
 *Materials and Methods
down arrowResults
 down arrowDiscussion
 down arrowReferences
 
A Biomathematical Model of an AVM
A biomathematical model based on network analysis25 26 27 was used to qualitatively and quantitatively investigate the altered hemodynamics of venous drainage impairment of an intracranial AVM and the corresponding risk of hemorrhage. The details of this AVM model are described elsewhere.28 In brief, the AVM network, nestled within a simulated circulatory network of the head and neck, consisted of four AFs, two DVs, and a nidal angioarchitecture with a randomly distributed array of 28 interconnected plexiform and fistulous components, as shown in Fig 1Down. This structure of the AVM model was developed to accurately characterize anatomic landmarks and features clinically observed in human AVMs. Twenty-four of the nidus vessels were plexiform, and four nidus vessels were fistulous. The plexiform vessels were held constant at a length of 2.0 cm and a radius of 0.05 cm,29 whereas the length and radius of the fistulous components were 4.0 cm and 0.10 cm, respectively. Two AFs (AF1 and AF2) were considered major feeders, and AF3 and AF4 (a simulated transdural supply) were minor feeders. Both DVs drained into the intracranial venous sinuses.



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  		Figure 1. Schematic diagram of the electrical network describing the biomathematical AVM hemodynamic model. CCA indicates common carotid artery; ECA, external carotid artery; ICA, internal carotid artery; SCA, subclavian artery; VA, vertebral artery; PCA, posterior cerebral artery; ACA, anterior cerebral artery; MCA, middle cerebral artery; E, electromotive force; N, node; and L, loop. The fistulous vessels are designated by the following components of the circuit: i13, i18, i25, and i36. Reproduced with permission.28

The hemodynamics within an AVM can be described by Poiseuille's formula:

(1)
where Q is the volumetric flow rate, {Delta}P is the pressure gradient, R is the inner radius of the vessel, L is the length of the vessel, and {eta} is the blood viscosity ({eta}=3.5 cp). Equation 1Up can be further simplified as

(2)
where Rv is the vascular resistance by the vascular bed given by

(3)
To determine the hemodynamic quantities within each vessel of the vascular array from a simulation, network analysis of the loops and nodes constituting the AVM model circuit was performed to yield 41 linear equations corresponding to the 41 vessels and distinct values of volumetric flow rate. The 41 derived linear equations were solved simultaneously by expanding Equation 2Up into matrix form. The matrices corresponding to pressure and resistance were created using a spreadsheet application (Microsoft Excel) and transported to an advanced mathematical computation program (Mathematica) for solution of the flow rate values for all 41 vessels.

Once the volumetric flow rate (Q) was determined for each simulation, it was then possible to calculate other hemodynamic parameters such as IVP gradient ([{Delta}P]nv) and biomechanical stress (S). Using the resistance for each nidus vessel, the IVP gradient was quantified according to ({Delta}P)nv=(Q)nv(Rv)nv, where nv refers to the particular nidus vessel. The IVP gradient was used to calculate the biomechanical stress from the relation S={Delta}P R/t, where R is the radius of the vessel and t is the vessel wall thickness.

DV Occlusion Schemes
The occlusion schemes adopted for the AVM simulations consisted of systematic impairment of the DVs, assuming underlying normal flow through a normal AVM nidus. Hemodynamic characteristics were first assessed through the AVM with both DVs patent. After this normal simulation, four sets of hemodynamic simulations were performed for various stages of venous drainage impairment. In the first set, with DV2 fully patent, hemodynamic simulations were performed with the progressive occlusion of DV1 by 25%, 50%, 75%, and 100%. Each simulation is described in Fig 2Down, with corresponding circles denoting the stage of occlusion for each DV. The shaded portion of the DV refers to vessel occlusion, and the unshaded area refers to vessel patency. Each stage of occlusion was represented by its calculated value of resistance with the exception of 100%. Total occlusion of a vessel corresponded to an infinite resistance and was represented in the calculations by the largest value of resistance that could be registered in its corresponding cell of the matrix. The second set of simulations involved keeping DV1 fully patent and performing similar simulations for DV2. In the third set, DV1 was 100% occluded and DV2 was occluded systematically by 25%, 50%, 75%, and 100%; the final set consisted of hemodynamic simulations with DV2 totally occluded and DV1 progressively occluded by 25%, 50%, 75%, and 100%.



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  		Figure 2. Tabular description of the state of venous drainage occlusion for each simulation. Shaded area of the circles representing the DVs corresponds to vessel occlusion, and the unshaded area corresponds to vessel patency. The value ranges for the risk of rupture correspond to those seen in all nidus vessels of the AVM. Risk values >=100% represent nidus rupture.

Risk of AVM Hemorrhage
The highly tortuous, structurally weak intranidal vessels coupled with the continual impingement of large hemodynamic forces make the AVM highly susceptible to hemorrhage. The precise location or region of rupture is extremely difficult to observe angiographically and to detect histologically, and it remains a source of speculation in the assessment of AVMs. It is commonly believed that, on the basis of biomechanical properties of the intranidal vessels, rupture occurs when the cumulative hemodynamic stresses of the vessel wall exceed its elastic modulus.

In static equilibrium, the distribution of forces acting on the cylindrical vessel can be explained by Laplace's law. In effect, Laplace's law equates a radial force, P, produced by the transmural blood pressure over the cross-sectional area of the lumen, which distends it to a circumferential force, T, that compensates for the distension. In mathematical form, Laplace's law for a cylinder is

(4)
where R is the radius of curvature. The stress within the vessel wall can be determined by

(5)
where t is the wall thickness. Equation 5Up is only valid, however, for thin-walled vessels, ie, t R. On the basis of biomechanical reasoning, a relation most likely exists between the wall thickness and radius of the vessel, eg, wall thickness decreases for a corresponding increase in radius. The exact relation is unknown and, therefore, in this study, has been approximated by a fixed value. The integral factors that influence the wall stress can be shown by taking the differential of Equation 5Up:

(6)
The circumferential wall stress, given in Equation 5Up, can also be expressed as

(7)
where {varepsilon} is the circumferential strain and E is the elastic modulus of the vessel under physiological conditions. The circumferential strain can be expressed as the ratio {Delta}R/R, where {Delta}R is the change in radial length between the relaxed and strained states and R is the radial length in the unstrained state. Assuming that changes in the radial length are infinitesimal, Equation 7Up can be expressed in differential form as

(8)
equating Equations 6 and 8 as

(9)
The condition of vessel rupture or blowout can be derived by determining the differential relation between the radius and pressure: dR/dP. Dividing both sides of Equation 9Up by dP and solving for dR/dP,

(10)
Expressing Equation 10Up in terms of volume can be accomplished by

(11)
where dV/dP is referred to as volume distensibility and V is the volume of a cylinder ({pi}R2L). Thus,

or

(12)

Assuming that E is constant, the rate of volume expansion will continually increase with increasing pressure to the state where E/R=P/t. Solving for R yields the critical radius, Rc,

(13)
Equation 13Up describes the critical radius of the blood vessel. Any increase or decrease in the state variables that would upset this equality could possibly induce rupture. Evaluation of the risk of rupture is based on the functional distribution of the critical radius with respect to the theoretical blood pressure extremes encountered by the nidus and is given by

(14)

where Pmin and Pmax are the CVP and the "maximum intranidal pressure," respectively, and Pexp is the pressure of the nidus vessel determined at simulation. In this study, Pmin=6.6x103 dyne/cm2 (equivalent to a CVP of 5 mm Hg). The upper limit of arterial pressure experienced by the nidus microvessels before rupture is likely to occur during considerable systemic hypertension (ie, blood pressure that is then transmitted to the AFs and the nidus). It has been observed that systemic hypertension to a mean value of 118 mm Hg does not precipitate AVM hemorrhage.4 The influence of higher levels of systemic hypertension on the propensity of AVMs to rupture is unknown. Therefore, in our calculations it is assumed that the normally low-pressure AFs may reach a maximum value of 74 mm Hg during mean systemic hypertensive levels of 118 mm Hg (derived by assuming a linear relationship between these two parameters30 ). Therefore, 74 mm Hg (equivalent to 9.8x104 dyne/cm2) was chosen as the upper limit of blood pressure (Pmax) possibly encountered by nidus vessels before rupture. This value of pressure is acknowledged as a conservative estimate, since it has not been observed clinically or determined experimentally. In reality, selection of this pressure value is of secondary importance for the risk calculations because this does not affect the principles conveyed by our theoretical study. We consider the value of 74 mm Hg to be an adequate approximation for the purpose of our calculations. It is noted that maximal AF pressures obtained by Young et al30 on systemic hypertensive challenges reached values close to 74 mm Hg.

Because the variables E and t cannot be determined quantitatively from in vivo imaging techniques, they are assumed constant and factored from the equation for risk of rupture. Evaluation of Equation 14Up yields the following expression for risk of AVM nidus rupture:

(15)
The expression given in Equation 15Up represents the normalized probability or risk of rupture and is multiplied by 100% to present the results as a percentage of risk of rupture. The denominator or normalization constant is the integrated distribution of critical radii for the maximum possible transnidal pressure gradient. It can be seen that, on a qualitative basis, as the IVP of the nidus vessels reaches that of the "maximum intranidal pressure," the risk of rupture approaches 100%, implying certain rupture. Conversely, for IVPs closer to that of CVP, the risk of rupture decreases accordingly. Fig 3Down shows the influence of intranidal vessel pressure on the risk of rupture. Risk of rupture is lowest at values closest to those of CVP and increases in an exponential fashion to a maximum value at pressures equal to or greater than the maximum intranidal pressure.



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  		Figure 3. Graph showing risk of rupture versus IVP gradient for the nidus vessels.


*    Results
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 up arrowAbstract
 up arrowIntroduction
 up arrowMaterials and Methods
 *Results
down arrowDiscussion
 down arrowReferences
 
The range of risk of rupture for all nidus vessels during all simulations are displayed in Fig 2Up. The wide range of values observed for each simulation is a reflection on the high variability of risk within the multitude of interconnected nidus vessels. The upper limit of each range is of paramount importance because, should this risk value exceed 100%, then the whole nidus is deemed to have ruptured.

Fig 4Down displays the results of the hemodynamic simulations with the AVM in its normal state, ie, both DVs patent. Fig 5Down depicts the IVP gradient, biomechanical stress, and risk of rupture displayed in each column occurring as a result of the systematic occlusion of DV1 with DV2 patent displayed according to row. This particular series of simulations was chosen for graphical display primarily because of the dramatic hemodynamic effects induced by the occlusion of the DV fed by the fistulous component. The hemodynamic results from the AVM simulations presented in Figs 4Down and 5Down show the individual values of IVP gradient, biomechanical stress, and risk of rupture within each nidus vessel mapped onto the AVM model network. In addition, regions or areas of abnormal hemodynamics induced by the systematic occlusion of the DVs can readily be observed. For example, in the case of normal flow through the AVM shown in Fig 4Down, one would expect values of low pressure, biomechanical stress, and risk of rupture within the plexiform vessels of the nidus. Conversely, high values of pressure gradient, biomechanical stress, and risk of rupture would be expected through the fistulous component of the nidus. For the venous occlusion simulations represented in Fig 5Down, a region of low hemodynamic significance can be seen in the immediate area of the minor AFs, since they are small contributors of blood flow to the AVM. This region is generally consistent throughout all simulations and all stages of occlusion. In contrast, a region of high hemodynamic values can be seen around DV1 where the fistula makes a direct connection. As one DV becomes occluded, the nidus vessels constituting the opposite side of the nidus compensate with increasing hemodynamic values. As DV1 becomes occluded, IVP in the upper portion of the nidus increases in response. The same behavior is observed in the occlusion of DV2 with DV1 patent.



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  		Figure 4. Schematic diagram of the nidus portion of the biomathematical AVM model depicting the intranidal values of IVP gradient (A), biomechanical stress (B), and risk of rupture (C) with both DVs patent (simulation 1).



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  		Figure 5. (facing page). Schematic diagram of the nidus portion of the biomathematical AVM model depicting the intranidal values of IVP gradient (A), biomechanical stress (B), and risk of rupture (C) with DV1 occluded 25%, 50%, 75%, and 100% and DV2 patent. The numbers along the left-hand column refer to the simulation numbers.

Simulation series A (consisting of simulations 2 through 5 and 10 through 13) and series B (consisting of simulations 6 through 9 and 14 through 17) represent a continuous series of systematic venous drainage occlusion in which series A begins with the occlusion of DV1 (simulations 2 through 5) followed by the occlusion of DV2 with DV1 totally occluded (simulations 10 through 13), and series B begins with the occlusion of DV2 (simulations 6 through 9) followed by the occlusion of DV1 with DV2 totally occluded (simulations 14 through 17). Each simulation and its corresponding range of risk of rupture values are summarized in Fig 2Up. Fig 6Down shows the variation of the IVP gradient, biomechanical stress, and risk of rupture for each AVM nidus vessel with respect to stage of occlusion for the systematic impairment of DV1 with a patent DV2, represented by simulations 2 through 5. This set of simulations was chosen for display in Fig 6Down because of the definitive observation of AVM rupture. In Fig 6Down, increases in pressure gradient, biomechanical stress, and risk of rupture are observed at the nidus vessels in close proximity to the DVs and, on a much smaller scale, at nidus vessels near the major AFs (AF1 and AF2). In fact, at 100% occlusion of DV1, rupture was shown to occur at nidus vessel 34. Decreases in pressure gradient, biomechanical stress, and risk of rupture are noted at the nidus vessels near the minor AFs (AF3 and AF4) and the vessels composing the central core or region of the AVM nidus. In comparison with the results in Fig 6Down, similar trends in the hemodynamic values are observed in the remaining set of simulations in series A, with increases in the nidus vessels near the DVs and rupture continuing to be seen in nidus vessel 34. In contrast to DV1, occlusion of DV2 produces relatively minor increases of pressure gradient, biomechanical stress, and risk of rupture at the nidus vessels closest to the DVs, with little to no decrease in parameters elsewhere. The pressure gradient, biomechanical stress, and risk of rupture revealed a gradual increase in the nidus vessels directly connected to the venous drainage (DV1 and DV2). Risk of rupture in nidus vessel 35 increases to approximately 88.7%, whereas the risk in nidus vessel 37 remains at 91.4% before reaching a plateau, even throughout total occlusion of DV1, before sharply decreasing to zero for both DVs occluded. For both DV1 and DV2 totally occluded, the pressure gradient, biomechanical stress, and risk of rupture dropped dramatically to extremely low values, presumably because of a theoretical stagnant flow through the nidus; however, nidus rupture occurs before this possibility.



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  		Figure 6. Three-dimensional line graphs showing the variation of the intranidal values of IVP gradient (A), biomechanical stress (B), and risk of rupture (C) for each AVM nidus vessel with DV1 occluded 0%, 25%, 50%, 75%, and 100% and DV2 patent.

The critical value of stenosis is defined in the literature as the degree of occlusion that shows a pronounced decrease in flow. To elucidate the critical value of stenosis at which this drop in flow occurs and reasons for a substantial decrease in risk of rupture for total occlusion of both DVs, the risk of rupture was plotted against stage of occlusion for the six nidus vessels adjoining the DVs (nidus vessels 32 through 37) for series A and is presented in Fig 7Down. The risk of rupture for the only ruptured nidus vessel (nidus vessel 34) exceeded 100% with the combination of 80% occlusion of DV1 and DV2 patent (point 1) and remains high before decreasing and reaching 100% on the opposing end of this range (point 2) at the combination of 62.5% occlusion of DV2 and DV1 totally occluded. Points 1 and 2 are represented in Fig 7Down by the line segments bisecting the line at 100% risk of rupture. These values for stage of occlusion were obtained by extrapolating from the points defining the above range downward to the x axis. The risk of rupture past this range sharply decreases to values of 0%. Thus, the range of critical venous drainage impairment (causing rupture) for this specific AVM model occurs as a result of combinations that yield 80% to 100% stenosis in DV1 plus 0% to 62.5% stenosis in DV2 (ie, higher stenoses or even occlusion in DV1 are poorly tolerated by the nidus and thus induce rupture).



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  		Figure 7. Graph showing the variation of risk of rupture for the nidus vessels adjacent to the DVs (nidus vessels 32, 33, 34, 35, 36, and 37) over the gradual stages of occlusion in which DV1 is occluded before DV2.


*    Discussion
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 up arrowAbstract
 up arrowIntroduction
 up arrowMaterials and Methods
 up arrowResults
 *Discussion
down arrowReferences
 
Venous Drainage Impairment
Several studies over the last decade have brought attention to the possibility that characteristics of venous drainage in cerebral AVMs may be important factors in the pathophysiology of spontaneous nidus rupture.13 15 19 31 However, not all are in agreement regarding the role of venous factors in the generation of AVM hemorrhage, and this topic remains controversial. Other recent accounts suggest that the importance of the venous system in the natural history of cerebral AVMs may have been overstated.32 In this study, we have adopted a novel biomathematical AVM model for further theoretical characterization and study of the influences of venous drainage impairment on an adjacent AVM nidus, particularly in relation to its consequent risk of rupture.

Characteristics of venous drainage that have been reported previously to influence the risk of spontaneous AVM nidus rupture include (1) venous drainage impairment (stenosis or occlusion of DVs), (2) number of DVs, (3) the location of DVs (deep, superficial, or a combination of both), and (4) the presence of venous aneurysms or varices. The role of the number of veins draining an AVM in relation to the risk of nidus rupture remains unclear, with conflicting accounts in the literature.11 13 14 31 In our theoretical analysis, we chose not to investigate the effect of varying the number of DVs because this would have also added considerably to the complexity and number of the AVM models required and the mathematical calculations entailed. This important analysis will nevertheless be the subject of a future communication. Regarding the location of the DVs, previous results indicate that despite the high improbability that the hemodynamics of deep venous drainage are significantly different from that of more superficial drainage,33 there may be a higher risk of bleeding from AVMs with deep venous drainage.10 14 These factors cannot be examined using our model until the data resulting from direct intraoperative or endovascular pressure sampling of both superficial and deep veins become available.

Venous drainage impairment results from stenoses or occlusions in the veins draining an AVM. Previously, a stenosis has been considered to be a reduction of greater than 50%15 31 or greater than three fifths34 of any AVM DV in two angiographic views. Because of this nonuniform definition of a venous stenosis, we investigated the effects of various (0% to 100%) degrees of narrowing in the DVs on the risk of nidus rupture. Stenoses are most often seen at the junction of major DVs as they enter a dural sinus, eg, at the junction of the vein of Galen and the straight sinus or the junction of a cortical vein with the superior sagittal sinus.31 Complete obliteration of the galenic system was also observed by Viñuela et al19 in 10 patients with AVMs. The exact etiology of a venous stenosis/occlusion is unclear, although several suggestions have been made. Albert et al13 have explained these stenoses as mere common variations in the width of veins or as a consequence of radiographic projection. Local damage to the venous endothelium due to turbulent flow has been proposed by Viñuela et al19 on the basis of previous observations by Fry35 that such intimal damage results from the mechanical action of pressure and shearing stress on the endothelial surface and the change in the electrochemical characteristics of the endothelial membrane produced by convective properties of flow. This may result in subsequent thrombosis and luminal narrowing. Quisling and Mickle36 suggested that some instances of venous narrowing may not be rigidly atretic or fibrotic but rather represent a vasoconstrictive response to increased shunt volume or elevated venous pressure; it is known, for instance, that cerebral veins possess both myogenic and neurogenic vasoconstrictive properties.37 Willinsky et al12 have also postulated that mechanical kinking of veins at the edge of the tentorium may be one of the precipitating factors in the high incidence of hemorrhage from temporal and posterior fossa AVMs.

The site of a typical AVM rupture has never been demonstrated histologically but has been postulated to occur at the venous end of the nidus.20 The mechanisms responsible for this in the presence of venous drainage impairment have been thought to occur because of retrograde venous hypertension yielding regional intranidal hypertension with consequent rupture of delicate AVM vessels.9 19 The presence of significant prestenotic DV hypertension has been confirmed by Miyasaka et al16 during intraoperative measurements in three patients with marked segmental stenoses and a history of AVM rupture. They found an average prestenotic mean pressure of 34 mm Hg (ie, considerably above normal cortical pressures and normal AVM DV pressures28 ), whereas pressure measurements in poststenotic sites approached normal values. From a clinical standpoint, however, conflicting results are available regarding the association of venous drainage impairment and a history of hemorrhage in patients harboring AVMs. Thus, in four series,15 18 19 31 hemorrhage was found to occur in 48% to 94% of AVMs with venous drainage impairment. Willinsky et al12 found that 52% of patients with AVM hemorrhage had venous stenosis. Young et al,30 on the other hand, found no relationship between venous factors and AVM hemorrhage. Similarly, Patel et al18 showed no clear evidence that venous restrictive changes promote hemorrhage. Our own recent experience in correlating angioarchitectural features of cerebral AVMs with a clinical presentation of intracranial hemorrhage has demonstrated no significant association between the presence of venous drainage impairment and nidus rupture: stenosis was present in 25 of 40 patients (62%) with AVM hemorrhage versus 33 of 60 patients (55%) without AVM hemorrhage.11

Risk of AVM Nidus Rupture
To adequately and properly investigate the previously mentioned conflicting clinical observations, a biomathematical AVM model was used to theoretically study the changes in AVM intranidal hemodynamics in response to progressive occlusion of its DVs. In addition, the risk of AVM nidus rupture was determined on the basis of the critical radii of intranidal microvessels. With use of a biomathematical model of an AVM, the results of theoretical simulations are well within expected values from clinical observations and biophysical principles. However, there are several issues involving the AVM model that bear further discussion.

The expression for the risk of rupture used in this research is based on the transnidal distribution of pressure and normalized to the pressure gradient between Pmax (maximum intranidal pressure) and Pmin (CVP). One could raise the following questions: (1) Is pressure the sole variable responsible for risk of rupture? and (2) Why is the risk of rupture normalized to a pressure gradient spanning the two extremes encountered by the nidus? The first issue can be addressed by considering the possible influence of an autoregulatory capacity of an AVM. The lack of autoregulation, which is not implemented in this AVM model, assumes that nidus vessels are viewed as fixed vascular conduits and that increases in arterial blood pressure are transmitted directly to the AVM nidus vessels and to the DVs.38 Because the basis for risk of rupture is the critical radius of a cylindrical vessel described by biomechanical constants typical of a nidus vessel, pressure is at least a complicating if not precipitating factor in the stability and prediction of rupture of the nidus vessels. An increase in pressure translates to increased biomechanical stress of the vessel. According to Laplace's law, the biomechanical stress increases to the point of the elastic limit of the vessel, beyond which the vessel ruptures. The simulation results of normal flow in Fig 5Up reflect a range of risk of rupture for nidus components from 4.4% to 91.2%. Because risk of rupture values do not exceed 100%, rupture does not always occur under normal circumstances. However, as venous drainage becomes impaired, the risk of rupture changes as the intranidal pressure redistributes itself to compensate for the venous occlusion. Increases in risk of AVM nidus rupture are observed particularly in close proximity to the DVs. In the event of nidus rupture, vessels closest to the venous drainage were shown to exhibit a substantial increase in the risk of rupture, and nidus vessel 34 was shown to rupture. The occurrence of rupture at the DV end of the nidus is due presumably to the increased pressure buildup in the nidus vessels feeding into the stenosed DVs experienced at the site of occlusion and is supported by current clinical hypotheses. These findings are clearly visualized from simulation 5 of Fig 6CUp. Interestingly, these observations appear to provide a theoretical biophysical basis for the commonly held supposition that AVMs rupture near the origin of their DVs.20

With regard to the second issue, it is tempting to describe the risk of rupture in terms of the baseline transnidal pressure gradient (ie, that in the presence of normal systemic blood pressure) between the AFs and DVs or to state that any nidus vessel whose IVP gradient exceeds the baseline normal pressure of the AF is prone to rupture. However, AVM physiology, particularly within the nidus, is much more complex. In an AVM with multiple AFs and DVs, as simulated in our model, pressures from each feeding and draining pedicle contribute to the distribution of pressure within the nidus. Thus, the total pressure entering the nidus is starkly different in both magnitude and direction from the pressure exiting the nidus. Because pressure is a physical vector quantity, it stands to reason that the pressure from multiple feeders translates into an intranidal pressure that could possibly exceed the pressure within some AFs, without rupture. It is, however, impossible that the intranidal pressures will exceed those within the feeder with the highest blood pressure. Therefore, for the purpose of our calculations, the maximum intranidal pressure was assumed to be equivalent to the highest values achievable for blood pressure in adjacent AFs, ie, those occurring during systemic hypertension.

The range in risk of rupture for the simulations progressed in continuous trends depending on the state of intranidal and transnidal hemodynamics, with the exception of the simulation for DV1 occluded 50% with DV2 patent. In the preceding simulations where DV1 is patent or occluded 25%, the risk-of-rupture ranges are 4.4% to 91.2% and 4.7% to 90.5%, respectively. When DV1 was occluded 50%, a decrease in the range of values for risk of rupture was observed before the continual progression of these values with further venous drainage impairment. DV1 is fed by the fistula and assumes the majority of the hemodynamic forces. As DV1 is occluded, this reduces the biomechanical stress and hence risk of rupture on adjacent nidus vessels. Although this change in stress translates into an increase in pressure in surrounding nidus vessels, it is well within the upper limit of biomechanical stress of these nidus vessels and can thus easily be accommodated without the incidence of rupture.

A primary limitation of the AVM simulations is that a constant (nonpulsatile) blood pressure was assumed corresponding to the absence of a temporal effect. Because hemodynamic alterations are a time-dependent phenomenon, the risk of nidus rupture will depend on the rate of venous drainage occlusion, ie, abrupt or gradual. Since intracranial AVMs are also drained by a myriad of angiographically occult venules of less than 50 µm,34 it is hypothesized that gradual occlusion may more readily accommodate the hemodynamic and vasodilatatory adaptation of these AVM veins and thus translate into a reduced risk of nidus rupture compared with abrupt occlusion. These important temporal/dynamic factors will be addressed in future analyses using this model.

Critical Stenosis of AVM DVs
The remaining points of interest raised by the results from this study are interrelated in that they are based on the hemodynamic consequences of a stenosis, and they bear further discussion. With respect to this research, the hemodynamics at a DV stenosis (and particularly a critical stenosis) are important primarily for two reasons: (1) to quantitatively and qualitatively investigate the resulting alterations in intranidal hemodynamics and (2) to help explain why rupture was evident in series A, in which DV1 was occluded first, yet was not observed in series B, in which the reverse was performed. The basis for the hemodynamics described in the model simulations was Poiseuille's law, which in effect states a linear relationship between volumetric flow rate and pressure gradient under normal circumstances. One is then left to consider quantitative values of flow rate as the pressure continually increases in the nidus. The linear relationship holds true only as a first-order effect and is valid only to a point where the hemodynamic relationship becomes nonlinear or drastically changes in form. The reasons for this change in form include non-Newtonian viscosity of blood, turbulence, pulsatile driving pressures, kinetic energy transformations, and distensibility of vessels.39 This point corresponds to the transition from laminar flow to turbulent flow. As the stenotic region of a vessel becomes pronounced, the flow rate decreases, the pressure gradient across the stenosis decreases, and the flow velocity increases, as illustrated by Bernoulli's principle. In addition, there is a buildup of excess pressure proximal to the stenosis. The trends of these hemodynamic parameters continue until a critical stenosis is reached. The critical stenosis is unique to vessel geometry and hemodynamics but has been shown to generally occur at about 75% to 80% obstruction of the major vessels in the human vasculature.40 41 42 43 At the point of critical stenosis, a sharp decrease of flow rate is observed that is due to the increased turbulence prior to the stenosis.

As the stenosis totally occludes the vessel, the pressure drop across the stenosis reaches 100% of the maximum and the flow rate is zero. In addition, the prestenotic pressure is equal in magnitude to that at the source or the location just prior to the vessel with the occlusion. As an illustrative example of Newton's third law, the stenosis exerts an equal yet opposite force against the hemodynamic forces generated by the AF driving the blood through the nidus. At 100% stenosis, the pressure gradient is not exactly zero because of the additional energy losses. However, no relations, experiments, or adequate explanations exist that elucidate and accurately characterize these energy losses, leading one to approximate the pressure gradient.43

Hemodynamics before, within, and beyond a critical stenosis are crucial to understanding intranidal hemodynamics during venous drainage impairment. However, the most important issue is the use of this knowledge to explain the reasons behind the observation of rupture during occlusion of DV1 followed by DV2 and nonrupture of the AVM nidus in the reverse case. It is tempting to believe that, on the basis of basic hydrodynamics and clinical experience, total occlusion of either DV would induce definite nidus rupture. This was shown to not necessarily always be the case. Occlusion of the DV fed by the fistulous component transfers an enormous hemodynamic burden to the high-resistance plexiform vessels for exit through DV2. Interestingly, it was noted that as DV1 was totally occluded a sharp increase in pressure was observed at vessels 35 and 26, both of which feed into vessel 34 via N26. It was the physical combination of the marked pressure increases from these two vessels that contributed to the substantial increase in pressure and thus rupture of vessel 34 (ie, rupture at the venous end of the nidus). This can readily be seen in Fig 6Up. In contrast to DV1, occlusion of DV2 (fed entirely by plexiform nidus vessels) places a relatively minor hemodynamic burden on the vessels feeding into DV1. The only increase in pressure and hence risk of rupture was noted in nidus vessel 35, together with the relatively constant high risk of rupture in nidus vessel 37 at 91.4%, but within the biomechanical stresses of the vessel wall. Thus, although the IVP gradient was high in both nidus vessels, the two vessels were able to withstand the increased hemodynamic burden in concert without rupture in either vessel.

One would expect that as both DVs became totally occluded the IVP gradient and hence the risk of nidus rupture would increase accordingly. This is reflected in the well-established clinical/angiographic finding that interruption of the venous outlet of a plexiform pial AVM causes nidus rupture and not simple stagnation of flow within the nidus.44 45 However, the model displays results from simulations at discrete points; therefore, once rupture has occurred, simulations beyond that stage of occlusion become meaningless. One might speculate that if the nidus vessels were strong enough to withstand the increasing risk of rupture, as shown by the sharp slope increase in Fig 7Up, then intranidal flow stagnation would eventually occur (as depicted by the simulation results) when the values for the critical-stenosis range in the DVs are surpassed. Indeed, this may well provide the hemodynamic rationale for the safety and efficacy of the transvenous (endovascular or surgical46 47 48 ) treatment (ie, total occlusion) of single-hole or multichannel dural arteriovenous fistulas. It is hypothesized that under these circumstances the much stronger site of shunting (direct fistula surrounded by dura, as opposed to delicate nidus microvessels) is able to withstand the upslope of the rupture-risk curve in Fig 7Up, without rupturing, to be followed by the precipitous drop in pressure gradient, biomechanical stress, and risk of rupture, signifying blood stagnation and effective therapeutic interruption of the fistula.

Nidus Rupture as a Surgical Complication
The above findings may also help explain the biophysical and hemodynamic mechanisms for some complications related to AVM surgery. Occlusion of AFs before DVs is a conventional technique that is followed by most neurosurgeons during operative removal of intracranial AVMs. This respect for the venous drainage of AVMs dates from Dandy's49 description of catastrophic nidus rupture that occurred when DVs were ligated initially. More recently, however, the idea has emerged50 that, under certain circumstances, it may be safe to occlude, transect, and mobilize a DV (using it as a handle) as a first step toward dissection of the AVM core (Malis procedure51 ). It has been stated that this is permissible (safe) only in the presence of multiple veins draining an AVM, so as not to impair its total drainage with consequent nidus rupture. The results of our study concur in part with this concept, and they provide a theoretical hemodynamic rationale for the safety of the Malis procedure but only under certain circumstances. In the particular theoretical model we have chosen, total occlusion of one AVM vein that drains the fistulous vessels does result in nidus rupture. On the other hand, occlusion of the vein draining the plexiform vessels did not cause rupture. Therefore, our theoretical simulations demonstrate that the Malis procedure may not be safe if any single vein is transected. Instead, careful angiographic analysis may be necessary beforehand to avoid initial occlusion of the vein carrying the most drainage, lest AVM rupture does occur. Should more than one DV be totally occluded before AFs during surgery, then the risk of immediate rupture would depend on the ratio of number of DVs to AFs for that particular AVM. However, adopting the Malis procedure as originally conceived, ie, occlusion of only one vein (the most hemodynamically appropriate) in the presence of multiple veins, would ensure as low a risk as possible of rupturing the AVM. Emerging from our results also is the proviso that, should the Malis procedure be performed using one vein, careful angiographic scrutiny of the remaining DVs would then be necessary to detect any naturally occurring stenosis(es) that may help tip the balance of overall venous drainage impairment toward critical nidus rupture.

Venous restrictive disease has also been implicated recently in the hyperemic complications associated with endovascular or surgical treatment of AVMs.52 Thus, Al-Rodhan et al52 have proposed the concept of "occlusive hyperemia" to explain the morbidity that may occur after resection of high-flow intracranial AVMs. It was suggested that these complications may occur on surgical resection because of outflow obstruction in veins draining an AVM, which also implicates the normal drainage from brain surrounding the AVM, thus resulting in passive engorgement in normal brain tissue. Should a portion of the nidus be retained after surgery, this venous overload (occlusive hyperemia) may become "malignant" because of a concurrent arterial overload within the retained nidus.20 Our experimental simulations therefore also provide a theoretical rationale for events leading to possible retained nidus rupture in the presence of venous drainage impairment and occlusive hyperemia. The results of this study demonstrate that, for the particular configuration and hemodynamics of AVM that were simulated, nidus rupture occurs within a certain range of critical venous drainage impairment. It is conceivable that overall equivalent degrees of impairment may follow incomplete surgery, resulting in rupture of retained nidus.

Conclusions
In conclusion, we have shown using our biomathematical AVM model that AVM rupture as a result of venous drainage impairment occurs in nidus vessels at the venous end of the AVM and is dependent on the nidus angioarchitecture and hemodynamics. Specifically, these results, based on theoretical analysis, suggest that the risk of nidus rupture depends on the relative distribution of hemodynamic load within the DVs before stenosis/occlusion. Stenosis/occlusion of a high-flow DV induces a significant redistribution of blood into the weak plexiform vessels of the opposing region of the nidus, causing a hemodynamic overload and an increased risk of rupture. These findings should be carefully considered among all factors affecting the natural history of intracranial AVMs and the mechanisms implicated in their spontaneous rupture. They may also provide a theoretical rationale for some of the hemorrhagic complications that occur during and after surgical treatment.


*    Selected Abbreviations and Acronyms
 
AF = arterial feeder
AVM = arteriovenous malformation
CVP = central venous pressure
DV = draining vein
IVP = intravascular pressure


*    Acknowledgments
 
This work was supported in part by National Institutes of Health grant 1-RO1-HL/NS52352-01A1. The authors gratefully acknowledge the assistance of Lynne Olson for the artistic preparation and presentation of the illustrations and of Kelly Hademenos and Susan Massoud for the management of the data for the AVM model simulations.


*    Footnotes
 
Reprint requests to George J. Hademenos, PhD, Division of Medical Imaging-172115, Department of Radiological Sciences, UCLA School of Medicine, 10833 Le Conte Ave, Los Angeles, CA 90024-1721. E-mail hademeno@endeavor.radsci.ucla.edu.

Received November 20, 1995; revision received February 22, 1996; accepted February 28, 1996.


*    References
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