The purpose and role of theoretical models is instrumental in the understanding of AVM physiology under normal hemodynamic conditions and during various types and stages of therapy. Models are necessary to achieve reproducibilityÐan essential component of scientific experimentation. An overall drawback inherent to all theoretical modeling is the lack of biological traits and biovariability, as may be available in naturally occurring or constructed in vivo models. However, biomathematical modeling and simulations are useful as a means of providing: (1) a systematic and effective way of assembling existing knowledge about a system; (2) identification of important parameters and determination of the overall system sensitivity to variation in each parameter; (3) calculation of quantitative values of variables that are difficult or impossible to measure; (4) a method to test hypotheses rapidly, efficiently, and inexpensively; (5) identification of specific elements or information gaps that must be further quantified, thus leading to the development of important experiments or quantitative measures; and (6) an effective model which can be utilized to predict the behavior of a real system.[2] Each one of these factors are relevant in the development of an AVM model.

Theoretical models have been used previously to study the hemodynamics of AVMs and their risk of hemorrhage.[3-12] These models represent a theoretical method of investigating AVM hemodynamics but currently provide limited information due to the simplicity of simulated anatomic and physiologic characteristics in available models. Previous AVM models have typically involved multi-compartmental analysis of nidus vessels arranged in a parallel and independent fashion with one AF and one DV. Although a brain AVM with one AF and one DV is somewhat uncommon, it should be the first type of model to be considered due to its simplicity. However, clinical and pathological examination of AVMs have shown that they much more often are fed and drained by multiple AFs and DVs. Also, the tightly interwoven microvessels contained within the nidus are typically interconnected with many branches, at least bifurcations and trifurcations, making them extremely dependent on each other and on the hemodynamics at the feeding and draining pedicles.

Previous models have simulated hemodynamics in AVMs with elementary feeding and draining pedicle anatomy and a nidus typically composed of a single or multiple array of parallel, compartmentalized vessels. The AVM model employed by Lo and colleagues[3 5] consisted of three linked compartments representing AFs, shunting arterioles, and the core vessels of the AVM with flow draining into the central venous drainage. They have simulated hemodynamics within small and large AVMs with results comparable to those observed clinically but neglected the appearance of DVs. Hecht et al[6] expanded upon this concept with computer simulations in an AVM nidus composed of 1,000 nidus vessel compartments or microchannels. As each microchannel was occluded at a constant rate, they observed an exponentially increasing change in the rate of flow reduction.

Ornstein et al[7] introduced a more complex AVM model by considering the influence of inductance, conductance, and autoregulation. They employed an electrical circuit analogue to develop a simulated model of an AVM. Their AVM model consisted of a single AF and DV, and a capillary bed with an encased fistula. The study modeled the hemodynamic effects of AVM occlusion simulating neurosurgical resection and embolotherapy. Gao et al[11] expanded the complexity of the AVM in their theoretical model to include regional cerebral hemodynamics. This model incorporated autoregulation and flow-induced conductance vessel dilatation to investigate mechanisms for severe hyperemia following treatment.

All of these previous models present with common underlying limitations: (1) single AF and either a single DV or neglect of venous drainage, (2) compartmentalized nidus vessels, (3) omission of a fistulous component which is a common occurrence in large AVMs, and (4) representation of the nidus as a single resistance or series of resistances. Modeling flow within the nidus by a single resistance or an array of resistances is probably justified since: (1) it is virtually impossible to acquire any quantitative intranidal hemodynamic measurements at angiography or MRI, and (2) intranidal flow is visually displayed at angiography as a rapid traversal of contrast agent through the AVM without any reference to anatomy or geometry, due primarily to the limited resolution of current imaging systems.

In an attempt to qualitatively and quantitatively investigate blood flow through an AVM and assess its risk of rupture, Hademenos et al[8] developed an electrical network (EN) model of a brain AVM based on morphological, histopathological, clinical, and biophysical observations from human AVMs. The network represents a replica of the cerebral circulation with an AVM nidus where the vessels are represented by interconnected electrical components. Using electrical analogies, the circulatory network can be represented by a complex electrical circuit of connected wires with variable resistance through which the current or flow, powered by an electric voltage source or pressure gradient, will traverse. Hemodynamic information will be acquired through an analogous interpretation of Ohm's law of electricity (Voltage is Pressure Gradient, Current is Blood Flow, Resistance is Vessel Resistance) within an EN.

In a recent study, Gao et al[12] developed a hybrid AVM model implementing a more realistic nidus from the model by Hademenos et al[8] into a pre-existing model. This advanced model was used to investigate mechanisms of AVM rupture.

Given any theoretical model, one must subject the model to ranges of typically observed values for each of the hemodynamic and biophysical parameters to monitor the qualitative and quantitative behavior of the AVM.[10] The structure of an AVM is complex, thereby requiring the definition of a number of parameters prior to the simulation. These parameters include the number and size of the AFs, nidus vessels, and DVs as well as the intravascular pressure at the AFs and DVs. As a simulation of an AVM model is performed, the user must define each of these parameters within values that have been observed at histopathological investigations but must also yield typical values of volumetric blood flow rate. Thus one is confronted with the task of performing simulations based on combinations and permutations of all of these parameters yet arriving at typical values of flow rate. This forms the basis for parameter sensitivty analysis identifying the influence of a combination of parameters on the stability of the model.[13] Parameter sensitivity analysis is critical in defining the limits of the parameter values and is a necessary procedure in validating the corresponding utility of the AVM model.

Theoretical models are only as accurate as their inherent physical and mathematical complexity but provide the unique advantage of investigating the influence of biophysical and hemodynamic factors on clinically relevant issues involving physiology of the AVM in its normal state, at stages during and after therapeutic procedures, and subsequent abnormal conditions predisposing the AVM to rupture. Two examples of applications for a theoretical AVM model include the study of: (1) AVM hemodynamics and (2) AVM pathophysiology.

Example 1: AVM Hemodynamics
One advantage of biomathematical models is the ability to theoretically construct an AVM to represent the structural or angioarchitectural characteristics of a unique or specific AVM based on pre-existing knowledge from clinical, angiographic, histologic, and biophysical analysis. The EN model described in the previous section was developed to simulate the hemodynamic characteristics of a high-flow/low-resistance medium to large AVM with an intranidal fistula. However, in reality, this represents only a fraction of AVM patient cases observed clinically. The question thus arises, "How, if possible, can a biomathematical model be adapted to represent a particular AVM seen in any given patient?" The various hemodynamic and biophysical parameters unique to an AVM, including the radii of intranidal vessels (plexiform and fistulous), systemic pressure, pressures at the AFs and DVs, elastic modulus, and wall thickness vary considerably within an AVM and among a given subset of AVMs.

The volumetric blood flow through the 2-D BM AVM model of Hademenos et al was 814 ml/min.[8] Hemodynamic analysis of the AVM showed increased flow rate through the fistulous component. The flow rate varied from 5.5 to 57.0 ml/min for the plexiform vessels and from 595.1 to 640.1 ml/min for the fistulous vessels.[8] It is impossible to verify the flow rate through an individual nidus vessel in a given patient but the flow rate can be determined quite easily and the value obtained from the theoretical AVM model is in good agreement with clinical observations.

Example 2: AVM Pathophysiology
A possible risk factor for AVM hemorrhage is increased resistance in the venous drainage of cerebral AVMs.[14-17] Venous drainage impairment may result from naturally occurring stenoses/occlusions, or if draining veins undergo occlusion before feeding arteries during surgical removal, or following surgery in the presence of "occlusive hyperemia.[18,19]" Venous drainage impairment is simulated in the model developed by Hademenos et al[8] by gradually increasing the vascular resistance of the impaired DV, thereby restricting flow and introducing an obstruction.[9] For example, let us consider the hemodynamic simulations through the AVM with both DVs patent and with the progressive occlusion of each DV by: 25%, 50%, 75%, and 100% with the other DV fully patent. 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 an extremely large value of resistance. As the DVs are occluded, a resultant pressure buildup and transfer of pressure to nidus regions opposite the DVs will occur. Given this, the questions that need addressing are: "By how much have the biomechanical stresses of the nidus vessels increased as a result of this pressure redistribution?" and "Is this sufficient to induce nidus rupture?"

As one draining vein becomes occluded, the nidus vessels comprising the opposite side of the nidus compensate with increasing hemodynamic values. As the DV fed by the fistula becomes occluded, intravascular pressure in the opposite portion of the nidus increases in response. The nidus vessels in this region assume the majority of the hemodynamic load and undergo a corresponding shift in intravascular pressure in response to the occlusion of the DV.The increase in pressure could be enough to cause rupture of a nidus vessel. In general, venous drainage impairment was predictive of AVM nidus rupture which could occur before or after treatment although rupture is highly dependent on AVM anatomy and physiology.

An important application of the AVM model is adaptation to human AVMs prompting one to consider the question, "How can the model be applied to simulate accurately AVM hemodynamics on a patient-by-patient basis?" In order to do so, one must improve upon the current theoretical AVM models in terms of geometry and complexity of the AVM nidus. A human AVM typically consists of thousands of intertwined and branching vessels contained within an irregularly-shaped nidus. Monte Carlo techniques or other similar computational techniques that employ random number generators to represent physical interactions or phenomena can be employed, not only to accommodate these constraints, but also to generate a more realistic, more complex AVM model using anatomic and physiologic data, maintaining the 3-D spatial arrangement of nidus vessels.

It is fully acknowledged that the limitations of current AVM models are present because of the inability to obtain precise intranidal values of hemodynamic and biophysical parameters on a patient-by-patient basis. These limitations include: (1) pulsatile nature of blood flow; (2) vascular compliance; (3) turbulence; (4) thrombus formation; and (5) venous loading. All of these factors are relevant, to some degree, in the modeling of AVM hemodynamics and its inherent risk of rupture. Pulsatile blood flow plays a prominent role through the course of model development and validation toward ultimate clinical implementation of the AVM model. This was evidenced by Nornes et al[20] who made blood flow measurements at arterial feeders and draining veins of various cerebral angiomas, revealing the unique blood flow waveforms. The influence of pulsatile blood flow can be implemented by considering an electrical network where the cerebral vessels are connected to the heart, represented by an alternating-current (AC) generator. Information for the AC waveform, such as peak pressure/flow and frequency, can be easily obtained from a given patient at the heart as well as the major arterial feeders and draining veins of an AVM via catheter measurements.

Although the importance of the remaining factors can be debated, the inclusion of these factors into a working theoretical AVM model presents with two problems:

(1) There currently exist no experimental measurements/observations of either of these factors with respect to AVM physiology and hemodynamics. Because these phenomena are in vivo processes, it is extremely impossible to acquire reliable information to implement into the theoretical AVM model. If these factors were to be implemented into an AVM model, they would have to be based on crude approximations which will translate into possible large degrees of error.

(2) In addition, because these measurements/observations cannot be obtained for a typical AVM, efforts to validate the behavior of the AVM model following implementation of these factors becomes futile and hopeless.

It is these limitations that prompts one to emphasize the potential role of the AVM model in the management of patients with AVMs. The role of the theoretical AVM model may serve as an additional objective measure upon which to base decisions regarding therapy, particularly in problematic cases where the AVM is either surgically inaccessible, large, or located in a deep or eloquent region of the brain. It would not be intended that the model supercede any and all factors employed currently in the decision process regarding therapy but to provide additional support to a particular decision regarding treatment options.

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