Simulation Analysis for Non-invasive Measurement for Hemodynamics in Cerebral Arteriovenous Malformation Types

The Cerebral Arteriovenous Malformation (CAVM) is a complex neurovascular malformation. It contains different types of complex nidus structures combinations. Each CAVM classification is of different clinical procedure and management of disease. The clinical procedure to measure hemodynamics for each CAVM classification is risky, due to complex vessel geometry may get ruptured. In this paper, we address this problem for different AVM classification and nidus complex structures combinations, by measuring hemodynamics non-invasively. The simulation of model is extended for organs other than Cerebral AVM, such as spine and lungs. The modeling results are validated with mechanical measurements. 15 CAVM patients and 20 simulated scenarios of different AVM types are studied to measure hemodynamic indices non-invasively.


INTRODUCTION
The literature describes different classification of Cerebral Arteriovenous Malformation (CAVM) [1]. The CAVM classification is based on different angioarchitectural parameters. The parameters includes some of the combinations, such as CAVM size, location of AVM, number of AVM in the human body and distribution of feeding vessels, pattern of venous drainage, low and the amount of blood steal from the surrounding brain considers different clinical parameters In this paper, we proposed non-invasive methodology to model CAVM types with different Nidus geometric structures combinations. The model is based on the Windkessel model to create different lumped model for various geometric structures. We have simulated different types of AVM geometric structures of different organs and validated results with mechanical simulations. The modeling for AVM types is generic, can be extended for other organs.

Study Population
The CAVM data is obtained from KMC Manipal. The subjects have given informed consent to participate in this study, which was approved by the KMC Manipal.

CAVM Structure Type Simulation
The reporting standards on CAVM by neurosurgeons [9] analyzed various types of CAVM attributes and different structures of Nidus structures. The AVM's are located in various part of human body such as spine, lungs etc. The AVM consists of various combinations of Nidus complex vessel structure. The formation of Nidus structures is different in each type of CAVM. The invasive method to acquire hemodynamics measurements for each type of AVM for various organs is difficult to clinicians and risky to patients. The proposed non-invasive method to measure hemodynamics is based on lumped model. The details of each type of CAVM structures as per Cho [10] classification are as follows:

Type 1
The cerebral vessel structure has feeding arteries, capillaries and veins but in CAVM patients arteries are shunted to veins leading to complex structure. The Type 1 CAVM structure type consists of at most 3 separate arteries shunted to a single draining vein. The three individual feeding arteries are combined to form Nidus complex structures (Fig. 1). The lumped element for Type 1 CAVM structure is based on the Nidus path starting from feeding arteries ending in a draining veins. The lumped element creation is based on the adaptive segmentation and modeling [11].

Type II
The Type II CAVM structure consists of multiple arterioles shunted into a single draining vein. This structure is the complex in nature. As this type of CAVM consists of multiple feeding arteries and each feeding arteries path may not end in a draining veins, sometimes it ends in a loop structures (Fig. 2). This type of CAVM structures is more prominent in patients. The lumped model creation for these types of CAVM structures is based on the patient specific geometric structure paths. In this type of CAVM, there are multiple paths, each path is modelled independently. The steps to model each path are as follows: • Each path is modelled separately.

Type IIIa
The Type IIIa is a plexiform arteriovenous shunts which contains multiple combinations of shunts between arterioles and venules. It consists of multiple individual paths for each shunts and each shunts consists of complex structures. This type of CAVM are non-dilated vessels. The clinical investigation to these CAVM is easier compared to other types as it is too fine to be punctured directly. Each individual path of each shunt is modelled separately. Each path contains complex structures in between the arterioles and venules (Fig. 3). The individual path with complex structures is subdivided in to smaller substructures. The substructures are modelled independently. The individual path is modeled by combining each substructure lumped model. The complete model for Type IIIa is formed by combination of lumped model for each path interconnected in parallel combination.

Type IIIb
The Type IIIb is the type of CAVM which are dilated in nature. This type of CAVM has multiple combinations of arteries which are dilated and interconnected to multiple dilated draining veins (Fig. 4). The dilation causes multiple complexities. The dilated vessels change the cerebral hemodynamics and result in increased pressure at Nidus. The invasive procedure for these CAVM types is risky. In this type of CAVM, the lumped model is created for each path. Each vessel path from artery to vein through nidus complex structures are modelled independently. The complete vessel structures for Type IIIb are formed by connecting each vessel path lumped model in parallel combinations. The challenge in this CAVM is variations in a flow rate and pressure variations are higher, due to dilation of vessels. This is addressed by simulation of lumped model using different input signal combinations. The Fig. 5 shows example for CAVM complex vessel structure of Type II. The feeding arteries are represented as A, B, C. Each feeding arteries is modelled separately. The lumped models for each vessel path are as follows:

Feeding Arteries Vessel A
The Vessel "A" starts from junction 10 and ends in Nidus formation. The vessel path consists of different substructures using adaptive segmentation and modeling technique. Each substructure is modelled independently for the vessel path 10-A (Fig. 6). The lumped elements are reported in (Fig. 7) for Node 10 to Node 11 and in Fig. 8 for Node 11 to Node A. The lumped elements model for vessel path Node 10 to Node A reported in (Fig. 9).

Feeding Arteries Vessel C
The vessel "C" is modelled for the location starts from Node 10 to Node "C" (Fig. 10). The vessel structure for Node10-Node"C" is of bending geometry. The lumped model for Node 10-Node "C" (Fig. 11).

Feeding Arteries Vessel B
The vessel "B" is modelled for location Node 10-Node B (Fig. 14). The lumped model for Node 10-Node B (Fig. 15). The complete lumped model for complete path combinations is the feeding arteries bifurcation junction to Nidus complex structure formation. The lumped network is formed by parallel combination of (Node 10-Node A) || (Node 10-Node C) || (Node 10-Node D) || (Node 10-Node B) (Fig. 16).The lumped network equations are explained in Appendix A.

Signal Simulations
The different types of CAVM is simulated by different signal combinations. This will simulate actual patient conditions of CAVM with different Nidus structure types and geometric vessels combinations. The signal combinations replicate flow and pressure variations inside CAVM types. In case of Type II and Type IIIb, the complexity of vessel structures is comparably higher than other types. The signal simulations for these types of CAVM patients, helps to analyze each specific part of CAVM using lumped element for specific geometric variations. The input signals to lumped model are varied for both magnitude and phase signals (Fig. 17).

Simulation of Complex Path Structures
The different combinations of vessel complex structures is derived from different types of AVM from various anatomical location such as spine AVM, pulmonary AVM [13]. The vessel path creation is based on vessel path navigation, obtained from clinical data. In some cases, vessel path are derived with help of clinicians by visual inspection. The path structures are derived based on adaptive segmentation and modeling methodology (Fig. 18). The lumped model is created for patient specific nidus structure path with different complex geometric structures. The proposed non-invasive method to measure hemodynamics can be extended for other AVM in other organs. However, we didn't get Type IIIa CAVM types. The data is combination of Type I and Type II. The Type IIIb is subset of Type II. The validation is performed using mechanical simulation only not with clinical measurements.

Validation with Mechanical Simulation
The CAVM types are modeled using lumped models and results are compared with mechanical simulations. The mechanical software used is Comsol Pipe flow module software. The extracted nidus path of any vessel network is modelled using adaptive segmentation and lumped modeling. The results are compared with mechanical simulation and comparison is performed for the amount of percentage deviation. Fig. 19 shows the Type I feeding artery vessel structure. This CAVM has three feeding arteries inputs namely vessels path -3, 7, 6. Each vessel path is modelled separately. The lumped model network is created independently for each path and interconnected to form a complete vessel path for CAVM Type I structure (Fig. 20). The complete lumped model is simulated for signal variations (Fig. 21). The Table 1 shows that for various node locations in Fig. 21, pressure values are measured. The node is simulated for different pressure signals such as 80 mmHg, 100 mmHg. The node outputs for each input represents the output voltage/pressure for given input. The node output is compared with mechanical simulations. The amount of percentage deviation between them is calculated.   The complex structure of AVM is common in all its types, irrespective of organ or location. Since network formation using Windkessel is generic, it can be applied for any type of AVM of any organ. The proposed logic to create model and its analysis for flow variations of any type of AVM of other organs still hold good.

CONCLUSION
The objective of this study is to measure hemodynamics for different CAVM types and Nidus structures variations. The proposed methodology based on non-invasive method, using lumped model address the problem faced by clinicians. The present clinical procedure to measure AVM hemodynamics measurement by invasive procedure is risky as Nidus may get rupture, which is evident in Type II and Type IIIb patients. In this paper, we have created a lumped model for Nidus for different CAVM structures and non-cerebral organs such as spine and lungs to measure hemodynamics non-invasively. The modeling results are validated with mechanical results, however, the limitation of this study is the absence of clinical validation for modeling results for spine and pulmonary AVMs.

ETHICAL APPROVAL
All authors hereby declare that all experiments have been examined and approved by the appropriate ethics committee and have therefore been performed in accordance with the ethical standards laid down in the 1964 Declaration of Helsinki.

APPENDIX A
The lumped model are solved by Runge Kutta Method 4th order method. The lumped model creates a set of differential equations, which are solved are using Runge Kutta Method. Runge-Kutta method, which is based on formulas derived by using an approximation to replace the truncated Taylor series expansion.
The steps to solve differential equations are as follows: 1. Apply KVL, KCL to the created lumped network models. 2. The differential equations are generated for each loop as per KVL. 3. Solve KVL equations using Runge Kutta Method The following example shows solution for simple lumped model.
Using Ohm's and Kirchhoff's laws, we get the formula for current i: i = -1/(R C) q Because i = dq/dt, we obtain the first order differential equation: Knowing the charge q0 at the instant t0, we would like to find the time variation of the charge, which is an unknown function q = q (t). Despite the fact that an analytical formula for this function exists, we want rather to get numerical approximations of q = q (t). Among many methods invented to solve such initial problems, fourth-order Runge-Kutta method is more applicable for our analysis. Having a function of the first derivative and an initial condition: dy/dt = f (t, y), y (t0) = y0 The Runge-Kutta method finds the next value yn+1 from the present value yn with the help of following equation: yn+1 = yn + h/6 (k1 + 2 k2 + 2 k3 + k4) Where h is a selected time interval and coefficients k1 to k4 are: k1 = f(tn, yn) k2 = f(tn + h/2, yn + h/2 k1) k3 = f(tn + h/2, yn + h/2 k2) k4 = f(tn + h, yn + h k3) _________________________________________________________________________________