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PERSONALIZED MATHEMATICAL MODELING OF CEREBRAL ARTERIAL ANEURYSMS
Aleksandr Kancherovich KHE12, Aleksandr Pavlovich CHUPAKHIN12, Aleksandr Aleksandrovich CHEREVKO1,2, Daniil Vasilevich PARSHIN1, Aleksey Leonidovich KRIVOSHAPKIN3, Kirill Yurevich ORLOV3
1 Lavrentyev Institute of Hydrodynamics 630090, Novosibirsk, Lavrentyev av., 15
2 Novosibirsk State University 630090, Novosibirsk, Pirogov str., 2
3 Meshalkin Novosibirsk Research Institute of Circulation Pathology 630055, Novosibirsk, Rechkunovskaya str., 15
The mathematical and computational framework for personalized modeling of the cerebral hemodynamics in the presence of arterial aneurysms has been described. As an example we study nonlocal hydrodynamic properties of cerebral arterial aneurysms. The area of the vessel aneurysm influence on the surrounding blood flow and the changes caused by pressure increase and decrease have been determined using numerical simulations, as well as the flow features in the case of multiple aneurysms presence have been investigated. The personalized clinical data obtained during intraoperative endovascular measurements have been used for computations.
Keywords: hemodynamics, arterial aneurysm, aneurysm rupture factor, liquid 3D-flow modeling.
Cerebral aneurysms are local enlargement of the arterial wall due to the wall damage and weakening. In most cases aneurysms occur in the places of anatomic variations and pathological conditions or high-flow arteriovenous malformations which cause locally increased flow in the cerebral circulation, and, at points of flow bifurcation [7]. Aneurysm is one of the most frequent and dangerous diseases of the cerebral arteries. The most serious consequences of the presence of the aneurysm are their rupture and intracranial hemorrhage which could be lethal. According to statistics, up to 5 % of all deceased people being in autopsy have cerebral aneurysms. Treatment of aneurysms is a challenging task, as often there are no visible symptoms of aneurysms before its rupture. At the same time, treatment car-
ries a risk which often exceeds a risk of having an aneurysm rupture on the early stage. So, at the moment when neurosurgeon recognizes that patient has aneurysm, he meets a problem of determining the time when it is better to treat the patient. To start a surgery on time, a surgeon must know how aneu-rysm is growing and when it will rupture. Despite an aneurysm is a frequent disease, mechanisms of its formation, evolution, and rupture are not well understood yet. Thus, the planning of an effective surgery is a very difficult task for neurosurgeons.
There are several factors involved in the aneu-rysm formation, growth, and rupture, such as histo-logical, hemodynamic, and genetic factors. Modeling of aneurysms is a complex multi-parametric problem. There are several theories of aneurysm
Khe A.K. - candidate ofphysical mathematical sciences, senior researcher of laboratory for differential equation, e-mail: alekhe@hydro.nsc.ru
Chupakhin A.P. - doctor ofphysical mathematical sciences, head of laboratory for differential equation Cherevko A.A. - candidate ofphysical mathematical sciences, senior researcher of laboratory for differential equation
Parshin D.V. - candidate ofphysical mathematical sciences, junior researcher of laboratory for differential equation, e-mail: danilo.skiman@gmail.com
Krivoshapkin A.L. - corresponding member of RAS, chief researcher, neurosurgeon
Orlov K. Yu. - candidate of medical sciences, neurosurgeon, head of the center for angioneurology and neurosurgery 114 СИБИРСКИЙ НАУЧНЫЙ МЕДИЦИНСКИЙ ЖУРНАЛ, ТОМ 36, № 1, 2016
formation and growth. For example, in [7] it is stated that the main factor influencing a dynamics of the aneurysm is the wall shear stress (WSS) provoked by the blood flow. It is considered that endo-thelium could sense WSS and effect the wall structure. At the same time, Sforza et al. doesn't assert that it is exactly the WSS increase that leads to aneurysm, or to its growth, or rupture. According to them, there are two main hypothesis: «low-flow» and «high-flow» theories (low and high velocities of the blood flow). We are not able to decide which of these theories is correct, as each of them has its pros and cons.
Treating an aneurysm requires a surgery. There are several types of endovascular surgery that could be used. The main idea of such treatment is to cut off an aneurysm from the blood flow by an introduction of the agents provoking thrombosis inside aneurysm, or, in other words, to do embolisation of the aneurysm.
The main difficulty of the endovascular treatment is coming from a wide-neck aneurysm, as it is often too difficult to occlude an entire aneurysm sac. In that case, neurosurgeons usually resort to put a stent either to redirect flow and reduce cross-neck flow into the aneurysm or to bring a vessel geometry to its initial state. Usually neurosurgeons don't try to coil such aneurysms because of the risk of the coil falling out after a surgery even with the stent support.
In our previous papers [8, 9] we studied local characteristics of flow in the vicinity of aneurysm (both hydrodynamic and mechanical parameters). Our focus was on the analysis of the main hemody-namic parameters and elastic stresses for the clinical data analysis of a specific patient. To understand how the fluid flows in the aneurysmal sac and how the flow changes after a surgery, we examined streamlines' behavior. Furthermore, we examined pressure distribution, its gradient, and wall shear stresses.
In this work our goal is to determine how far the aneurysm affects the hemodynamics in the circulation network. We carried out several numerical experiments for different case studies: effect of the presence of the aneurysm, influence of the pressure change, presence of the multiple aneurysms.
Real-world data play crucial role in mathematical modeling and it is the main ingredient for verification and justification of the theory being developed. Designing an experimental setup and a measurement system is a challenging task in engineering sciences. And it is even more difficult in natural and medical sciences. In this work we describe a measurement system we use in the framework of mathematical modeling of cerebral hemodynamics. Clini-
cal data are collected during endovascular neurosur-gical treatments of cerebral blood vessel diseases (arteriovenous malformations or aneurysms).
A widely used method to assess cerebral blood flow is the transcranial Doppler ultrasonography [3]. However it is applicable only for relatively large vessels-carotid arteries and their first level branches: a., m., and p. cerebral arteries. Moreover this method does not allow one to measure the pressure. In our work we use endovascular flow and pressure measurement system Volcano ComboMap (Volcano Corp., USA). Measurements are performed using a 0.36 mm diameter wire ComboWire. In Fig. 1, 2 an example of the blood flow monitoring is shown.
Using analog-to-digital convertor we collect the data being measured in real time. The data is being processed and displayed with a home-made software (Fig. 3). The software displays real-time velocity-pressure and flow rate-energy flow rate diagrams which can be used to evaluate the operation [0, 4, 5].
Fig. 1. Measurement locations
Fig. 2. ComboMap screen showing pressure and velocity
Fig. 3. Clinical data acquisition and real-time diagrams
Data post-processing includes noise filtering, extraction of the signal segments and mapping it to a measurement location. After that, the data can be used for patient-specific numerical modeling.
The measurement system presented in this work can be used as an additional instrument used in en-dovascular surgery for assessment and monitoring of the operation [6].
MATERIAL AND METHODS
In this work we perform nonstationary 3D numerical simulations of relatively large cerebral arterial circulation areas with aneurysms (Fig. 4). Computations are carried out at the Informational and Computational Center of the Novosibirsk State University using the ANSYS commercial software. Hy-drodynamic properties of the flow are simulated using the ANSYS CFX solver, while the deformations and stresses in the vessel wall are computed with the ANSYS Mechanical.
Fig. 4. Computational geometry with multiple aneurysms
For our simulations we use patient specific MRI and CT data obtained at the Meshalkin Novosibirsk Research Institute of Circulation Pathology and the Burdenko Research Institute of Neurosurgery. The patients underwent minimally invasive (endovascu-lar) or open surgery. To reconstruct the flow domains preoperative scans are used. The segmentation was performed with ITK-SNAP [10] and VMTK [1].
Blood flow parameters (velocity and pressure) are taken from intraoperative monitoring with Volcano ComboMap endovascular blood flow measurement system [0].
When modeling such complex and multifactorial objects one always has to find a balance between a mathematical model describing the phenomenon the most accurately, however being computationally very expensive, and a simpler one, but describing the main features of the case. In our work we consider blood as a viscous incompressible Newtonian fluid governed by the Navier-Stokes equations. The vessel wall is considered a linearly elastic isotropic material. However the elasticity parameters of the healthy vessel wall and the one of the aneurysms can be different.
RESULTS
In our first series of numerical simulations we studied the effect of the presence of the aneurysm of the flow. The «treatment» of the aneurysm was performed numerically with an appropriate software surface editor. The calculations show the localness of such an influence. The influence area spreads out no far than several diameters of the aneurysm. In Fig. 5 a comparison of the pressure distribution is
Fig. 5. Comparison of the pressure distribution on the vessel walls
shown: 1 - original configuration with aneurysm, 2 - with aneurysm numerically removed, 3 - the pressure difference between the two calculation (where applicable) is shown.
In the second series of the experiments we gradually increased and decreased pressure at the outlet of the flow domain. These changes can be considered as hypertension (high blood pressure) and hypotension (low blood pressure). In Fig. 6 a diagram of relative change in velocity and pressure is shown. The computations show localness of the changes in flow structure and linear quantitative changes of flow parameters.
In the last series of our work we studied hemo-dynamic properties of the blood flow in a vessel net with multiple aneurysms. The purpose of these simulations was to determine if there is an influence of the aneurysms on each other. An example of the numerical simulations is shown in Fig. 7.
ACKNOWLEDGMENTS
This work was financially supported by the Russian Foundation for Basic Research (Project No. 14-01-00036).
REFERENCES
1. Antiga L., Steinman D. The vascular modeling toolkit. URL: http://www.vmtk.org/
2. Chupakhin A.P., Cherevko A.A., Khe A.K. et al. Measurement and analysis of cerebral hemodynamic
parameters in the presence of brain vascular anomalies // Patologiya krovoobrashcheniya i kardiokhirur-giya [Circulation Pathology and Cardiac Surgery]. 2012. 4. 27-31. [In Russian].
3. Gaidar B.V., Semenyutin V.B., Parfenov V.E., Svistov D.V. Transcranial Doppler ultrasonography in
Boundary Pressure Pressure Velocity conditions max min max
□ +30% increase Q Initial value H -30 % decrease
Fig. 6. Relative velocity and pressure changes
Fig. 7. Wall shear stress
neurosurgery. Saint-Petersburg: ELBI-SPb, 2008. 288 p. [In Russian].
4. Krivoshapkin A.L., Panarin V.A., Orlov K.Yu. et al. Hemodynamic hemorrhage prevention algorithm during cerebral arteriovenous malformations emboli-zation. Byulleten' Sibirskogo otdeleniya Rossiyskoy akademii meditsinskikh nauk [Bulletin of Siberian Branch of Russian academy of medical sciences]. 2013. (6). 65-73. [In Russian].
5. Panarin V.A., Orlov K.Yu., Krivoshapkin A.L. et al. Hydrodynamic calculations for embolization strategy of cerebral arteriovenous malformations with fis-tulous component. Patologiya krovoobrashcheniya i kardiokhirurgiya [Circulation Pathology and Cardiac Surgery]. 2012. 3. 39-43. [In Russian].
6. Patent 2511235 RF. Method for intraoperative Doppler control of radicality of arteriovenous malfor-
mation embolisation / Orlov K.Yu., Panarin V.A., Be-restov V.V. et al; published 10.04.2014. [In Russian].
7. Sforza D.M., Putman C.M., Cebral J.R. Hemodynamics of Cerebral Aneurysms // Annu Rev Fluid Mech. 2009. 41. 91-107.
8. Vorobtsova N.A., Yanchenko A.A., Cherevko A.A. et al. Modelling of cerebral aneurysm parameters under stent installation // Rus. J. Numer. Anal. Math. Model. 2013. 28. 505-516.
9. Yanchenko A.A., Cherevko A.A., Chupakhin A.P. et al. Nonstationary hemodynamics modelling in a cerebral aneurysm of a blood vessel // Rus. J. Numer. Anal. Math. Model. 2014. 29. 307-317.
10. Yushkevich P.A., Piven J., Hazlett H.C. et al. User-guided 3D active contour segmentation of anatomical structures: Significantly improved efficiency and reliability // Neuroimage. 2006. 31. (3). 1116-1128.
ПЕРСОНАЛИЗИРОВАННОЕ МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ ЦЕРЕБРАЛЬНЫХ АРТЕРИАЛЬНЫХ АНЕВРИЗМ
Александр Канчерович ХЕ1,2, Александр Павлович ЧУПАХИН12, Александр Александрович ЧЕРЕВКО12, Даниил Васильевич ПАРШИН1, Алексей Леонидович КРИВОШАПКИН3, Кирилл Юрьевич ОРЛОВ3
1 Институт гидродинамики им. М.А. Лаврентьева СО РАН 630090, г. Новосибирск, пр. Академика Лаврентьева, 15
2Новосибирский государственный университет 630090, г. Новосибирск, ул. Пирогова, 2.
3 Новосибирский НИИ патологии кровообращения им. Е.Н. Мешалкина 630055, г. Новосибирск, ул. Речкуновская, 15
В работе представлен математический и компьютерный подход к моделированию церебральной гемодинамики при наличии артериальной аневризмы. С использованием численного моделирования определяются: воздействие аневризмы на окружающий ее поток крови, изменения, обусловленные ростом или падением давления в сосуде, а также исследуются характеристики потока крови в случае наличия множественных аневризм. Для численных расчетов были использованы клинические данные, собранные в ходе проведения эндоваску-лярного вмешательства.
Ключевые слова: гемодинамика, артериальная аневризма, критерий разрыва аневризмы, моделирование трехмерного потока жидкости.
Хе А.К. - к.ф.-м.н., лаборатория дифференциальных уравнений, старший научный сотрудник, e-mail: alekhe@hydro.nsc.ru
Чупахин А.П. - д.ф.-м.н., лаборатория дифференциальных уравнений, заведующий лабораторией Черевко А.А. - к.ф.-м.н., лаборатория дифференциальных уравнений, старший научный сотрудник Паршин Д.В. - к.ф.-м.н., лаборатория дифференциальных уравнений, младший научный сотрудник, e-mail: danilo.skiman@gmail.com
Кривошапкин А.Л. - чл.-корр. РАН, д.м.н., главный научный сотрудник, врач-нейрохирургч Орлов К.Ю. - к.м.н., врач-нейрохирург, руководитель Центра ангионеврологии и нейрохирургии
