The brain is one of the most peculiar, fascinating, and yet undiscovered human organs. Compared to the whole human body, it weighs less than 2% of its mass, while receiving almost 20% of the resting cardiac output blood and 25% of the total oxygen supply 1]. Cerebral blood flow occurs on various scales through vessels of different types and physical properties ranging from the arteries, veins, and capillaries. The circulation as a whole forms a very complex biophysical phenomenon that is crucial to support the proper functioning of the brain. Many serious clinical events can occur when blood flow is altered or blocked leading to a ischemic stroke. Other cerebrovascular diseases include aneurysm, haemorrhage, and moyamoya disease apart from others [2]. All of them can be considered serious leading to long-term repercussions [3]. Therefore, there is a need for better understanding and diagnosis of various abnormalities in brain circulation. This requires a broad multidisciplinary approach that involves medicine, mathematics, physics, and engineering.
The main circulatory system that supplies the brain with blood is called the Willis circle (see Fig. 1). It consists of anterior cerebral arteries (left and right), anterior communicating artery, internal carotid arteries (left and right), posterior cerebral arteries (left and right), and posterior communicating arteries (left and right). The Circle of Willis provides a self-regulatory flow system that creates redundancy and can provide collateral circulation in the event of a narrowing or obstruction of one blood vessel [4]. In effect, in the case of a blocked flow, the brain can still function, however, some serious consequences can occur. A possible treatment of these conditions can be performed with bypass surgery that increases blood flow by placing an extracranial artery near the location of the obstruction [5]. Given the importance of this form of treatment and a large anatomic variance of the circle of Willis, it is important to be able to simulate, understand, and predict the pattern of cerebral circulation in silico with the use of carefully designed mathematical model, data acquisition, and accurate computer simulations.
We have been able to construct a specific end-to-end computer simulation model that begins with real patient data extraction and provides a neurosurgeon a complete cerebral flow simulation in various clinical settings, see Fig. 2. Additionally, the model can provide information on the circulation before and after bypass surgery or the effectiveness of stent implantation, see Figs. 3 and 4. The scientist can explore various situations and surgery scenarios in silico before supplementing insights for the neurosurgeon, which provides a cheap and, most importantly, safe aid for decision-making.
The first step in building the model is the acquisition of data from which the arterial geometry can be constructed. The primary medical imaging techniques on which the model is build are: magnetic resonance angiography, computed tomography angiography and subtraction angiography. Using the appropriate software to create and process three-dimensional models (SpaceClaim, RadiAnt, 3D Slicer), one builds a 3D model of blood vessels, see Fig. 5. Next, the model is transformed to the CAD geometry (Fig. 6), which is tailor-made for mathematical and numerical computation.
Having the well-prepared geometry one can use it in computational fluid dynamics software such as ANSYS® Fluent in which the mathematical equations representing mass and momentum conservation, the so-called Navier-Stokes equations, are solved on this complex domain resulting in both steady or transient flow simulation. One of the most important steps in the model construction is a proper choice of blood viscosity constitutive law, since this fluid is very different from water. Due to its chemical composition, the blood behaves more like a pseudoplastic fluid in which the viscosity changes with changing velocity. This type of fluid is called non-Newtonian. Solving all of the corresponding equations, we are able to predict various hemodynamical properties of the flow such as the pressure and velocity fields, different fluxes, and changing geometry. An exemplary plot of an output is given in Fig. 2 where a steady flow velocity field is depicted in the form of pathlines. The main advantage of this approach is the ability to test different scenarios of arterial flow without the need to perform actual surgery. In particular, it is possible to verify how the use of bypasses or stents will affect blood perfusion.
This project aims to develop valuable and safe computational tools, thanks to which a neurosurgeon will be able to explore different situations without taking any medical risk to fully treat and help the patient. Our results are already being used by neurosurgeons to support the diagnosis of diseases of the nervous system and to analyse and verify different scenarios for brain surgeries.
Literature
[1] Perdikaris, Paris, Leopold Grinberg, and George Em Karniadakis. “Multiscale modeling and simulation of brain blood flow.” Physics of Fluids 28.2 (2016): 021304.
[2] Powers, William J. “Cerebral hemodynamics in ischemic cerebrovascular disease.” Annals of Neurology: Official Journal of the American Neurological Association and the Child Neurology Society 29.3 (1991): 231-240.
[3] Portegies, M. L. P., P. J. Koudstaal, and M. A. Ikram. “Cerebrovascular disease.” Handbook of clinical neurology 138 (2016): 239-261.
[4] Purves, Dale Ed, et al. “Neuroscience.” (2008).
[5] Wessels, L., N. Hecht, and P. Vajkoczy. “Bypass in neurosurgery—indications and techniques.” Neurosurgical Review 42 (2019): 389-393.
By Marcin Magdziarz and Łukasz Płociniczak (Faculty of Pure and Applied Mathematics), and Dariusz Szarek (Faculty of Medicine), Wrocław University of Science and Technology, Poland