Cerebral arteriovenous malformation (AVM) is a congenital brain vessels pathology, in which the arterial and venous blood channels are connected by tangles of abnormal blood vessels. It is a dangerous disease that affects brain functioning causing the high risk of intracerebral haemorrhage. One of AVM treatment methods is embolization—the endovascular filling of the AVM vessel bundle with a special embolic agent. This method is widely used, but still in some cases is accompanied by intraoperative AVM vessels rupture. In this paper, the optimal scenario of AVM embolization is studied from the safety and effectiveness of the procedure point of view. The co-movement of blood and embolic agent in the AVM body is modelled on the basis of a one-dimensional two-phase filtration model. Optimal control problem with phase constraints arising from medicine is formulated and numerically solved. In numerical analysis, the monotone modification of the CABARET scheme is used. Optimal embolization model is constructed on the basis of real patients' clinical data collected during neurosurgical operations. For the special case of embolic agent, input admissible and optimal embolization scenarios were calculated.

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A shallow water model on a rotating attracting sphere is proposed to describe large-scale motions of the gas in planetary atmospheres and of the liquid in the world ocean. The equations of the model coincide with the equations of gas-dynamic of a polytropic gas in the case of spherical gas motions on the surface of a rotating sphere. The range of applicability of the model is discussed, and the conservation of potential vorticity along the trajectories is proved. The equations of stationary shallow water motions are presented in the form of Bernoulli and potential vorticity integrals, which relate the liquid depth to the stream function. The simplest stationary solutions that describe the equilibrium state differing from the spherically symmetric state and the zonal flows along the parallels are found. It is demonstrated that the stationary equations of the model admit the infinitely dimensional Lie group of equivalence.Introduction. Description of hydrodynamic phenomena that occur in the atmosphere and in the ocean is a non-trivial problem. Important factors that affect the dynamics of the liquid or gaseous shell of a planet are gravitation and rotation. It is interaction of these two forces that retains the medium as a whole in an equilibrium state, with motions of different scales in the atmosphere.For brevity, in what follows, we understand the atmosphere hydrodynamics as the motion of a continuous medium (liquid or gas) located on the surface of a rotating sphere in the field of a gravity force directed toward the sphere center with a constant acceleration. The model being developed is equally applicable to gas motions in planetary atmospheres and to large-scale oceanic flows.Hydrodynamic phenomena in the atmosphere are characterized by a large variety of scales. On one hand, these are large-scale (planetary) phenomena (circulation cells), in particular, global vortices, such as cyclones and anticyclones, and oceanic flows; on the other hand, these are small-scale motions, which essentially depend on the surface relief. As the phenomenon under study depends on many factors, the universal model of atmosphere hydrodynamics is extremely complicated and difficult for investigations. It seems natural to identify the characteristic scales of motion to be studied and to use an appropriate approximate hydrodynamic model.The specific feature of atmosphere hydrodynamics problems is the compactness of the manifold on which the corresponding mathematical model is determined. Available results on the behavior of the vector fields on a sphere, on one hand, and an empirical idea that all areas of the planet cannot have an identical weather, on the other hand, allow us to conclude that solutions that describe the motion of the atmosphere as a whole have singularities, such as sources and sinks, discontinuities and fronts separating air masses with different characteristics of motion.