Nyquist diagram bifurcations for generalized Van der Pol

Nyquist diagram bifurcations for generalized Van der Pol Duffing equation describing local cerebral hemodynamics Bord E. E. Novosibirsk State University, Cherevko A. A. Lavrentyev Institute of Hydrodynamics of SB RAS

Aim of work • The pathologies of cerebral vascular system break proper functioning of blood circulation. This creates a serious danger for patient’s health. In this work we simulate the bloodstream in the surroundings of anomalies based on generalized Van der Pol – Duffing equation. We learn how the behaviour of equation solutions depends on the parameters. The aim of work is to obtain the information about characteristic bloodstream behaviour in the surroundings of anomalies.

Pathologies of cerebral vascular system • The cerebral blood vessel network is a complex system with blood circulating. The walls of a blood vessel are viscoelastic. Besides, the vessel is surrounded by the brain substance. • Vascular pathologies are aneurysms and AVM. Aneurysm AVM • Pathologies disturb normal blood supply. • Intravascular methods are successfully applied in treatment such pathologies. Stenting or embolization in case of aneurysms and embolization in case of AVM.

Formulation of the problem • Problem: Prediction of probable complications after surgery • Solution: Developing the way of modeling vessels behaviour during surgery

Monitoring of hemodynamic parameters • To study the vascular pathologies hemodynamic parameters are monitored during the neurosurgical operations. It is carried out in collaboration with Novosibirsk Research Institute Of Blood Circulation Pathology E. N. Meshalkin. • The blood velocity and pressure inside of vessels are measured with the Combomap device. • Data collection frequency is about 200 Hz. Velocity sensor Pressure sensor

Arterial vessels structure • Arterial vessels have complicated multilayer active structure • Smooth round muscle contracts with the each heartbeat tracking the pulse wave. “Understands” the increasing of blood velocity in the vessel. Artery Connective tissue Smooth round muscle Elasic layer Endothelium

System study • We have a system «blood flow + active vessel walls + vessel surroundings» . From mathematical point of view the processes in this system are described by the large scale system Vein Artery • We are able to measure and study only the small dimension projection (velocity-pressure) of phase space for this complicated system.

Hypothesis validation • Is the projection on «velocity-pressure» plane good enough? • There are slow and fast velocity and pressure changes. Slow changes are caused by the reaction to medical intervention. Fast changes occur within one cardiac cycle. • Slow variables: their behaviour is defined by mechanisms of autoregulation • Fast dimensionless variables: their behaviour is defined by local vessel reaction to the pulse wave. q(t), u(t) [0, 1] • The behaviour of fast variables can be predicted with our model.

Generalized Van der Pol – Duffing equation • To identify the characteristic behaviour of hemodynamic parameters in the surroundings of vascular pathologies the model of generalized Van der Pol - Duffing equation is suggested. • Dimensionless velocity u is a given value, dimensionless pressure q is founded as the solution of equation. • The coefficients are constructed based on the experimental data by methods of inverse problems. Especially bi respond for the elastic properties of the vessels, ai respond for the viscous friction, ε corresponds to the relaxation oscillation character.

Experimental confirmation of the model • The model was confirmed experimentally with large amount of clinical data obtained during surgeries in Novosibirsk Research Institute Of Blood Circulation Pathology E. N. Meshalkin. • It is shown that the equation reproduces the clinical data well enough. computed u u u initial q q q

Prediction of blood flow parameters behaviour during surgery • The equation is built for the concrete point of vessel network based on clinical data for 5 cardiac cycles. • Pressure q(t) is reproduced quite good. • Equation connecting dimensionless values q and u does not change in this point. So it is possible to predict solution behaviour during surgery within several dozens of minutes

“Sounding” of the system with harmonic signal • From the medical point of view it seems important to know how would the vessels react to the bloodstream changing. • Variation of the right side amplitude and frequency helps to define the dependence of solution behavior from these two parameters. • We suppose that easy dynamics corresponds to health vessels, whereas complicated dynamics corresponds to sick vessels.

Nyquist diagram (APFR) • To analyse how the solution behaviour depends on the right side amplitude and frequency we use nonlinear generalization of Nyquist diagrams (ND) [Journal of Physics: Conference Series]. • In the nonlinear case maximum and minimum of solution do not necessarily coincide. The blue and red curves represent solution minimum and maximum modulus respectively. • The solution plot is represented in polar coordinates. 1. 8 Гц 3. 5 Гц 10 Гц Right side frequency 0. 5 Гц is the parameter along the curve. Polar coordinates mean phase shift and solution amplitude. 50 Гц 0. 01 Гц

Nyquist diagram (APFR) • In the farthest points of solution maximum and minimum we find frequencies and build the graphs of solution (in yellow) and right side (in blue). 1. 8 Гц 0. 5 Гц 3. 5 Гц 10 Гц 50 Гц 0. 01 Гц

Range of physiologically significant frequencies • The range of physiologically significant frequencies is about 1 Hz. Bifurcation of ND is usually found in this range.

Measurements in arteries Patient Gender Pathology Amount of points Gu m AA 15 Ko f AA 15 Pi m AA 6 Po f AA 10 Si m AA 14 Ch f AVM 14 300 ND on the whole. • In this work data of six patients were used. Five of them had aneurysm and the last one had AVM. For all of them measurements in arteries were carried out with the various device positions. • The each point is defined by the device position. On the whole the are 74 points handled. For the each point we built the set of diagrams and plot solutions for different meanings of amplitude and frequency.

ND classes chain for the patients with aneurysms • Loops and gaps • Nyquist diagrams are show the influence divided into classes. of the anomalies. • With the increasing of the amplitude diagrams move from one class to another in definite order. gap case • In normal case ND resembles the diagram for linear system divergent case normal case break case loop case Increasing of the right side amplitude from 0 to 1.

Continuous solution changing • Solution changes • Subharmonics and disruptions continuously show the • Only stable influence of the periodic cycles anomalies. are acceptable for the living system. break case gap case loop case divergent case normal case Increasing of the right side amplitude from 0 to 1.

Influence of the anomalies on ND class • ND runs the whole chain of classes not for all device positions. There are cases when ND stays in first two classes. • Complicated nonlinear oscillations which characterize the influence of the anomalies are mostly peculiar to the last three classes.

Results for the patients • It was founded that for the each concrete point ND runs the chain in one direction with the increasing of outer force amplitude. For two neighboring amplitude meanings the diagram may move to the next class or stay in current. ND can not turn to the previous class. normal case -> divergent case -> loop case -> gap case -> break case • Furthere are shown results for five patients with aneurysms.

Patient Gu • Saccular aneurysm • Right internal carotid artery • 42 years, m. [M] (42 Y) 2 5 13 6, 15, 1611, 12 8, 9 3, 4 7 18, 19 Before During After

Classes Changing of ND classes Amplitude

Classes Changing of ND classes Amplitude

Patient Ko • Giant aneurysm • Right internal carotid artery • 64 years, f. Before During After 1 15 3, 4, 16 6, 11 8, 19 5, 13

Classes Changing of ND classes Amplitude

Patient Pi • Aneurysm • Bifurcation of the left middle cerebral artery • 55 years, m. [M] (55 Y) 3 8 7 1 2 6 Before During After

Classes Changing of ND classes Amplitude

Patient Po • Giant aneurysm • Left ascending artery • 55 years, f. 1, 2 9 4, 5 10 6, 7, 8 3 Before During After

Classes Changing of ND classes Amplitude

Patient Si • Saccular aneurysm • Bifurcation of the basilar artery • 40 years, m. 1, 2, 11 5, 6 12 3 13 4, 8 14, 15 9, 10 Before During After

Classes Changing of ND classes Si Amplitude

Classes Diagram for all patients Amplitude

Results and conclusion • ND for six patients are considered. Five patients had aneurysms and one had AVM. ND for the 300 cases were explored. • It is shown that NG for the patients with aneurysms are divided into several classes and move from one to another in definite order with the increasing of outer force amplitude. • The dependence of the characteristic solution behaviour on the form of ND is found out • Now we work on the more detailed classification of ND aiming to find out the dependence of ND type on the position and farness measurement place from the aneurysm. Later we intend to investigate in details ND for the patients with AVM. • This classification would be useful for describing the vessels reaction to the changes in bloodstream. • The range of physiologically significant frequencies is about 1 Hz. Bifurcation of ND is usually found in this range.

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