 Alexander Cherevko from Novosibirsk, and he will give us a talk about simple equation describing brain hemodynamics. Alexander Cherevko works in the Lavrentiev Institute of Hydrodynamics there in Novosibirsk, Russia. Alexander, please, you can share a screen and start your talk. I share my screen, or you can see my screen. Yes, we see the screen. Yes. Can I start? Yes, please begin. Okay. I will speak to this term, simple equation describing brain hemodynamics, and I am an old speaker, and with this equation of give dynamics, with measure of give dynamics works many, many people, and I speak about the speaker within my speech. Yes. First of all, the medicine, what, with what medicine problem we work. In human brain, there are some abnormalities, vascular abnormalities. There are many abnormalities, but there are two general abnormalities. This is the anemeurysm, yes, and the arterial venous malformation. This pathology is very bad. This pathology is very bad because the pathology may imply a hemorrhage, and what is this pathology? First of all, this is the arterial venous malformation. Arterial venous malformation is a pathological connection between vein and arterius without any capillars. This is the schematic of this malformation, and this is the angiography of this malformation. This is the crane, and this is the pathology. We can see this pathology because in arterial bed, we have the special contrast, regent contrast. The treatment of this pathology is surgical, and this is embolization. There is an elective cell in a blood vessel with some special medicine composition. This is a before surgery. We can see this pathology, and this after surgery. We cannot see the blood in this pathology. Another abnormality is the anemeurysm, arterial anemeurysm. This is the schemosycin anemeurysm, and this is the trempine way. One way of treatment is embolization or coiling with many micro-spirals coils, and this micro-spirals embolized this pathology. This is an instrument. Another way is the stent installation, and this stent installation changed the velocity, the blood velocity, and this is a foot too. But there is a problem. What is the problem? When surgical doing this operation, the patient can have hemorrhage. This is the hemorrhage operation, this catheter, this some pathology, and we develop and study a simple model of blood flow within the pathology, and we wish to use this equation to predict this problem. First part is the measurement. With the measurement work many people. This is the nearest surgical, this Alexandr Kanchevich here. Alexandr Kanchevich participates in our collaboration. This is the portrait Alexandr Kanchevich here. This is the people who work many with measurements. What's the measurement device? Measurement device is a combo map device with combo wire sensor. This is a combo-wise device, special device for measurement blood flow parameters. This is the tip of combo wire sensor. It's a very thin sensor, and on the tip of the sensor this is a human figure, and this is the sensor. This is the ultrasonic speed sensor, and pressure sensor. This is the screen of this device. This is the velocity profile, this yellow. This is the pressure profile, and right, this is the pressure profile, not in brain, but system pressure. This is the operation room. This is the nearest surgical, this patient, this x-ray scanner, and when the nearest surgical is doing this operation, he doesn't look to this percent, but look on screen. In the screen, the nearest surgical can see all that's doing in the brain. This, for example, this is a combo wire parameter, small diameter, and the length approximately two meters, and this is an interoperational screenshot. This is the sensor. This is the tip of the sensor. This embolized arterial mass modification, and this is the screenshot of the device. This sensor, nearest surgical can shift the sensor, as this, for example, the typical operation, and in this screen shot, we can see the positions of this sensor before embolization, during embolization, and after embolization. Yes. Now, this is another full screen of ComboMap device, and this is a measure, the data that we have. This is a noise data, for example, velocity is a noise, very noise pressure, not so noise, but noise. It was a clinical data, but when I use this ComboMap device, I have two problems. First of all, when I use this device first time, I don't have any data, and I spent this nearest surgical approximately one year to study what can we do to take this data. But another problem is that this data is taken as a frequency of this frequency, and, oh, excuse me, and we have many, many very, very big, very low data, and when we look to these graphics, we can't understand what we're doing. Another good investigation is the PV diagram. The PV diagram, we, some kind of, approximately, it's not in a really strong sense, fast plane, but some things, approximately, as a fast plane. This velocity, this pressure, oh, excuse me, in this picture, I am a Russian later. This velocity in meter and second, and then the pressure in the hydrographic millimeter. And because this is a periodical data, we have this cycle, and when I look on this cycle, it's good because we can see more information, this cycle, as the information, as the information with this graphics. And when I look to this PV diagram, we did PV chat, there are two different, there are the difference with, there are the difference between two types of pathology, of AVM, arterial venous malformation, and arterial aneurysm. This is a typical picture for, in this picture, we have a red and black lines, and the red lines, we now ignore red lines because red lines is a numerical approximation. I speak, I speak about black. This is a typical picture for arterial venous malformation, and this is a, maybe yellow back, is a bird's, some bird's picture. This is a picture of arterial venous malformation, this is typical, this is a difference. Another difference is that when we measure the pressure and velocity in artery, we have the diagram. Well, it appears there's some connection problem. Yeah, I think you're connected. Yes, yes. I can see him among the speakers. So, for now, it looks fine. So, if you recollect, we can... Yes, but I don't know how many times this will look fine, maybe not long. Excuse me, yes, yes, yes. When we have arterial venous malformation, this first diagram rotates counterclockwise, and when we have venous malformation in venous malformation, it rotates clockwise. There are two hypotheses why this rotates different. First hypothesis in publishing this paper, in this paper, we investigate some model of this brain, of the brain, with a porous layer, layer. This layer is some model cut-ups. And is this model with some... And this model in artery is first counterclockwise rotation, and then clockwise rotation. The second hypothesis is because the artery, as venous, are quite different things. In artery, we have the musculoskeletal layer. This layer is active. In the... When we heard work, we have the pulse wave in the arterial bed, and this musculoskeletal layer contracts in each pulse wave. And in the artery, we have not... In the vein, we have not this musculoskeletal layer. I... With my PhD student, I study this diagram in silicon tube. We use the computer pump, and we program this computer pump. And we use the result of measurement to program this computer pump. And in silicon tube, we have the rotation of this diagram as in vein. And this is an open question. Why is this different? Yes. This diagram is very useful for neurosurgeons. Aleksandr Kancherich here writes some program. This program analyses the data of interoperational monitor in real time, and plot this diagram in real time in the operational room. And where this plot is diagramed in real time, we can see what the shifting to the diagram, what is the perturbation of this diagram, is good for the additional predictor for neurosurgeons. What about... This is... This is about the measurement and the medicine. What about the equation? This equation, we can see to this equation as some kind of approximation of our clinical data or regression of our clinical data. These are the people who work in this domain, in the domain of regression. Now, these are the people who don't work now. And these people work. This is a Yuri Bugay, my student, and Mikhail Shishlein, the avocalabraic. What can we speak about this equation? First of all, we study the blood flow is artery. It's more interesting that in the vein, we have the very knowledge of the data, and the blood flow in the vein is not so octal for neurosurgeons as the blood flow is artery. Oh, Russian words, excuse me. Arterial vessel is a multilayer structure, this endothelial layer, this elastic layer, this is the muscular layer, this muscular is this pulsate, and this is an X-trigger layer. And this artery is placed in the brain, yes, in the parenchyma. And the process of this complex system, the blood flow, the artery wall multilayer, and the parenchyma is a very, very complicated process, and this process is described by a multidimensional system. Oh, for, if we want a very good describing, we might use the, for example, list of equations for flow, and so on. This is a system, complicated system, also a question, a special differential. But we can measure and study only the two-dimensional projection of this multidimensional system, the projection of other space of velocity and pressure plate. The layer is not thinned, yes, this layer is not thinned, this is a muscular layer, not thin layer. We can't use the, the approximation to thin layer. But the mechanism, mechanism of reaction of muscular wall to the pulse is a local, this is physiologically, because when we have the pulse way of the flow in the artery, there are some biological sensors on the endotelium, endotelium layer, and this sensor takes the signal that we have the pulse flow and get the signal to the muscular layer to contract. And the other process, controllable central nervous system is very, very complicated. And is a question, is a projection of velocity pressure plane good? Can we take some information from this two-dimensional production? The answer is, is a good production. If we split the pressure and speed to slow variables and fast variables, as we can see, this is a slow variables. This is a medium pressure, medium velocity. This is the amplitude of pressure, amplitude of velocity. And this is a dimensionless, fast variables. Blue variables is a slow variables. And this is a fast. Slow variables change approximately minutes under the fast variables. This is the variables of pulse, pulse variables. And we write the equations just for these q and u variables, for fast dimensional variables. This equation, I write this equation in the form of a Vandals-Poll equation and I use the ordinary differential equation because we wish to use this equation interoperational, who predict some bad things and interoperational vaccines. And then this equation must be simple. The numerical method for this equation must be fast. And then I use this question. I test many forms of equations, but this form is now, I think, a good form. But maybe this a2 and b2 with a good form. But now with Yuri and Mikhail Shishlenin, I improve this equation. It does not change this general form. But we have some improving. But I don't speak now about this improvement because it is in the work. Now, to you at a dimensional velocity, two-dimensional pressure. And for each patient, we determine this coefficient. One, two, three, four, five, six, seven coefficient. Use the clinical data. And this epsilon, this is a fixed coefficient, is one divided by 100. For this equation, we for determine this coefficient, we solve inverse problem. And when I begin the study of this equation, I solve inverse problem for discretization of this equation. And use the Matlab toolbox for the term that coefficients, for the term that coefficient. I solve, I take the saved inverse solution in this sense. Now, with Mikhail Shishlenin, we use more sophisticated algorithm. And this algorithm not have more good residue. What about approximation clinical data with the equation? This is a chart for three percent, the typical chart. Black line is the clinical data. Red line is the solution of this equation. Yes. Or we can see in this chart that in this position we, excuse me, I all see my mouse. Or can I use the other point device? Yes, you can use it. Excuse me. I, oh, this is not good position. But now with Mikhail Shishlenin, Mikhail Shishlenin, this is a specialist in solve and correct and inverse problem. We don't see this, this we can see the good approximation. But this is a good approximation to take it to training data. Yes. What about the prediction? This is, this is a pressure, this is a time, I'm asking, this is three minutes, six minutes, nine minutes in the, this is a mean value for blue, the mean value of velocity, red, the mean value of pressure, this maximum and minimum for velocity and for pressure. And this, in this interval of time, we have operation of embolization. We train our model for on data with five second lengths. Yes. And we predict the behavioral pressure and pressure on this length, this second length, night, hundred lengths, five minutes. And this is a good prediction. And this is, this prediction on the velocity and pressure plane, this is a position, Russian, Russian, I have many Russian letter on graphics. This, in this position is a training data. Yes. We train our model, we take the coefficient and we see that black, the interoperational data, a red calculation data is good, approximately, approximately, and this, this, this is a prediction data. We use our equation to calculate the velocity pressure for this, this, this is data. In this position, the embolization is, this is the end of embolization. We have very low velocity and the very noisy data. And this is not so beautiful graphics. And we can speech the normalized equation. This is the equation that we use. Is invariant with the, with the, with this, within the embolization. Not only, with not only training this equation, we can use this equation for prediction. This is the mathematical, some mathematical property of this equation. The stability of, not based, we have the stability of the equation on the point of view of the initial data. The initial data forgotten in time interval, approximately, cardiac cycle. This is the phase plane of this equation. There are some structural stability when we change the coefficient of equation by two persons. This diagram change, but not destroy. Do we test this equation of in the 177 cases? In all of this, this is a passion. This passion, this passion have a different position of measured sensor. This, we study this 77 cases and in all of these cases, the equation work well. And we study, we study the different cases. This, we study, we use this equation to predict the, the velocity in the venous. This is the artery, A, D and venous, C on the sinuses of the crane. And D, this is the prediction for elastic silicone model. This is a laboratory. Not these cases. When we adapt equation for these cases, the question is good. This good equation, what about the property of this equation? What, what is property? We can solve this equation numerically, but I see that I think that when you study some equation, non-linear equation, numerically, we take many, many, many data and I won't take some, some law. Yes. And for when many people doing this some type of equation, the non-linear equation, this non-linear, non-autonomous equation, this right hand depends on the type. We can use the harmonic right hand with the frequency omega and B that, the amplitude. And we can look our equation in the black box. Yes. And we test that black box with a harmonic signal. And we ask the question, what is the property of this box? When I, when we test this black box with harmonic signal, that's the first question. As the second question, what is the relation of this property of this equation and the medical property of this pattern? Now I, I speak about property of this equation. We have the hypothesis where the in healthy vessel with this equation have the simple dynamics is the left picture. Yes. The blue line, this is the right side as the yellow line is the solution of this equation. And the, when we have thick vessel, we have complex dynamics is the right picture. Yes. Blue line is the right side and the yellow, yellow line will have some not harmonic behavior of the solution of this equation. And this, what can we study this behavior, behavior of this solution? First, first technique is a generalized, generalized diagram of Nyquist or Nyquist diagram. Nyquist diagram, this is a very, very old, not very, but very old technique. But this technique in general, people use for linear system. For linear system, Nyquist diagram describe all this system. And I I propose some generalization, a little bit of the generalization of this diagram, and I named this generalized diagram. What is Nyquist diagram? First of all, we fix the amplitude or right side on this right side. Yes. And then we plot in polar coordinate some line. The parameter along the line is the frequency or right side and the polar coordinate for each point of this line is the amplitude, amplitude, that is this radius of this line, this polar radius and the fast shift. Fast shift right side, right side of the equation versus solution of equation. For generalized, this is in linear case. In non-linear case, we use two lines. The blue line is the minimum modulus solution and the red line is the maximum modulus solution. Oh, this in this picture, we interchange this, excuse us, we interchange these two lines. Yes. Oh, for example, if we look this case, we must draw some line, some two, we must look to these two points. With this, this is a fast shift, the polar angle and this is green and red, this is amplitude. This, what about real clinical data for the, in term of this Nyquist diagram? For example, first patient, this patient have little aneurysm, this patient not affect, vaguely affect the blood flow on its vicinity. And we have many this Nyquist diagram, each Nyquist diagram correspond to fix it, amplitude of right hand of our equation and the coefficient for this equation, determine it for this patient. And we have a pseudo-linear behavior. This more complicated aneurysm. In this situation, we have different Nyquist diagram. Yes. And we have the solution, not so, not harmonical, is trying, not sinusoid, but triangle oscillation. And this guantaneurysm, this aneurysm, very, very big change there, blood flow. And this situation, we see this very complicated Nyquist diagram. For points, for different points of Nyquist diagram, we have different behavior to our solution. For this, for example, very, very complicated. The complication on Nyquist diagram implies a complication to solution. But on Nyquist diagram, we can look to all solutions in completeness. We study about 300 Nyquist diagram, we analyzed it. And we have this behavior. Nyquist diagram for real percent divide into classes. As amplitude of right hand increase, the diagram move from one place to another in certain order. First, the normal case, divergent case, yes, this is a distance, big distance. Loop case, some loops on the diagram. Gap case, in this, in this diapason, we have not, we have not a periodical solution of our equation. And this is a break case, some very complicated. Normal mod oscillator, similar to this, if we use not our equation, but equation of linear selection. And the look on breaks shows the influence of anomalies. This is, we can look behavior of solution, yes. No, no, this is a result. I know, speak very slow, but I, all results for that, this is a chain of this perturbation of this diagram. And some patients, when the patient have, have not very big or complicated anomalies, this patient, the nice Nyquist diagram of this patient life, this class. As a way to study our equation and to, to understand the relation of the equation and the abnormalities, is a study the resonance curve of this equation. We study this equation, we study this resonance curve with my student, Yuri Bugay. We use the analytical approximation of the equation and use the Galerkin method. Some people speak about the Galerkin method as, as the method of harmonic balance, and this method have many, many names. We study one-dimensional approximation, two-dimensional, three-dimensional, and we study this Yuri Bugay, the approximation with this ultra-harmonic, this ultra-harmonic, this ultra-harmonic. What about, first of all, what about the more simple case, the approximation equations that have only one frequency, but maybe kind of shit? We have the Russian, Russian. This is the amplitude, amplitude frequency plot. This is fast frequency plot. The typical, this is the typical plots. Yes, this is a, for some, for real percent, this is a real percent, use the real percent data. There is a teleaxis, this is frequency of right height, right hand, and this is the amplitude of Galerkin approximation of equation. We have this, each line corresponds to fixed amplitude of right hand. We can look this multimodal equation, multimodal solution, yes, this is this equation. When we look at different patients, we can see the different frequency response. This is a very interesting picture. We have the patient that, this patient have the gamma-RH within the measurement. And what about these plots? Before the gamma-RH, we can look this, not this, amplitude frequency plot, and this amplitude frequency plot. But after the gamma-RH, we have what we have, we have the defecation of this amplitude frequency plot. And this fast frequency plot is also close to this equation. We can see that the, this picture can be, have the relation with the patient status. And in this picture, we connect the Nyquist diagram and the Uttar-Gharmonik analysis. Type of line, solid line, dotted line, dashed line is a symbolize different Nyquist diagram class, normal case, divergent case, loop case. This line symbolize the patient and the distance. Symbolize the ratio of the maximum amplitude of fundamental harmonic and the ultragharmonic of secondary. The left picture is a numerical calculation. This is a not calculation in, not a numerical calculation of full equation with used not Nyquist or not Galerkin approximation. And the right plot is a Galerkin approximation. We can see the quality and quantity and some quality relation. And this picture is for, not second order, the third order. We can see good, approximately good relation of these graphics. And we can speak that the Galerkin approximation is simple, but Galerkin approximation take some important information from our equation. And then the last part is the work of Daniel Pasha. I don't work. This part is not the result of my work. This is Daniel Pasha and take photo of Daniel Pasha from Daniel Pasha avatar from WhatsApp. Excuse me, Daniel, if it is not a good picture. We have our equation is the equation with the, with small coefficient. Yes, this type of equation in this type of equation, when we want to study this equation, when the absolute, like, limit to zero, we can take some information. For this, Daniel, rewrite this equation of some special form. Alexander, I want to tell the, to warn you that we have five minutes to go. Five minutes. Okay. Okay. Very, very, very, I speak very fast. Just rewrite this equation, split this equation to fast subsystem, a slow subsystem. And from slow subsystem extract slow surface of this slow surface, this low surface is in a special space. And then study the projection of this equation on slow subs, on slow surface. And Daniel, look, when we can see that normal vessels, simple dynapses are these as a way of the complex dynamics as a, as a previous study or in this slow surface, there are many singular, singular points in cases. And Daniel classifies the singular. This is the classification singular point. This point may be, may be stable on unstable node, stable, unstable focus. And when he studied hyperbolic singularity, he take the results, these results that these singularities can classify the type of, the type of pathology, type of pathology. Is the pathology intact, or is this pathology after this, after the operation? And that's all. I finished my speech. All right. Thank you. So let's give two, five minutes for questions. So please, whoever have questions, whoever has questions, ask this. I don't see at the moment anyone asking question. Let me ask then, if you could go back to the equation where you introduced for the first time the model. Go, go, go, go, go. Yes, yes, yes, yes. Go, go, go, go to this equation. Where is my question? One moment. It was no, no, no, no. You passed, I think. Oh, yes. Yes. So, so this equation is for the q pressure. Yes, yes, for the q pressure. You, this is velocity, velocity. So is it considered in this equation as external force? Or is it a function of q? Excuse me. I don't understand the question. Well, so on the left hand side of this equation, we have variable q. Yes, this will be velocity. Yes, this is the pressure, only pressure equation. Yes. And on the right side, you have. This is a velocity. Yes. Yes. So this velocity in this equation, do you assume to be a given function? Yes. So it's external force. This is the given function. And this function will take from the measurements. From the measurements. Yes. And for instance, I just wonder for instance, could this be assumed to be, for instance, if in fact, a pressure could be a linear function of u, but it would, its self-sustained oscillations could be actually produced by the forcing function. Could be the case. One moment. First of all, this equation is the equation for error. If we use a similar equation in v, we must change the role, change this variable. Yes. First of all, this, because we use this form of the equation, this form of the equation, we use this form of the equation because we have in artery, fast shift. First of all, change the velocity and then change the pressure. Yes. In vans, this is a different thing. Yes. I, what about the question? I study many types of this equation. I study linear equation for this prediction and different nonlinearity. This is can see experimental form of the equation and study linear equation don't work. Yes. But if we write equation with more higher over power, for example, if we add q, for example, yes, this equation will be unstable. Yes. And when you add the terms with higher power of q, and when we use the experiment, a clinical data for right hand, we have the problem of stability of solution of this equation. We have not served a What you say, so for instance, if we had only linear terms, so without q2 and q3 terms in the equation, you would not be able to reproduce so well there. Yes. I see. Thank you. Other questions? Okay. Well, no, well, it appears that no one wants to ask questions. Yes. All right. Well, then, at this point, then I think Alexander, yeah, for giving a late seminar today also, because in Novosibirsk it's already dinner time, but Alexander kindly responded to our request to put the seminar later so that our Brazilian colleagues could join. All right. I thank Alexander. Thank you very much.