",isbn:"978-1-83968-760-0",printIsbn:"978-1-83968-759-4",pdfIsbn:"978-1-83968-761-7",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"cc49d6034d85f8f2e2890c6acc3cc629",bookSignature:"Dr. Abhijit Biswas",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/10285.jpg",keywords:"Mott Insulators, Semi Metals, Polycrystals, Single Crystals, Electronic Properties, Magnetic Properties, PLD, MBE, Topological Insulators, Topological Hall Effect, Devices Applications, Catalysis",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:null,numberOfDimensionsCitations:null,numberOfTotalCitations:null,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"September 9th 2020",dateEndSecondStepPublish:"October 7th 2020",dateEndThirdStepPublish:"December 6th 2020",dateEndFourthStepPublish:"February 24th 2021",dateEndFifthStepPublish:"April 25th 2021",remainingDaysToSecondStep:"5 months",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:null,kuFlag:!1,biosketch:"A pioneering researcher in the field of tailoring metal oxide crystal surfaces and growth as well as engineering of thin films for various emergent phenomena and energy applications. Dr. Biswas received his Ph.D. from POSTECH, South Korea.",coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"194151",title:"Dr.",name:"Abhijit",middleName:null,surname:"Biswas",slug:"abhijit-biswas",fullName:"Abhijit Biswas",profilePictureURL:"https://mts.intechopen.com/storage/users/194151/images/system/194151.png",biography:"Dr. Abhijit Biswas is a research associate at the Indian Institute of Science Education and Research (IISER) Pune, in India. His research goal is to design and synthesize highest quality epitaxial heterostructures and superlattices, to play with their internal degrees of freedom to exploit the structure–property relationships, in order to find the next-generation multi-functional materials, in view of applications and of fundamental interest. His current research interest ranges from growth of novel perovskite oxides to non-oxides epitaxial films, down to its ultra-thin limit, to observe unforeseeable phenomena. He is also engaged in the growth of high quality epitaxial layered carbides and two-dimensional non-oxide thin films, to exploit the strain, dimension, and quantum confinement effect. His recent work also includes the metal-insulator transitions and magneto-transport phenomena in strong spin-orbit coupled epitaxial perovskite oxide thin films by reducing dimensionality as well as strain engineering. He is also extremely interested in the various energy related environment friendly future technological applications of thin films. In his early research career, he had also extensively worked on the tailoring of metal oxide crystal surfaces to obtain the atomic flatness with single terminating layer. Currently, he is also serving as a reviewer of several reputed peer-review journals.\nDr. Biswas received his B.Sc. in Physics from Kalyani University, followed by M.Sc in Physics (specialization in experimental condensed matter physics) from Indian Institute of Technology (IIT), Bombay. His Ph.D., also in experimental condensed matter physics, was awarded by POSTECH, South Korea for his work on the transport phenomena in perovskite oxide thin films. Before moving back to India as a national post-doctoral fellow, he was a post-doc at POSTECH working in the field of growth and characterizations of strong spin-orbit coupled metal oxide thin films.",institutionString:"Indian Institute of Science Education and Research Pune",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"2",totalChapterViews:"0",totalEditedBooks:"0",institution:{name:"Indian Institute of Science Education and Research Pune",institutionURL:null,country:{name:"India"}}}],coeditorOne:null,coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"20",title:"Physics",slug:"physics"}],chapters:null,productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"205697",firstName:"Kristina",lastName:"Kardum Cvitan",middleName:null,title:"Ms.",imageUrl:"https://mts.intechopen.com/storage/users/205697/images/5186_n.jpg",email:"kristina.k@intechopen.com",biography:"As an Author Service Manager my responsibilities include monitoring and facilitating all publishing activities for authors and editors. From chapter submission and review, to approval and revision, copyediting and design, until final publication, I work closely with authors and editors to ensure a simple and easy publishing process. I maintain constant and effective communication with authors, editors and reviewers, which allows for a level of personal support that enables contributors to fully commit and concentrate on the chapters they are writing, editing, or reviewing. I assist authors in the preparation of their full chapter submissions and track important deadlines and ensure they are met. I help to coordinate internal processes such as linguistic review, and monitor the technical aspects of the process. As an ASM I am also involved in the acquisition of editors. 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\n
1. Introduction
\n
This chapter uses the publications of Shubina M.V.:
Exact Traveling Wave Solutions of One-Dimensional Parabolic-Parabolic Models of Chemotaxis, Russian J Math Phys., Maik Nauka/Interperiodica Publishing (Russian Federation), 25(3), 383–395, 2018.
The 1D parabolic-parabolic Patlak-Keller-Segel model of chemotaxis: The particular integrable case and soliton solution, J Math Phys., 57(9), 091501, 2016.
\n\n
Chemotaxis, or the directed cell (bacteria or other organisms) movement up or down a chemical concentration gradient, plays an important role in many biological and medical fields such as embryogenesis, immunology, cancer growth, and invasion. The macroscopic classical model of chemotaxis was proposed by Patlak in 1953 [1] and by Keller and Segel in the 1970s [2, 3, 4]. Since then, the mathematical modeling of chemotaxis has been widely developed. This model is described by the system of coupled nonlinear partial differential equations. Proceeding from the study of the properties of these equations, it is concluded that the model demonstrates a deep mathematical structure. The survey of Horstmann [5] provides a detailed introduction into the mathematics of the Patlak-Keller-Segel model and summarizes different mathematical results; the detailed reviews also can be found in the textbooks of Suzuki [6] and Perthame [7]. In the review of Hillen and Painter [8], a number of variations of the original Patlak-Keller-Segel model are explored in detail. The authors study their formulation from a biological perspective, summarize key results on their analytical properties, and classify their solution forms [8]. It should be noted that interest in the Patlak-Keller-Segel model does not weaken and new works appear devoted to the study of various properties of equations and their solutions [9, 10, 11, 12] and the links below.
\n
In this chapter we investigate a number of different models describing chemotaxis. The aim of this paper is to obtain exact analytical solutions of these models. For one-dimensional parabolic-parabolic systems under consideration, we present these solutions in explicit form in terms of traveling wave variables. Of course, not all of the solutions obtained can have appropriate biological interpretation since the biological functions must be nonnegative in all domains of definition. However some of these solutions are positive and bounded, and their analysis requires further investigation. Despite the large number of works devoted to the systems under consideration and their properties, as well as the properties of their solutions, it seems to us that the solutions obtained in this paper are new.
\n
The Patlak-Keller-Segel model describes the space–time evolution of a cell density \n\nu\n(\nt\n,\n\nr\n→\n\n)\n\n and a concentration of a chemical substance v\n\n\n(\nt\n,\n\nr\n→\n\n)\n\n. The general form of this model is:
where \n\n\nδ\n1\n\n>\n0\n\n and \n\n\nδ\n2\n\n≥\n0\n\n are cell and chemical substance diffusion coefficients, respectively, and \n\n\nη\n1\n\n\n is a chemotaxis coefficient; when \n\n\nη\n1\n\n>\n0\n\n, this is an attractive chemotaxis (“positive taxis”), and when \n\n\nη\n1\n\n<\n0\n\n, this is a repulsive (“negative”) one [13, 14]. \n\nϕ\n\nv\n\n\n is the chemosensitivity function, and \n\nf\n\nu\nv\n\n\n characterizes the chemical growth and degradation. These functions are taken in different forms that correspond to some variations of the original Patlak-Keller-Segel model. We follow the reviews of Hillen and Painter [8] and of Wang [15] and consider the models presented therein.
\n
This paper is concerned with one-dimensional simplified models when the coefficients \n\n\nδ\n1\n\n\n, \n\n\nδ\n2\n\n\n, and \n\n\nη\n1\n\n\n are positive constants, \n\nx\n∈\nℜ\n,\nt\n≥\n0\n\n, \n\nu\n=\nu\n\nx\nt\n\n\n, and \n\nv\n=\nv\n\nx\nt\n\n\n.
\n
\n
\n
2. Signal-dependent sensitivity model
\n
Let us start with a model that allows nonnegative bounded solutions that may be of interest from a biological point of view. Now consider the “logistic” model, one of versions of signal-dependent sensitivity model [8] with the chemosensitivity functions \n\nϕ\n\nv\n\n=\n\n\n1\n+\nb\n\n\nln\n\n\nv\n+\nb\n\n\n\n, where \n\nb\n=\nconst\n\n, and \n\nf\n\nu\nv\n\n=\n\nσ\n˜\n\nu\n−\n\nβ\n˜\n\nv\n\n. In the review [5] one can see a mathematical analysis of this model. When \n\nb\n=\n0\n\n and \n\n\nβ\n˜\n\n=\n0\n\n, the existence of traveling waves was established in [16, 17]. The replacements of \n\nt\n→\n\nδ\n1\n\nt\n\n and \n\nu\n→\nσ\n\n\nσ\n˜\n\n\nδ\n1\n\n\nu\n\n give \n\n\nδ\n1\n\n=\n1\n\n, \n\nα\n=\n\n\nδ\n2\n\n\nδ\n1\n\n\n\n, \n\nβ\n=\n\n\nβ\n˜\n\n\nδ\n1\n\n\n\n, and \n\nσ\n=\n±\n1\n\n. We also set \n\nη\n=\n\n\n\nη\n1\n\n\n\n1\n+\nb\n\n\n\n\nδ\n1\n\n\n\n, \n\n1\n+\nb\n>\n0\n\n, as well as \n\nϕ\n\nv\n\n=\nln\n∣\nv\n+\nb\n∣\n\n. It should be noted that a sign of \n\nσ\n\n may effect on the mathematical properties of the system. So, \n\nσ\n=\n1\n\n corresponds to an increase of a chemical substance, proportional to cell density, whereas \n\nσ\n=\n−\n1\n\n corresponds to its decrease. And as we shall see later, various solutions correspond to these two cases.
If we introduce the function \n\nυ\n=\nv\n+\nb\n\n, in terms of traveling wave variable \n\ny\n=\nx\n−\nct\n\n, where \n\nc\n=\nconst\n\n, this system has the form:
where \n\n\nν\n2\n\n=\n\n1\n\nα\n2\n\n\n\n\n1\n+\n\n\n4\nαβ\n\n\nc\n2\n\n\n\n\n\n and \n\nΛ\n=\n−\n\n\n4\nβb\n\n\nα\n\nc\n2\n\n\n\n\n\n\n\n4\nσ\n\nC\nu\n\n\n\nα\n\nc\n2\n\n\n\n\n\n1\n4\n\n\n\n. Eq. (6) is the Lommel differential equation [18, 19] with \n\nμ\n=\n−\n1\n−\n\n1\nα\n\n\n, and we consider \n\nσ\n\nC\nu\n\n>\n0\n\n. Since this is a linear inhomogeneous second-order differential equation, one can integrate it by the method of variation of parameters. We assume a solution in the form:
where \n\n\nJ\nν\n\n\nz\n\n\n and \n\n\nY\nν\n\n\nz\n\n\n are Bessel functions and \n\n\nC\nJ\n\n\nz\n\n\n and \n\n\nC\nY\n\n\nz\n\n\n are the functions of \n\nz\n\n that satisfy the equations:
where \n\n\nc\nJ\n\n\n and \n\n\nc\nY\n\n\n are constants. If both of the numbers \n\n−\n\n1\nα\n\n±\nν\n\n are positive, the lower limits in the integrals may be taken to be zero. Then a particular integral of Lommel equation “proceeding in ascending powers of \n\nz\n\n” is \n\n\ns\n\nμ\n,\nν\n\n\n\nz\n\n\n [19]; if one considers a solution of Lommel equation “in the form of descending series,” one obtains the function \n\n\nS\n\nμ\n,\nν\n\n\n\nz\n\n\n [19] [see Eq. (8)]. Thus, quoting Watson [19] “...and so, of Lommel’s two functions \n\n\ns\n\nμ\n,\nν\n\n\n\nz\n\n\n and \n\n\nS\n\nμ\n,\nν\n\n\n\nz\n\n\n, it is frequently more convenient to use the latter.” Then the general solution of Eq. (6) has the form:
are Lommel functions, and \n\n\n\n\n1\n\n\nF\n2\n\n\n is the generalized hypergeometric function [18, 19]. Further, substituting the initial variable \n\ny\n\n and the function \n\nv\n\n [see Eq. (5)] into Eq. (7), we obtain a formal solution.
\n\n\nb\n=\n0\n\n\n
\n\n
We first consider the case \n\nb\n=\n0\n\n. Then \n\nυ\n=\nv\n≥\n0\n\n and \n\n\nC\nu\n\n>\n0\n\n. Eq. (6) becomes homogeneous, and for \n\nσ\n=\n1\n\n, its general solution is:
However one can check that the function \n\nu\n=\nu\n\n\ny\n\n\n diverges as \n\ncy\n→\n−\n∞\n\n for all \n\nν\n\n.
\n
Consider now \n\nσ\n=\n−\n1\n\n. For \n\nv\n\n\ny\n\n\n to be real, let \n\nα\n=\n2\n\n. Then Eq. (6) becomes the modified Bessel equation; the analysis of solution behavior at \n\n±\n∞\n\n leads to suitable solutions for \n\nv\n\n\ny\n\n\n and \n\nu\n\n\ny\n\n\n:
with restrictions \n\nν\n≤\n\n1\n2\n\n\n and \n\nβ\n≤\n0\n\n. So one can see that \n\nv\n\n\ny\n\n→\n0\n\n as \n\ncy\n→\n−\n∞\n\n for all \n\nν\n≤\n\n1\n2\n\n\n; \n\nv\n\n\ny\n\n→\n0\n\n for \n\nν\n<\n\n1\n2\n\n\n and \n\nv\n\n\ny\n\n→\n\n\n\n\nπ\n2\n\n\nc\n2\n\n\n\n8\n\nC\nu\n\n\n\n4\n\n\n for \n\nν\n=\n\n1\n2\n\n\n as \n\ncy\n→\n∞\n\n and \n\nu\n\n\ny\n\n→\n0\n\n as \n\ny\n→\n±\n∞\n\n for all \n\nν\n≤\n\n1\n2\n\n\n. The curves of these functions are presented in Figures 1 and 2, and the plots for \n\nc\n=\n5\n\n are thicker than for \n\nc\n=\n1\n\n. Thus, the solution obtained may be considered as a biologically appropriated one, and this requires further investigation.
Let us return to Eq. (6) with \n\nΛ\n≠\n0\n\n. The analysis of solution asymptotic forms at \n\n±\n∞\n\n [18, 19] gives the following expressions for \n\nv\n\n\ny\n\n\n and \n\nu\n\n\ny\n\n\n:
with \n\nσ\n\nC\nu\n\n>\n0\n\n and \n\nν\n<\n\n1\nα\n\n\n. The latter condition leads to the requirement \n\n−\n\n\nc\n2\n\n\n4\nα\n\n\n≤\nβ\n<\n0\n\n. The \n\nv\n\n\ny\n\n→\n−\nb\n\n, \n\nu\n\n\ny\n\n→\n−\n\nβb\nσ\n\n\n as \n\ncy\n→\n−\n∞\n\n and \n\nv\n\n\ny\n\n→\n0\n\n and \n\nu\n\n\ny\n\n→\n0\n\n as \n\ncy\n→\n∞\n\n. Thus, one can see that for \n\nb\n>\n0\n\n, \n\nσ\n=\n1\n\n, and \n\n\nC\nu\n\n>\n0\n\n, \n\nu\n\n\ny\n\n≥\n0\n\n is satisfied but \n\nv\n\n\ny\n\n<\n0\n\n. These functions are presented in Figures 3 and 4. It should be noted that \n\nν\n≠\n\n1\nα\n\n\n or \n\nβ\n≠\n0\n\n because of the pole in \n\nΓ\n\n function.
Using the analysis of Eq. (11), one can see that the condition \n\nb\n<\n0\n\n along with \n\nσ\n=\n−\n1\n\n and \n\n\nC\nu\n\n<\n0\n\n (\n\nσ\n\nC\nu\n\n>\n0\n\n) leads to the fact that the function \n\nu\n\n\ny\n\n\n has not changed, but \n\nv\n\n\ny\n\n\n becomes positive on all domains of definition. This function is presented in Figure 5.
To integrate this equation, we use the Lie group method of infinitesimal transformations [20]. We find a group invariant of a second prolongation of one-parameter symmetry group vector of (12), and with its help, we transform Eq. (12) into an equation of the first order. It turns out that nontrivial symmetry group requires some conditions:
Then we find the solutions of Eq. (15) in parametric form [21] with the parameter \n\nt\n\n. Now we consider the case \n\n2\nα\n+\nη\n≠\n1\n\n. A combination of substitutions leads to:
where \n\n\n\nC\n˜\n\nϑ\n\n\n and \n\n\nC\nϑ\n\n\n are constants and \n\n\n\n\n2\n\n\nF\n1\n\n\n is the hypergeometric Gauss function. Further we obtain the solutions of initial Eqs. (3)–(4) in parametric form:
where the constant \n\n\n\nC\n˜\n\nϑ\n\n\n is chosen so that \n\n\n\n2\nα\n+\nη\n−\n1\n\n\n\n\nC\n˜\n\nϑ\n\n<\n0\n\n, which is consistent with Eq. (17). Using the asymptotic representation of hypergeometric Gauss function as \n\nt\n→\n±\n∞\n\n [18], we can take:
in order for \n\ny\n,\nv\n\n, and \n\nu\n\n to be real. Then one can see that all functions in Eq. (19) are continuous bounded ones for \n\nt\n∈\nℜ\n\n and \n\nv\n,\nu\n\n are positive. Hence, one may try to biologically interpret the functions \n\nv\n\n\ny\n\n\n and \n\nu\n\n\ny\n\n\n, and this requires further investigation. In Figure 6 one may see the different curves \n\nv\n\n\ny\n\n\n for \n\nη\n=\n0.1\n\n and different \n\nα\n\n. Figure 7 demonstrates \n\nv\n\n\ny\n\n\n and \n\nu\n\n\ny\n\n\n for two values \n\nη\n:\nη\n=\n0.1\n\n and \n\nη\n=\n0.01\n\n, see Figure 7. Further, for larger values of \n\nα\n\n and \n\nη\n\n, it seems more convenient to present the curves \n\ny\n\nt\n\n\n, \n\nv\n\nt\n\n\n, and \n\nu\n\nt\n\n\n to analyze them (see Figures 8–10). One can see from Eq. (13) that \n\nβ\n≷\n0\n\n when \n\nα\n≷\n2\n\n, and the case of \n\nβ\n=\n0\n\n and \n\nα\n=\n2\n\n is presented in Figure 11.
The model with logarithmic chemosensitivity function \n\nϕ\n\nv\n\n∼\nln\nv\n\n is also studied. For the case of \n\nf\n\nu\nv\n\n=\n−\n\nv\nm\n\nu\n+\n\nβ\n˜\n\nv\n\n, where \n\n\nβ\n˜\n\n=\nconst\n\n, an extensive analysis is performed in [15]. This survey is focused on different aspects of traveling wave solutions. When \n\nm\n=\n0\n\n, this model coincides with Eq. (1) for \n\nb\n=\n0\n\n. When \n\n\nβ\n˜\n\n=\n0\n\n and \n\nm\n=\n1\n\n, the system was studied in [22, 23]. The complete analysis for \n\n\nβ\n˜\n\n=\n0\n\n is performed in [15]. An existence of global solution is established in [24].
\n
Now we consider the system with \n\nϕ\n\nv\n\n=\nln\nv\n\n and \n\nf\n\nu\nv\n\n=\n\nσ\n˜\n\nvu\n−\n\nβ\n˜\n\nv\n\n. Similarly, a replacement of \n\nt\n→\n\nδ\n1\n\nt\n\n and \n\nu\n→\nσ\n\n\nσ\n˜\n\n\nδ\n1\n\n\nu\n\n gives \n\n\nδ\n1\n\n=\n1\n\n, \n\nη\n=\n\n\nη\n1\n\n\nδ\n1\n\n\n\n, \n\nα\n=\n\n\nδ\n2\n\n\nδ\n1\n\n\n\n, \n\nβ\n=\n\n\nβ\n˜\n\n\nδ\n1\n\n\n\n, and \n\nσ\n=\n±\n1\n\n. Then the model has the form:
where \n\nu\n=\nu\n\n\ny\n\n\n, \n\nυ\n=\nυ\n\n\ny\n\n\n, and \n\nλ\n\n is an integration constant. To integrate Eq. (23), we tested this system on the Painlevé ODE test. One can show that for \n\nη\n>\n0\n\n, it passes this test only if \n\nα\n=\n2\n\n with the additional condition \n\nλ\n=\n−\nσcβ\n\n\n1\n+\n\nη\n2\n\n\n\n\n [25]. If we express \n\nu\n\n\ny\n\n\n as \n\nυ\n\n\ny\n\n\n from Eq. (23), we obtain an equation only for \n\nυ\n\n\ny\n\n\n; for \n\nα\n=\n2\n\n, it has the form:
For \n\nλ\n=\n−\nσcβ\n\n\n1\n+\n\nη\n2\n\n\n\n\n, this equation can be linearized. It becomes equivalent to the following linear equation for \n\nF\n\n:
where \n\n\nC\nF\n\n=\nconst\n\n. If we rewrite Eq. (26) in terms of the variable \n\nξ\n=\n\ne\n\n−\n\ncy\n2\n\n\n\n\n for the function \n\nΨ\n\nξ\n\n=\n\ne\n\n−\n\nη\n2\n\nυ\n\n\n\n, we obtain an equation similar to Eq. (12) with zero right-hand side:
where \n\nϑ\n\nt\n\n\n is given in Eq. (18) and \n\nu\n\n\ny\n\n\n may be expressed from Eq. (23). However one may see that \n\nv\n→\n∞\n\n as \n\nt\n→\n±\n∞\n\n, and this solution is unacceptable as a biological function.
\n
Another possibility to solve this equation exactly is to put \n\n\nC\nF\n\n\n equal to zero. When \n\n\nC\nF\n\n=\n0\n\n, that means \n\nF\n\n\ny\n\n=\n0\n\n, for \n\nβ\n≠\n0\n\n; Eq. (27) can be linearized by \n\nξ\n=\n\ne\nτ\n\n\n [21]. Its solution has three forms according to a sign of the expression \n\nD\n=\n\n\n2\n\nη\n2\n\nβ\n\n\nc\n2\n\n\n+\n1\n\n. Since \n\nv\n\n should be a nonnegative and bounded function as \n\ncy\n→\n±\n∞\n\n, the only suitable solution is:
where \n\n\nC\n±\n\n\n are positive constants and \n\nβ\n>\n0\n\n. Unfortunately, the corresponding solution for \n\nu\n\n\ny\n\n\n is alternating and has the form:
It is easy to see that \n\nσu\n\n\ny\n\n→\n\n\n\nc\n2\n\n\n\nη\n+\n2\n\n\n\n\n2\n\nη\n2\n\n\n\n\n\n−\n1\n±\n\nD\n\n\n\n\n as \n\ncy\n→\n±\n∞\n\n. These functions are presented in Figures 12 and 13.
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Figure 12.
\n\n\nv\n\n\ny\n\n\n; \n\nc\n=\n1\n\n.
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Figure 13.
\n\n\nσu\n\n\ny\n\n\n; \n\nc\n=\n1\n\n.
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\n
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4. Linear sensitivity
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Let us consider the system with linear function \n\nϕ\n\nv\n\n∼\nv\n\n. When \n\nf\n\nu\nv\n\n=\nu\n−\nv\n\n, the system is called the minimal chemotaxis model following the nomenclature of [26]. This model is often considered with \n\nf\n\nu\nv\n\n=\n\nσ\n˜\n\nu\n−\n\nβ\n˜\n\nv\n\n (\n\n\nσ\n˜\n\n\n and \n\n\nβ\n˜\n\n\n are constants), and it is studied in many papers. As was proved in [27, 28], the solutions of this system are global and bounded in time for one space dimension. The case of positive \n\n\nσ\n˜\n\n\n and nonnegative \n\n\nβ\n˜\n\n\n is studied in [29, 30, 31, 32, 33]. As we noted earlier, a sign of \n\n\nσ\n˜\n\n\n may effect on the mathematical properties of the system, which changes its solvability conditions [34].
\n
Now we consider the linear chemosensitivity function \n\nϕ\n\nv\n\n=\nv\n\n and \n\nf\n\nu\nv\n\n=\n\nσ\n˜\n\nu\n−\n\nβ\n˜\n\nv\n\n. The replacement of \n\nt\n→\n\nδ\n1\n\nt\n\n, \n\nv\n→\n\n\nη\n1\n\n\nδ\n1\n\n\nv\n\n, and \n\nu\n→\nσ\n\n\n\nσ\n˜\n\n\nη\n1\n\n\n\nδ\n1\n2\n\n\nu\n\n leads to \n\n\nδ\n1\n\n=\n\nη\n1\n\n=\n1\n\n, \n\nα\n=\n\n\nδ\n2\n\n\nδ\n1\n\n\n\n, \n\nβ\n=\n\n\nβ\n˜\n\n\nδ\n1\n\n\n\n, and \n\nσ\n=\n±\n1\n\n. Then the system has the form:
where \n\nu\n=\nu\n\n\ny\n\n\n, \n\nv\n=\nv\n\n\ny\n\n\n, and \n\nλ\n\n is an integration constant. As shown in [35], this system passes the Painlevé ODE test only if \n\nα\n=\n2\n\n and \n\nβ\n=\n0\n\n. Let us focus on this case.
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It is convenient to solve Eq. (32) in terms of variable:
The function \n\n\nZ\nν\n\n\nz\n\n\n satisfies the modified Bessel’s equation and can be present as a linear combination of Infeld’s and Macdonald’s functions.
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Using the series expansion of the Infeld’s function, as well as theirs asymptotic behavior [36], one may obtain the following asymptotic forms for \n\n\ne\n\n\nv\nν\n\n\nz\n\n\n\n\n and \n\n\nu\nν\n\n\nz\n\n\n:
where \n\nκ\n>\n0\n\n, \n\nA\n\n, and \n\nB\n\n are arbitrary constants and the functions \n\n\nI\nν\n\n\n and \n\n\nK\nν\n\n\n are Infeld’s and Macdonald’s functions, respectively. This solution is not satisfactory from the biological point of view, since \n\nv\n\n\ny\n\n\n is an alternating function for any \n\nν\n\n. However it seems interesting because of the following: in the case of \n\nν\n=\n\n1\n2\n\n\n and \n\nB\n=\n\n\n2\n+\nπ\n\n\n2\nπ\n\n\n\n in terms of \n\n\ne\n\n−\n\ncy\n2\n\n\n\n\n, its form coincides with the well-known Korteweg-de Vries soliton.
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Consider now the class of solutions with half-integer index \n\nν\n=\nn\n+\n\n1\n2\n\n\n, when \n\n\nZ\nν\n\n\nz\n\n\n can be expressed in hyperbolic functions. The requirement of absence of divergence \n\nu\n→\n−\n∞\n\n for finite \n\nz\n\n leads to the following form for \n\n\nZ\n\nn\n+\n\n1\n2\n\n\n\n\nz\n\n\n:
At first let us consider the solutions obtained for \n\n\ne\n\nv\n\nn\n+\n\n1\n2\n\n\n\n\n\n and \n\n\nu\n\nn\n+\n\n1\n2\n\n\n\n\n as functions of \n\nz\n\n. We begin with \n\nn\n=\n0\n\n or \n\nν\n=\n\n1\n2\n\n\n. It is interesting to present the expressions for \n\n\ne\n\n\nu\n\n1\n2\n\n\n\nz\n\n\n\n\n and \n\n\nu\n\n1\n2\n\n\n\nz\n\n\n:
where Eq. (40) appears the one-soliton solution exactly the same as the well-known one of the Korteweg-de Vries equation. Returning to the variable \n\ny\n\n:
One can see that for \n\nσ\n=\n1\n\n (an increase of a chemical substance), the cell density \n\nu\n\n\ny\n\n≥\n0\n\n for \n\nB\n≥\n\n1\nπ\n\n\n and that for \n\nB\n>\n0\n\n\n\n\nu\n\n\ny\n\n\n is the solitary continuous solution vanishing as \n\ny\n→\n±\n∞\n\n, whereas for \n\nB\n<\n0\n\n\n\n\nu\n\n\ny\n\n\n has a point of discontinuity. One can say that when \n\nB\n<\n0\n\n, we obtain “blow-up” solution in the sense that it goes to infinity for finite \n\ny\n\n, and this is true for different \n\nν\n\n.
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The expressions for \n\nn\n≥\n1\n\n become more complicated, and one can see the solitonic behavior of \n\n\ne\n\n\nv\n\nn\n+\n\n1\n2\n\n\n\n\nz\n\n\n\n\n and the curves for \n\n\nu\n\nn\n+\n\n1\n2\n\n\n\n\nz\n\n\n in Figures 14 and 15.
The explicit form of our solution in terms of the variable \n\ny\n\n can be obtained by direct substitution of Eq. (33) into Eq. (39), where \n\n\nλ\nc\n\n=\n−\n\nc\n2\n\nn\n\n\nn\n+\n1\n\n\n\n. The resulting formulae are complicated and slightly difficult for analytic analysis; it seems to be more convenient to present the plots.
\n
For \n\nn\n=\n0\n\n in the function \n\n\ne\n\n\nv\n\n1\n2\n\n\n\n\ny\n\n\n\n\n, we have the “step” whose altitude depends on the values of velocity \n\nc\n\n and arbitrary constant \n\nκ\n\n. One may see that these curves become higher and shift to the right with different rates for the rising \n\nκ\n\n. The \n\n\nu\n\n1\n2\n\n\n\n\ny\n\n\n is the positive function whose altitude and sharpness of peak depend on c (see Figures 16 and 17).
For \n\nn\n≥\n1\n\n we can see that the solitonic behavior of \n\n\ne\n\n\nv\n\nn\n+\n\n1\n2\n\n\n\n\n\ny\n\n\n\n\n is retained for different values of \n\nc\n\n and \n\nκ\n\n; the curves become higher and more tight, and they shift to the right also with an increase of \n\nc\n\n and \n\nκ\n\n. For the cell density \n\n\nu\n\nn\n+\n\n1\n2\n\n\n\n\n\ny\n\n\n, the obtained solution has the negative section converging to zero for \n\ncy\n→\n−\n∞\n\n (Figures 18–21).
The curves for the concentration of the chemical substance \n\n\nv\n\nn\n+\n\n1\n2\n\n\n\n\n\ny\n\n\n are presented in Figure 22. Since \n\n\nv\n\nn\n+\n\n1\n2\n\n\n\n\n\ny\n\n\n has to be positive (nonnegative), we see that these functions do not satisfy this requirement in all domains of definition.
In conclusion it seems interesting to present the plots for \n\n\ne\n\n\nv\nν\n\n\n\ny\n\n\n\n\n and \n\n\nu\nν\n\n\n\ny\n\n\n for different values of \n\nν\n\n (Figures 23–25). It is interesting to see that there are irregular solutions for \n\n\ne\n\n\nv\nν\n\n\n\ny\n\n\n\n\n; however, the corresponding solutions for \n\n\nu\nν\n\n\n\ny\n\n\n are regular [see Eqs. (35)–(37)].
We investigate three different one-dimensional parabolic-parabolic Patlak-Keller-Segel models. For each of them, we obtain the exact solutions in terms of traveling wave variables. Not all of these solutions are acceptable for biological interpretation, but there are solutions that require detailed analysis. It seems interesting to consider the latter for the experimental values of the parameters and see their correspondence with experiment. This question requires further investigations.
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\n\n',keywords:"parabolic-parabolic system, exact solution, soliton solution, Patlak-Keller-Segel model, chemotaxis",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/71181.pdf",chapterXML:"https://mts.intechopen.com/source/xml/71181.xml",downloadPdfUrl:"/chapter/pdf-download/71181",previewPdfUrl:"/chapter/pdf-preview/71181",totalDownloads:137,totalViews:0,totalCrossrefCites:0,totalDimensionsCites:0,hasAltmetrics:0,dateSubmitted:"December 13th 2019",dateReviewed:"January 15th 2020",datePrePublished:"March 31st 2020",datePublished:null,dateFinished:null,readingETA:"0",abstract:"In this chapter we consider several different parabolic-parabolic systems of chemotaxis which depend on time and one space coordinate. For these systems we obtain the exact analytical solutions in terms of traveling wave variables. Not all of these solutions are acceptable for biological interpretation, but there are solutions that require detailed analysis. We find this interesting, since chemotaxis is present in the continuous mathematical models of cancer growth and invasion (Anderson, Chaplain, Lolas, et al.) which are described by the systems of reaction–diffusion-taxis partial differential equations, and the obtaining of exact solutions to these systems seems to be a very interesting task, and a more detailed analysis is possible in a future study.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/71181",risUrl:"/chapter/ris/71181",book:{slug:"mathematical-theorems-boundary-value-problems-and-approximations"},signatures:"Maria Vladimirovna Shubina",authors:null,sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Signal-dependent sensitivity model",level:"1"},{id:"sec_3",title:"3. Logarithmic sensitivity",level:"1"},{id:"sec_4",title:"4. Linear sensitivity",level:"1"},{id:"sec_5",title:"5. Conclusion",level:"1"}],chapterReferences:[{id:"B1",body:'\nPatlak CS. Random walk with persistence and external bias. The Bulletin of Mathematical Biophysics. 1953;15(3):311-338\n'},{id:"B2",body:'\nKeller EF, Segel LA. Initiation of slime Mold aggregation viewed as an instability. Journal of Theoretical Biology. 1970;26(3):399-415\n'},{id:"B3",body:'\nKeller EF, Segel LA. Model for Chemotaxis. Journal of Theoretical Biology. 1971;30(2):225-234\n'},{id:"B4",body:'\nKeller EF, Segel LA. Traveling bands of chemotactic bacteria: A theoretical analysis. Journal of Theoretical Biology. 1971;30(2):235-248\n'},{id:"B5",body:'\nHorstmann D. From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Preprints: Max-Planck-Institut für Mathematik in den Naturwissenschaften. Niedersächsische Staats- und Universitätsbibliothek. 2003\n'},{id:"B6",body:'\nSuzuki T. Free Energy and Self-Interacting Particles. Boston: Birkhuser; 2005\n'},{id:"B7",body:'\nPerthame B. Transport Equations in Biology. Basel: Birkhuser; 2007\n'},{id:"B8",body:'\nHillen T, Painter KJ. A users guide to PDE models for chemotaxis. Journal of Mathematical Biology. 2009;58:183-217\n'},{id:"B9",body:'\nPerthame B. PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic. Applications of Mathematics. 2004;49(6):539-564\n'},{id:"B10",body:'\nBlanchet A, Dolbeault J, Perthame B. Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions. Electronic Journal of Differential Equations. 2006;44:1-32\n'},{id:"B11",body:'\nBlanchet A, Laurenot P. The parabolic-parabolic Keller-Segel system with critical diffusion as a gradient flow in \n\n\nℜ\nd\n\n\n\n, d ≥ 3. Communications in Partial Differential Equations. 2013;38(4):658-686\n'},{id:"B12",body:'\nBlanchet A, Carrillo JA, Kinderlehrer D, Kowalczyk M, Laurenot P, Lisini S. A hybrid variational principle for the Keller-Segel system in \n\n\nℜ\n2\n\n\n. ESAIM: M2AN. 2015;49(6):1553-1576\n'},{id:"B13",body:'\nNi W-M. Diffusion, cross-diffusion, and their spike-layer steady states. Notices of the American Mathematical Society. 1998;45(1):9-18\n'},{id:"B14",body:'\nLi T, Wang ZA. Nonlinear stability of large amplitude viscous shock waves of a generalized hyperbolic-parabolic system arising in chemotaxis. Mathematical Models and Methods in Applied Sciences. 2010;20:1967-1998\n'},{id:"B15",body:'\nWang ZA. Mathematics of traveling waves in chemotaxis. Discrete and Continuous Dynamical Systems - Series B. 2013;18(3):601-641\n'},{id:"B16",body:'\nNagai T, Ikeda T. Traveling waves in a chemotactic model. Journal of Mathematical Biology. 1991;30(2):169-184\n'},{id:"B17",body:'\nEbihara Y, Furusho Y, Nagai T. Singular solution of traveling waves in a chemotactic model. Bulletin of the Kyushu Institute of Technology, Mathematics, Natural Science. 1992;39:29-38\n'},{id:"B18",body:'\nBateman H, Erdélyi A. Higher Transcendental Functions. Vol. 2. New York, Toronto, London: McGHAW-HILL Book Company, INC; 1953\n'},{id:"B19",body:'\nWatson GN. A Treatise on the Theory of Bessel Functions. Cambridge, England: Cambridge University Press; 1944\n'},{id:"B20",body:'\nOlver PJ. Applications of Lie Groups to Differential Equations (Springer, 1986) (In translation). Moscow: Springer; 1989\n'},{id:"B21",body:'\nZaitsev VF, Polyanin AD. Handbook on Ordinary Differential Equations. Moscow: Physmatlit; 2001\n'},{id:"B22",body:'\nNossal R. Boundary movement of chemotactic bacterial populations. Mathematical Biosciences. 1972;13:397-406\n'},{id:"B23",body:'\nRosen G. Theoretical significance of the condition delta = 2mu in bacterial chemotaxis. Bulletin of Mathematical Biology. 1983;45(2):151-153\n'},{id:"B24",body:'\nWinkler M. Global solutions in a fully parabolic chemotaxis system with singular sensitivity. Mathematical Methods in the Applied Sciences. 2011;34:176-190\n'},{id:"B25",body:'\nShubina M. Painlevé Analysis for Two 1D Parabolic-Parabolic Models of Chemotaxis: Some Travelling Wave Solutions. arXiv:1607.00349 [nlin.SI]\n'},{id:"B26",body:'\nChildress S, Percus JK. Nonlinear aspects of chemotaxis. Mathematical Biosciences. 1981;56:217-237\n'},{id:"B27",body:'\nOsaki K, Yagi A. Finite dimensional attractor for one-dimensional Keller-Segel equations. Funkcialaj Ekvacioj. 2001;44:441-469\n'},{id:"B28",body:'\nHillen T, Potapov A. The one-dimensional chemotaxis model: Global existence and asymptotic profile. Mathematical Methods in the Applied Sciences. 2004;27:1783-1801\n'},{id:"B29",body:'\nJäger W, Luckhaus S. On explosions of solutions to a system of partial differential equations modelling chemotaxis. Transactions of the American Mathematical Society. 1992;329:819-824\n'},{id:"B30",body:'\nNagai T, Senba T, Yoshida K. Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis. Funkcialaj Ekvacioj. 1997;40:411-433\n'},{id:"B31",body:'\nCorrias L, Escobedo M, Matos J. Existence, uniqueness and asymptotic behavior of the solutions to the fully parabolic Keller-Segel system in the plane. Journal of Differential Equations. 2014;257:1840-1878\n'},{id:"B32",body:'\nTao Y, Winkler M. Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity. Journal of Differential Equations. 2012;252(1):692-715\n'},{id:"B33",body:'\nFatkullin I. A study of blow-ups in the Keller-Segel model of chemotaxis. Nonlinearity. 2013;26(1):81-94\n'},{id:"B34",body:'\nTupchiev VA, Fomina NA. On a correctness of two-dimensional boundary value problem for a system of equations of chemotaxis. Matematicheskoe Modelirovanie. 2001;13(12):95-106\n'},{id:"B35",body:'\nShubina M. The 1D parabolic-parabolic Patlak-Keller-Segel model of chemotaxis: The particular integrable case and soliton solution. Journal of Mathematical Physics. 2016;57:091501\n'},{id:"B36",body:'\nNikiforov AF, Uvarov VB. The Special Functions of Mathematical Physics. Moscow: Nauka; 1978\n'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Maria Vladimirovna Shubina",address:"yurova-m@rambler.ru",affiliation:'
Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow, Russian Federation
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1. Introduction
In recent years, technological advances in the field of echocardiography have allowed for a faster acquisition of images with an improved spatial and temporal resolution. As part of these advances, the advent of speckle tracking imaging has resulted in an explosion of investigations into myocardial deformation, as evidenced by more than 5000 articles on PubMed, increasing exponentially since 2005 (https://pubmed.ncbi.nlm.nih.gov/?term=speckle+tracking, accessed 7th of May 2020). The past two decades has also seen in a shift in “stress echocardiography” from being dominated by acute drug-based interventions to primarily exercise challenges. Therefore, this chapter focuses on the current knowledge related to myocardial deformation during acute exercise stress. Instead of just summarizing the current literature, a careful selection of articles is presented that is then used to provide the reader with a narrative that highlights important general principles of cardiac physiology, including the responses to exercise. To achieve this aim, first a brief overview of the principles and mechanisms governing myocardial deformation will be provided summarised and the key terminology will be defined. Then, the general role of exercise stress testing will be discussed, before the benefits of obtaining myocardial deformation during exercise in health and disease will be reviewed.
2. Principles of myocardial deformation
During contraction of the heart, deformation of the whole muscle occurs in four quantifiable dimensions. In general, these have been identified as: longitudinal shortening (=longitudinal strain, %), circumferential shortening (circumferential strain, %), radial lengthening (=radial strain, %) and rotation (apical − basal rotation = net twist angle, degrees), as well as the diastolic reversal of all of these indices. In addition, the rate of systolic shortening and diastolic lengthening can be measured, which is referred to as strain rate, twisting rate, and untwisting rate. An important distinction must be made between myocardial deformation and pure “velocities”, which do not consider the relative shortening (contraction) or lengthening (relaxation) of heart muscle itself but only consider the linear displacement of single myocardial points. Although myocardial velocities can also be measured, they are not representative of the contraction and relaxation of heart muscle. For these reasons, parameters such as E’ (“E prime”), which typically represent myocardial velocities in a single location on the mitral annulus, are not discussed in this chapter.
The conventional categorizations of deformation into strain and twist are logical from a biophysics and bioengineering perspective, since deformation of the heart can indeed be detected in these distinct 2-dimensional echocardiographic imaging planes. However, as will be reviewed in the following section on the anatomy and electrical conductance, the structure of the heart is far from symmetrical and—to achieve the final coordination of all components with each heartbeat—important functional differences in the various regions within the heart are present. These intricate deformational patterns can be conceptually simplified by considering the region-specific deformation in a 2-dimensional plane, allowing for easier evaluation of cardiac mechanics in both the laboratory and the clinic. However, one must consider the 3D deformation of the heart muscle, where the deformation of the four imaging planes occur simultaneously and with many of these aspects anatomically and functionally interwoven. This anatomical complexity is the focus of the next section.
2.1 Anatomy
Historical reviews have often credited Leonardo da Vinci’s observations in the 15th century as some of the first to describe the gross anatomy of the heart and his speculations about the resulting function. In his drawings1, da Vinci refers to the importance of vortices, which necessitate the presence of helical structures and/or motions that were apparent as “clockwise and counterclockwise spirals within the aorta as the outlet of the left ventricle” [1]. More than a century after da Vinci’s death, William Harvey published his seminal book Exercitatio Anatomica De Motu Cordis Et Sanguinis In Animalibus (An Anatomical Study on the Motion of the Heart and Blood in Living Beings, 1628 [2]), in which he established the circulation—including the anatomy and motion of the heart—as we mostly know it today, thereby also popularizing the previous work by Ibn al-Nafis [3]. In 1669, Richard Lower provided remarkable detail on the anatomy of the heart in his publication of Tractatus de Corde… (Treatise on the Heart. … [4]). Despite these early discoveries, it wasn’t until the contributions by McCallum and then Mall in the early twentieth century that there were new advancements in this field [5, 6]. During the second World War, Robb & Robb provided an exceptionally detailed overview of the accumulated knowledge that covered five centuries of discoveries [7]. Then, 27 years later, in 1969, Streeter et al. published the much-cited myocardial fiber distribution of the left ventricle (LV) in dogs, and Greenbaum et al. confirmed the observations in human cadavers [8, 9].
Today, after centuries of observations, there is still debate on the exact origins and arrangements of the heart [10]. However, general consensus exists that the mammalian LV consists of oblique fibers in the endocardium that gradually change into circumferential fibers in the midwall and continue to oblique fibers in the subepicardium, orientated in the opposite direction to those in the endocardium, thus creating what is often referred to as a helical arrangement [11, 12, 13, 14]. Noteworthy insight has also been provided by the description of sheets and laminae, which may not only impact the effect of individual myofibres but also the electrical propagation across the myocardium [15, 16]. With regard to the latter, the coordinated sequence of electrical propagation and activation of the LV occurs in a specific apex-to-base and endocardial-to-epicardial order during systole [17]. Due to these different electrical activation times, each part of the heart muscle is activated for different durations, therefore shortening and lengthening velocities (or systolic and diastolic “strain rates”) vary significantly in the different regions of the LV and are not associated with the overall heart rate [18]. A significant addition to the longstanding knowledge on oblique and circumferential fibers was provided by Lunkenheimer et al., who provided evidence for the existence of transmural myofibres that may be of fundamental relevance to the regulation of forces associated with normal myocardial contraction and relaxation [19]. Finally, there is important structural diversity on the myocyte level that contributes to the overall elasticity of the cardiomyocyte, as revealed by different isoforms of the giant protein titin, which may influence myocardial deformation in systole and diastole, not least during exercise [20, 21]. Collectively, the current knowledge indicates a non-uniform, complex mesh of diverse cardiac myofibre arrangements which may be grouped in sheets and laminae, influencing the electrical activation sequence of the heterogeneously distributed autonomic nerves in the heart (Figure 1, [22]). In comparison to the LV, the macro-structure of the right ventricle (RV) is not cone-shaped but resembles that of a crescent, almost wrapping around the LV. Yet, the underlying micro-structure is similar to the LV, albeit with some key differences. Like the LV, the epicardial and endocardial fibers are arranged helically, but with a smaller range of oblique angles [23]. The main difference to the LV seems to be in the myofiber arrangement of the midwall. Here, “the circumferentially arranged middle fibres are confined to the LV and septum” [8] and “without such beneficial architectural remodeling […] seem unsuited structurally to sustain a permanent increase in afterload” [23]. It is probably because of the overall crescent shape (that makes echocardiographic image acquisition in any plane other than the longitudinal challenging), and the lack of an obvious torsional motion, that the assessment of right ventricular deformation has largely focused on longitudinal strain.
Figure 1.
LV anatomy, strain and twist. (A) Although the detailed anatomy of the heart is still a matter of debate, the most comprehensive, evidence-based model includes a mesh of oblique, circumferential and transmural fibers (1–5). (B) LV strain is typically assessed in three planes, the longitudinal plane (from the apex to the base, L), the circumferential plane, C, and the radial plane (from the endocardium to epicardium, R). Owing to the specific anatomy, contraction of the LV results in a twisting motion around the long-axis, with an opposing rotational movement at the base compared with the apex that is rapidly released in diastole. Resultant twist and twist velocity curves produce a clear signal for peak LV twist and early diastolic untwisting rate (red arrows). Please see further details and the original figures in Refs. [14, 24].
2.2 Definitions and selection of myocardial deformation parameters
Because of the increasing number of studies focused on myocardial deformation mentioned in the introduction to this chapter, it has been inevitable that some inconsistencies exist regarding the nomenclature in the literature. Here, a summary of the most common definitions is provided and the reader is also referred to previous review articles for further details on the terminology [24, 25, 26].
With regard to the LV, three strain components have been established: longitudinal, circumferential and radial strain [25]. Systolic strain rate was once thought to reflect contractility; however, these hopes have not been sustained. Furthermore, the anatomy of the heart does not support the measurement of radial strain since there are no radial fibers in the LV or RV. Although the transmural fibers may somewhat relate to this type of strain, they maximally constitute ~20% to overall deformation and do not seem to run strictly in the radial direction. Second, the classification of twist or torsion as a “shear strain” or fourth dimension of deformation does not fit the underlying anatomy of the heart either. There is currently no empirical evidence for the existence of a meaningful number of longitudinal fibers that could determine longitudinal deformation of the ventricles. Instead, the oblique fibers that make up most of the fibers within the left ventricular walls are likely responsible for deformation in the longitudinal direction. Consequently, it does not seem appropriate to calculate twist or torsion from the longitudinal and circumferential shear angle, also because this approach does not capture the potential regional differences that exist between the base and apex in both the LV and RV. Despite these drawbacks to the radial and longitudinal parameters, it must be acknowledged that longitudinal strain has become the most established measure as a clinical marker with diagnostic potential [27]. For these reasons, in the context of this chapter, it seems appropriate to ignore LV radial strain but include LV longitudinal and circumferential strain as well as twist and untwisting rate. Since no clear circumferential fibers or twisting motion have been detected in the RV, the focus for that chamber will be exclusively on longitudinal strainFigure 2.
Figure 2.
RV strain. The measurement of RV strain at rest (left) and during exercise (right) in a patient with hypertrophic cardiomyopathy. Because of the anatomical arrangement of the RV, longitudinal strain is the most commonly investigated parameter, although further clarity is required whether to always include or exclude the septum [28]. From a functional perspective, there is strong evidence that the septal deformation is more similar to that of the LV than the RV free wall, as supported by evidence of a shared morphology [29, 30]. Please see further details and the original figure in: Wu et al. [31].
Parameter (unit)
Description
Circumferential strain (%)
Percentage shortening of the circumference
Global longitudinal strain (%)
Typically, the average strain of multiple walls obtained from different echocardiographic windows (4-chamber, 2-chamber, 3-chamber)
Longitudinal strain (%)
Shortening along the long-axis of the ventricles in a single 2-dimensional imaging plane (for example a 4-chamber view)
Shear strain
The strain resulting from two different normal strains, for example “longitudinal-circumferential shear strain”
Strain (rate) imaging
Generic term that can refer to strain data obtained with either tissue Doppler or speckle tracking echocardiography
Strain rate (/s)
The rate of shortening (strain) or lengthening (strain) of each strain
Tissue Doppler strain (%)
Strain obtained with tissue Doppler echocardiography, which is more angle-dependent than speckle tracking echocardiography
Tissue velocity imaging (%)
Echocardiographic imaging based upon Doppler modality, often synonymous with tissue Doppler strain
Twist (degrees)
Also called the net twist angle, obtained from the net difference in rotation between the left ventricular base and apex. Not to be confused with torsion or rotation, the latter referring to the local angular deformation at the base and apex
Untwisting rate (°/s)
The maximal early diastolic rate of reversal of twist
Table 1.
Deformation parameters.
3. Echocardiographic assessment of myocardial deformation during exercise
3.1 Why exercise?
Even if all humans were elite athletes, we would spend most of the time in a day in a biological state of rest—or certainly in a state of low physical activity that only constitutes a fraction of the total capacity of our cardiovascular system. Accordingly, the routine clinical practice of examining cardiac function at rest is a good representation of the condition we find ourselves in most of the time. However, when a person requires an echocardiographic examination, it is typically for clinical reasons initiated by the presence of negative symptoms, often presenting as “exertional dyspnea” or angina. If an echocardiographic examination then detects structural and functional abnormalities of the heart that are congruent with the individual’s symptoms, the diagnosis of heart disease is likely. However, resting assessment of cardiac function often fails to recapitulate conditions of exertional dyspnea, and thus can sometimes lead to misdiagnosis. Equally, waiting until the emergence of symptoms postpones clinical treatment. For this reason, “stress testing” has been suggested to offer the opportunity of a “window into the future”. By taking the person out of their typical state of rest or low physical activity and stressing the full range of their cardiovascular system until maximum effort, underlying abnormalities may be detected that remain otherwise unknown. Examples for the benefit of exercise testing have been presented in relation to “unmasking masked hypertension” [32, 33]. Similarly, in pregnancy it has been proposed that the cardiovascular responses to exercise tests prior to conception may be indicators of the presence or absence of complications during future pregnancies [34, 35, 36, 37]. Furthermore, the complex etiology of heart failure has justified detailed exercise testing to identify the most important contributors out of the numerous cardiac or peripheral factors that may be involved in the development and/or the state of heart failure [38, 39, 40].
It is now recognized among clinical practitioners that the investigation of myocardial deformation during exercise can provide additive value, since previous research studies have revealed new (and sometimes surprising) insight into the behavior of the heart during exercise. As will be discussed in detail in Section 3.3, these findings have informed our basic understanding of cardiac function and sometimes guided future clinical investigation. Since myocardial function, including parameters of myocardial deformation, are influenced by the general loading state of the heart, any exercise responses must be seen in the context of general cardiovascular responses, as discussed in the next section.
3.2 General cardiovascular responses to exercise
In the context of myocardial deformation, the most relevant cardiovascular and cardiopulmonary responses to a standardized exercise test pertain to stroke volume, cardiac output, end-diastolic volume, blood pressure, arterial resistance, lactate, and maximal oxygen consumption (VO2max). In healthy individuals, a clear change in these parameters can be expected at the onset of low intensity dynamic exercise that should continue to change linearly up to moderate intensities. Importantly, dynamic exercise tests cause a disproportionate peripheral vasodilation in relation to the increase in cardiac output, and hence total peripheral resistance drops sharply at the onset of exercise and then remains constant across moderate and high exercise intensities [41]. From a diastolic perspective, end-diastolic volume has been shown to increase in some studies while others have not observed any change with exercise. This is not trivial since an acute increase in end-diastolic volume has been associated with an increased stroke volume, an effect also known as the Frank-Starling mechanism [42]. However, the overall contribution of end-diastolic volume to stroke volume is still relatively low because most of the increase in stroke volume has been attributed to the enhanced contractility that reduces the end-systolic volume.
At workloads above moderate intensity, several important physiological changes occur in healthy individuals. Blood lactate concentrations increase exponentially and CO2 production rises above O2 consumption, both reflecting the greater contribution of anaerobic metabolic pathways to overall energy utilization and causing a strong stimulus for vasodilation not least in the cerebral circulation. During the highest effort, stroke volume and VO2 have been reported to plateau and even decrease, but the exact pattern and the underlying mechanisms to this response remain a matter of debate [43]. Fortunately, this does not seem to impact the interpretation of cardiovascular responses to exercise in patients, since the sub-maximal data are currently thought to be of sufficient clinical value to determine whether exercise performance is normal or impaired [44].
One important distinction between the LV and RV responses to exercise is the potential for a “disproportionate load” on the RV [45], which is perhaps explained by both a greater relative rise in pulmonary blood pressure compared with that in the aorta, and differences in RV intrinsic factors such as force development. The differences between the LV and RV responses to exercise highlight the specific impact exercise has on the cardiovascular system. Consequently, determining the true origin of exercise limitations is challenging because many components of the cardiovascular system may be affected. For example, studies have shown that an exaggerated rise in blood pressure during exercise may be associated with negative outcomes, but whether this is caused by the heart or the periphery may be more difficult to determine [46, 47, 48]. Even in heart failure, the reduced exercise tolerance has been suggested to be a result of both central and non-cardiac limitations [38, 39, 40, 49]. Consequently, assessing myocardial deformation in relation to conventional exercise responses is essential for the quantification of the contributions of the heart muscle itself.
3.3 Myocardial deformation during exercise
Whatever myocardial parameter one chooses to examine during exercise, the interpretation of the responses can be tricky. For example, an increase in myocardial deformation with sub-maximal cycle exercise along with a typical drop in arterial resistance and concomitant reductions in end-systolic volume, in the presence of no adverse structural remodeling would be reflective of a “healthy” response. Equally, it is theoretically possible that the absence of a clear increase in myocardial deformation—which could be interpreted to represent myocardial dysfunction—may be a normal response if the increase in blood pressure and peripheral resistance were excessively high (or the exercise test did in fact create a condition of increased afterload). In this case, it is conceivable that the origin of the exercise limitation may not be cardiac despite the attenuated deformation, but perhaps peripheral in nature causing an exercise failure before the cardiac reserve is fully used [39]. Therefore, this section provides an overview of the general trend of myocardial deformation during exercise, but the reader is alerted that a qualitative interpretation must be performed after consideration of the wider physiology. Articles in this section were included if the studies had obtained data with echocardiography during exercise (tissue Doppler and tissue velocity imaging data were mostly excluded because both techniques are angle-dependent and typically represent only data from a single segment within the mitral annulus). Although a promising and exciting alternative to echocardiography, myocardial deformation during exercise obtained using MRI is not the focus of this chapter [50, 51]. Studies were also excluded if they obtained data immediately following exercise effort, as discussed in more detail in the section on methodological considerations. Finally, the avid reader is referred to some excellent review articles that cover more of the literature than this book chapter can accommodate [52, 53, 54, 55].
3.3.1 Physiological insight from healthy individuals
The physiology of myocardial deformation during exercise in healthy people is the fundamental basis upon which to interpret the responses in patient populations. Although many clinical research studies also include a healthy control group, sometimes these are matched to the patient groups in their demographics and, therefore, may not represent truly “healthy” individuals. Wherever possible, the data presented here will be from populations purposefully recruited as young healthy reference groups. To date, studies have revealed a variety of new perspectives that may be of great importance for the interpretation of clinical populations.
A decade ago, two studies revealed the strain and twist responses during incremental exercise. First, Doucende et al. showed that left ventricular twist and circumferential strain increased linearly up to moderate exercise intensities, while longitudinal strain increased initially but then plateaued at low exercise efforts [56]. This study also highlighted the interdependence of systolic and diastolic deformation, the role of untwisting rate in LV filling during exercise and the contribution of the LV apex to the overall myocardial response. Second, it was shown that LV twist and untwisting rate increased linearly up to near-maximal efforts, correlating with stroke volume and, thus, perhaps contributing to maximal exercise capacity in humans [57]. The importance of regional LV deformation, at the LV apex, was again highlighted. Several other studies have revealed similar patterns of LV twist during exercise in pre- and postmenopausal women, in athletes and of humans ascending to high altitude [58, 59, 60, 61, 62]. Consequently, it is now generally accepted that an increase in LV twist with exercise up to moderate intensities can be expected as a normal response (Figure 3). Surprisingly few studies dedicated to healthy individuals have measured LV strain during exercise, but they agree in general that longitudinal strain also increases with exercise [56, 61, 62, 63, 64]. Because of the risk of potential confounders, it is not possible to directly compare the response in LV twist and strain obtained in different studies. But in general, it is of great importance to note that the patters of the responses to exercise are not always the same for the two parameters, reminding us that they do not represent the same myocardial deformation. In agreement with the general physiological response to incremental exercise, LV twist increases linearly while longitudinal strain seems to plateau at low exercise efforts. This was more recently confirmed by Williams et al., who reported the same disparity between parameters in young healthy men [62]. Interestingly, in the same study, women seemed to have more of a linear response in longitudinal strain akin to LV twist. The disparity between LV twist and longitudinal strain has also been noted in studies on aging where LV twist consistently increases, but longitudinal strain does not change or decreases. Considering the well-established progression of aortic stiffness with aging [65], longitudinal strain appears to be at odds again with general physiology. Future studies should not only examine the parameters in relation to their sensitivity as a clinical marker but also consider the fit with general physiology.
Figure 3.
Myocardial deformation to incremental exercise. LV twist curves during incremental exercise, revealing a linear increase up to 70–80% of maximal individual exercise effort for both peak systolic LV twist (highest value in black lines top row) and peak diastolic untwisting rate (lowest value within black lines bottom row). Red lines represent myocardial deformation at the LV apex, blue lines at the LV base. Black lines are the composite of apical and basal data. Please see further details and the original figure in Ref. [57].
Studying the acute effects of exercise on myocardial deformation may be influenced by the chronic remodeling that humans have experienced. In this regard, Burns et al. showed that aging seems to be associated with a reduced LV twist reserve during exercise in a population of 60-year old individuals [66]. Similarly, “female aging”, as represented by the menopause, seems to impact the myocardial response to exercise, which may be further altered by exercise training [58]. One of the more surprising observations has been that of Cooke et al. who proposed that endurance trained athletes with enlarged “athlete’s heart” and a greater stroke volume had a similar systolic LV function, including LV twist, during submaximal exercise compared to untrained humans with smaller stroke volume [67]. Similar to the results presented by Doucende and Williams discussed above [56, 62], this particular exercise response strongly suggests that the mechanical2 systolic function of the heart may not be strictly associated with its output (stroke volume). Some mathematical calculations support the potentially poor linear association between systolic LV mechanical function and ejection fraction while others suggest a strong relationship [68]. In any case, the previous findings suggest that future investigations into the interaction between systolic deformation and ejection, and diastolic deformation and filling are needed to clarify the current uncertainty. One reason for the existing disagreement between mechanical function and associated hemodynamics may be the technical limitations causing restricted views from a 2D echocardiographic window. In the case of exercise responses, this may be particularly evident at the LV apex, since the apex has been proposed as an important contributor to exercise responses (in particular in diastolic function) [56, 57, 69]. However, in the echocardiographic images relevant for the measurement of global longitudinal strain, the representation of the apical segments is proportionately small and their contribution to longitudinal strain and strain rate may be underestimated compared with short-axis views [18]. Thus, some of the insight provided by myocardial deformation during exercise in healthy people relates to our more general understanding of cardiac function.
Compared with LV strain, RV longitudinal strain seems to be ~10 percentage points higher in healthy young humans at rest, likely reflecting the different anatomy combined with a lower pulmonary resistance compared with the aorta. Most studies reporting RV strain in healthy individuals during exercise have done so by including healthy controls as comparators to cardiac patients. From those studies, some patterns have emerged that suggest a consistently increased RV longitudinal strain during submaximal exercise in healthy individuals [28, 31, 70]. The mechanisms for this are probably similar to those of the LV, where an increased sympathetic state increases contractility while peripheral (pulmonary) vasodilation decreases downstream resistance [71]. However, during intense exercise, it seems that right ventricular myocardial deformation increases perhaps less than the LV, and it has even been shown to decrease. Given that both ventricles should produce approximately the same stroke volume under stable conditions, the lower RV strain during exercise is another indicator that the interaction between the mechanical function of the ventricles and the circulation may depend as much on the local arterial resistance as it may depend on the muscular performance (and therefore health) of the ventricles, and thus fitting the long-standing concept of a greater afterload-sensitivity of the RV. Recent studies in advanced heart failure patients who were surgically implanted with left ventricular assist devices (LVAD) may support this, since the mechanical pumps “unload” the LV and shift blood volume to the rest of the circulation, maybe creating “A Different Kind of Stress Test for the RV” [72, 73]. The accurate measurement of pulmonary and aortic resistance beyond the measurement or estimation of blood pressure is certainly going to elucidate the differential exposure and performance of the two ventricles [74]. At present, it seems that exercise does indeed cause a greater afterload challenge for the RV compared with the LV. In fact, it is worth noting that the exercise modalities used in the studies presented so far in this section have mostly employed “dynamic” exercise (see Section 3.4). In this context, it is essential to point out that this type of exercise increases sympathetic activation of the myocardium and reduces arterial resistance compared with the resting state, therefore creating an environment for the LV (and at low intensities for the RV) that is characterized by reduced afterload. During higher exercise intensities, pulmonary resistance can increase during dynamic exercise and create an augmented afterload challenge [45]. Strength exercise, also called resistance exercise, and isometric handgrip exercise are two other modalities that can provide an afterload challenge for the LV [75]. Interestingly, studies employing these exercise modalities in a number of different populations have consistently shown that the reduced systolic deformation is in part compensated for by an increase in heart rate, but can also be uncoupled from diastolic function [76, 77, 78, 79]. Given that resistance exercise produces a very different challenge to dynamic exercise, and that strength training is an important addition to rehabilitation, future research should consider incorporating responses during high resistive efforts [80, 81].
3.3.2 Exercise responses in patients with cardiovascular disease
In a seminal study, Notomi et al. provided mechanistic insight into the complex interdependence between systolic and diastolic function in hypertrophic cardiomyopathy [20]. Although the study used Tissue Doppler Imaging, it is a landmark study that has provided new insight and has popularized the use of exercise testing for both basic science and new insight into cardiac performance in patients. The study revealed that LV twist during exercise was significantly reduced in patients with hypertrophic cardiomyopathy. Similarly, two other studies concluded that systolic deformation reserve is reduced in patients with hypertrophic cardiomyopathy [82, 83]. However, one challenge in patient populations is that the change in heart rate is often different compared with control groups, and therefore it is possible that the groups experienced different physiological stimuli. This is a recurring problem in exercise studies that currently reduces the confidence in some conclusions. Equally, sometimes the matching of the change in heart rate between groups may lead to unequal workloads or changes in blood pressure, highlighting again the need to interpret myocardial deformation during exercise in the context of general physiological responses. Notwithstanding, the overall trend is that LV myocardial deformation in patients is reduced in response to an acute exercise challenge, including in cardiac amyloidosis, hypertension, cancer, coronary artery disease, as well as in patients with valve disease before and after surgical correction [84, 85, 86, 87, 88, 89]. Some subtle observations, however, are worthy of discussion. For example, in patients with microvascular angina, only the subendocardial strain was reduced, and diastolic function during exercise was more severely affected than systolic reserve [90]. Similarly, myocardial regions can respond differently during exercise in coronary artery disease patients, as shown by differential basal vs. apical rotational mechanics [89]. In an elegant study in patients with hypertrophic cardiomyopathy, Soullier et al. showed that there was significant heterogeneity in the response of the different deformation parameters to exercise, and that resting twist was even increased in patients while diastolic untwisting rate was less affected [83]. In patients with a prior heart transplant, the age of the recipients and donors seem to influence the longitudinal and circumferential strain response to exercise [91, 92]. All these observations highlight the very subtle changes that can occur between parameters, and between systolic and diastolic function. To determine the full significance of such differences should be the focus of future investigations. Furthermore, it will be essential to relate myocardial deformation more often to parameters like cardiac output, to enable the meaningful interpretation of deformation indices and their contribution to the overall capacity of the heart. When this was done in previous studies, the myocardial deformation during exercise provided a clear advancement of our general understanding of the etiology and/or progression of cardiac disease [93].
Because of the prevalence and importance of pulmonary hypertension, and the exercise limitations of heart failure patients, myocardial deformation of the RV during exercise has received heightened attention [94]. Similar to the LV response, the expected increase in RV myocardial deformation during exercise is generally blunted, not just in pulmonary hypertension but also in tetralogy of Fallot, systemic sclerosis, and hypertrophic cardiomyopathy [31, 95, 96, 97]. Most often, there is clear evidence that pulmonary artery pressures increased disproportionately in the groups that had a blunted increase in RV longitudinal strain during exercise. Importantly, these patients often have normal pulmonary artery pressures at rest, which not only emphasizes the diagnostic value of exercise testing, it also highlights the possibility that patients with suspected LV pathology should be tested for the RV myocardial response to exercise.
3.4 Important practical considerations
Any echocardiographic examination consists of two main parts: (1) the acquisition of standardized echocardiographic images, and (2) the analysis of images for the quantification of relevant parameters [98, 99]. When conducting echocardiography during exercise, both parts require modified approaches to ensure that the conclusions drawn remain valid. Here, based upon our extensive practical experience, we present some “take-home-messages” that we consider essential for the echocardiographic assessment of myocardial deformation during exercises.
Typically, exercise tests are performed in a stepwise (constant intensity for some minutes, then increasing) or incremental (gradually increasing intensity with every second) manner. Because different protocols provoke different physiological responses, the correct protocol must be selected carefully.
Exercise responses depend on the relative workload of an individual. Therefore, exercise intensities should be adjusted to an individual’s anticipated capacity and patients’ myocardial deformation interpreted in relation to the relative workload [100].
The individual adjustment of workload increments during the test should also acknowledge fitness, age, sex, medical history, and acute or chronic injuries.
For the assessment of myocardial deformation during exercise, running or cycling modalities are the most common. For the reason of improved image quality and because it is relatively safe/feasible, the preferred choice for exercise echocardiography may be supine cycling.
While it is generally accepted that gentle end-expiratory breath holds can be performed to obtain images, it is preferable to obtain echocardiographic cine loops during free breathing and average some cardiac cycles during inspiration and expiration.
It is important to distinguish between the physiological demands of different exercise modalities, categorized as: dynamic, static, and impact [101]. Consequently, certain types of exercise can be considered more as an “afterload challenge” than others, and the responses of myocardial deformation may vary greatly between these types of exercise. In this context, the reader is reminded that exercise training interventions for health will need to consider the same complexities, as evidenced by the potential for differential effects of moderate continuous exercise training versus high-intensity interval training in some cardiac patients [102].
One concern with regard to exercise testing is the risk of triggering adverse events. Although this will depend on the specific individual being tested and must be decided by qualified personnel on a case-by-case basis, as evidenced by a comprehensive study performed by Rognmo et al. [103], the overall risk for serious adverse events seems to be relatively low. Particular health and safety precautions should be taken in patients with overt or suspected arrhythmia and the decision “not allowed to perform an exercise test” may have to be taken.
Standardization of echocardiographic data acquisition during exercise is absolutely necessary. Sonographers should minimize the sector width and depth, maximize imaging frame rates, only use one focal point and position this in the optimal location, and optimize the overall image to maximize the visibility of the endocardial border for speckle tracking analysis. Although 3D echocardiography may solve some of the limitations of 2D echocardiography, at present the frame rates are too low to obtain the necessary temporal resolution for quantification of myocardial deformation during exercise, although this is expected to change in the near future.
During exercise, when respiration and heart are increased, the quick location of the optimal echocardiographic window is necessary. Marking up the location on the chest after the resting assessment serves as a “quick help” during exercise. The sonographer must, however, still optimize the image and perhaps move the transducer slightly during exercise.
Since heart rate increases during exercise but imaging frame rates are already maximized, the effective frame rate (data points per cardiac cycle) decreases. Although this cannot be fully corrected, it seems advisable to perform cubic spline interpolation to attenuate some of these limitations [104]. Note that cubic spline interpolation will not only add points in time (for example for the more confident assessment of dyssynchrony), it also slightly adjusts the peak values.
Data acquisition immediately following exercise is not the same as “during” exercise. With the cessation of exercise, especially after a strenuous effort with strong muscular contractions, instant changes in whole-body hemodynamics set in [105]. Hence, these data do not reflect an exercise challenge but a “exercise recovery” state.
For the acquisition of LV twist, apical data must be obtained by moving the transducer close to the point of obtaining a 4-chamber view, otherwise severely misrepresentative data will be collected [24].
4. Summary and conclusions
The assessment of LV and RV myocardial deformation during exercise is feasible and has contributed unique insight into cardiac physiology in health and disease. Inherent methodological challenges require appropriate training and a careful approach to image acquisition, analysis and interpretation. However, ongoing technological advancements and an increasing knowledge suggest that the echocardiographic assessment of myocardial deformation during exercise will play an ever-increasing role in future research and the clinical examination of the cardiac patient.
Acknowledgments
The authors express their sincere gratitude to the publisher, IntechOpen, for their very kind and generous financial support of this chapter.
\n',keywords:"exercise, heart, stress testing, diagnostics, imaging, echocardiography, VO2max, CPET, strain, twist, torsion, untwisting rate, blood pressure, LVAD, heart failure, speckle tracking, hypertension",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/72623.pdf",chapterXML:"https://mts.intechopen.com/source/xml/72623.xml",downloadPdfUrl:"/chapter/pdf-download/72623",previewPdfUrl:"/chapter/pdf-preview/72623",totalDownloads:124,totalViews:0,totalCrossrefCites:0,dateSubmitted:"March 4th 2020",dateReviewed:"May 25th 2020",datePrePublished:"June 26th 2020",datePublished:null,dateFinished:null,readingETA:"0",abstract:"The human heart is an asymmetrical structure that consists of oblique, circumferential, and transmural fibers, as well as laminae and sheets. Sequential electrical activation of all the muscle fibers ultimately results in a coordinated contraction of the heart muscle also referred to as “deformation.” This is immediately followed by myocardial relaxation, when the preceding deformation is reversed, and the ventricles fill with blood. Given the complexity of these repetitive motions, it is not surprising that there is great diversity in the myocardial deformation between different individuals and between distinct populations. Exercise presents a natural challenge to determine the full capacity of an individual’s heart, and modern imaging technologies allow for the non-invasive assessment of myocardial deformation during exercise. In this chapter, the most relevant anatomical basis for myocardial deformation is summarized and definitions of the most relevant parameters are provided. Then, the general cardiac responses to exercise are highlighted before the current knowledge on myocardial deformation during exercise is discussed. The literature clearly indicates that the echocardiographic evaluation of myocardial deformation during exercise holds great promise for the identification of sub-clinical disease. Future studies should aim to determine the mechanisms of differential expression of myocardial deformation during exercise in health and disease.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/72623",risUrl:"/chapter/ris/72623",signatures:"Eric J. Stöhr and T. Jake Samuel",book:{id:"9581",title:"Endocarditis",subtitle:null,fullTitle:"Endocarditis",slug:null,publishedDate:null,bookSignature:"",coverURL:"https://cdn.intechopen.com/books/images_new/9581.jpg",licenceType:"CC BY 3.0",editedByType:null,editors:null,productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}},authors:null,sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Principles of myocardial deformation",level:"1"},{id:"sec_2_2",title:"2.1 Anatomy",level:"2"},{id:"sec_3_2",title:"2.2 Definitions and selection of myocardial deformation parameters",level:"2"},{id:"sec_5",title:"3. Echocardiographic assessment of myocardial deformation during exercise",level:"1"},{id:"sec_5_2",title:"3.1 Why exercise?",level:"2"},{id:"sec_6_2",title:"3.2 General cardiovascular responses to exercise",level:"2"},{id:"sec_7_2",title:"3.3 Myocardial deformation during exercise",level:"2"},{id:"sec_7_3",title:"3.3.1 Physiological insight from healthy individuals",level:"3"},{id:"sec_8_3",title:"3.3.2 Exercise responses in patients with cardiovascular disease",level:"3"},{id:"sec_10_2",title:"3.4 Important practical considerations",level:"2"},{id:"sec_12",title:"4. Summary and conclusions",level:"1"},{id:"sec_13",title:"Acknowledgments",level:"1"}],chapterReferences:[{id:"B1",body:'Buckberg GD. 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School of Sport and Health Sciences, Cardiff Metropolitan University, UK
Division of Cardiology, Advanced Heart Failure Program, Columbia University Irving Medical Center, USA
Department of Kinesiology, University of Texas at Arlington, USA
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