\r\n\tOne basic topic is that of expression manipulation: combining, expanding etc, and the applications of this scholar topic needs focusing on.
\r\n\r\n\tThe general topic of "polynomials" is very large, and here the focus is both on scholar/student basics of it, and on applications of some special polynomials in science and research.
\r\n\r\n\tAn important topic of the book is "algebraic curve". Here the approaches are multiple: basic/scholar on one hand, and applications on the other hand. It must be noticed the use of algebraic curves properties in the field of differential equations, for example for finding the singularities.
\r\n\r\n\tGrobner basis is a very modern and applied topic of algebra. Here we must outline the great importance of Grobner basis and polynomial ideals manipulation, in the differential equations field, an example being in fast finding normal forms of differential systems.
\r\n\r\n\tRelated to this last topic of the book, but applying to all specified topics, it must be noticed the importance of numeric algorithms. The importance of software algorithms in all fields of science is continuously increasing. Therefore, computational approach of the specified algebraic topics is very useful, with applications in other mathematical and scientific fields.
",isbn:null,printIsbn:"979-953-307-X-X",pdfIsbn:null,doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"2a81efb05ce334905cc672188033b15d",bookSignature:"Dr. Adela Ionescu",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/9907.jpg",keywords:"expand, factoring, combining, simplifying, random polynomials, special polynomials, orthogonal polynomials, polynomial factorization, two variables polynomials, homogenization, parameterization, singularity, monomial order, polynomial ideal, leading monomial, normal form",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:null,numberOfDimensionsCitations:null,numberOfTotalCitations:null,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"November 26th 2019",dateEndSecondStepPublish:"December 17th 2019",dateEndThirdStepPublish:"February 15th 2020",dateEndFourthStepPublish:"May 5th 2020",dateEndFifthStepPublish:"July 4th 2020",remainingDaysToSecondStep:"a year",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:null,kuFlag:!1,biosketch:null,coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"146822",title:"Dr.",name:"Adela",middleName:null,surname:"Ionescu",slug:"adela-ionescu",fullName:"Adela Ionescu",profilePictureURL:"https://mts.intechopen.com/storage/users/146822/images/system/146822.jpg",biography:"Dr. Adela Ionescu is a lecturer at the University of Craiova, Romania. She received her PhD degree from the Polytechnic University of Bucharest, Romania. Her research focuses on development and implementation of new methods in the qualitative and computational analysis of differential equations and their applications. This includes constructing adequate models for approaching the study of different industrial phenomena from a dynamical system standpoint and also from a computational fluid dynamics standpoint. By its optimizing techniques, the aim of the modeling is to facilitate the high understanding of the experimental phenomena and to implement new methods, techniques, and processes. Currently, Dr. Ionescu is working in developing new analytical techniques for linearizing nonlinear dynamical systems, with subsequent applications in experimental cases. The bifurcation theory and its applications in related fields is also a domain of interest for her. She has published six monographs and few scientific papers in high-impact journals. She is also a member of few scientific international associations and has attended more than 45 international conferences.",institutionString:"University of Craiova",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"1",totalChapterViews:"0",totalEditedBooks:"1",institution:{name:"University of Craiova",institutionURL:null,country:{name:"Romania"}}}],coeditorOne:null,coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"15",title:"Mathematics",slug:"mathematics"}],chapters:null,productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"287827",firstName:"Gordan",lastName:"Tot",middleName:null,title:"Mr.",imageUrl:"https://mts.intechopen.com/storage/users/287827/images/8493_n.png",email:"gordan@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. Whether that be identifying an exceptional author and proposing an editorship collaboration, or contacting researchers who would like the opportunity to work with IntechOpen, I establish and help manage author and editor acquisition and contact."}},relatedBooks:[{type:"book",id:"6217",title:"Computational Fluid Dynamics",subtitle:"Basic Instruments and Applications in Science",isOpenForSubmission:!1,hash:"0fb7b242fd063d519b361e5c2c99187b",slug:"computational-fluid-dynamics-basic-instruments-and-applications-in-science",bookSignature:"Adela Ionescu",coverURL:"https://cdn.intechopen.com/books/images_new/6217.jpg",editedByType:"Edited by",editors:[{id:"146822",title:"Dr.",name:"Adela",surname:"Ionescu",slug:"adela-ionescu",fullName:"Adela Ionescu"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"1591",title:"Infrared Spectroscopy",subtitle:"Materials Science, Engineering and Technology",isOpenForSubmission:!1,hash:"99b4b7b71a8caeb693ed762b40b017f4",slug:"infrared-spectroscopy-materials-science-engineering-and-technology",bookSignature:"Theophile Theophanides",coverURL:"https://cdn.intechopen.com/books/images_new/1591.jpg",editedByType:"Edited by",editors:[{id:"37194",title:"Dr.",name:"Theophanides",surname:"Theophile",slug:"theophanides-theophile",fullName:"Theophanides Theophile"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"3092",title:"Anopheles mosquitoes",subtitle:"New insights into malaria vectors",isOpenForSubmission:!1,hash:"c9e622485316d5e296288bf24d2b0d64",slug:"anopheles-mosquitoes-new-insights-into-malaria-vectors",bookSignature:"Sylvie Manguin",coverURL:"https://cdn.intechopen.com/books/images_new/3092.jpg",editedByType:"Edited by",editors:[{id:"50017",title:"Prof.",name:"Sylvie",surname:"Manguin",slug:"sylvie-manguin",fullName:"Sylvie Manguin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"3161",title:"Frontiers in Guided Wave Optics and Optoelectronics",subtitle:null,isOpenForSubmission:!1,hash:"deb44e9c99f82bbce1083abea743146c",slug:"frontiers-in-guided-wave-optics-and-optoelectronics",bookSignature:"Bishnu Pal",coverURL:"https://cdn.intechopen.com/books/images_new/3161.jpg",editedByType:"Edited by",editors:[{id:"4782",title:"Prof.",name:"Bishnu",surname:"Pal",slug:"bishnu-pal",fullName:"Bishnu Pal"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"72",title:"Ionic Liquids",subtitle:"Theory, Properties, New Approaches",isOpenForSubmission:!1,hash:"d94ffa3cfa10505e3b1d676d46fcd3f5",slug:"ionic-liquids-theory-properties-new-approaches",bookSignature:"Alexander Kokorin",coverURL:"https://cdn.intechopen.com/books/images_new/72.jpg",editedByType:"Edited by",editors:[{id:"19816",title:"Prof.",name:"Alexander",surname:"Kokorin",slug:"alexander-kokorin",fullName:"Alexander Kokorin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"1373",title:"Ionic Liquids",subtitle:"Applications and Perspectives",isOpenForSubmission:!1,hash:"5e9ae5ae9167cde4b344e499a792c41c",slug:"ionic-liquids-applications-and-perspectives",bookSignature:"Alexander Kokorin",coverURL:"https://cdn.intechopen.com/books/images_new/1373.jpg",editedByType:"Edited by",editors:[{id:"19816",title:"Prof.",name:"Alexander",surname:"Kokorin",slug:"alexander-kokorin",fullName:"Alexander Kokorin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"57",title:"Physics and Applications of Graphene",subtitle:"Experiments",isOpenForSubmission:!1,hash:"0e6622a71cf4f02f45bfdd5691e1189a",slug:"physics-and-applications-of-graphene-experiments",bookSignature:"Sergey Mikhailov",coverURL:"https://cdn.intechopen.com/books/images_new/57.jpg",editedByType:"Edited by",editors:[{id:"16042",title:"Dr.",name:"Sergey",surname:"Mikhailov",slug:"sergey-mikhailov",fullName:"Sergey Mikhailov"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"371",title:"Abiotic Stress in Plants",subtitle:"Mechanisms and Adaptations",isOpenForSubmission:!1,hash:"588466f487e307619849d72389178a74",slug:"abiotic-stress-in-plants-mechanisms-and-adaptations",bookSignature:"Arun Shanker and B. Venkateswarlu",coverURL:"https://cdn.intechopen.com/books/images_new/371.jpg",editedByType:"Edited by",editors:[{id:"58592",title:"Dr.",name:"Arun",surname:"Shanker",slug:"arun-shanker",fullName:"Arun Shanker"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"878",title:"Phytochemicals",subtitle:"A Global Perspective of Their Role in Nutrition and Health",isOpenForSubmission:!1,hash:"ec77671f63975ef2d16192897deb6835",slug:"phytochemicals-a-global-perspective-of-their-role-in-nutrition-and-health",bookSignature:"Venketeshwer Rao",coverURL:"https://cdn.intechopen.com/books/images_new/878.jpg",editedByType:"Edited by",editors:[{id:"82663",title:"Dr.",name:"Venketeshwer",surname:"Rao",slug:"venketeshwer-rao",fullName:"Venketeshwer Rao"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"4816",title:"Face Recognition",subtitle:null,isOpenForSubmission:!1,hash:"146063b5359146b7718ea86bad47c8eb",slug:"face_recognition",bookSignature:"Kresimir Delac and Mislav Grgic",coverURL:"https://cdn.intechopen.com/books/images_new/4816.jpg",editedByType:"Edited by",editors:[{id:"528",title:"Dr.",name:"Kresimir",surname:"Delac",slug:"kresimir-delac",fullName:"Kresimir Delac"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"38390",title:"Advances in Spatio-Temporal Modeling and Prediction for Environmental Risk Assessment",doi:"10.5772/51227",slug:"advances-in-spatio-temporal-modeling-and-prediction-for-environmental-risk-assessment",body:'Meteorological readings, hydrological parameters and many measures of air, soil and water pollution are often collected for a certain span, regularly in time, and at different survey stations of a monitoring network. Then, these observations can be viewed as realizations of a random function with a spatio-temporal variability. In this context, the arrangement of valid models for spatio-temporal prediction and environmental risk assessment is strongly required. Spatio-temporal models might be used for different goals: optimization of sampling design network, prediction at unsampled spatial locations or unsampled time points and computation of maps of predicted values, assessing the uncertainty of predicted values starting from the experimental measurements, trend detection in space and time, particularly important to cope with risks coming from concentrations of hazardous pollutants. Hence, more and more attention is given to spatio-temporal analysis in order to sort out these issues.
Spatio-temporal geostatistical techniques provide useful tools to analyze, interpret and control the complex evolution of various variables observed by environmental monitoring networks. However, in the literature there are no specialized monographs which contain a thorough presentation of multivariate methodologies available in Geostatistics, especially in a spatio-temporal context. Several authors have developed different multivariate models for analyzing the spatial and spatio-temporal behavior of environmental variables, as it is clarified in the following brief review.
In multivariate spatial analysis, direct and cross correlations for the variables under study are quantified by estimating and modeling the matrix variogram. The difficulty in modeling this matrix function, especially the off diagonal entries of the same matrix, has been first faced by using the linear coregionalization model (LCM), proposed by [Matheron, 1982].
For matrix covariance functions, [Gneiting et al., 2010] constructed a parametric family of symmetric covariance models for stationary and isotropic multivariate Gaussian spatial random fields, where both the diagonal and off diagonal entries are of the Matérn type. In the bivariate case, they provided necessary and sufficient conditions for the positive definiteness of the second-order structure, whereas for the other multivariate cases they suggested a parsimonious model which imposes restrictions on the scale and cross-covariance smoothness parameters. In the bivariate case, where the smoothness parameter is the same for both covariance functions, the Gneiting model is a simplified LCM. The Gneiting cross-covariance model also assumes that the scale parameter is the same for all the covariance functions and the cross-covariance functions. Both the LCM and Gneiting constructions for cross-covariances result in symmetric models; however, no distributional assumptions are required for using a LCM, which can easily incorporate components with compact support and multiple ranges and an unbounded variogram component.
Although models for multivariate spatial data have been extensively explored [Wackernagel, 2003, Gelfand et al., 2004, Ver hoef Barry, 1998], models for multivariate spatio-temporal data have received relatively less attention. In the literature, it is common to use classical techniques for multivariate spatial and temporal analysis [Wackernagel, 2003, De Iaco et al., 2001b]. Recently, canonical correlation analysis was combined with space-time geostatistical tools for detecting possible interactions between two groups of variables, associated with pollutants and atmospheric conditions [De Iaco, 2011]. In the dynamic modeling framework, there are some results in studying the spatio-temporal variability of several correlated variables: [Gelfand et al., 2005], for example, extended univariate spatio-temporal dynamic models to multivariate dynamic spatial models. Moreover, [Li et al., 2008] proposed a methodology to evaluate the appropriateness of several common assumptions, such as symmetry, separability and linear model of coregionalization, on multivariate covariance functions in the spatio-temporal context, while [Choi et al., 2009] proposed a spatio-temporal LCM where the multivariate spatio-temporal process was expressed as a linear combination of independent Gaussian processes in space-time with mean zero and a separable spatio-temporal covariance. [Apanasovich Genton, 2010] considered some solutions to the symmetry problem; moreover, they proposed a class of cross-covariance functions for multivariate random fields based on the work of [Gneiting, 2002]. The maximum likelihood estimation of heterotopic spatio-temporal models with spatial LCM components and temporal dynamics was developed by [Fassò Finazzi(2011)]. A GSLib [Deutsch Journel, 1998] routine for cokriging was properly modified in [De Iaco et al., 2010] to incorporate the spatio-temporal LCM, previously developed using the generalized product-sum variogram model [De Iaco et al., 2003]. Recently, in [De Iaco et al., 2012a] an automatic procedure for fitting the spatio-temporal LCM using the product-sum variogram model has been presented and some computational aspects, analytically described by a main flow-chart, have been discussed. In [De Iaco et al., 2012 b] simultaneous diagonalization of the sample matrix variograms has been used to isolate the basic components of a spatio-temporal LCM and it has been illustrated how nearly simultaneous diagonalization of the cross-variogram matrices simplifies modeling of the matrix variogram.
In the following, after an introduction of the theoretical framework of the multivariate spatio-temporal random function and its features (Section 2), a review of recent techniques for building admissible models is proposed (Section 3). Successively, the spatio-temporal LCM, its assumptions and appropriate statistical tests are presented (Section 4) and techniques for prediction and risk assessment maps are introduced (Section 5). Some critical aspects regarding sampling, modeling and computational problems are discussed (Section 6). Finally, a case study concerning particle pollution (
represents a multivariate spatio-temporal random function ( MSTRF), where
Afterwards, the MSTRF will be denoted with
The observations
Given a MSTRF Z, with
the expected value, or first-order moment of each component
the second-order moments,
1. the variance of each component
2. the cross-covariance for each pair of STRF\n\t\t\t\t\t
3. the cross-variogram for each pair of STRF\n\t\t\t\t\t
Note that for
the covariance of the STRF
with
\n\t\t\t\tthe direct variogram of the STRF
These moments describe the basic features of a MSTRF, such as the spatio-temporal correlation for each variable and the cross-correlation among the variables.
In multivariate Geostatistics, admissibility conditions concern both the cross-covariances and the cross-variograms, as described in the following.
Letfor any choice of the
where
As in the univariate case, the
under the constraint: 3mm
Stationarity hypotheses allow to make inference on the MSTRF. In particular, second-order stationarity and intrinsic hypotheses concern the first and second-order moments of the MSTRF.
A MSTRF Z, with p components, is second-order stationary if:
for any STRF\n\t\t\t\t\t\t\t\t
for any pair of STRF\n\t\t\t\t\t\t\t\t
where
There exist several physical phenomena for which neither variance, nor the covariance exist, however it is possible to assume the existence of the variogram.
A MSTRF Z, with
for any STRF\n\t\t\t\t\t\t\t\t
for any pair of STRF\n\t\t\t\t\t\t\t\t
where
\n\t\t\t\t\tSecond-order stationarity implies the existence of the intrinsic hypotheses, however the converse is not true. Intrinsic hypotheses imply that the cross-variogram can be expressed as the expected value of the product of the increments:
Given a second-order stationary MSTRF, the cross-covariance satisfies the properties listed below.
The cross-covariance is not invariant with respect to the exchange of the variables:
as well as it is not invariant with respect to the sign of the vector
However, the cross-covariance is invariant with respect to the joint exchange of the variables and the sign of the vector h:
Afterwards, the main properties of the cross-variogram for intrinsic MSTRF are given.
The cross-variogram vanishes at the origin, that is:
The cross-variogram is invariant with respect to the exchange of the variables:
The cross-variogram is invariant with respect to the sign of the vector h:
From (15) and (16) follows that the cross-variogram is completely symmetric, as it will be pointed out in the next sections.
The cross-covariance
where
hence the changes of the cross-covariance functions, with respect to the changes of the vector h, do not depend on the pair of the STRF
The cross-covariance
where
The cross-covariance
or, equivalently, if:
The cross-covariance
or, equivalently,
Atmospheric, environmental and geophysical processes are often under the influence of prevailing air or water flows, resulting in a lack of full symmetry [de Luna Genton, 2005, Gneiting, 2002, Stein, 2005].
Fig. 1 summarizes the relationships between separability, symmetry, stationarity and the LCM in the general class of the cross-covariance functions of a MSTRF Z. If a cross-covariance is separable, then it is symmetric, however, in general, the converse is not true. Moreover, the hypothesis of full separability is a special case of full symmetry.
Several tests to check symmetry and separability of cross-covariance functions can be found in the literature [Scaccia Martin, 2005, Lu Zimmerman, 2005a, Lu Zimmerman, 2005b, Li et al., 2008].
Relationships among different classes of spatio-temporal covariance functions
In the following, a brief review of the most utilized techniques to construct admissible cross-covariance models is presented.
For the intrinsic correlation model, the matrices
where the coefficients
In the kernel convolution method [Ver hoef Barry, 1998] the cross-covariance functions is represented as follows:
where the
In the covariance convolution for stationary processes [Gaspari Cohn, 1999, Majumdar Gelfand, 2007] the cross-covariance functions is represented as follows:
where
Recently, an approach based on latent dimensions and existing covariance models for univariate random fields, has been proposed; the idea is to develop flexible, interpretable and computationally feasible classes of cross-covariance functions in closed form [Apanasovich Genton, 2010].
LCM is based on the hypothesis that each direct or cross-variogram (covariogram) can be represented as a linear combination of some basic models and each direct or cross-variogram (covariogram) must be built using the same basic models [Journel Huijbregts, 1981].
LCM is utilized in several applications because of its flexibility, moreover it encouraged the development of algorithms able to estimate quickly the parameters of the selected model, assuring the admissibility conditions, even in presence of several variables [Goovaerts, 1997, Goulard, 1989, Goulard Voltz, 1992].
Let Z be a second-order stationary MSTRF with p components,
where
hence, in the LCM, it is assumed that:
with
\n\t\t\tThe matrix C for the second-order stationary Z is built as follows:
Analogously, it is also possible to introduce the LCM for a MSTRF which satisfies the intrinsic hypotheses. In such a case, the direct and cross-variograms are built as follows:
where each basic structure
Then, for a MSTRF which satisfies the intrinsic hypotheses the matrix
The necessary and sufficient conditions because the model defined in (17) and (20) are admissible are:
the matrices
The necessary, but not sufficient conditions, for the coefficients
The basic structures
As discussed in [De Iaco et al., 2003], each basic spatio-temporal structure,
where
is the parameter of generalized product-sum variogram model and it is such that
The inequality (23) represents a necessary and sufficient condition in order that each basic structure
Substituting (21) in (20), the spatio-temporal LCM can be defined through two marginals: one in space and one in time, i.e.:
Using the generalized product-sum variogram model it is possible:
to identify the different scales of variability and build the matrices
to describe the correlation structure of processes characterized by a different spatial and temporal variability.
Fitting a spatio-temporal LCM to the data requires the identification of the spatio-temporal basic variograms and the corresponding positive definite coregionalization matrices, however this is often a hard step to tackle. A recent approach [De Iaco et al., 2012b], based on the simultaneous diagonalization of a set of matrix variograms computed for several spatio-temporal lags, allows to determine the spatio-temporal LCM parameters in a very simple way.
In several environmental applications [Wackernagel, 2003], the cross-covariance function is not symmetric, as for example, in time series in presence of a delay effect, as well as in hydrology, for the cross-correlation between a variable and its derivative, such as water head and transmissivity [Thiebaux, 1990]. Hence, this assumption should be tested before fitting a spatio-temporal LCM.
A useful hint to verify the symmetry of the cross-covariance can be given by estimating all the pseudo cross-variograms [Myers, 1991] of the standardized variables
The appropriateness of the assumption of symmetry of a spatio-temporal LCM can be tested by using the methodology proposed by [Li et al., 2008], based on the asymptotic joint normality of the sample spatio-temporal cross-covariances estimators. Given a set
where a is the row rank of the matrix A, which is such that
Moreover, the choice of modeling the MSTRF Z by a spatio-temporal LCM is based on the prior assumption that the multivariate correlation structure of the variables under study is characterized by
where
Obviously, this last model is just a particular case (
Remarks
In the spatio-temporal LCM, each component is represented as a linear combination of latent, independent univariate spatio-temporal processes. However, the smoothness of any component defaults to that of the roughest latent process, and thus the standard approach does not admit individually distinct smoothness properties, unless structural zeros are imposed on the latent process coefficients [Gneiting et al., 2010].
In most applications, the wide use of the spatio-temporal LCM is justified by practical aspects concerning the admissibility condition for the matrix variograms (covariances). Indeed, it is enough to verify the positive definiteness of the coregionalization matrices,
The spatio-temporal LCM allows unbounded variogram components to be used [Wackernagel, 2003].
For prediction purposes, various cokriging algorithms can be found in the literature [Journel Huijbregts, 1981, Chilés Delfiner, 1999]. As a natural extension of spatial ordinary cokriging to the spatio-temporal context, the linear spatio-temporal predictor can be written as
where
The predicted spatio-temporal random vector
The matrices of weights
The new.
Similarly, for environmental risk assessment, the formalism of multivariate spatio-temporal indicator random function (
be a vector of p spatio-temporal indicator random functions (STIRF) defined on the domain
where
represents a MSTIRF. In other words, for each coregionalized variable
Multivariate geostatistical analysis for spatio-temporal data is rather complex because of several problems concerning:
a. sampling,
b. the choice of admissible direct and cross-correlation models,
c. the definition of automatic procedures for estimation and modeling.
Sampling problems
There exist several sampling techniques for multivariate spatio-temporal data, as specified herein. Let
be the sets of sampled points in the spatio-temporal domain for the p variables under study. It is possible to distinguish the following situations:
total heterotopy, where the sets of the sampled points are pairwise disjoint, that is
partial heterotopy, where the sets of the sampled points are not pairwise disjoint, that is
isotopy, where the sets of the sampled points coincide, that is
A special case of partial heterotopy is the so-called undersampling: in such a case, the points where a variable, called primary or principal, has been sampled, constitute a subset of the points where the remaining variables, called auxiliary or secondary, have been observed. The secondary variables provide additional information useful to improve the prediction of the primary variable.
Modeling problems
In Geostatistics, the main modeling problems concern the choice of an admissible parametric model to be fitted to the empirical correlation function (covariance or variogram). In particular, in multivariate analysis for spatio-temporal data it is important to identify an admissible model able to describe the correlation among several variables which describe the spatio-temporal process. In this context, it is suitable to underline that
it is not enough to select an admissible direct variogram to model a cross-variogram;
the direct variograms are positive functions, on the other hand cross-variograms could be negative functions;
only some necessary conditions of admissibility are known to model a cross-variogram, as the Cauchy-Schwartz inequality, however sufficient conditions cannot be easily applied [Wackernagel, 2003].
The use of multivariate correlation models well-known in the literature, such as the LCM [Wackernagel, 2003], the class of non-separable and asymmetric cross-covariances, proposed by [Apanasovich Genton, 2010], or the parametric family of cross-covariances, where each component is a Matérn process [Gneiting et al., 2010], requires the identification of several parameters, especially in presence of many variables.
Moreover, estimation and modeling the direct and cross-correlation functions could be compromised by the sampling plan.
Computational problems
For spatial multivariate data different algorithms for fitting the LCM have been implemented in software packages. [Goulard, 1989] and [Goulard Voltz, 1992] described an iterative procedure to fit a LCM using a weighted least-squares like technique: this requires first fitting the diagonal entries, i.e. the basic variogram structures must be determined first. In contrast, [Xie Myers, 1995] and [Xie et al., 1995] developed an alternative method for modeling the matrix-valued variogram by near simultaneously diagonalizing the sample variogram matrices, without assuming any model for the variogram matrix; [Kunsch et al., 1997] proposed estimating the range parameters of a LCM using a non-linear regression method to fit the range parameters; [Lark Papritz, 2003] used simulated annealing to minimize a weighted sum of squares of differences between the empirical and the modelled variograms; [Pelletier et al., 2004] modified the Goulard and Voltz algorithm to make it more general and usable for generalized least-squares or any other least-squares estimation procedure, such as ordinary least-squares; [Zhang, 2007] developed an algorithm for the maximum-likelihood estimation for the purely spatial LCM and proved that the EM algorithm gives an iterative procedure based on quasi-closed-form formulas, at least in the isotopic case. Significant contributions concerning estimation and computational aspects of a LCM can be found in [Emery, 2010]. Unfortunately, for multivariate spatio-temporal data there does not exist software packages which perform in an unified way a) the structural analysis, b) a convenient graphical representation of the covariance (variogram) models fitted to the empirical ones and c) predictions. One of the solutions could be to extend the above mentioned techniques to space-time. Indeed, some routines already implemented in the GSLib\n\t\t\t\tsoftware [Deutsch Journel, 1998] or in the module gstat of R, can only be applied in a multivariate spatial context.
In the last years, a new GSLib routine, called COK2ST, can be used to make predictions in the domain under study, utilizing the spatio-temporal LCM, based on the generalized product-sum model [De Iaco et al., 2010]. This routine could be merged with the automatic procedure for fitting the spatio-temporal LCM using the product-sum variogram model, presented in [De Iaco et al., 2012 a], in order to provide a complete and helpful package for the analyst who needs to obtain predictions in a spatio-temporal multivariate context. This is certainly the first step for other developments and improvements in this field.
In the present case study, the environmental data set, with a multivariate spatio-temporal structure, involves
Posting map and box plots of PM10 daily concentrations classified by typology of survey stations
Fig. 3 shows the temporal profiles of the observed values. It is evident that low (high) values of Temperature and Wind Speed are associated with high (low) values of
Time series plots ofPM10, Temperature and Wind Speed daily averages
Note that, during the period of interest, the
Spatio-temporal modeling and prediction techniques have been applied in order to assess
1. estimating and modeling spatio-temporal correlation among the variables; in the fitting stage of a spatio-temporal LCM, the recent procedure [De Iaco et al., 2012b] based on the nearly simultaneous diagonalization of several sample matrix variograms, has been applied and the product-sum variogram model [De Iaco et al., 2001a] has been fitted to the basic components;
2. spatio-temporal cokriging based on the estimated model, in order to obtain prediction maps for
3. generating and comparing, for two sites of interest (one close to an industrial area and the other one close to a heavy traffic area), the probability distributions that
Modeling the spatio-temporal correlation among the variables under study by using the spatio-temporal LCM, requires first to check the adequacy of such model. In particular, the symmetry assumption has been checked by
a. exploring the differences between the pseudo cross-variograms of the standardized variables (standardized by subtracting the mean value and dividing this difference by the standard deviation),
b. using the methodology proposed by [Li et al., 2008].
As regards point a), the largest absolute difference has been equal to 0.135 and has been observed among the differences between the two pseudo cross-variograms concerning
By using the recent fitting procedure [De Iaco et al., 2012b] based on the nearly simultaneous diagonalization of several sample matrix variograms computed for a selection of spatio-temporal lags, the basic independent components and the scales of spatio-temporal variability have been simply identified. In particular, the spatio-temporal surfaces of the variables under study have been computed for 7 and 5 user-chosen spatial and temporal lags, respectively (Fig. 4).
Spatio-temporal variogram and cross-variogram surfaces ofPM10, Temperature and Wind Speed daily averages
Then, the 35 symmetric matrices of sample direct and cross-variograms have been nearly simultaneous diagonalization in order to detect the independent basic components. In this way, 3 scales of spatial and temporal variability have been identified: 10, 18 and 31.5 km in space, and 2.5, 3.5 and 6 days in time.
Thus, the following spatio-temporal LCM has been fitted to the observed data:
where the spatio-temporal variograms
The spatial and temporal marginal basic variogram models,
where
Finally, the matrices
Fig. 5 shows the spatio-temporal variograms and cross-variograms fitted to the surfaces of
In order to obtain the prediction maps for
It is important to highlight that the highest
Spatio-temporal variograms and cross-variograms fitted to variogram and cross-variogram surfaces ofPM10, Temperature and Wind Speed daily averages
Prediction maps of PM10 daily concentrations and risk maps at the threshold fixed by National Laws, for the 24th, 25th and 26th of November 2009
Prediction maps of PM10 daily concentrations and risk maps at the threshold fixed by National Laws, for the 27th, 28th and 29th of November 2009
After producing predicted maps of
Two different sites, one close to an industrialized area, located at Brindisi district, and the other one close to a heavy traffic area, located at Lecce district, have been considered in order to generate and compare the probability distributions that
Note that at the site close to a heavy traffic area, the probability that
Probability distributions that PM10 daily concentrations overcome several risk levels during the period 24-29 November 2009
In this paper, some significant theoretical and practical aspects for multivariate geostatistical analysis have been discussed and some critical issues concerning sampling, modeling and computational aspects, which should be faced, have been pointed out. The proposed multivariate geostatistical techniques have been applied to a case study pertaining particle pollution (
Further analysis regarding the integration of land use and possible sources of pollution through an appropriate geographical information system could be helpful to fully understand the dynamics of
The authors are grateful to the Editor and the reviewers, whose comments contribute to improve the present version of the paper. This research has been partially supported by the 5per1000 project (grant given by University of Salento in 2011).
The advent of endovascular treatment determined the crisis of cerebral aneurysm surgery. Endovascular therapy is less invasive and its progression is rapid. Industries’ interests and investments potentiate the technological endovascular advancements. Surgical treatment by clipping of intracranial aneurysms is durable and stable in time. There are advances in making surgical treatment safer and offering treatment to the more complicated cases, not amenable to endovascular therapy. Advances and investments in the surgery of cerebral aneurysms are less deafening in the last decade. However, some silent innovative advances are made over the years. Here we present innovations in cerebral aneurysm surgery.
Indocyanine green (ICG) is a near-infrared (NIR) fluorescent dye initially approved by the Food and Drug Administration (FDA) in 1956 for the evaluation of the cardiocirculatory and liver function. FDA extended the approval for ophthalmic angiography in 1975. Nowadays, ICG fluorescence is routinely used in ophthalmology for the visualization of the retinal microcirculation. A specific optical setup for near-infrared (NIR) light is necessary for the visualization of the ICG fluorescence. The development of an ICG angiography new system allowed further implementation for the intraoperative visualization of the tissue perfusion in general surgery.
Raabe et al. gave a substantial contribution by implementing ICG angiography use in vascular neurosurgery [1].
The absorption peak of ICG is 805 nm, while the emission peak is 835 nm. Within these two peaks, the endogenous tissue chromophore absorption is low.
NIR light penetrates the tissue from several millimeters to a few centimeters. ICG is injected intravenously, and it bounds in 1–2 s predominantly to globulins (α1-lipoproteins). In the absence of vascular permeability damage, ICG bound to globulins remains intravascular. ICG has a plasma half-life of 3–4 min. It is only excreted by the liver with no metabolization.
Raabe et al. used a laser-fluorescence imaging device (IC-View; Pulsion Medical Systems AG, Munich, Germany), consisting of a NIR laser light source (0.16 W, λ = 780 nm) and a NIR-sensitive digital camcorder.
ICG was injected intravenously in a bolus (standard dose of 25 mg dissolved in 5 ml of water). ICG fluorescence was induced by the NIR light emitted by the laser light source. The digital video camera with optical filtering recorded only the ICG-induced fluorescence signal.
The near-infrared filter was commercially available for surgical microscopes routinely used in neurosurgery, and ICG-VA could be applied in vascular neurosurgery.
For example, intraoperative ICG-VA could be performed using a surgical microscope (OPMI® PenteroTM, The Carl Zeiss Co., Oberkochen, Germany) equipped with a microscope-integrated near-infrared ICG-VA (Carl Zeiss, Infrared 800TM, Meditec, Germany). ICG is injected intravenously in a bolus of 25 mg dissolved in 5 ml of water, and the operating field is illuminated with near-infrared light. Real-time angiographic images are visualized on a video screen and recorded. The images can be replayed. Only the illuminated field is recorded. ICG can be injected multiple times during surgery; thus ICG-VA is repeatable.
After Raabe’s first report [1], ICG-VA gradually became routinely used in aneurysm surgery. It is used after aneurysm clipping to access whether the aneurysm occlusion is complete. Catheter angiography is the gold standard for cerebral aneurysm diagnosis [2] and for confirming aneurysm occlusion. However, intraoperative catheter angiography requires a programmed setup; is invasive, expensive, and time-consuming, and requires a well-trained staff. It is reserved to particular complicated cases, and it is not intraoperatively routinely used in the surgery of intracranial aneurysms. Furthermore, the time required for an intraoperative angiogram may be sufficient for the establishment of irreversible ischemia. Postoperative residual aneurysms after clipping are reported in a variable range from 4 to 19% of cases [3, 4, 5, 6, 7, 8, 9, 10]. ICG-VA is fast, easily used, and not invasive. It allows immediate assessment of the aneurysm occlusion after clipping, and permits whether necessary clip repositioning or further clip positioning. Although, ICG-VA is inferior to catheter angiography in assessing aneurysm complete occlusion, it is intraoperatively easily used in the cases where intraoperative catheter angiography would not be used routinely. It does not substitute postoperative neuroradiological control of aneurysm complete occlusion but allows to have higher postoperative occlusion rates. Postoperative catheter angiography remains the gold standard for assessment of aneurysm occlusion. Generally, intraoperative ICG-VA is used to access clipping after apparent complete occlusion under the microscope light. Della Puppa et al. showed that despite apparent complete occlusion under microscope visualization, ICG-VA revealed unexpected residual aneurysms in 9% [11]. Roessler et al. in a study of 295 aneurysms clipped with the use of ICG-VA showed an intraoperative clip modification rate of 15% based on ICG-VA data [12]. Thus, ICG-VA is a complementary tool that increases aneurysm occlusion rate, but it does not substitute postoperative digital subtraction angiography (DSA) for the detection of aneurysm remnants [13]. Intraoperative aneurysm puncture, or opening whenever possible, remains the most reliable intraoperative measure to assess complete occlusion.
Different factors can determine false-negative or false-positive ICG-VA findings. Arteriosclerosis and wall thickening at the clipping site influence false-negative ICG-VA findings [14]. Repeated ICG can determine false-positive results. Also, a small remnant detected in ICG can undergo spontaneous thrombosis and thus may not present a real residual.
ICG-VA despite improvement of aneurysm occlusions rate can also show deceptive false-negative results. Della Puppa et al. described a surgical simple maneuver to detect false-negative ICG-VA results after clipping of a cerebral aneurysm [15]. The squeezing maneuver consists of a gentle pinch with bipolar/Cushing bayonet forceps of the dome of a clipped aneurysm when ICG-VA documents its apparent exclusion.
The maneuver is performed during the same ICG injection to confirm the aneurysm exclusion. It is considered positive when, after an initial ICG-VA shows the aneurysm exclusion, a gentle pinch of the slack aneurysm dome with a bipolar or Cushing bayonet forceps under ICG-VA visualization causes the prompt dyeing of the sac, suggesting that the aneurysm is still filling up. The maneuver is considered negative when, after pinching of the clipped dome, the sac does not fill up. The puncture and opening of the sac can confirm whether a flow is still filling the aneurysm. The squeezing maneuver can depict ICG-VA false-negative results.
This permits to readjust the clip or position a second clip to completely exclude the aneurysm during the same procedure. Calcification/atheroma of the wall/neck was predictive of a positive maneuver (P = 0.001). This is consistent with Gekka et al. findings several years later, which report false-negative ICG-VA results in atherosclerosis and wall thickening at the clipping site [14].
ICG-VA can also show false-positive results, if misinterpreted. When ICG is injected before the final aneurysm clipping, the dye might be entrapped within the sac by the clip’s blades, which would obstacle the dye washout. ICG-VA would show the dye entrapped in the sac. An erroneous interpretation of the data would be to consider the aneurysm unsecured. Della Puppa et al. introduced the ICG entrapment sign as the detection under infrared light of ICG remnants sequestered in the dome [16]. ICG entrapment sign detects dye stasis, and not active filling. It is considered a sign of aneurysm occlusion in the setting of ICG injection prior to final clipping. This may happen if ICG is injected prior to clipping for visualization of perforating arteries near to the sac or detection of atheromas of the neck/dome. This happens more commonly after clip repositioning based on ICG indication.
The squeezing maneuver can detect a false-negative ICG-VA (an unsecured aneurysm despite apparent occlusion after ICG), whereas the ICG entrapment sign can detect a false-positive ICV-VA result (a secured aneurysm under infrared light, despite ICG-VA showing dye).
ICG-VA can be used before dural opening in vascular arteriovenous malformations or fistulas to optimize the exposure of the malformation, perform a safe dural opening, and identify dural vascular connections of the lesion [17]. The cases where transdural ICG can help in aneurysm surgery are very rare. These are the cases of distal cortical, generally distal middle cerebral artery (M4) aneurysms. In a case of M4 ruptured aneurysm, ICG-VA allowed transdural aneurysm visualization [18]. This is particularly helpful in an emergency setting, when neuronavigation is not available, to localize the aneurysm and avoid damage while opening the dura.
Other exceptional ICG-VA applications reported are the transoptic aneurysm visualization and occlusion confirmation in a case of an optic splitting aneurysm [19]. An ophthalmic artery aneurysm medially and superiorly projecting, suspicious for an under optic growth, underwent surgery. Initially the aneurysm was not visible. ICG-VA permitted the transoptic aneurysm visualization and after clipping final occlusion.
ICG-VA application was extended in other pathologies [20, 21, 22, 23, 24, 25, 26].
Vascular micro-Dopplers are used in cerebral aneurysm surgery to indicate the flow velocity. They are easy to use and give the surgeon an acoustic signal feedback. The flow velocity is used as a surrogate of the flow quantity. Flow velocity is not the most reliable indicator for flow. Flow quantity is the most reliable flow measure.
Till the 1990s, the intraoperative ultrasonic blood flow probes have been used to quantitatively measure flow only in cardiac, vascular, and transplant surgery. Charbel et al. in the University of Illinois at Chicago first reported in 1997 the implantation of the ultrasonic perivascular micro blood flow probes in the clipping of cerebral aneurysms [27, 28, 29, 30, 31].
The first transit time flowmeters were described in 1962 and 1964 [32, 33]. Limitations in estimating vessel diameter, vessel misalignment, and an unstable zero calibration prevented medical applications [34]. In 1978 Drost et al. presented the theoretical basis for a flowmeter based on the transit time technique [35, 36]. The transit time flowmeter was introduced in 1983 [36].
The transit time blood volume flowmeter gives a direct measurement of volume flow through the acoustic window of its implanted sensor, independent of flow profile. In contrast, earlier Doppler and transit time ultrasonic flowmeters sense blood velocity, which makes volume flow measurements critically dependent on vessel diameter [35, 36].
The transonic perivascular flow-measuring device includes an electronic flow detection unit with enhanced frequency resolution and volume flow-sensing perivascular probes (Transonic Medical Flowmeter; Transonic Systems, Inc., Ithaca, NY, USA). The perivascular flow probes are manufactured in 1.5, 2, and 3 mm diameter and can be used to measure the average flow volume (mL/min) instantaneously in cerebral vessels.
The flowmeter uses ultrasonic transit time principle to sense liquid volume flow in vessels independent of flow velocity, hematocrit, and turbulence.
The electronic flow-detecting unit is a line-powered flowmeter that automatically identifies the scaling factor and individual calibration factor of the flow probe connected to it. The flow sensors are connected to the flow-detecting unit via a flexible cable.
The ultrasonic transducers transmit ultrasound which helps to sense the volume of blood flowing through the blood vessel in which the sensor is applied.
The flow probe consists of a probe body which houses two ultrasonic transducers and a fixed acoustic reflector. The transducer is positioned around the blood vessel, and then the flow in that vessel is displayed in the digital form. The flowmeter derives an accurate measure of the “transit time,” which is the time the wave of ultrasound has taken to travel from one transducer to the other [28].
Practically, a portion of the vessel of interest is dissected from the arachnoid, and the probe is hooked around the vessel under saline irrigation.
The flow appears as a digital display on the detection unit and is registered as positive or negative dependent on the direction of flow in relation to the orientation of the probe. The flow is detected as the volume (mL/min), and the flow volume over time of recording diagram can be printed.
Quantitative blood flow measurement became essential in blood flow preservation to avoid postoperative ischemic complication in cerebral aneurysm surgery. Amin-Hanjani et al. proposed a baseline evaluation of blood flow in the vessels at risk of flow compromise after clipping (generally the efferent arteries, distal to the aneurysm) and a second flow evaluation of the same vessels after clipping [31]. A reduction of the flow greater than 25% of baseline was considered at risk for ischemic complications, and the clip was repositioned. The data was reproduced by other studies [11, 37, 38]. Flow measurement by microflow probe was also used in other cerebrovascular diseases [39, 40, 41, 42].
IONM routinely used in oncological surgery, over the years, has become essential also in vascular surgery to avoid ischemic complications. Monitoring includes bilateral upper and lower limb motor evoked potentials (MEPs) and somatosensory evoked potentials (SSEPs). Generally, in aneurysm surgery, MEPs are monitored by transcranial electric stimulation, rather than by direct cortical mapping, because the motor area is not routinely exposed during standard pterional approaches for anterior circulation aneurysms, and it is never exposed with mini invasive approaches. Direct electrical cortical stimulation might also increase the risk of epileptic seizures. For MEP monitoring transcranial electric stimulation is made by electrodes positioned at C1 and C2 according to the 10–20 International System (IS). These electrodes continuously stimulate the motor area during surgery by train stimuli (the pulse trains used are different according to the users; there are reported 4–8 pulse trains) [37, 43]. MEPs are recorded by subcutaneous needle electrodes from the abductor pollicis brevis and abductor hallucis muscles. The baseline MEPs are recorded at the beginning of the surgery, and it is found a compromise with the surgeon between continuous MEP recording and the movements caused by the stimuli tolerated by the surgeon for the dissection and clipping procedure. To avoid false-negative results on MEPs, the stimulus applied for MEP acquisition should be minimal not to activate the distal motor pathways [44]. To avoid false-positive results, brain shift due to the cerebrospinal loss after cisternal opening should be considered.
The reduction of 50% in MEP amplitude or MEP disappearance is considered as an alarm criterion. The surgeon changes the strategy to tempt to recover the MEPs. Temporary clipping is interrupted, or the definitive clip is released. The arterial blood pressure may be increased; local irrigation with saline and papaverine may be tempted. The surgical procedure is temporarily stopped, whenever it is possible (the aneurysm is not intraoperatively ruptured, or it is not opened by the surgeon) to permit the MEPs to recover. The MEP deterioration can be reversible, when the MEPs return to more than 50% of baseline amplitude.
Another methodology of MEP monitoring is by direct cortical stimulation.
Upper limb SSEPs are recorded from C3 and C4 by electrically stimulating the contralateral median nerve at the wrist. Lower limbs are recorded by electric stimulation of the contralateral tibial nerve at the medial malleolus. Recordings from Cz′ and Fz are made according to the 10–20 International System [37].
A total intravenous anesthesia (TIVA) is used. Neuromuscular blockers are not used during surgery, unless extremely necessary. Muscle relaxants are used only for endotracheal intubation.
An early experimental study by Branston et al. in 1974 showed that there is a failure of neuronal function in the cortex when the local blood flow falls below about 16 ml/100 g/min. This failure becomes manifest as a progressive reduction in the amplitude of the surface recorded SSEP, and this results in the abolition of the SSEPs if the flow is below about 12 ml/l00 g/min. There is a close relationship between reduced cerebral blood flow (CBF: 12–16 ml/100 g/min) and a reduction in SSEP amplitude or SSEP abolition [45].
MEP monitoring has higher diagnostic accuracy than SSEPs in predicting the occurrence of a postoperative neurological deficit [46].
Li et al. analyzed 92 patients operated for cerebral aneurysms that showed intraoperative MEP deterioration. They found that a MEP deterioration duration greater than or equal to 13 min in intracranial aneurysm surgery was significantly associated with postoperative motor deficits [43].
The principles of neuro-oncological monitoring are being gradually transferred to cerebral aneurysm surgery. In neuro-oncological surgery, the best way to monitor the neurological and neurophysiological function is by awake surgery. Awake surgery of cerebral aneurysms is controversial for the potential consequences of intraoperative aneurysm rupture. Intraoperative aneurysm rupture does not impact the clinical outcome. This would not be true if the clipping was performed in awake surgery. However, in selected case of clipping of the aneurysms of the dominant hemisphere, the monitoring of the language function and thus awake surgery would be useful. Abdulrauf et al. reported awake surgery in 30 unruptured cerebral aneurysms [47]. In three patients the strategy affected the outcome, since the removal of the temporary clipping determined the reversal of the clinical neurological and neurophysiological changes. One patient developed a neurological damage depending on the clipping, but that could not be reversed by the clip repositioning. Three patients developed hemiparesis without changes in MEPs; thus they were false negatives. The important was the visual testing after final clipping in four patients with internal carotid artery ophthalmic segment aneurysms, and one of these patients required repositioning of the clip. Three patients underwent an intraoperative vessel occlusion test, since the vessel occlusion was part of the permanent treatment of the aneurysm.
The endoscope is mainly used as assistance during microsurgical clipping of intracranial aneurysms. The endoscopic-assisted microsurgery has been promoted by the father of the keyhole approaches Perneczky and by Fries [48]. The endoscope can be used for inspection before clipping; also clipping under endoscopic view and post clipping evaluation to observe the perforator integrity can be performed [49]. The endoscope allows the visualization of the blind spots to the microscope, allows thus the vision around corners, and enhances the visualization. It can potentially ameliorate the quality of treatment. The microscope enables vision in a straight line, while the endoscope enables the visualization of angles. The endoscope can be complementary to the microscope. However, the surgeon must be familiar with the endoscope use, not to cause iatrogenic damage [50].
Endoscopic ICG-VA is an important development that combines the benefits of the vision behind the corners of endoscopy and the vessel visualization of ICG-VA [51]. The combination of both is particularly important for the visualization of the perforating arteries hidden in blind spots.
Purely endoscopic approach to cerebral aneurysms is a potential method in its very beginning. There are case reports of endonasal clipping of aneurysms and of endoscopic transcranial pure approaches.
Radovanovic reports a cadaveric study of a purely endoscopic transpterional port craniotomy to access lesions involving the cavernous sinus and the anterolateral skull base [52]. In the illustration videos, the author includes clipping of a middle cerebral artery aneurysm through this approach. There are also strictly selected case reports or small series of endoscopic endonasal clipping of anterior circulation aneurysms [53]. Other case reports regard the pure endoscopic endonasal transclival approach for clipping of posterior circulation aneurysms [54]. These minimally invasive approaches constitute very limited experiences, and they need very deep expertise; otherwise they become dangerous. Safety must never be sacrificed for achieving minimal invasiveness.
Exoscopes are projected to combine the benefits of neurosurgical microscopes and endoscopes. With the exoscope, the surgeon looks at the monitor while operating, and the entire surgical team has the same view as the primary surgeon. The 3D 4K-HD exoscopes have favorable ergonomics to visualize angles maintaining the surgeon’s comfort, maneuverability, and immersive visual experience. The assistant positioning relative to the surgeon can be problematic during surgery. ORBEYE (Olympus, Tokyo, Japan) exoscope has been used in aneurysm surgery with reported excellent visualization of the arterial tree [55], but with a subjective disadvantage in the visualization of bleeding tissue particularly in the muscle or white matter [56]. This new technology is proposed with advantages and limits, and time will tell its exact role in the future.
The perforating arteries are small twigs of the main cerebral arteries that irrorate the paramedian region of the brainstem, the diencephalon, the basal ganglia, and the internal capsule [57]. Commonly, the perforators are small vessels of less than 1 mm in diameter, except for some lenticulostriate arteries (LSAs) and Heubner arteries larger than 1 mm. They may be multiple and sometimes anastomose. Although the same territory can be supplied by multiple perforators, the consequence of the occlusion of a perforator is unpredictable and more often than not results in a neurological deficit. Perforator infarction was shown as an independent risk factor of poor functional outcome in a series of anterior communicating aneurysms [58]. In the surgery of cerebral aneurysms, it is essential to preserve the perforating arteries. While the flow preservation of a larger artery is easier and eventually the flow can be replaced by revascularization, the perforator damage is more feared and less predictable. The monitoring of the perforators is fundamental. ICG-VA has the advantage of being able to visualize the perforating arteries. As a speculation, it is assumed that flowmetry and SSEPs account for the cortical gray matter function while the MEPs for the subcortical white matter function. Thus, MEPs can be used to evaluate the perforating artery function, and ICG-VA allows their visualization along with endoscopes, exoscopes, and endoscopic ICG-VA that permit the vision behind blind points.
None of the tools described is superior to the others, and their role in improving clinical results in aneurysm surgery is complementary. Della Puppa et al. have described the complementary role of enhanced visualization with ICG-VA and maneuvers and signs to better interpret the data, along with monitoring of function (IONM) and perfusion (flowmetry) [37].
Quantitative magnetic resonance angiography (QMRA) is an MRA that permits blood flow quantification of the major cerebral vessels [59, 60]. QMRA is implemented with a commercially available software called Noninvasive Optimal Vessel Analysis (NOVA) (VasSol, Inc., Chicago, Illinois). MRA creates the cerebral vascular tree. A double-oblique scan is performed using a gated two-dimensional phase-contrast MRA imaged perpendicular to the vessel of interest axis. The software generates a flow report with the mean volumetric flow rate (mL/min) of the vessels of interest. QMRA data have been validated in vivo and have shown proportional differences, around 10% to direct transit time flow measurements [60]. QMRA is reported to be used preoperatively to evaluate the flow of the major vessels in patients with cerebral aneurysm that would require a bypass, when vessel sacrifice is needed to treat the aneurysm [42]. Intraoperatively a flow-based algorithm can be used to determine the flow needed to replace the flow sacrificed. Transit time flow measurements are used for intraoperative measurements. These measurements indicate which is the more appropriate donor graft to be used for the anastomosis. This methodology has shown that superficial temporal artery is often sufficient to replace flow, which renders the surgery easier than using vein or radial artery grafts. QMRA is used in follow-up to detect the bypass flow. The hemisphere flows are calculated and are maintained over time. Details of the algorithm used for calculation of the flow needed to be replaced and the donor flow potential can be found in the paper by Rustemi et al. [42].
Lawton has rendered popular the intracranial-intracranial (IC-IC) bypass for flow replacement in complex aneurysms [61]. This type of bypass, although more elegant, has several pitfalls and requires very experienced surgeons. IC-IC bypass puts at dangers both the donor and receiving territories. The anastomosis is deep and more difficult to be performed. However, in selected cases and experienced hands, it represents an advancement.
The ELANA has been developed for intracranial bypass without the need for temporary recipient occlusion. A sutureless variant of the ELANA—the SELANA slide—showed a preclinical success and clinical application started. Unfortunately, it was not shown suitable for clinical applications [62].
Many technological innovations now assist the surgeon in the treatment of cerebral aneurysms. Also, different clips are used over the years. However, the clip principle has not changed. A change in occlusion strategy, based on principles other than clipping, might be desirable for the future. There is space for new ideas and new principles. The basic research studies for cerebral aneurysms are focused on wall analysis and flow stimulations. If advances will become more solid, in the future, cerebral aneurysms will not need neurosurgeons or neuroendovascular radiologists.
Surgery of cerebral aneurysms has advanced over the years. Innovations are rapid in this technological era. Many innovations are now routinely used in the clinical practice; others will be soon implemented. The technological innovations currently used in the surgery of cerebral aneurysms are summarized in this chapter. The comprehension of the biology and pathology might in the future render the aneurysm a medical disease.
Nothing to declare.
IONM | intraoperative neurophysiological monitoring |
ICG-VA | indocyanine green video angiography |
ICG | indocyanine green |
NIR | near-infrared |
FDA | Food and Drug Administration |
M4 | distal middle cerebral artery |
MEPs | motor evoked potentials |
SSEPs | somatosensory evoked potentials |
IS | International System |
TIVA | total intravenous anesthesia |
CBF | cerebral blood flow |
LSAs | lenticulostriate arteries |
QMRA | quantitative magnetic resonance angiography |
NOVA | Noninvasive Optimal Vessel Analysis |
IC-IC | intracranial-intracranial |
ELANA | excimer laser-assisted nonocclusive anastomosis |
SELANA | sutureless ELANA |
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