\\n\\n
More than half of the publishers listed alongside IntechOpen (18 out of 30) are Social Science and Humanities publishers. IntechOpen is an exception to this as a leader in not only Open Access content but Open Access content across all scientific disciplines, including Physical Sciences, Engineering and Technology, Health Sciences, Life Science, and Social Sciences and Humanities.
\\n\\nOur breakdown of titles published demonstrates this with 47% PET, 31% HS, 18% LS, and 4% SSH books published.
\\n\\n“Even though ItechOpen has shown the potential of sci-tech books using an OA approach,” other publishers “have shown little interest in OA books.”
\\n\\nAdditionally, each book published by IntechOpen contains original content and research findings.
\\n\\nWe are honored to be among such prestigious publishers and we hope to continue to spearhead that growth in our quest to promote Open Access as a true pioneer in OA book publishing.
\\n\\n\\n\\n
\\n"}]',published:!0,mainMedia:null},components:[{type:"htmlEditorComponent",content:'
Simba Information has released its Open Access Book Publishing 2020 - 2024 report and has again identified IntechOpen as the world’s largest Open Access book publisher by title count.
\n\nSimba Information is a leading provider for market intelligence and forecasts in the media and publishing industry. The report, published every year, provides an overview and financial outlook for the global professional e-book publishing market.
\n\nIntechOpen, De Gruyter, and Frontiers are the largest OA book publishers by title count, with IntechOpen coming in at first place with 5,101 OA books published, a good 1,782 titles ahead of the nearest competitor.
\n\nSince the first Open Access Book Publishing report published in 2016, IntechOpen has held the top stop each year.
\n\n\n\nMore than half of the publishers listed alongside IntechOpen (18 out of 30) are Social Science and Humanities publishers. IntechOpen is an exception to this as a leader in not only Open Access content but Open Access content across all scientific disciplines, including Physical Sciences, Engineering and Technology, Health Sciences, Life Science, and Social Sciences and Humanities.
\n\nOur breakdown of titles published demonstrates this with 47% PET, 31% HS, 18% LS, and 4% SSH books published.
\n\n“Even though ItechOpen has shown the potential of sci-tech books using an OA approach,” other publishers “have shown little interest in OA books.”
\n\nAdditionally, each book published by IntechOpen contains original content and research findings.
\n\nWe are honored to be among such prestigious publishers and we hope to continue to spearhead that growth in our quest to promote Open Access as a true pioneer in OA book publishing.
\n\n\n\n
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A wide variety of natural objects can be described mathematically using fractal geometry as, for example, contours of clouds, coastlines, turbulence in fluids, fracture surfaces, or rugged surfaces in contact, rocks, and so on. None of them is a real fractal, fractal characteristics disappear if an object is viewed at a scale sufficiently small. However, for a wide range of scales the natural objects look very much like fractals, in which case they can be considered fractal. There are no true fractals in nature and there are no real straight lines or circles too. Clearly, fractal models are better approximations of real objects that are straight lines or circles. If the classical Euclidean geometry is considered as a first approximation to irregular lines, planes and volumes, apparently flat on natural objects the fractal geometry is a more rigorous level of approximation. Fractal geometry provides a new scientific way of thinking about natural phenomena. According to Mandelbrot [1], a fractal is a set whose fractional dimension (Hausdorff-Besicovitch dimension) is strictly greater than its topological dimension (Euclidean dimension).
In the phenomenon of fracture, by monotonic loading test or impact on a piece of metal, ceramic, or polymer, as the chemical bonds between the atoms of the material are broken, it produces two complementary fracture surfaces. Due to the irregular crystalline arrangement of these materials the fracture surfaces can also be irregular, i.e., rough and difficult geometrical description. The roughness that they have is directly related to the material microstructure that are formed. Thus, the various microstructural features of a material (metal, ceramic, or polymer) which may be, particles, inclusions, precipitates, etc. affect the topography of the fracture surface, since the different types of defects present in a material can act as stress concentrators and influence the formation of fracture surface. These various microstructural defects interact with the crack tip, while it moves within the material, forming a totally irregular relief as chemical bonds are broken, allowing the microstructure to be separated from grains (transgranular and intergranular fracture) and microvoids are joining (coalescence of microvoids, etc..) until the fracture surfaces depart. Moreover, the characteristics of macrostructures such as the size and shape of the sample and notch from which the fracture is initiated, also influence the formation of the fracture surface, due to the type of test and the stress field applied to the specimen.
After the above considerations, one can say with certainty that the information in the fracture process are partly recorded in the "story" that describes the crack, as it walks inside the material [2]. The remainder of this information is lost to the external environment in a form of dissipated energy such as sound, heat, radiation, etc. [30, 31]. The remaining part of the information is undoubtedly related to the relief of the fracture surface that somehow describes the difficulty that the crack found to grow [2]. With this, you can analyze the fracture phenomenon through the relief described by the fracture surface and try to relate it to the magnitudes of fracture mechanics [3, 4, 5, 6, 7, 8, 9 - 11, 12, 13]. This was the basic idea that brought about the development of the topographic study of the fracture surface called fractography.
In fractography anterior the fractal theory the description of geometric structures found on a fracture surface was limited to regular polyhedra-connected to each other and randomly distributed throughout fracture surface, as a way of describing the topography of the irregular surface. Moreover, the study fractographic hitherto used only techniques and statistical analysis profilometric relief without considering the geometric auto-correlation of surfaces associated with the fractal exponents that characterize the roughness of the fracture surface.
The basic concepts of fractal theory developed by Mandelbrot [1] and other scientists, have been used in the description of irregular structures, such as fracture surfaces and crack [14 ], in order to relate the geometrical description of these objects with the materials properties [15 ].
The fractal theory, from the viewpoint of physical, involves the study of irregular structures which have the property of invariance by scale transformation, this property in which the parts of a structure are similar to the whole in successive ranges of view (magnification or reduction) in all directions or at least one direction (self-similarity or self-affinity, respectively) [36]. The nature of these intriguing properties in existing structures, which extend in several scales of magnification is the subject of much research in several phenomena in nature and in materials science [16, 17 and others]. Thus, the fractal theory has many contexts, both in physics and in mathematics such as chaos theory [18], the study of phase transitions and critical phenomena [19, 20, 21], study of particle agglomeration [22], etc.. The context that is more directly related to Fracture Mechanics, because of the physical nature of the process is with respect to fractal growth [23, 24, 25, 26]. In this subarea are studied the growth mechanisms of structures that arise in cases of instability, and dissipation of energy, such as crack [27, 28] and branching patterns [29]. In this sense, is to be sought to approach the problem of propagation of cracks.
The fractal theory becomes increasingly present in the description of phenomena that have a measurable disorder, called deterministic chaos [18, 27, 28]. The phenomenon of fracture and crack propagation, while being statistically shows that some rules or laws are obeyed, and every day become more clear or obvious, by understanding the properties of fractals [27, 28].
In this part will be presented the development of basic concepts of fractal geometry, analogous to Euclidean geometry for the basic elements such as points, lines, surfaces and fractals volumes. It will be introduce the measurement fractal theory as a generalization of Euclidean measure geometric theory. It will be also describe what are the main mathematical conditions to obtain a measure with fractal precision.
It is possible to draw a parallel between Euclidean and fractal geometry showing some examples of self-similar fractals projected onto Euclidean dimensions and some self-affine fractals. For, just as in Euclidean geometry, one has the elements of geometric construction, in the fractal geometry. In the fractal geometry one can find similar objects to these Euclidean elements. The different types of fractals that exist are outlined in Figure 1 to Figure 4.
An example of a fractal immersed in Euclidean dimension
Fractal immersed in the one-dimensional space where D≅0,631.
This fractal has dimension
For a fractal immersed in a Euclidean dimension
Fractal immersed in dimension d = 2. rugged fractal line.
For a fractal immersed in a Euclidean dimension,
Irregular or rugged surface that has a fractal scaling with dimension D between 2≤D≤3.
Comparison between Euclidean and fractal geometry. D,d and Df represents the topological, Euclidean and fractal dimensions, of a point, line segment, flat surface, and a cube, respectively
Making a parallel comparison of different situations that has been previously described, one has (Figure 4)
An object has a fractal dimension,
where
Based on this definition of fractal dimension it can be calculates doing:
taking the logarithm one has
From the geometrical viewpoint, a fractal must be immersed into a integer Euclidean dimension,
For a fractal the space fraction filled with points is also invariant by scale transformation, i.e.:
Thus,
where
Therefore, the fractal dimension can be calculated from the fllowing equation:
If it is interesting to scale the holes of a fractal object (the complement of a fractal), it is observed that the fractal dimension of this new additional dimension corresponds to the Euclidean space in which it is immersed less the fractal dimension of the original.
Now will be described how to process a general geometric measure whose dimension is any. Similarly to the case of Euclidean measure the measurement process is generalized, using the concept of Hausdorff-Besicovitch dimension as follows.
Suppose a geometric object is recovered by
Choosing from all the sets
The smallest possible value of the summation in (8) is calculated to obtain the adjustment with best precision of the measurement performed. Finally taking the limit of
The interpretation for the function
For,
Measuring, MD(δ) of an area A with a dimension, D=2 made with different measure units uD for D=1, 2, 3.
Therefore, the function,
That is, the function
Therefore, for a generalized measurement there is a generalized dimension which the measurement unit converge to the determined value,
where
Again the value of a fractal measure can be obtain as the result of a series.
One may label each of the stages of construction of the function
the first is the measure itself. Because it is actually the step that evaluates the extension of the set, summing the geometrical size of the recover units. Thus, the extension of the set is being overestimated, because it is always less or equal than tthe size of its coverage.
The next step is the optimization to select the arrangement of units which provide the smallest value measured previously, i.e. the value which best approximates the real extension of the assembly.
The last step is the limit. Repeat the previous steps with smaller and smaller units to take into account all the details, however small, the structure of the set.
As the value of the generalized dimension is defined as a critical function,
In this part we will define the dimension-Hausdorf Besicovicth and a fractal object itself. The basic properties of objects with "anomalous" dimensions (different from Euclidean) were observed and investigated at the beginning of this century, mainly by Hausdorff and Besicovitch [32,34]. The importance of fractals to physics and many other fields of knowledge has been pointed out by Mandelbrot [1]. He demonstrated the richness of fractal geometry, and also important results presented in his books on the subject [1, 35, 36].
The geometric sequence, S is given by:
represented in Euclidean space, is a fractal when the measure of its geometric extension, given by the series,
where:
Therefore, fractal is any object that has a non-integer dimension that exceeds the topological dimension (
According to the previous section, it is said that an object is fractal, when the respective magnitudes characterizing features as perimeter, area or volume, are homogeneous functions with non-integer. In this case, the invariance property by scaling transformation (self-similar or self-affinity) is due to a scale transformation of at least one of these functions.
The fractal concept is closely associated to the concept of Hausdorff-Besicovitch dimension, so that one of the first definitions of fractal created by Mandelbrot [36] was:
“Fractal by definition is a set to which the Haussdorf-Besicovitch dimension exceeds strictly the topological dimension".
One can therefore say that fractals are geometrical objects that have structures in all scales of magnification, commonly with some similarity between them. They are objects whose usual definition of Euclidean dimension is incomplete, requiring a more suitable to their context as they have just seen. This is exactly the Hausdorff-Besicovitch dimension.
A dimension object,
One of the most fascinating aspects of the fractals is the extremely rich variety of possible realizations of such geometric objects. This fact gives rise to the question of classification, and the book of Mandelbrot [1] and in the following publications many types of fractal structures have been described. Below some important classes will be discussed with some emphasis on their relevance to the phenomenon of growth.
Fractals are classified, or are divided into: mathematical and physical (or natural) fractals and uniform and non-uniform fractals. Mathematical fractals are those whose scaling relationship is exact, i.e., they are generated by exact iteration and purely geometrical rules and does not have cutoff scaling limits, not upper nor lower, because they are generated by rules with infinity interactions (Figure 6a) without taking into account none phenomenology itself, as shown in Figure 6a. Some fractals appear in a special way in the phase space of dynamical systems that are close to situations of chaotic motion according to the Theory of Nonlinear Dynamical Systems and Chaos Theory. This approach will not be made here, because it is another matter that is outside the scope of this chapter.
Example of branching fractals, showing the structural elements, or elementary geometrical units, of two fractals. a) A self-similar mathematical fractal. b) A statistically self-similar physical fractal.
Real or physical fractals (also called natural fractals) are those statistical fracals, where not only the scale but all of fractal parameters can vary randomly. Therefore, their scaling relationship is approximated or statistical, i. e., they are observed in the statistical average made throughout the fractal, since a lower cutoff scale,
where
It is true that the physical or real fractals can be deterministic or random. In random or statistical fractal the properties of self-similarity changes statistically from region to region of the fractal. The dimension cannot be unique, but characterized by a mean value, similarly to the analysis of mathematical fractals. The Figure 6b shows aspects of a statistically self-similar fractal whose appearance varies from branch to branch giving us the impression that each part is similar to the whole.
The mathematical fractals (or exact) and physical (or statistical), in turn, can be subdivided into uniform and nonuniform fractal.
Uniforms fractals are those that grow uniformly with a well behaved unique scale and constant factor,
Non-uniform fractals are those that grow with scale factors
Thus, the fractal theory can be studied under three fundamental aspects of its origin:
From the geometric patterns with self-similar features in different objects found in nature.
From the nonlinear dynamics theory in the phase space of complex systems.
From the geometric interpretation of the theory of critical exponents of statistical mechanics.
In this section one intends to describe the main methods for measuring the fractal dimension of a structure, such as: the compass method, the Box-Counting method, the Sand-Box Method, etc.
It will be described, from now, how to obtain a measure of length, area or fractal volume. In fractal analysis of an object or structure different types of fractal dimension are obtained, all related to the type of phenomenon that has fractality and the measurement method used in obtaining the fractal measurement. These fractal dimensions can be defined as follows.
A fractal dimension
In the literature there is controversy concerning the relationship between different fractal dimensions and roughness exponents. The term "fractal dimension" is used generically to refer to different fractional dimensions found in different phenomenologies, which results in formation of geometric patterns or energy dissipation, which are commonly called fractals [1]. Among these patterns is the growth of aggregates by diffusion (DLA - Diffusion Limited Aggregation), the film growth by ballistic deposition (BD), the fracture surfaces (SF), etc. The fractal dimensions found in these phenomena are certainly not the same and depend on both the phenomenology studied as the fractal characterization method used. Therefore, to characterize such phenomena using fractal geometry, a distinction between the different dimensions found is necessary.
Among the various fractal dimensions one can emphasize the Hausdorff-Besicovitch dimension,
Therefore, the phenomenological equation of the fracture phenomenon can also, in theory, provide a relationship between fractal dimension and roughness exponent of a fracture surface, as happens to other phenomenologies. In this study, there was obtained a fractal model for a fracture surface, as a generalization of the box-counting method. Thus, will be discussed the relationship between the local and global box dimension and the roughness dimension, which are involved in the characterization of a fracture surface, and any other dimension necessary to describe a fractal fracture surface.
The divider dimension
Compass method applied to a rugged line.
For a fractal rough line the divider dimension can be defined as:
where
Compass method applied on a line noise or a rough self-affine fractal.
Several methods for determining the fractal dimension based on the compass method, among them stand out the following methods: the Coastlines Richardson Method, the Slit Island Method, etc.
There are basically two ways to recover an object with boxes for fractal dimension measuring. In the first method, boxes of different sizes extending from a minimum size
The Box-Counting method, comes from the theory of critical phenomena in statistical mechanics. In statistical mechanics there is an analogous mathematical method to describing phenomena which have self-similar properties, permitting scale transformations without loss of generality in the description of physical information of the phenomenon ranging from quantities such as volume up to energy. However, in the case described here, the Box-Counting method is performed filling the space occupied by a fractal object with boxes of arbitrary size
Plotting the data in a
In the Box-Counting method (Figure 9), a grid that recover the object is divided into
Fragment of a crack on a testing sample showing the variation of measurement of the crack length L with the measuring scale, εk=δk/L0 for a partition, δk=variável and Lk=L0 (fixed), with sectioning done for counting by one-dimensional Box-Counting scaling method.
Therefore, the number
In the Figure 9 is illustrated the use of this method in a fractal object. Are present different grids, or meshes, constructed to recover the entire structure, whose fractal dimension one wants to know. The grids are drawn from an original square, involving the whole space occupied by the structure. At each stage of refinement of the grid
From the viewpoint of experimental measurement, one can consider using different methods of viewing the crack to obtain the fractal dimension, such as optical microscopy, electron microscopy, atomic force microscope, etc.., Which naturally have different rules
The fractal dimension is usually calculated using the Box-Counting shown in Figure 9, i.e. by varying the size of the measuring ruler
The description of a crack according to the Box-Counting method follows the idea shown in Figure 9, which results in:
The same result can be obtained using the Box-Sand method, as shown in Figure 10.
The Sand-box method consists in the same way as the Box-Counting method, to count the number of boxes,
Fragment of a crack on test specimen showing the variation of measurement of the crack length L with the measuring scale, εk=δk/L0 for a partition Lk=variável, and δk=l0 (fixed), with sectioning done for counting by one-dimensional Sand-Box scaling method.
In the Sand-Box method (Figure10), the figure is recovered with boxes of different sizes
To define the box dimension,
countting the number of boxes
At this point, there are two ways to obtain the actual value of the measure, or taking the limit when
In the case of numerical estimation, one can not solve the limit indicated in the equation (21). Then,
Self-affine fractals requiring different variations in scale length for different directions. Therefore, one can use the Box-Counting method with some care being taken, in the sense that the box dimension
for a global measurement
where
For measurement, generally, for any self-affine fractal structure the local fractal dimension is related to the Hurst exponent,
At this point, one observes that for a profile the relationship
Some textbooks on the subject show an example of calculation of local and global fractal dimension of self-affine fractals, obtained by a specific algorithm [18, 22, 23, 26, 38,39].
In crossing the limit of fractal dimension local
The mathematical definition of generalized dimension of Haussdorff-Besicovitch need a method that can measure it properly to the fractal phenomenon under study. Some authors [23, 40, 44, 45] have discussed the possibility of using the Box-Counting method as one of the graphical methods which obtains a box dimension
In this sense the box dimension,
where
The analytical calculation of the Hausdorff dimension is only possible in some cases and it is difficult to implement by computation. In numerical calculation, is used another more appropriate definition, called box dimension,
All the definitions related to fractal exponents that are shown here, and all numerical evaluation of these, always calculates the inclination of some amount
The two definitions of, Hausdorff-Besicovitch Dimension,
by analogy with equation (13), i.e. approximating to the geometric extension of the object by the number of boxes (of the same size) necessary to recover it. But, since the definition of the box dimension there is no optimization step, and its value is directly dependent on
Considering from (28) that:
and that
Therefore, dividing (29) by (30) has:
taking
From as early as (31)
Substituting (33) in (28) has:
This equation is analogous to the fundamental Richardson relationship for a fractal length.
The two main problematics of mathematical description of Fracture Mechanics are based on the following aspects: the surface roughness generated in the process and the field stress/strain applied to the specimen. This section deals with the fractal mathematical description of the first aspect, i.e., the roughness of cracks on Fracture Mechanics, using fractal geometry to model its irregular profile. In it will be shown basic mathematical assumptions to model and describe the geometric structures of irregular cracks and generic fracture surfaces using the fractal geometry. Subsequently, one presents also the proposal for a self-affine fractal model for rugged surfaces of fracture. The model was derived from a generalization of Voss [48] ([1] -) equation and the model of Morel [49] for fractal self-affine fracture surfaces. A general analytical expression for a rugged crack length as a function of the projected length and fractal dimension is obtained. It is also derived the expression of roughness, which can be directly inserted in the analytical context of Classical Fracture Mechanics.
The objectives of this section are: (i) based geometrical concepts, extracted from the fractal theory and apply them to the CFM in order to (ii) construct a precise language for its mathematical description of the CFM, into the new vision the fractal theory. (iii) eliminate some of the questions that arise when using the fractal scaling in the formulation of physical quantities that depend on the rough area of fracture, instead of the projected area, in the manner which is commonly used in fracture mechanics. (iv) another objective is to study the way which the fractal concept can enrich and clarify various aspects of fracture mechanics. For this will be done initially in this section, a brief review of the major advances obtained by the fractal theory, in the understanding of the fractography and in the formation of fracture surfaces and their properties. Then it will be done, also, a mathematical description of our approach, aiming to unify and clarify aspects still disconnected from the classical theory and modern vision, provided by fractal geometry. This will make it possible for the reader to understand what were the major conceptual changes introduced in this work, as well as the point from which the models proposed progressed unfolding in new concepts, new equations and new interpretations of the phenomenon.
In this section one intends to do a brief history of the fractography development as a fractal characterization methodology of a fractal fracture surface.
The technique used for geometric analysis of the fracture surface is called fractography. Until recently it was based only on profilometric study and statistical analysis of irregular surfaces [50]. Over the years, after repeated observations of these surfaces at various magnifications, was also revealed a variety of self-similar structures that lie between the micro and macro-structural level, characteristic of the type of fracture under observation. Since 1950 it is known that certain structures observed in fracture surfaces by microscopy, showed the phenomenon of invariance by magnification. Such structures recently started to be described in a systematic way by means of fractal geometry [51, 25]. This new approach allows the description of patterns that at first sight seem irregular, but keep an invariance by scale transformation (self-similarity or self-affinity). This means that some facts concerning the fracture have the same character independently of the magnification scale, i.e. the phenomenology that give rise to these structures is the same in different observation scales.
The Euclidean scaling of physical quantities is a common occurrence in many physical theories, but when it comes to fractality appears the possibility to describe irregular structures. The fracture for each type of material has a behavior that depends on their physical, chemical, structural, etc. properties. Looking at the topography and the different structures and geometrical patterns formed on the fracture surfaces of various materials, it is impossible to find a single pattern that can describe all these surfaces (Figure 11), since the fractal behavior of the fracture depends on the type of material [52]. However, the fracture surfaces obtained under the same mechanical testing conditions and for the same type of material, retains geometric aspects similar of its relief [53] (see Figure 12).
This similarity demonstrates that exist similar conditions in the fracture process for the same material, although also exist statistically changinga from piece to piece, constructed of the same material and under the same conditions [54; 55]. Based on this observation was born the idea to relate the surface roughness of the fracture with the mechanical properties of materials [50].
Various aspects of the fracture surface for different materials: (a) Metallic B2CT2 sample, (b) Polymeric, sample PU1.0, with details of the microvoids formation during stable crack propagation, (c) Ceramic [56].
Let us now identify the fractal aspects of fracture surfaces of materials in general, to be obtained an experimental basis for the fractal modeling of a generic fracture surface. The description of irregular patterns and structures, is not a trivial task. Every description is related to the identification of facts, aspects and features that may be included in a class of phenomena or structure previously established. Likewise, the mathematical description of the fracture surface must also have criteria for identifying the geometric aspects, in order to identify the irregular patterns and structures which may be subject to classification. The criteria, used until recently were provided by the fractográfico study through statistical analysis of quantities such as average grain size, roughness, etc. From geometrical view point this description of the irregular fracture surface, was based, until recently, the foundations of Euclidean geometry. However, this procedure made this description a task too complicated. With the advent of fractal geometry, it became possible to approach the problem analytically, and in more authentic way.
Fracture surfaces of different parts mades with the same material, a) Lot A9 b) Lot A1 [56\n\t\t\t\t\t\t\t\t1999].
Inside the fractography, fractal description of rugged surfaces, has emerged as a powerful tool able to describe the fracture patterns found in brittle and ductile materials. With this new characterization has become possible to complement the vision of the fracture phenomenon, summarizing the main geometric information left on the fracture surface in just a number, "
A fracture surface is a record of information left by the fracture process. But the Classical Fracture Mechanics (CFM) was developed idealizing a regular fracture surface as being smooth and flat. Thus the mathematical foundations of CFM consider an energy equivalence between the rough (actual) and projected (idealized) fracture surfaces [57]. Besides the mathematical complexity, part of this foundation is associated with the difficulties of an accurate measure of the actual area of fracture. In fact, the geometry of the crack surfaces is usually rough and can not be described in a mathematically simple by Euclidean geometry [52]. Although there are several methods to quantify the fracture area, the results are dependent on the measure ruler size used [56]. Since the last century all the existing methods to measure a rugged surface did not contribute to its insertion into the analytical mathematical formalism of CFM until to rise the fractal geometry. Generally, the roughness of a fracture surface has fractal geometry. Therefore, it is possible to establish a relationship between its topology and the physical quantities of fracture mechanics using fractal characterization techniques. Thus, with the advent of fractal theory, it became possible to describe and quantify any structure apparently irregular in nature [1]. In fact, many theories based on Euclidean geometry are being revised. It was experimentally proved that the fracture surfaces have a fractal scaling, so the Fracture Mechanics is one of the areas included in this scientific context.
The mathematical formalism of the CFM was prepared by imagining a fracture surface flat, smooth and regular. However, this is an mathematical idealization because actually the microscopic viewpoint, and in some cases up to macroscopic a fracture surface is generally a rough and irregular structure difficult to describe geometrically. This type of mathematical simplification above mentioned, exists in many other areas of exact sciences. However, to make useful the mathematical formalism developed over the years, Irwin started to consider the projected area of the fracture surface [57] as being energetically equivalent to the rugged surface area. This was adopted due to experimental difficulties to accurately measure the true area of the fracture, in addition to its highly complex mathematics. Although there are different methods to quantify the actual area of the fracture [56], its equationing within the fracture mechanics was not considered, because the values resulting from experimental measurements depended on the "ruler size" used by various methods. No mathematical theory had emerged so far, able to solve the problem until a few decades came to fractal geometry. Thus, modern fractal geometry can circumvent the problem of complicated mathematical description of the fracture surface, making it useful in mathematical modeling of the fracture.
In particular, it was shown experimentally that cracks and fracture surfaces follow a fractional scaling as expected by fractal geometry. Therefore, the fractal modeling of a irregular fracture surface is necessary to obtain the correct measurement of its true area. Therefore, fracture mechanics is included in the above context and all its classical theory takes into account only the projected surface. But with the advent of fractal geometry, is also necessary to revise it by modifying its equations, so that their mathematical description becomes more authentic and accurate. Thus, it is possible to relate the fractal geometric characterization with the physical quantities that describe the fracture, including the true area of irregular fracture surface instead of the projected surface. Thought this idea was that Mandelbrot and Passoja [58] developed the fractal analysis by the "slit island method ". Through this method, they sought to correlate the fractal dimension with the physical well-known quantities in fracture mechanics, only an empirical way. Following this pioneering work, other authors [3, 4, 5, 6, 7, 8, 11, 12, 13, 59 ] have made theoretical and geometrical considerations with the goal of trying to relate the geometrical parameters of the fracture surfaces with the magnitudes of fracture mechanics, such as fracture energy, surface energy, fracture toughness, etc.. However, some misconceptions were made regarding the application of fractal geometry in fracture mechanics.
Several authors have suggested different models for the fracture surfaces [60-63]. Everyone knows that when it was possible to model generically a fracture surface, independently of the fractured material, this will allow an analytical description of the phenomena resulting the roughness of these surfaces within the Fracture Mechanics. Thus the Fracture Mechanics will may incorporate fractal aspects of the fracture surfaces explaining more appropriately the material properties in general. In this section one propose a generic model, which results in different cases of fracture surfaces, seeking to portray the variety of geometric features found on these surfaces for different materials. For this a basic mathematical conceptualization is needed which will be described below. For this reason it is done in the following section a brief bibliographic review of the progress made by researchers of the fractal theory and of the Fracture Mechanics in order to obtain a mathematical description of a fracture surface sufficiently complete to be included in the analytical framework of the Mechanics Fracture.
Mosolov [64] and Borodich [3 ] were first to associate the deformation energy and fracture surface involved in the fracture with the exponents of surface roughness generated during the process of breaking chemical bonds, separation of the surfaces and consequently the energy dissipation. They did this relationship using the stress field. Mosolov and Borodich [64, 3 ] used the fractional dependence of singularity exponents of this field at the crack tip and the fractional dependence of fractal scaling exponents of fracture surfaces, postulating the equivalence between the variations in deformation and surface energy. Bouchaud [62] disagreed with the Mosolov model [64] and proposed another model in terms of fluctuations in heights of the roughness on fracture surfaces in the perpendicular direction to the line of crack growth, obtaining a relationship between the fracture critical parameters such as
By observing a crack, in general, one notes that it presents similar geometrical aspects that reproduce itself, at least within a limited range of scales. This property called invariance by scale transformation is called also self-similarity, if not privilege any direction, or self-affinity, when it favors some direction over the other. Some authors define it as the property that have certain geometrical objects, in which its parts are similar to the whole in in successive scales transformation. In the case of fracture, this takes place from a range of minimum cutoff scale,
Self-similarity present in a pine (fractal), with different levels of scaling, k.
To understand clearly the statements of the preceding paragraph, one can use the pine example shown in Figure 13. It is known that any stick of a pine is similar in scale, the other branches, which in its turn are similar to the whole pine. The relationship between the scales mentioned above, in case of pine, can be obtained considering from the size of the lower branch (similar to the pine whole) until the macroscopic pine size. Calling of
where an intermediate scale
The magnitude
Similarly it is assumed that the cracks and fracture surfaces also have their scaling relations, like that represented in equations (35) and (36). In the continuous cutoff scale levels, lower and upper (minimum and maximum), are thus defined as follows:
Note that the self-similarity of the pine so as the crack self-affinity, although statistical, is limited by a lower scale
From the concepts described so far, it is verified that the measuring scale
Which is the value of scale
There is a minimum fracture size that depends only on the type of material?
Surely the answer to this question lies in the need to define the smallest size of the fractal structure of a crack or fracture surface, so that its size can be used as a minimal calibration measuring ruler([1] -).
Since an fracture surface or crack, is considered a fractal, first, it is necessary to identify in the microstructure of the material which should be the size as small as possible a of a rugged fracture, i.e. the value of
Mandelbrot [58] pointed out in his work that the fracture surfaces and objects found in nature, in general, fall into a regular hierarchy, where different sizes of the irregularities described by fractal geometry, are limited by upper and lower sizes, in which each level is a version in scale of the levels contained below and above of these sizes. Some structures that appear in nature, as opposed to the mathematical fractals, present the property of invariance by scale transformation (self- similarity of self-affinity) only within a limited range of scale transformation (
One must not confuse this mathematical recursive construction way, with the way in which fractals appear in nature really. In physical media, fractals appears normally in situations of local or global instability [24], giving rise to structures that can be called fractals, at least within a narrow range of scaling (
Considering that the fracture surface formed follows a fractal behavior necessarily also admits the existence of a geometric pattern that repeats itself, independent of the scale of observation. The existence of this pattern also shows that a certain degree of geometric information is stored in scale, during the crack growth. Thus, for each type of material can be abstract a kind of geometric pattern, apparently irregular with slight statistical variations, able to describe the fracture surface.
Moreover, for the same type of material is necessary to observe carefully the enlargement or reduction scales of the fracture surface. For, as it reduces or enlarges the scale of view, are found pattern and structures which are modified from certain ranges of these scales. This can be seen in Figure 14. In this figure is shown that in an alumina ceramic, whose ampliation of one of its grains at the microstructure reveals an underlying structure of the cleavage steps, showing that for different magnifications the material shows different morphologies of the surface of fracture.
Changings in pattern of irregularities with the magnification scale on a ceramic alumina, Lot A8 [56].
To approach this problem one must first observe that, what is the structure for a scale becomes pattern element or structural element to another scale. For example, to study the material, the level of atomic dimensions, the atom that has its own structure (Figure 15a) is the element of another upper level, i.e., the crystalline (Figure 15b). At this level, the cleavage steps formed by the set of crystalline planes displaced, in turn, become the structural elements of microsuperfície fracture in this scale (Figure 15c). At the next level, the crystalline, is the microstructural level of the material, where each fracture microsurface becomes the structural member, although irregular, of the macroscopic rugged fracture surface, as visible to the naked eye, as is shown diagrammatically in Figure 15d. Thus, the hierarchical structural levels [69] are defined within the material (Figure 15), as already described in this section.
Different hierarchical structural levels of a fracture in function of the observation scale; a) atomic level; b) crystalline level (cleavage steps); c) microstructural level (fracture microsurfaces) and d) macrostructural level (fracture surface).
Based on the observations made in the preceding paragraph, it is observed that the fractal scaling of a fracture surface should be limited to certain ranges of scale in order to maintain the mathematical description of the same geometric pattern (atom, crystal, etc.), which is shown in detail in section - 4.2.3. Although it is possible to find a structural element, forming a pattern, at each hierarchical level, it should be remembered that each type of structure has a characteristic fractal dimension. Therefore, it is impossible to characterize all scale levels of a fracture with only a single fractal dimension. To resolve this problem one can uses a multifractal description. However, within the purpose of this section a description monofractal provides satisfactory results. For this reason, it was considered in the first instance, that a more sophisticated would be unnecessary.
Considering the analytical problem of the fractal description, one must establish a lower and an upper observation scale, in which the mathematical considerations are kept within this range. These scales limits are established from the mechanical properties and from the sample size, as will be seen later. Obviously, a mathematical description at another level of scale, should take into account the new range of scales and measurement rules within this other level, as well as the corresponding fractal dimension.
As already mentioned, the description of the rough fracture surface can be performed at the atomic level, in cleavage steps level (crystalline) or in microstructural level (fracture microsurfaces), depending on the phenomenological degree of detail that wants to reach. This section will be fixed at the microstructural level (micrometer scale), because it reflects the morphology of the surface described by the thermodynamic view of the fracture. This means that the characteristics lengths of generated defects are large in relation to the atomic scale, thus defining a continuous means that reconciles in the same scale the mechanical properties with the thermodynamic properties. Meanwhile, the atomic level and the level of cleavage steps is treated by molecular dynamics and plasticity theory, respectively, which are part areas.
To answer the previous question, about the minimum fracture size, Mishnaevsky [70] proposes a minimum characteristic size,
where
where
Mishnaevsky equates with mathematical elegance, the crack propagation as the result of a "physical reaction" of interaction of a crack size,
where
Mishnaevsky proposes a fractal scaling for the fracture process since the minimum scale, given by the size
As a consequence for the existence of a minimum fracture size, recently has arosen a hypothesis that the fracture process is discrete or quantized (Passoja, 1988, Taylor et al., 2005; Wnuk, 2007). Taylor et al. (2005) conducted mathematical changes in CFM to validate this hypothesis. Experimental results have confirmed that a minimum fractures length is given by:
where
A mathematical relationship between the extension of the self-similar contour and a extension of its projection is calculated as follows.
Being
where,
An illustration of the relationship (43), (44) and (45) can be seen in Figure 16.
Rugged surface formed by a homogeneous function A, with frational degee D, whose planar projection, A0 is a homogeneous function of integer degree d, showing the unit surface area Au.
The rugged fracture surface, may be considered to be a homogeneous function with frational degree,
and its planar projection, may be considered as a homogeneous function with integer degree
The index
Considering that, for
The equation (48), means that the scaling performed between a smooth and another irregular surface, must be accompanied by a power term of type
From now on will be obtained a relationship between the rugged and the projected profile of the fracture in analogous way to equation (250) for a thin flat plate (Figure 17a and\n\t\t\t\t\t\tFigure 17b) with thickness
Scaling of a rugged profile of a fracture surface or a crack, using the Mishnaevsky minimum size as a "measuring ruler"; a) in the case of a crack is a non-fractal straight line, where D=d=1; b) in the case of tortuous fractal crack, with its projected crack length, where d≤D≤d+1.
and the area of the projected surface as
According to the equation (48) the valid relationship is:
where,
The fracture is characterized from the final separation of the crystal planes. This separation has a minimum well-defined value, possibly given by theory Mishnaevsky Jr. (1994). If it is considered that below of this minimum value the fracture does not exist, and above it the crack is defined as the crystal planes moving continuously (and the formed crack tip penetrates the material), so that an increasing number of crystal planes are finally separated. One can in principle to use this minimum microscopic size as a kind of ruler (or scale) for the measurement of the crack as a whole([1] -), i.e. from the start point from which the crack grows until its end characterized by instantaneous process of crack growth, for example.
The above idea can be expressed mathematically as follows:
dividing the entire expression (52) above by the minimum Mishnaevsky size one has:
or
where
and yet:
where:
Within this context the number of microcracks that form the macroscopic crack is given by:
In this context (in Mishnaevsky model), the above expression is volumetric and admits cracks branching generated in the fracture process with opening and coalescence of microcracks. However, he continue equating the process in a one-dimensional way reaching an expression for the crack propagation velocity. A complete discussion of this subject, using a self-affine fractal model to be more realistic and accurate, can be done in another research paper.
The answer to the question about what should be the best scale to be used for fractal fracture scaling is then given as follows: being the limit of the crack length
where
Therefore, the statistical self-similarity or self-affinity of a fracture surface, or a crack is limited by a cutoff lower scale
In two dimensions, the problem of existence of a minimum scale size (possibly given by the Mishnaevky minimum size), leads to abstraction of a microsurface with minimum area, whose shape will be investigated further, in Appendices, in terms of the number of stress concentrators nearest existing within a material.
In this section one intend to present the development of fractal models of self-similar surfaces. From a rough fracture surface can be extracted numerous profiles also rough on the crack propagation direction. However, in this section is considered only one profile, which is representative of the entire fracture surface (Figure 18). The plane strain condition admits this assumption. Because, although the fracture toughness varies along the thickness of the material to a plastic zone reduced in relation to material thickness, it can be considered a property. This means that it is possible to obtain a statistically rough profile, equivalent to other possible profiles, which can be obtained within the thickness range considered by plane strain conditions.
Statistically equivalent profiles along the thickness of the material
In order also equivalent to this, it is also possible to obtain an average projected crack length as a result of an average of the crack size along the thickness of the material thickness within the range considered by plane strain, for the purpose of calculations in CFM, it is considered this average size as if it were a single projected crack length, as recommended by the ASTM – E1737-96 [71]. Therefore, inwhat follows, is effected by reducing or lowering the dimensional degree of relationship from the two-dimensional case, shown above, for the one-dimensional case, as follows:
thus, for a self-affine fractal one has:
where
is the Hurst exponent measuring the profile ruggedness. In one-dimensional case the fracture surface is a profile whose length
Considering a profile of the fracture surface as a self-affine fractal, analogous to the fractal of Figure 19, which perpendicular directions have the same physical nature the Voss [48] equation to the Brownian motion can be generalized([1] -) to obtain rugged crack length
Figure 19 illustrates one of the methods for fractal measuring. This measure can be obtained by taking boxes or rectangular portions, based
Assume that the rectangular little boxes (or recovering units) of microscopic size,
Self-affine fractal of Weierstrass-Mandelbrot, where εk=1/4 and Dx=1.5 and H=0.5, used to represent a fracture profile (Family, Fereydoon; Vicsek, Tamas Dynamics of Fractal Surfaces, World Scientific, Singapore, 1991, p.7).
where
Considering that the self-affine fractal extends in the horizontal direction along
where
Therefore, for the corresponding rugged crack length (real)
where
where
substituting (61) in (65), one has:
Since that in the fracture process, the scales in orthogonal directions are the same physical nature, one can choose
being necessarily
rewriting the equation (66), one has:
writing
Eliminating in (69) the dependence of
The curve length in the stretch,
whose the plot is shown in Figure 20. Note that the lengths
Applying the logaritm on the both sides of equation (72) one obtains an expression that relates the fractal dimension with the projected crack length:
Graph of the rugged length L in function of the projected length L0, showing the influence of height, H0, of the boxes in the fractal model of fracture surface: a) in the upper curves is observed the effect of H0 as it tends to unity (H0→1.0), b) in the lower curves, that appearing almost overlap, is observed the effect of H0 as it tends to zero (H0→0).
The graph in Figure 20 shows the influence of the boxes height
Counting boxes (or strechts) with rectangular sizes Lo x Ho where the boxes that recovers the profile have different extensions in the horizontal and vertical directions.
Making up the counting boxes (or stretch) with rectangular sizes
which plot is shown in Figure 22.
The graph in Figure 22 shows the influence of the roughness dimension on the rugged crack length,
Graph of the rugged length, L, in function of the projected length, L0, showing the influence of the Hurst exponent H, in the fractal model of the fracture surface.
Note that for
which is a self-similar relation between the projected crack length,
It important to observe that
In the study of a self-affine fractal there are two extremes limits to be verified. One is the limit at which the boxes height is high in relation to its projected length,
Case. 1 : The self-similar or local limit of the fractality
Taking the local limit of the self-affine fractal measure as given by (72), i. e. for the case where,
where
This equation is analogous to self-similar mathematical relationship only that the exponent is
According to these results it is observed that the relation (77) has a commitment to the Hurst exponent of the profiles on the considered observation scale
although
Case. 2: The self-affine or global limit of fractality
Taking the global limit of the self-affine fractal measure given by (72), i.e. for the case in which:
It must be noted that the ductile materials by having a high fractality have a crack profile which can be better fitted by the equation (76), while brittle materials by having a low fractality will be better fitted by the equation (79) corresponding the classical model, i.e., a flat geometry for the fracture surface. Furthermore, the cleavage which occurs on the microstructure of ductile materials tend to produce a surface, where
Defining the local roughness of a fracture surface, as:
where
using (74) in (81), one has that:
From (81) note that when there is no roughness on surfaces (flat fracture) one has that:
The quantity
Schematization of a rugged surface which is inclined with respect to its projection.
The ruggedness must depend on infinitesimally of the projected length and its relative orientation to it. In this case, the surface roughness by the usual definition adds an error equal to the angle secant
However, by the definition proposed herein, when only one inclines a smooth surface against to the horizontal, one has:
Within this philosophy will be considered as rugged any surface that presents in an infinitesimal portion a variation of their contour such that
Considering a fractal model for the fracture surface given by the equation:
it is possible to describe its ruggedness in order to include it in the mathematical formalism CFM to obtain a Fractal Fracture Mechanics (FFM). This derivative of equation (86) defines a fractal surface ruggedness, which for the case of a self-affine crack which grows with,\n\t\t\t\t\t\t
such modifications were added to equations of the Irregular Fracture Mechanics to obtain a Fractal Fracture Mechanics as described below.
In Figure 24 and Figure 25, a good agreement is observed in the curve fitting of equation (72) and equation (73) to the fractal analyses of the mortar specimen
a) Fractal analysis of mortar specimen A2 side 1 – Fractal dimension x Projected length, L0; b)Fractal analysis of mortar specimen A2 side 1 - rugged length L x projected length, L0
a) Fractal analysis of red clay A8 side 1 – Fractal dimension x Projected length, L0; b) Fractal analysis of red ceramic specimen A8 - rugged length L x projected length, L0
It is possible, in principle, mathematically distinguish a crack in different materials using geometric characteristics which can be portrayed by different values of roughness exponents in the relations (72) and (74).
The fractal model of the rugged crack length,
The rugged crack length is a response to its interaction with the microstructure. of the material. Therefore, mathematically is possible to portray the rugged peculiar behavior of a crack using fractal geometry
The mathematical model presents a wealth (mathematical richness) that can still be explored in terms of determining the minimum crack length,
The mathematical model is sensitive to variations in the behavior of the crack length it is a linear or logarithmic with the projected crack length.
Comparing the experimental results with the model proposed in this chapter, it is concluded that one of the more important results obtained here are the equations (72), (74) and (82) leading to finding that the fracture surfaces of the materials analyzed are indeed (actually) self-affine fractals. Starting from this verification it becomes feasible to consider the fractal model of rugged fracture surface and its ruggedess inside the equations of the classical fracture mechanics, according to equation (74) and (82). As there is a close relationship between phenomenology and structure formed by virtue of its fractal geometry, the understanding of the formation processes of these dissipative structures, as the cracks, should be derived from their mathematical analysis, as the close relationship between the phenomenology of the formation process of dissipative structures and their fractal geometry. Therefore, the mathematical description of fractal structures must exceed a simple geometrical characterization, in order to correlate the pattern formed in the process of energy dissipation with the amount of energy dissipated in the process that generated it. Thus, it is possible to use the fractal geometry in order to understand other more and more complex processes inside the fracture mechanics. Therefore, the various mechanisms responsible by the crack deviation and by the formation of the rugged fracture surface can then, from the fractal model, be quantified in the fractal analysis of this surface.
The idea of obtaining a relationship between
On the other hand, we are interested in developing a Fractal Thermodynamic for a rugged crack that will be related to the CFM and the Classical Fracture Thermodynamics when the crack ruggedness is neglected or the crack is considered smooth.
A normal spine alignment including coronal and sagittal balance is essential for optimal biomechanical function [1, 2, 3, 4, 5, 6, 7]. The spine, which allows for simultaneous stability and mobility, also has the inherent role of housing and protecting the brain and spinal cord. The alignment of the spine is critical in the context of allowing normal function of the central nervous system (CNS); that is, by not impeding its function by various loading mechanisms (i.e. overstretching the nervous tissues) [8, 9]. Clinical trials [10, 11, 12, 13, 14, 15, 16] and case reports [17, 18, 19, 20, 21, 22, 23, 24, 25] have demonstrated that corrections in patient posture have resulted in relief of neurological symptoms including for example, cervical spondylotic and discogenic radiculopathy, cervical spondylotic myelopathy (CSM), lumbosacral discogenic radiculopathy, trigeminal neuralgia (TN), dystonia, Parkinson’s disease (PD), carpal tunnel syndrome (CTS), and Tourette’s syndrome (TS). Although the precise mechanisms underlying improved neurological function in patients having improved postural alignment are not fully understood, they are thought to lie in the biomechanics of the CNS and in normalization of load sharing across tissues innervated by mechanoreceptors which are integral in sensorimotor control through somatosensory potentials.
In 1960, a monograph was published by Alf Breig documenting for the first time, the most comprehensive illustrative demonstrations of the biomechanics of the central nervous system (CNS) [8]. This seminal work laid the groundwork for the comprehensive understanding of how spine movement affects the CNS; that is, how physiologic deformation of the cord and brainstem simultaneously accompanies normal postural movements of the spine (i.e. ‘neurodynamics’). In 1978, Breig published a second book expanding on the concepts outlined in 1960, and focused on ‘adverse mechanical tension’ in the CNS and how this produces common neurological symptoms and signs [9]. An exciting development by Breig was his invention of the ‘cervicolordodesis’ surgical procedure that increased the cervical lordosis and prevented cervical flexion to relieve tension within the cord, brainstem and nerve roots demonstrating dramatic improvements of neurological disorders including nerve root compression syndromes, TN, multiple sclerosis (MS) and other neuromusculoskeletal conditions [26].
A second prevailing theory on how normalization of spine/posture alignment can dramatically alter patient pain, disability, function, and neurophysiology is through cervical spine sensory afferent input (so called afferentation) and its influence on the motor system termed sensorimotor control. As a result of activation of mechanoreceptors contained in the various ligaments, discs, muscles and skin, changes in spine position-alignment has a major influence on motor control [27]. Intimate connections exist between afferent input (from the proprioceptive, visual and vestibular systems) and stable upright postures of the head and neck [28]. The mechanoreceptors in the cervical spine soft tissues provide necessary neurophysiological input in a feed forward and feedback system for sensorimotor control via connections to the vestibular, visual and central nervous systems [29]. Furthermore, a complex network of neurophysiological connections between cervical spine mechanoreceptors and the sympathetic nervous system exists [30, 31, 32]. Though the effects of autonomic system activity on musculoskeletal function has been extensively studied, there is a paucity of research demonstrating that the autonomic nervous system is intimately responsive to changes in the afferent articular input due to spine joint dysfunction [33]. Alterations in afferent articular input driven by spine joint aberrant movement (altered kinematics) and subtle or overt tissue damage is generally referred to as ‘dysafferentation’ in the literature. The assumption that restoring normal posture and cervical spine alignment is important for a better afferentation process and improved autonomic nervous system function has some preliminary evidence in the recent literature [10, 34].
Today there are non-surgical evidence-based techniques known to improve posture and spine alignment; in essence to accomplish what Breig was able to do, only without surgery (e.g. increase cervical lordosis). One of these methods is Chiropractic BioPhysics® (CBP®) technique which is a full-spine and posture treatment that utilizes mirror image® (i.e. ‘corrective’) exercises, adjustments and spinal traction procedures to restore normal spine alignment [35, 36, 37, 38, 39]. Due to the implicit interconnectedness of spine alignment and neurologic function, these methods are proving to be particularly effective in treating patients with neurological and sensorimotor control disorders, where perhaps unknowingly, poor spine alignment is a causative factor in patients suffering from neurologic ailments in which their symptoms are exacerbated and/or directly caused by the adverse nerve tensions placed upon them and by dysafferentation caused by spine and postural deviations (i.e. adult spinal deformity/subluxation).
This chapter reviews the Harrison normal spinal model [40, 41, 42, 43, 44, 45, 46, 47] that is used to assess a patient’s spine alignment as compared to the normal/ideal position (i.e. gold standard), how the central nervous system is housed in and biomechanically functions within the skeletal structure under normal and pathologic conditions, including mechanisms for neurologic symptom generation under pathologic biomechanical tensions, and altered sensorimotor control from dysafferentation driven by altered load sharing and spine kinematics. Simultaneously, the CBP structural rehabilitation approach to realigning the spine and postural position in order to treat patients who present with spinal subluxation that is suspected to be pathognomonic for their pain, disability, and generalized neurologic sensorimotor disorders will be a main theme.
Any contemporary discussion about the normal/ideal human spinal configuration is regarding its precise orientation (i.e. precise shape of the different spinal regions). Although many research groups have attempted to model the shape of the normal human spine, few have done so as comprehensively and systematically as the Harrison group [40, 41, 42, 43, 44, 45, 46, 47]. In a series of studies, elliptical shape modeling of the path of the posterior longitudinal ligament was performed on radiograph samples of asymptomatic subjects. Computer iterations of spine shape modeling was used to determine a best-fit geometric spinal shape by fitting various ellipses of altered minor-to-major axis ratios to the digitized posterior vertebral body corners of the cervical [40, 41, 42], thoracic [43, 44], and lumbar spinal regions [45, 46, 47] (Figure 1).
The Harrison normal sagittal spine model as the path of the posterior longitudinal ligament. The cervical, thoracic and lumbar curves are all portions of an elliptical curve having a unique minor-to-major axis ratio. The cervical curve is circular meaning the minor and major axes are equal.
The Harrison normal spine model (Figure 1) features a circular cervical lordosis, and portions of an elliptical curve for both the thoracic kyphosis (more curvature cephalad), and lumbar lordosis (more curvature caudad). Consequently, features of the normal human spine reveal that the opposite thoracic and lumbar curves meet together at the thoraco-lumbar junction being essentially straight; the upper, deeper curve of the upper thoracic spine reflects oppositely at the cervico-thoracic junction (between T1 and T2) and continues into the cervical lordosis; the lower lumbar spine increases its lordotic alignment having two-thirds of its curve between L4-S1 as it meets the forward tilted sacral base. The spine is modeled as vertical in the front view. The spine alignment is easily quantified by repeatable and reliable methods from measuring its position from standing X-rays [48, 49, 50, 51, 52] (Figure 2).
Harrison posterior tangent method involves lines drawn contiguous with the posterior vertebral body margins. Intersegmental as well as regional sagittal curves are easily quantified having a standard error of measurement within about 2° (Courtesy CBP seminars).
The Harrison normal spine model has been validated in several ways. Simple analyses of alignment data of normal asymptomatic populations have been done [40, 41, 42, 43, 44, 45, 46, 47, 53]. Comparison studies between normal samples to symptomatic samples [40, 41, 53]; as well as between normal samples to theoretical ideal models have been done [40, 41, 43, 44, 45, 46]. The statistical differentiation of asymptomatic subjects from symptomatic pain group patients based on alignment data have been performed [42, 47]. The demonstration of paralleled spine alignment improvements with reduction in pain and disability, versus no change in untreated control groups in pre-post clinical trials have been performed [54, 55, 56, 57, 58, 59]. The demonstration in randomized clinical trials that only patient groups achieving lordosis improvement (lumbar or cervical) and hyper-kyphosis (thoracic) reduction achieve long-term improvements in various outcome measures versus comparative treatment control groups not getting spine alignment improvement who experience regression in multiple outcome measures at follow-up have also been done [10, 11, 12, 13, 14, 15, 16, 60, 61, 62, 63, 64].
Chiropractors practicing Chiropractic BioPhysics® (CBP®) structural rehabilitation techniques have used this spine model as the goal of care for over 20 years; and more recently physical therapists and other manual medicine rehabilitation specialists have adopted components of this system as well. It is noted that this model serves as the baseline for patient comparison; specific patient comparisons, however, must include patient-specific considerations related to thoracic inlet parameters [65] as well as pelvic morphology [66] as these may dictate a structural modification to the model for a given patient. There are software programs (i.e. PostureRay Inc., Trinity FL, USA) that aid in the ability for practitioners to assess spine alignment quickly in daily practice (Figure 3). It must also be mentioned that proper assessment of the spine includes the whole spine, that is, the cervical, thoracic and lumbar regions and femur heads. This is because spine balance and compensation mechanisms involve the whole spine; thus, regional X-rays to the ‘problem area’ can mislead treatment and not account for distal spinopelvic compensations that need to be considered prior to initiating a trial of spine care by these methods.
Three patients demonstrating dramatically different spine alignment patterns. Left: Excessive lumbar hyperlordosis, L4 anterolisthesis, and excessive anterior sagittal balance in a mid-aged female with disabling low back pain; Middle: Excessive thoracolumbar kyphosis and early degenerative changes in a mid-aged male; Right: Excessive thoracic hyperkyphosis in a young male with Scheuermann’s disease. Red line is contiguous with posterior vertebral body margins; green line represents Harrison normal spinal model (Courtesy PAO).
The brainstem (mesencephalon, pons, medulla oblongata), cranial nerves V-XII, spinal cord, cauda equina, and nerve roots may collectively be referred to as the pons-cord tissue tract. The static and dynamic characteristics of the pons-cord tract constitutes a self-contained compartment of biomechanics [8, 9, 67]. This results from the way the cord is maintained within the canal by its many attachments: from above (being continuous with the brainstem), from below (sacral and coccyx attachment through the cauda equina and filum terminale), as well as throughout its length (intermittent dural attachments to the posterior longitudinal ligament, ventral attachments of the nerve root sleeves exiting the intervertebral foramina, and bilateral dentate ligament attachments ranging from the upper cervical region down to the level of L1). Under relatively normal static posture without pathological processes, spine dynamics produce normal or ‘physiologic’ tension as transmitted by its constraining elements, and without neurological compromise (Figure 4) [8, 9, 68].
Left: Physiologic folding of the cord and nerve roots in normal lordosis. Right: A forward flexion in those with normal lordosis causes normal unfolding and elasticity of the pons, cord and nerve roots that remain ‘physiologic’ or within tolerable tensions that do not overload the nervous system (Courtesy CBP Seminars).
Only when normal neurodynamic aspects of the so-called pons-cord tract are understood can neuropathology from adverse tensions be fully understood. As Breig states “Internal deformation of the tissue cannot be ruled out as a factor in any disease of the nervous system even in inflammatory and degenerative conditions of the hindbrain, cord and associated nerves, and in some cases it will be of primary pathologic significance” [8] (p. 12). A key concept is that under normal circumstances, normal movements of the spine involve physiologic unfolding and folding of the cord and nerve roots. Head flexion causes instantaneous unfolding and normal elasticity of the neural tissues and head extension causes elastic rebound and a re-folding of the cord and nerve roots (Figure 4). In this way the CNS can preserve normal function while accommodating differing spinal positions. Breig also found that movements of the cord occur at the location of movement as well as throughout the entire pons-cord tract; cervical motions produce strains (deformations) caudally down to the cauda equina and movements of the lower spine cause strains as far up as the cervical cord and brainstem. In fact, deformation of the brain tissue below the tentorium (which can affect cranial nerves V-XII [8]) within the cerebellum occurs to accommodate spine movements (particularly maximal functional positions).
All ventral flexion movements throughout the spine (i.e. cervical, thoracic, lumbar) cause a lengthening of the spinal canal, and therefore, a transmission of axial tension onto the cord. Pathological processes, such as disc herniations and bone spurs, if severe enough, interfere with the pons-cord tract biomechanics [69, 70], where the normal tension transmitted by the pons-cord restraining elements may then be referred to as ‘pathological’ tension [71]. Independent but equally as significant, abnormal spinal postures may create adverse tension within the neural elements as well. For instance, Stein found that “in a deformed kyphotic cervical spine, even a ‘normal’ amount of movement in the cervical spine may cause compression of the spinal cord” [72]. This is because the spinal cord adopts the length of the bony canal [73].
Further, as suggested by Harrison et al. [74] static neutral postures or dynamically adopted combinations of postures; that is, rotations and translations of the head, thorax, or pelvis (Figures 5 and 6) [36], exert larger stresses and strains onto the pons-cord tissue tract. Thus, it can be deduced that the combination of pathological processes (bone spurs, disc herniations, etc.) and aberrations in posture (forward head posture, thoracic hyperkyphosis, etc.) may disrupt normal CNS biomechanics, and at levels below that at which either factor acting alone would elicit neurological symptoms. As it can be presumed when a patient has an accumulation of forward flexed spine positions, such as severe thoracic hyperkyphosis (THK) posture for example, the amount of spinal canal lengthening can be great and supersede the ‘normal’ or physiologic amount of unfolding and elastic deformation available within the pons-cord system. In this situation, normal physiologic tensions transition to become pathologic tensions causing intermittent over-stretching and over-straining of the tissues and ultimately, causing or exacerbating neurologic symptoms.
If the head, thoracic cage, and pelvis are considered rigid bodies, then the possible rotations in 3-dimensions are illustrated. Flexion and extension are rotations on the x-axis, axial rotation is about the y-axis, and lateral flexion is rotation about the z-axis (Courtesy CBP Seminars).
If the head, thoracic cage, and pelvis are considered rigid bodies, then the possible translations in 3-dimensions are illustrated. Lateral translations occur along the x-axis, vertical translations occur along the y-axis, and anterior–posterior translations (protraction-retraction) occurs along the z-axis (Courtesy CBP Seminars).
Understanding the normal biomechanics of the CNS lays the foundation for the understanding of postural-induced neurological signs and symptoms. As discussed, two main events may individually, or in combination, lead to excessive stresses (longitudinal, torsional, pure bending, shear) and strains (longitudinal cross-sectional) that are sufficient to produce symptoms. In words, poor postures (lengthened spinal canal via forward flexed spinal positions) and space occupying lesions (bone spurs, intervertebral disc prolapse, etc.) combine to produce symptomatology. The greater the spinal canal is flexed, or as discussed, the presence of combinations of rotations and translations in posture, the greater the forced unfolding and elastically stretched pons-cord tract (Figure 7). With the addition of space occupying lesions, patients having deviations in postural alignment become much more likely to succumb to various pressure mechanisms, or how the nervous tissue is compressed upon certain positions and movements.
Left: Cervical kyphosis subluxation in neutral posture results in the unfolding and elastic pre-tension present prior to flexion. Right: Forward flexion of a kyphotic neck may result in ‘pathologic’ or pons-cord-nerve root tensions that exceed physiologic limits and results in neurologic symptoms; particularly in the presence of a space-occupying lesion such as a bone spur or intervertebral disc prolapse (Courtesy CBP Seminars).
It is important to realize that those patients with poor spinal posture may at times be in positions that are tolerable by the pons-cord tract (i.e. not over-stretched), and at other times perform movements that dynamically lengthen the spinal canal causing a pivotal transition to over-stress and over-strain the system (i.e. dynamic stress and strain). Therefore, successful symptomatic relief resulting from postural correction to a patient suffering from neurological complaints may be elucidated. Although some spinal pathologies will not change (e.g. bone spurs), the reduction of forward flexion of the neutral postural position (e.g. increasing cervical/lumbar hypo-lordosis; reducing thoracic hyper-kyphosis) will change the resting, and therefore the dynamic tensions throughout the pons-cord tract sufficiently enough to reduce the tensions from surpassing some pathological tension threshold (maintaining physiologic or normal tensions), and therefore alleviate neurologic symptomatology [74, 75].
How does adverse mechanical tensions within the CNS produce symptoms? Ultimately, pathological CNS tensions affect the vascular supply and therefore the perfusion of the neural tissues or they may affect the actual nerve conduction ability of the nerve cells (causing hyper or hypo function). Mathematically, perfusion = mean arteriole pressure (MAP) – cord interstitial pressure (CIP) [76]. Thus, for perfusion to remain adequate, the MAP must remain greater than CIP. However, as discussed by Harrison et al. [77], an increase in CIP can be caused by at least two forces, a longitudinal force causing unfolding and elastic elongation of the cord, and a transverse force usually by the cord being thrust into the posterior margin of the vertebral body at the anterior portion of the spinal canal. As stated, Stein found that cervical kyphosis, posture subluxation alone is enough to interfere with cord conduction [72], but with an accompanying space occupying lesion the likelihood for a transverse cord/nerve compression pressure mechanism to limit perfusion and compromise neural function is much greater.
There have been several clinical controlled trials documenting the association with improved postural parameters (e.g. increasing cervical lordosis, increasing lumbar lordosis, decreasing thoracic hyperkyphosis) translating into improved physiological measures including specific tests indicative of neurologic function [10, 11, 12, 13, 14, 15, 16]. These measures include:
Central somatosensory conduction time N13-N20 (Figure 8);
Dermatomal somatosensory evoked potentials (DSSEPs) (Figures 9 and 10);
H-reflex;
Sensorimotor control measures (Figures 11 and 12);
Sympathetic skin resistance response (Figure 13).
Central conduction time (N13-N20) also known as spinal cord velocity. In the top figure, a representative example of central conduction time (N13-N20) at three intervals of measurement: baseline, following 10-weeks of treatment, and 1-year follow-up. This is from the study by Moustafa et al. [13] on symptomatic patients with cervical spine disc herniation. Follow correction of the cervical lordosis, a 20% change in central conduction speed is shown in milliseconds (m sec) indicating a faster more efficient response. In the bottom graph a representative sample from the study of Moustafa et al. [78] is shown. Here, in asymptomatic participants, correction of the cervical lordosis and anterior head posture was found to result in a 10% faster response in the central conduction time potential. Comparative and placebo control groups not attaining spine correction showed no improvement in central conduction time.
Dermatomal somatosensory evoked potential (DSSEPs) set up for C6, C7, and C8 nerve root assessment. In (A) Sites of recording: (1) active recording electrode at c3’, (2) reference electrode at Fz, and (3) grounding electrode at Fbz. Location of stimulation sites are indicated above: (B) for C6 dermatome, (C) for C7 dermatome, (D) C8 dermatome. Sites of recording. Courtesy of Moustafa et al. [16].
Example of Dermatomal somatosensory evoked potential (DSSEPs) of C6 dermatome at three intervals of measurement. Courtesy of Moustafa et al. [16]. The Pre-treatment DSSEPs from the RCT by Moustafa et al. [16] is shown. In the 10-week post treatment (Follow-up), the DSSEPs following cervical curve correction is shown. Finally, the 3-month (Post) follow-up, DSSEPs are shown where correction was stable over time.
Postural stability characteristics were evaluated with a Biodex Balance System SD (BBS) (Biodex Medical Systems, Inc., Shirley, NY). Dynamic balance testing was performed on the unlocked platform to allow free movement concurrently both in the anterior–posterior (AP) and medial-lateral (ML) directions. The platform permits variable levels of resistance to movement perturbation ranging from one to eight (1 being the most restrictive). BBS measures the deviation of each axis during dynamic balance assessments. The BBS software measures an overall stability index (OSI) and is a representative index of balance performance. OSI is the best indicator of the overall ability of the subject to balance the platform whereby a reduced balance or stability correlates with large variation or large value of OSI [79] [80]. From the RCT by Moustafa et al. [10] participants randomized to and achieving cervical spine correction obtained statistically significant improvements in the OSI compared to a standard care group (Pre vs. 10-weeks post vs. 1-year follow up). Courtesy of Moustafa et al. [10, 34].
In A, an example of the smooth pursuit neck torsion test (SPNT) is shown neutral, right turn, and left turn of the head indicated by the 45° torso rotated position. Middle and right image: A SPNT test eye velocity is shown as the uneven high amplitude curves. The ciphers indicate the length of the vertical part of the curve between two marks, i.e., the saccades. The forward head posture group (FHP group) has larger errors (≈30%) as compared to a match control group with normal head alignment (10% average error). Courtesy of Moustafa et al. [34].
Sympathetic skin resistance response (SSR). For measurement of the SSR, EMG equipment was used [10]. Active surface electrodes were attached on the palmar side, and the references were placed on the dorsum of the hand. The stimulus was given at the wrist contralateral to the recording side. Measurements were taken from left and right arms. Latencies were measured from the stimulation artifact to the first deflection from the baseline. The amplitude is measured from the peak of the first deflection to the peak of the next one (peak to peak) as shown. Depicted are the results for the study group receiving cervical spine correction (lordosis and forward head posture). On the Top: the study group receiving spine corrective extension traction is shown at pre-study; Middle: after 10 weeks of treatment (30 sessions); and Bottom: at 1-year follow up with no further treatment. Only the group receiving extension traction obtained sagittal plane cervical correction and statistically significant improvement in SSR latency and amplitude. Courtesy of Moustafa et al. [10].
Here we briefly summarize the particulars of exemplary trials that demonstrate how improvement in targeted spinal and postural parameters have led to improved neurophysiological outcomes.
Using standardized clinical procedures for median nerve stimulation at the wrist, a subjects’ central somatosensory conduction time measurement, N13-N20, can be determined. Differences in peak latencies between N13 and N20 is measured as central conduction time or similarly called ‘cervical spinal cord velocity’. In 2016, Moustafa et al. reported on a randomized controlled trial using a cervical extension traction orthotic device (Denneroll™; Denneroll Ptdy, limited, Sydney, Australia) in a multi-modal rehabilitation program for treating patients with discogenic radiculopathy [13]. Sixty patients were randomized to a treatment and comparison (control) group where both groups received TENS, thoracic spine manipulation, soft tissue mobilization and strengthening exercises. Only the treatment group performed the additional Denneroll orthotic device to increase cervical lordosis and reduce forward head posture. After 30 treatment applications over the course of 10-weeks, only the treatment group demonstrated significant improvements in N13-N20 (20% gain in velocity). Also, at a 1-year follow-up without further intervention, again only the treatment group demonstrated a statistically improved N13-N20 potential. Importantly, only the treatment group receiving the Denneroll had statistically improved cervical lordosis and reduced forward head posture at the 10-week and 1-year follow-up. Figure 8 (top) demonstrates the improvement in the N13-N20 potential in this trial.
In another trial, Moustafa et al. reported on the unique treatment of asymptomatic volunteers who had strictly defined anterior head translation and cervical hypolordosis [78]. Eighty persons were randomized into a treatment group who performed cervical extension traction on the Denneroll traction orthotic or a control group who lied on a rolled hydrocollator towel (placebo control). Both groups were treated for 30 sessions over 10-weeks and were then re-assessed after 12 further weeks of no further treatment. Central somatosensory conduction time latency (N13-N20) and amplitudes of spinal (N13), brainstem (P14), parietal (N20 and P27), and frontal (N30) potentials were measured at baseline (prior to treatment), 10-weeks and 22-weeks. After the 10-weeks of treatment, the treatment group had significantly better amplitudes of N13, P14, N20, N27 and N30 as well as central conduction time (10% faster conduction velocity of N13-N20). All significant differences between groups favouring the treatment group remained at the 12-week post-treatment follow-up. Lastly, a statistically significant multiple regression model to predict central conduction time changes, N13-N20, from correction in cervical lordosis and anterior head translation (AHT) was identified for the intervention group receiving the Denneroll orthotic at both the 10-week mark (p < .001) and the 3-month follow-up (p < .001) [78]. Figure 8 (bottom) demonstrates the improvement in the N13-N20 potential in this trial.
In 2011, Moustafa et al. reported on the results of a pilot trial that showed patients with cervical spondylotic radiculopathy randomized to a rehabilitation program including cervical spine stretching exercises, infrared radiation and 3-point bending cervical extension traction had significantly improved peak-to-peak amplitude measures of DSSEPs after both 10-weeks of treatment (30 treatment sessions) and at a 12-week follow-up [16]. The comparison (control) group receiving the same treatment less the neck traction did show an initial improvement in DSSEPs after the 10-week treatment period, however, this difference disappeared at the 12-week follow-up. Only the treatment group showed a statistically significant increase in cervical lordosis. Most importantly, Moustafa et al. identified a linear correlation between initial DSSEPs and cervical lordosis magnitude for both groups (r = .65; p < 0.0001), whereas this relationship was only maintained in the study group receiving 2-way traction at final follow-up (r = .55; p = 0.033). This indicates that cervical spine lordotic correction linearly correlates to improvements in DSSEPs [16]. Figures 9 and 10 depict the experimental setup for the DSSEPs of C6, C7, and C8 as well as the changes in C6 DSSEPs in the study group receiving the curve corrective traction.
As reported in the previous section (5.1), the 2016 trial reported by Moustafa et al. [13] treating patients with discogenic radiculopathy, both groups showed improvements in latency of DSSEPs at the 10-week post-treatment assessments, however only the treatment group showed statistically improved amplitude of DSSEPs. At a 1-year follow-up without intervention, only the treatment group demonstrated statistically improved latency and amplitude of DSSEPs. Also, only the treatment group had improved cervical lordosis and reduced forward head posture at the 10-week and 1-year follow-up.
In a unique randomized trial, Moustafa and colleagues [14] investigated the hypothesis that improving the cervical lordosis and reducing forward head translation would improve low back pain, disability, and neurophysiology in a sample of 80 (35 female) patients between 40 and 55 years suffering signs and symptoms from chronic discogenic lumbar radiculopathy (CDLR). Both groups received TENS therapy and hot packs; additionally, the study group received the Denneroll cervical traction orthotic. All treatment interventions were applied at a frequency of 3 x per week for 10 weeks. Both groups were followed for 6 months after their 10-week re-evaluation. Statistically significant differences between the study groups and the control group’s postural measures were found favouring improved posture in the Denneroll group for: lumbar lordotic curve (p = .002), thoracic kyphosis (p = .001), trunk inclination (p = .01) and imbalance (p = .001), pelvic inclination (p = .005), and surface rotation (p = 0.01). The two radiographic measures of cervical lordosis (p = .001) and forward head posture (p = .002), and H reflex amplitude (p = .007) and H reflex latency (p = .001) were likewise statistically different between the groups at 10 weeks favoring improvement in the Denneroll study group. Restoring cervical lordosis and reduction of forward head posture with Denneroll traction was found to have a positive impact on 3D posture parameters, leg and back pain scores, back disability, and H reflex latency and amplitude. Thus, improvement of sagittal cervical spine posture and alignment benefited the pain, disability, and postural imbalances in patients with CDLR.
In 2013, Moustafa et al. reported the results of a trial employing lumbar extension traction to increase the lumbar lordosis in patients suffering from MRI-verified lumbosacral radiculopathy [15]. Sixty-four patients were randomly allocated to the treatment or comparison (control) group who both received hot packs and interferential therapy; only the treatment group received the lumbar extension traction. After 30 treatment sessions over 10-weeks, only the treatment group showed statistically improved latency and amplitude of the H-reflex. At the 6-month follow-up, again, only the treatment group showed statistically improved H-reflex outcomes. Only the treatment group demonstrated improved lumbar lordosis after the 10-week treatment period and at the 6-month follow-up.
Improvement in head repositioning accuracy (HRA) as a result of sagittal plane spine alignment correction has been assessed in three recent randomized trials by the Moustafa et al. group; two of these trials assessed cervical lordotic correction and anterior head translation reduction [10, 79], whereas, one trial assessed improvement in thoracic hyper-kyphosis [11]. In 2017, Moustafa et al. reported on the improvement in HRA, a measurement of cervicocephalic kinesthetic sensibility [79]. Seventy-two patients suffering from cervicogenic dizziness were randomized to a treatment or comparison (control) group and received TENS, hot packs, mobilization, myofascial and suboccipital release, and therapeutic functional exercises. Only the treatment group also received the Denneroll cervical extension traction orthotic device. The cervical range of motion (CROM) device was used to assess cervicocephalic kinesthetic sensibility by measuring the head repositioning average error (HRA). The participants (blindfolded) started with their head in the neutral head position (NHP) and were asked to actively move to the midpoint of their maximum rotation range, which was called the “target position.” After returning to the NHP, they were then asked to rotate their head to the target position. The difference between the target position and the achieved position was recorded 3 times and averaged. The midpoint position was used rather than the NHP because it was considered a non-learned position. The CROM device has good criterion validity (r = 0.89–0.99) and reliability (intra-class correlation coefficient [ICC] = 0.92–0.96) [80].
After 30 treatment session over 10-weeks, both groups improved on the HRA test [79]. However, at the 1-year follow-up, the treatment group’s HRA to the left and right was statistically significantly better than the comparison group. Again, only the treatment group had a statistically improved cervical lordosis and improved forward head posture at both the 10-week and 1-year follow-up. In their more recent [10] randomized trial similar results were identified where improved HRA resulted from improved cervical lordosis and forward head translation reduction; herein, the improvement in the HRA was identified to be linearly correlated to the improvement in both cervical lordosis and reduction in forward head posture. The linear correlation between improved HRA and improved forward head posture magnitude is further supported by the results of a cross-sectional case control investigation which found a linear relationship between worsening HRA and increased magnitudes of forward head posture. [34]
In 2020, Moustafa et al. reported on the improvements in various sensorimotor control measures in patients treated for thoracic hyper-kyphosis [11]. Eighty patients with thoracic hyper-kyphosis were randomized to a treatment or comparison (control) group. Both groups were treated for 30 treatment sessions over a 10-week time period with TENS and hot packs, soft tissue mobilization, thoracic spine manipulation, and functional exercises. Only the treatment group also performed the Denneroll thoracic traction orthotic designed to reduce the thoracic curve. At the 10-week post-treatment assessment, no significant differences were found for left-sided HRA whereas, significant differences favouring the treatment group were found for the right sided HRA. At the 1-year follow-up without intervention, sensorimotor control measurement of HRA, bilaterally, was significantly superior for the intervention group. Also, only the treatment group experienced a reduction in thoracic hyper-kyphosis at the 10-week assessment that was maintained at the 1-year follow-up [11].
Posture stability efficiency is a key measurement or performance variable of sensorimotor control. In recent randomized trials [10, 11] and case control [34] investigations by Moustafa and colleagues, postural stability characteristics were evaluated with a Biodex Balance System SD (BBS) (Biodex Medical Systems, Inc., Shirley, NY) (Figure 11). Dynamic balance testing was performed on the unlocked platform to allow free movement concurrently in both the anterior–posterior (AP) and medial-lateral (ML) directions. The platform permits variable levels of resistance to movement perturbation ranging from one to eight (1 being the most restrictive). BBS measures the deviation of each axis during dynamic balance assessments. The BBS software measures an overall stability index (OSI) and is a representative index of balance performance. OSI is the best indicator of the overall ability of the subject to balance the platform whereby a reduced balance or stability correlates with large variation or large value of OSI [81, 82]. From the RCT by Moustafa et al. [10], participants randomized to and achieving correction of both cervical lordosis and anterior head posture obtained statistically significant improvements in the OSI compared to a standard care group (pre vs. 10-weeks post vs. 1-year follow up). Likewise, in the RCT looking at thoracic hyper-kyphosis reduction, it was found that OSI was statistically improved only in the group achieving reduction of thoracic kyphosis and that this result was stable at 1-year follow-up [11].
The fact that posture stability, as measured with OSI, improves due to correction of the sagittal cervical and thoracic spine alignments seems to make sense; as previously a linearly correlation between worsening OSI and increased magnitudes of forward head posture has been found [34].
The smooth pursuit neck torsion test (SPNT) is used to quantify alterations in and improvement in a person’s visual-motor control using electro-oculography equipment [83]. Figure 12 demostrates the SPNT procedure. First, participants perform the SPNT with the head and trunk in the neutral, forward facing posture. Next, while keeping the head facing forward, the torso is placed in a 45° rotation (about a vertical y-axis) position to each side in a consecutive manner. Participants typically perform three blinks of their eyes and are instructed to follow the path of a light source as close as possible with their eyes without movement of their head or neck. The accuracy of the SPNT is determined as the difference between the average increase/decrease in the participants NHP vs. the torsioned positions; errors are termed ‘corrective saccades’ and are reported as a percentage difference from perfect. In a recent case control cohort sample, Moustafa et al. [34] identified that the forward head posture group (FHP group) had larger SPNT errors (≈30%) as compared to a matched control group with normal head alignment (10% average error). In fact, a linear correlation was identified between the magnitude of forward head posture subluxation and the percent error in SPNT.
Importantly, in the Moustafa et al. RCTs [10, 11], SPNT test eye velocity was shown to improve in the group receiving spine correction as compared to the comparison group not receiving and not achieving spine correction. The average SPNT errors in both the cervical spine [10] and the thoracic spine [11] correction groups improved down to bench-mark values for healthy persons (≈10%).
Sympathetic skin resistance response (SSR) is a measurement of autonomic nervous system function or dysfunction. For measurement of the SSR, EMG equipment is typically used [10, 34]. Active surface electrodes are attached on the palmar side, and the references are placed on the dorsum of the hand. A stimulus is given at the wrist contralateral to the recording side. Measurements should typically be taken from left and right arms. The SSR is assessed as: (1) a latency measurement from the stimulation artifact to the first deflection from the baseline; and (2) an amplitude is measured from the peak of the first deflection to the peak of the next one (peak to peak). Figure 13 depicts a typical measurement of SSR latency and amplitude.
In a recent case–control cohort investigation of 160 asymptomatic volunteers, Moustafa and colleagues [34] investigated the SSR and its relationship to the severity of forward head posture; a strong linear correlation was identified between the magnitude of forward head posture and increased amplitude and latency of the SSR evoked potentials. Thus, increased magnitudes of forward head posture have a negative impact on the autonomic nervous system in essence leading to a state of hyperactivity or increased excitability. Only one RCT on the effects of spine correction on SSR latency and amplitude could be identified. In the study by Moustafa et al. [10], the group receiving spine corrective extension traction obtained sagittal plane cervical correction and statistically significant improvement in SSR latency and amplitude; the results indicated a linear correlation between the amount of correction of the cervical lordosis and FHP and the concomitant improvement of the SSR potentials [10].
The above review of sagittal plane spine alignment and its impact on neurophysiology has several strengths. First, there is strong biomechanical evidence indicating that altered and sustained sagittal plane spine and posture alignment results in increasing the stresses and strains acting on the pons-cord tract system and that this impairs directly or indirectly neurophysiology; this evidence has existed since 1960 [8, 9, 26, 68, 69, 71, 72, 73, 74, 75, 77]. Second, considering the results of the recent randomized trials reviewed above, it is clear, that rehabilitation techniques that increase the cervical lordosis and lumbar lordosis, have a profound and sustained effect of improving measurements of neurophysiology as measured with DSSEP’s, H-Reflex, and central conduction times [12, 13, 14, 15, 16, 78, 79]. Similarly, reducing the magnitudes of thoracic hyper-kyphosis [11] and FHP [12, 13, 14, 15, 16, 78, 79] has been found in RCT’s to result in improved neurophysiological measurements. Finally, clinical management and improvement of several complex neurological disorders have been documented in multiple case reports where spine correction was suggested to be the important variable which improved the patient’s neurophysiological disorder [17, 18, 19, 20, 21, 22, 23, 24, 25]. Thus, considering the facts that biomechanics studies, randomized trials, and case reports all point to the same finding, clinically and scientifically one would need to concede that improved neurophysiology following correction of the abnormal spine towards normal values is an evidence-based and logical approach to pursue with appropriate patients.
Similarly, the known intimate connections between afferent input (from the proprioceptive, visual and vestibular systems) and stable upright postures of the head and neck [28] and the fact that there exists a plethora of mechanoreceptors in the cervical spine soft tissues providing necessary neurophysiological input in a feed forward and feedback system provides a strong fundamental physiological basis for the concept that altered spine alignment can have a profound effect on sensorimotor control via connections to the vestibular, visual and central nervous systems [29]. Furthermore, and as explicitly stated in the introduction to this chapter, a complex network of neurophysiological connections between cervical spine mechanoreceptors and the sympathetic nervous system exists [30, 31, 32]. This information coupled with the findings of both case control investigations [34] and randomized trials [10, 11, 63, 79] provides strong clinical evidence that restoring normal sagittal plane posture and cervical spine alignment is important for a better afferentation process, improved sensorimotor control, and improved autonomic nervous system function.
Clinically, the astute reader should recognize the need to radiographically assess the full spine alignment, in particular the sagittal plane, to identify if a patient is a candidate and in need of true spine correction; that is, structural rehabilitation of the spine and posture. A comparison of the patient’s spine and posture should be made against tested normal alignment values such as the Harrison full spine model and posture displacement models discussed herein. Furthermore, the addition of fundamental neurophysiological testing and the basic parts of sensorimotor control measurements should be considered as important assessments during patient evaluations. Once an indication for corrective care has been identified, the clinical administration of specific spine mirror image corrective exercises and extension traction methods should be employed. Previously, we have discussed the techniques, indications and contraindications, timing of, and applications for several known spine corrective methods and the clinician should be willing to add these to their armamentarium for patient care; we refer the reader to this source [35]. Adding the goal and methods of true spine correction to the clinical outcomes of patient care should not be foreign, it should not be in disregard to traditional strength and functional conditioning; it should simply be part of the basic, fundamental treatment approach for abnormalities of the human frame in the effort to improve a variety of spine related and neurophysiological disorders.
This chapter has explored the hypothesis and evidence that restoring normal posture and spine alignment has important influences on neurophysiology, sensorimotor control and autonomic nervous system functionality. There is limited but high-quality research identifying that sagittal spine alignment restoration plays an important role in improving neurophysiology, sensorimotor control, and autonomic nervous system function. Within the limitations of the fact that only a handful of clinical trials exist on the topics discussed in this chapter, the unique contribution and importance of this review is that it demonstrates that radiographic determined re-alignment of the sagittal spine and posture plays a significant role in long-term management outcomes in people suffering from a variety of musculoskeletal, and health related disorders. Improved neurophysiological function as measured via dermatomal somatosensory evoked potentials, spinal cord velocity (N13-N20 potential), sensorimotor control, and sympathetic nervous system activity is directly influenced by and improved by full spine sagittal alignment in general and, more specifically, to cervical posture and spine alignment. This review identifies main issues that warrant further investigations to elucidate primary interactions and to identify ideal populations that would benefit from structural rehabilitation of the spine and posture techniques as discussed herein.
PAO is a paid consultant to CBP; DEH teaches spine rehabilitation methods and sells products related to the treatment of spine deformities. IMM has nothing to declare.
AP | Anterior–posterior |
BBS | Biodex Balance System SD |
CBP | Chiropractic BioPhysics® |
CDLR | Chronic discogenic lumbar radiculopathy |
CIP | Cord interstitial pressure |
CNS | Central nervous system |
CROM | Cervical range of motion |
CSM | Cervical spondylotic myelopathy |
CTS | Carpal tunnel syndrome |
DSSEPs | Dermatomal somatosensory evoked potentials |
EMG | Electromyography |
FHP | Forward head posture |
HRA | Head repositioning accuracy |
MAP | Mean arteriole pressure |
ML | Medial-lateral |
MRI | Magnetic resonance imaging |
NHP | Neutral head position |
N13-N20 | Central conduction time |
OSI | Overall stability index |
PD | Parkinson’s disease |
RCT | Randomized controlled trial |
SPNT | Smooth pursuit neck torsion |
SSR | Sympathetic skin resistance |
TENS | Transcutaneous electrical nerve stimulation |
THK | Thoracic hyper-kyphosis |
TN | Trigeminal neuralgia |
TS | Tourette’s syndrome |
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\\n\\nWe have adopted the Protocol to increase the number of readers of our publications. All our Works are more widely accessible, with resulting benefits for scholars, researchers, students, libraries, universities and other academic institutions. Through this method of exposing metadata, IntechOpen enables citation indexes, scientific search engines, scholarly databases, and scientific literature collections to gather metadata from our repository and make our publications available to a broader academic audience.
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\n\nAs a Data Provider, metadata for published Chapters and Journal Articles are available via our interface at the base URL:http://www.intechopen.com/oai/?.
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