Final Thoughtson the Control of Placement in Anatomical Systems

The Role of Placement in Anatomical Movements

We started by noting that anatomical movements are aboutlocation and orientation. The combination of location andorientation has been called placement. For the eye, gaze is equivalent to placement.

Much of this book has dealt with the properties of placementand anatomical movements that change placement. One must consider both location and orientation because theyare distinctly different attributes of an anatomical object and they change indifferent ways with anatomical movements. The nervous system must deal with both, because, to not do so, is tofail in the tasks of everyday living, which is often lethal in a competitiveworld. The relevance of bothlocation and orientation to survival makes both relevant to us as we consideranatomical movement.

Rotations that occur in a single plane about a single centerof rotation may occur in any order without affecting the final placement.  However, if rotations pass out of aplane, then there is a concurrent twisting of the moving object and the orderof the rotations is critical to the final placement. Most anatomical movementsare not confined to a single plane of rotation about a single axis of rotation,therefore, placement must be a consideration in their planning andexecution. If we are to understandthose movements, we must consider not only location, but also orientation.  In addition, if we are to understandthe neural control of movement, we must consider placement.

Description of Anatomical Movements and Their Control

While the approach developed in this book uses highlyabstract mathematics, it must be made quite clear from the beginning that it isvery unlikely that the nervous system operates by explicitly performing suchcalculations. The abstractapproach to description of anatomy and the calculation of consequent movementsis a tool for examining the movements and determining what logical tasks mustbe performed by the nervous system. Often the descriptions and calculations show that the movements have adeep irreducible complexity and computational richness.  Even the simplest movements may besurprisingly complex. Thatcomplexity tells us something of the problems that the nervous system mustsolve in order to perform the movements.

The mathematical analysis is directed at understanding themovement, but understanding the movements may tell us something of theprocesses that control the movement. The nervous system does not perform quaternion analysis to controlplacement, but it must implicitly solve the same problems.  Once we understand the problem thatmust be solved then it is necessary to turn to anatomical and physiological methodsto discover how the body solves the problem.

In the chapter on the eye we explored the nature of eyemovements and discovered that a system that guarantees that the orientation ofgaze during fixation is appropriate to the direction of gaze will also ensurethat saccadic eye movements land the eye on the new target with the properorientation. Put that way, itseems obvious that such should be the case, but our analysis of gaze shows thatan obvious solution, moving directly from the initial target position to thefinal target position, will generally lead to an incorrectly oriented eye uponarrival. The mathematics helps tounderstand the movement and, in this case, it indicated a possiblephysiological solution,

Model Systems

In order to develop this approach, it is necessary to applyit in a number of actual anatomical structures. However, it is important to choose problems where theanatomy does not overwhelm the mathematics. To that end, a number of model systems have beenexplored. Those introduced in thisbook have been an eyeball in its orbit, the upper and lower cervical spine, andflows in gels submitted to simple compression/tension or shear.  Many of these model systems have beensimplified or intrinsically less systems where we could begin to develop anunderstanding without the considerable detail that is characteristic of mostanatomical systems.

It is my hope that the eye movement system will point theway to exploring motor systems in that it is a bounded system with a limitednumber of well defined elements and definite purposes that may be expressed asa clear set of criteria. All ofthese features make it an excellent model system.

By contrast, the hand/arm movement system evokes patternedactivity in dozens of muscles in the hand, forearm, arm, and shoulder girdleduring the simplest reaching movement. The movements themselves are complex, involving movements in differentdirections in several joints with interdependent spatial relations between theactions of the different joints. Avery simple treatment of a part of that system was considered in the last partof the chapter on spin and swing.

In addition to the muscles that move the bones, there aremany muscles in the trunk and other extremities involved in establishing thefoundation for such a movement. Even if we reduce our consideration of reaching to the actual movers,there are often not clear criteria for the movement and there is certainly nota unique solution to the execution of the movement. There are many ways that the final placement may beachieved. It is almost certainthat such criteria do exist for the nervous system and there are certainlymechanical and neurological reasons for a particular execution of the movementthat actually occurs, but we are seldom privy to that information.   Part of the process during theneural planning of the movement must be the meshing of these possibilities andconstraints to attain both a particular location and a particular orientationin a smooth coordinated fashion. At this time it is unlikely that we will be able to give a definitivesolution to the problem. Analysisof the movement possibilities may give insight into what it is that we shouldbe looking for in the nervous system.

Even if we were aware of all the constraints on a movementin a system as complex as reaching with a hand and could write down an adequateanatomical description of the system, the calculations for the movements wouldbe horrendous. Therefore, let usstart with simple systems and simple movements and gradually expand ourexplorations as our grasp of the concepts is increases and methods aredeveloped for dealing with complex systems of anatomical description.

Clearly, much more detailed and complex analyses arepossible. The examples in thisbook have been kept simple enough that they can be fairly readilyfollowed. With a good computer andsufficient time and effort, one can address very complex anatomies.  However, as models become more complexthe numbers of possible variables grows to the point where one can spend yearsexploring all the possibilities. That is why the consideration of the lower cervical spine assumed a verysimilar anatomy and movements in all the elements.

Neural Networks Control Movements

When creating these types of models, it becomes rapidlyapparent that most movements are complex. Even in a system in which the anatomy is simple, such as the eyemovements, the movements are difficult to fully understand when one considerstheir full expression. And yet,the nervous system controls movement with considerable competence and inexquisite detail. It seems that in order to handle that task the nervous systembuilds internal models of the process to be controlled and by passing thecurrent state of the system and the anticipated action through that model itcreates a pattern of efferent activity that induces the desired and necessaryactions in the physical plant of muscles and bones and occasionally otheranatomical objects, such as an eyeball or a gland. These internal models are built in patterns of neuralconnections, which may be called neural networks. These networks are apparently built initially according togenetic/developmental programs, but they are subsequently modified and tuned byexperience. Such adaptive orlearning networks are able to accomplish the actual calculation and performanceof the necessary movements by manipulating placement.

It is my personal intuition that as we study other systemswe will find that a great deal of the computation is handled by neural networksthat generate virtual surfaces subject to the particular constraints of theaction and anatomical movement reflects movements in that logical surface.  The gaze-muscle length surface for eyemovements is an example of such a logic surface generated by a neuralnetwork. It seems that theproblems that need to be solved on the fly during routine, everyday, anatomicalmovements are so computationally demanding that they can be accomplished onlywith such an adaptive system of anatomical connectivity within the nervoussystem. Examination of thecontrolled movements may suggest many interesting properties of the neuralnetwork, without actually exploring the physical plant of the movement system.

The Role of Context in the Nature of Anatomical Movements

One of the principles that emerge from a carefulconsideration of placement and movement is that the description of a movementmay depend upon the context in which it occurs. Spin and swing are prime examples of that principle.  Orientation and favored axes arecentral to most traditional considerations of anatomical movement.  However, they are usually onlyimplicitly defined and seldom dealt with explicitly. They are central to the approach introduced here.

Even the descriptors of the cardinal movements assume aparticular context. Flexion andextension are generally unambiguous, because the sagittal plane is usuallyapparent and the axis of rotation is perpendicular to the sagittal plane.  Right sideflexion of a vertebraimplicitly assumes that the point of reference is the rostral face of thevertebra, that the rotation is about an axis pointing dorsoventrally, and thatthe direction of rotation is the direction that one’s fingers curl if one’sthumb points in the direction of the axis of rotation.  Similarly, left lateral rotation isreferenced to the ventral surface of the vertebra and an axis pointingrostrally.

The situation becomes more ambiguous when the movement is nolonger in a cardinal direction. Rotation about an axis tilted midway between rostrally and ventrallypointing axes in the midsagittal plane is considered left lateral rotation andright sideflexion, whereas rotation about an axis tilted midway between arostrally directed axis and a dorsally directed axis is a combination of leftsideflexion and left lateral rotation. If the vertebra is tilted so that its ventral face looks caudally, thenrotation about a strictly rostrally pointing axis is also considered rightsideflexion and left lateral rotation. In fact, the situation is more complex than is implied by thosedescriptions. In order to see thefull implications of such rotations it is necessary to introduce orientationframes and to be more specific about the axis of rotation.

Orientation of the vertebra is implicit in all of thesemovements. That is particularlytrue when we consider combined movements, such as lateral rotation withsideflexion. It makes no sense tospeak in such terms unless one has orientation in mind.  However, orientation is not usuallymentioned when discussing these types of movements.

The context is usually the vertebral orientation, ratherthan the movement relative to the entire body. However, the interpretation of an anatomical movement maydepend upon the assumed frame of reference. Turning the head from side to side in a horizontal planewhile sitting in normal resting position will be a different combination ofmovements when referenced to the external world, the axis vertebra, or the bodyat the T1 vertebra. It is yet adifferent movement in the context of the semicircular canals, actually ratherlike the movement referenced to the T1 vertebra, which is also tilted about 30°ventrally or negatively about a transverse axis directed ipsilaterally.

As has been stated several times in the course of this book,it makes no sense to talk of left lateral rotation being in the same directionas left side flexion, as is often done. In fact, they are as little in the same direction as it is possible tobe; they are orthogonal to each other. Also, in a right-handed coordinate system, left lateral rotation is apositive rotation and left side flexion is a negative rotation.  In a left-hand coordinate system, thepolarities would be reversed, but still opposite.

Spin and Swing

Spin and swing assume a prime anatomical axis for the objectthat is moving. When the axis ofrotation of the object coincides with that anatomical axis, then the movementis called a spin. Otherwisethe movement is a swing. If thereference point for the center of rotation lies in the plane of the rotation,then the rotation is said to be pure swing. Pure swing implies an axis that isorthogonal to the axis of rotation, and the movement occurs in a planeperpendicular to the axis of rotation. Since any unitary conical rotation can be rewritten as a planar rotation,by shifting the center of rotation along the axis of rotation, there is not aninvariant spin or swing.

Often, the designated prime anatomical axis is notexplicitly stated, nor is the axis of rotation or the center of rotation.  Without that information, it is notpossible to differentiate a spin from a swing. Whether a movement is spin or swing depends entirely uponthe context in which it is being viewed. That is not to say that they are not useful concepts, only that one mustbe explicit about the context in which the movement is being interpreted.

Shortcomings of the Traditional Approach

The shortcomings of the traditional approach should havebecome evident from the previous sections. Too much is normally left unsaid, left to the reader’sassumptions about the reference orientation and the default axes ofrotation. The descriptions areexceedingly qualitative and minimally informative once we move beyond aconsideration of the cardinal movements.

It adds little to my understanding to say that a movement isa combination of right sideflexion and right lateral rotation.  It is not even clear what thatmeans. Sideflexion and lateralrotation have definite meanings, if we agree on the directions of the cardinalrotations, but precisely what does it mean to have a combination of the tworotations. Rotation can occur inonly one direction at any given time and if one is going to resolve it intocomponent rotations, what is the protocol? Is there a single protocol and, if not, will the differentprotocols give the same result? Ifthey give different results, how do we judge which one is the most appropriate?

Rotations can combine, as anyone knows who has riddencarnival rides that move about two of more centers of rotation at the sametime. However, at any moment, therotation is about a single axis of rotation. Also, one can move through a sequence of simple rotations toattain a particular location and orientation, but the experience is not thesame as a single rotation about a particular center of rotation of even amoving center of rotation. Thechapter on the application of the models of the lower cervical spine to finelydivided complex movements illustrated how different the sequence of simplemovements is from the smooth continuous movement.

A Language for Describing Anatomical Movements: Frames and Framed Vectors

The approach considered here was developed to circumvent themanifold limitations of the traditional approach and to introduce a languagethat allows an intuitive approach of great power and efficiency.  It is a language that moves smoothlyfrom anatomical description to the calculation of its consequences formovement.

The approach allows one to describe location, extension, andorientation unambiguously and rotations in such a manner that one literallycomputes with the anatomical description. The central element of the approach introduced here is the descriptionof the anatomy in a manner that leads easily and naturally to computation ofmovement. The basis of thatapproach is distinguishing the different spatial attributes of an anatomicalobject and codifying them in terms sets of vectors. The framed vector is the principal tool.  Location is differentiated fromextension and both are differentiated from orientation.  The distinctions are subtle, butimportant. Each transformsdifferently with anatomical movement.

A critical component of the approach is the use ofquaternions to express the axes of rotation and the rotations of anatomicalmovement. Quaternions operate uponthe vectors of the anatomical description to determine the manner in which theanatomy is transformed by movement and by including the axes of rotation aspart of the anatomical descriptions, one can concatenate multiple anatomicalobjects into a chain of interacting objects that move in particular wayssubject to the constraints imposed by their anatomy. The cervical spine provides a number of tractable problemsthat can be readily studied by such modeling and computation.

In the case of the lower cervical spine, it was possible towrite down a description of the elements of the spine and concatenate them intoa kinetic chain by specifying how they move relative to each other.  Simply by specifying the amount of movementin each joint one automatically obtains the configuration of the entirespine. Describing the uppercervical spine allowed us to ask questions about the anatomy of the region thatbore on the strain in the vertebral artery in any anatomical situation and therole of the alar ligament in restricting lateral rotation in the atlanto-axialjoint. The description of theeyeball with its attached extrinsic muscles allowed us to ask several questionsabout the actions of the muscles, the roles of the fascial slings in thesuspensory ligament and the control of eye movements.

The language of frames, framed vectors and quaternionsprovides a natural, intuitive, powerful language that allows analysis onmultiple levels from literal hand waving to very complex and exactcalculations. The precision of theoutput reflects the level of detail of the input. The anatomical description implies the anatomical movementsassociated with the anatomy.

While being quantitative can be daunting to many, theclarity of the answers is well worth the effort of creating thedescription. In addition, theresults can be expressed in three-dimensional images of the anatomy.  Since the conformation of the relevantanatomy is implicit in its description, one can move directly from thenumerical description of the anatomy to an image.

Soft Frames and Flow in Anatomical Materials: The Deformation of Boxes

In the latter part of the book, we considered a ratherdifferent use of frames to describe a different type of anatomical movement,strain and flow. By attachingdeformable frames to locations in a medium it is possible to describe theconsequences of particular anatomies for flow in the medium.  Because these frames are ratherdifferent from the rigid, non-localizable, frames of the first part of the bookthey are given a different name, boxes.

At this time, this area of inquiry is less developed thanthat concerned with rigid frames and I am still feeling my way by exploring thepossibilities of particular anatomies and constraints.  It is less formed, but worthconsidering as a different application of the basic principles that started theinquiry into anatomical movement. I think that it will give some interesting insights into the nature ofanatomical movements as we explore more situations.

This approach seems to lead naturally from a description oflocal properties associated with strain to global properties, such as flow andflow lines, changes in contour, and probable loci for fracture.  As with all quantifiable computationalmodels, the exercise of creating the description of the anatomy forces one toconsider many features of the situation that would otherwise slip below theradar. Often, those features thatare assumed to be understood in a qualitative model are highly relevant to anunderstanding of the anatomy. Notinfrequently, the assumptions turn out to be incorrect or the anatomy notprecisely as assumed. Forinstance, it is only when examined closely that one discovers that spin andswing are contingent upon unstated assumptions about the structure of themoving object.

In Conclusion

There are great benefits to be realized by being carefulabout the concepts that one uses and about the language in which one framesquestions. Often, once thosefactors have been considered, the solution of the problem flows readily fromits statement. The approachintroduced here is an attempt to develop those skills.

The process of developing a mathematical model is oftenilluminating in itself. It forcesone to consider the details that may be critical to understanding the processesinvolved. Hand waving as a way ofdealing with difficult concepts is no longer acceptable.  Mathematical modeling may eliminatepossible models that do not stand up to close scrutiny or which fail only whenrealistic numbers are substituted for its variables.

If the situation is described in an appropriate mathematicallanguage, then the conclusions may be expressed much more clearly.  A quantitative description will usuallylead to a quantitative solution. Aquantitative solution may be necessary to a clear understanding of a phenomenonunder study.

It is by actually computing the changes in frames ofreference that one comes to see that the process is more complex than isimplied by a statement that there is a combination of right lateral rotationsand left sideflexion. One isforced to think about what it means to say that there is sideflexion when theaxis of rotation is not in a cardinal direction or the object is not alignedwith the cardinal axes. One isforced to come up with a means of expressing the relative amounts ofsideflexion and lateral rotation. Qualitative labels may prove inadequate for fully expressing the natureof the movements.

We have clearly just scratched the surface in dealing withthe implications of this approach and have only the broadest notions of what itmeans to control placement or how a flesh and blood nervous system might goabout doing so in a flesh and blood organism. That is where the excitement lies in the future applicationsof these ideas to anatomical structure and its movements.