Anatomical Descriptions ThatCompute Functional Attributes

Goal: To write adescription of an anatomical structure that leads directly to the calculationof its functional attributes. For instance, an anatomical description of ajoint may embody its range of motion or a description of the muscles andligaments around a joint yields the relationships between muscle length andjoint configuration. The functionsof interest are expressed in terms of movements. The movement may be an internal motion, such as strain, orand external motion, such as ranges of motion and movement trajectories.

The Concept of Placement of Anatomical Objects

At the foundation of this analysis is the fact thatanatomical structures are orientable. They may be described so that theirarrangement in space and relative to space are unambiguously determined.  A right hand can be differentiated froma left hand and the spatial location and orientation of either can becompletely specified.

Anatomical description is expressed in terms of location,extension, and orientation. Eachof these attributes has a precise meaning, a manner of specification, anddefinite rules for its manipulation during movements of all types.

Location is wherethe object is in space. Generally,it is specified for some particular point in the object, such as a bonyprominence or the center of a joint. Location is changed by translation, rotation, and re-scaling. It is expressed as a vector relative toan origin.

Orientation is howan object is aligned with its surroundings. It is expressed as three mutually orthogonal unit vectorsthat form a basis. Such a set oforientation vectors is called a frame of reference. Orientation is changed by rotation, but not bytranslation or re-scaling.

Orientation can be specified by only two quaternion vectors,with a prior agreement about the handedness of the coordinate system.  The third vector can always be computedas the ratio of the other two unit vectors, if one knows which way the ratio isto be assessed.

Location does not have orientation and orientation does nothave location. While all theelements are vectors and they may be expressed in terms of a common set ofcoordinates, they occupy different spaces.

The lumbar vertebra is described by a framed vector, , where the location, l,is relative to an origin, O, theextension from the location of the vertebra to the right facet joint is e, and the frame of reference for the orientation is given bythe ordered set of vectors [r, s, t]. All of the vectors of the framedvector and the coordinates of the space {x, y, z} may be defined in terms of the basis vectors {i, j, k}.

Extension is theinternal organization of an object, its length, depth, and width relative tosome reference point, usually the object’s location. While extension shares attributes with location andorientation and can be expressed in terms of locations and in terms oforientation, it is different from both. An object’s shape and/or conformation of its internal landmarks can beexpressed by a set of extension vectors. Extension is changed by rotation and re-scaling, but not by translation.

Placement is theterm used to indicate both the location of an anatomical object and itsorientation. Any non-null movementchanges placement.

In addition to placement, an anatomical object may haveextension. Placement and extensionmay be expressed as a framed vector. A framed vector is a locationvector, a set of extension vectors, and a frame of reference for theorientation. However, it is aloosely defined entity that can change to meet almost all situations fordescription.

The Concept of Joints

We develop a concept of joints, a concept similar in itsusual usage, but with a more general meaning. Some of the objects studied as examples of joints are notusually called joints. Forinstance, the eye in its orbital socket will be a particularly useful joint forexploring a number of lines of reasoning and developing some analytic methods.

A joint is ajunction of two orientable elements with a common axis of rotation. The jointis defined by a transformation between the orientation of the pre-joint elementand the post-joint element (Scapular Frame and Humeral Frame 1). There is also a transformation betweena joint element before moving and after moving (Humeral Frame 1 and HumeralFrame 2). Both types of transformations may be expressed as a rotation about anaxis of rotation, therefore they can be expressed by a quaternion.  To fully define the joint it is alsonecessary to compute the location of the axis of rotation.  That is a somewhat more complexcalculation that requires finding the axis of rotation that also transforms thelocation of the moving element before the movement into its location after themovement.

Usually the location of a functional joint is its axis ofrotation, which is seldom in the anatomical joint. It is often within one of the moving anatomical objects thatform the joint. However, inmulti-joint systems, like the cervical spine, the location of the functionaljoint may be outside all of the constituent elements.

The Concept of Quaternions

In general, the ratio of two frames of reference is aquaternion. This is equivalent to saying that theratio of two orientations is a rotation. It is also the case that the ratio of two oriented planesin their intersection,which is a quaternion.

A change of placement cannot always be expressed as aquaternion, but it may be expressed as a combination of a translation and arotation, therefore by a fixed vector and a quaternion.  Such movements are called compoundmovements.

A quaternion is a hypercomplex number, like a complexnumber, but with three imaginary components, i, j, and k. It is written as follows, all the coefficients being real numbers.

Quaternions add and multiply algebraically with the caveatthat i, j, and kare three different imaginary numbers that multiply as follows.

Despite the weirdness of number composed of three differentimaginary numbers and a real number, quaternions turn out to be ideal for thedescription of rotations in three-dimensional space. The three orthogonal directions of space are labeled withthe three imaginary numbers so that vectors formed by combining multiples ofthose three imaginary numbers are vectors in the space.

It is worth noting that quaternions include real numbers,that is, the scalar of a quaternion, and complex numbers and they add vectors,that is, quaternions with a null scalar. The concept of vectors started with quaternions and it was simplifiedwhen vector analysis was created. Quaternion vectors are subtly different from vector analysis vectors,which means that there are some things that can be done with quaternion vectorsthat are not legal with vector analysis vectors. One of the most powerful concepts of quaternion analysis isthe ratio of two vectors. Itunderlies almost everything that is presented here.

The ratio of two vectors is a quaternion

It is convenient to operationally define a quaternionas the ratio of two vectors. The vector ofa quaternion is the vector perpendicularto the plane that contains the two vectors that points in the direction of thethumb on a right hand that has the fingers curled in the direction that carriesthe denominator of the ratio into the numerator. The tensor of a quaternion is the ratio of the length of the numerator to thelength of the denominator. The angleof a quaternion is the angular excursionfrom the denominator to the numerator. The unit vector of the quaternion is thedirection of an axis of rotation. The angle of the quaternion is the angular excursion of a rotation aboutthat vector. The tensor of thequaternion is a re-scaling .

If a quaternion is defined in a space with the basis vectors{i, j, k} then the quaternion may bewritten as follows.

It follows from its definition of a quaternion that thequaternion operating uponthe vector is the vector .

The cosine term is the scalar of the quaternion, . It is a realnumber.

The sine term is the vector of the quaternion, , and it may be expressed as a sum of multiples of the basisvectors.

This means that the quaternion may be written in thefollowing form.

This will be called the rectangular form of the quaternionand the expression in terms of a tensor, angle, and unit vector will be thetrigonometric form. As withcomplex numbers, the two forms are useful in different contexts, so each willbe used frequently and interchangeably. Also, as with complex numbers, there is an exponential form, but therewill be little occasion to sue it.

Conical Rotations

Rotation of the vector aaround the vector of the quaternion R sweeps out a conical surface to yield the vector a’.

The power of quaternions is that any vector rotated about anarbitrary axis of rotation through anangular excursion is equal to the following expression.

Such a rotation is called a conical rotation because the rotatingvector sweeps out a conical surface. Conical rotations are the more usual type of rotation in most contexts,because if the axis of rotation and the rotating vector are chosen at random,the chances of their beng orthogonal are essentially nil.

Still, there are situations in which the definition of aquaternion as the ratio of two vectors, both orthogonal to the vector of thequaternion, will be extremely useful. For instance, when computing the ratio of two orientations it isnecessary to break the operation down into two component rotations, each ofwhich is precisely the ratio of two vectors in a plane.

Anatomical Descriptions That Embody Functional Consequences

These are the concepts that form the foundation foranatomical descriptions that compute functional implications.  The placements and extensions ofanatomical objects are expressed as framed vectors that express the relevantattributes of an anatomical object or objects. Often, we are interested in describing multi-joint systemsin which there are multiple elements that have definite constraints upon theirmovements because of the geometry of the joints that join them.  We can express each of these objectsand the relations between them in a concise language that embodies theinformation and the operations that are required to compute the movements thatmay occur in the system.

Although a detailed exact calculation of the consequences ofthe anatomy may be computationally complex, the concepts that are developedhere can be used in an intuitive, qualitative, manner to address many questionswith no calculation at all or simple back of the envelope types of estimation.

It may be noted that no actual calculations have beenpresented here. To do so effectively requires a bit more background, but allthe necessary foundations are laid out elsewhere along with a great manyapplications.