Muscle Set Surfaces as a Function of Joint Anatomy

Let us envision a generalized joint, as illustrated in theabove figure. It is a universaljoint with a spherical joint surface, so that all types of movements arepotentially possible, rather like the shoulder and hip joints.  Most joints are not that free.  In fact, the hip and shoulder jointsare not as free to move. However,we can consider most joints as special cases of this general joint.

We introduce a muscle that connects two bony offsets, one oneach bone (O_A onbone A and O_B on bone B). The offsets are attached to the bones are S_A and S_B, respectively.  The functional joint, J,is located at the center of the spherical facet. The value of each of these points is a variable and specificranges of values are characteristic of a particular type of joint. The geometryof a joint determines the special features of that joint.  A large part of the fascination instudying joint lies in seeing how their geometry determines their functionalcharacter. A detailedconsideration of the many possible variants is not appropriate here, becauserelationships between joint anatomy and joint function are complex, but a few specificexamples of different types of joints will be considered below.

Our present objective is to illustrate how a muscle setsurface may be computed for a set of muscles crossing a joint.  To start, we need to describe theanatomy with a set of framed vectors. The first defines the three points that are illustrated on bone A along with a frame of referencefor that bone. It contains alocation for the bone, L_A,and an attachment site, S_A,of the offset, O_A.   For Bone B we will use the location of the functional joint, J, as the location of the bone.

Muscle length is . It is thedependent variable of the muscle set surface. As described earlier in this chapter, the muscle moment isthe ratio of the origin to the insertion, relative to the joint.  In this instance, the muscle moment andits unit vector are as follows.

The axis of rotation for the muscle at the joint, , is the second element of the frame of reference for themuscle relative to the joint. Thefirst element is the unit vectorin the direction of the muscle, . Thethird element of the frame of reference, the perpendicular to the muscle fromthe joint, is the ratio of these two unit vectors.

Frame of Reference for a Bone Relative to a Muscle Attachment

The frame of the bone relative to a muscle (M) is an ordered set of three unitvectors: 1.) in the direction ofthe bone shaft (a),  2.) in the direction of the axis ofrotation from the bone shaft to the muscle attachment (b), and 3.)in the direction of the perpendicular from the bone shaft to the muscleattachment (g).

In this system, let us define the direction of a bone as theunit vector in the direction of the root of the offset.  In a multi-muscle system, we would haveto decide on a common direction for all the muscles, but any direction that ispicked may be expressed as a simple ratio to a particular direction and theroot of the offset has been chosen to lie on the ‘shaft’ of the bone in the modelthat we are considering here. Theplacement of a bone will be its location and its orientation, which will be setto include the direction of the bone, the axis of rotation from the bone to theoffset, and the perpendicular direction from the shaft of the bone to theoffset. In the current model, theframe of reference for a bone will be three ordered vectors.

The first component is the direction of the bone, the secondis the plane of the muscle attachment relative to the bone, and the third isthe perpendicular direction of the muscle attachment relative to the bone.

The Muscle Set Surface is an Invariant

The muscle set surface is an invariant for the geometry of ajoint/muscle system. It may beexpressed in a form that does not depend upon the particular placement of thesystem because one can always rotate and translate the joint configuration sothat one bone, say bone A, is in astandard location and orientation and then the same transformation applied tothe moving bone, bone B, will bringit along in its original relation to bone A.  All possible configurations of bones A and Bcan be realized in such a standardized musculoskeletal system.  It is a canonical image of the system.

In such a canonical system, a muscle set surface can beexpressed with complete generality as function of joint movements relative to aneutral placement. Muscle lengthis not affected by moving the system into a standard form, because internal spatialrelationships are unchanged by a rotation of the system as a whole.  On the other hand, muscle length variesas a function of movements from a neutral placement of the joint, irrespectiveof the orientation of the musculoskeletal system as a whole.

Since the movement will be the same for all muscles crossingthe joint, the complete muscle set surface may be computed and plotted againstthe same independent variables. The complete muscle set surface is the combination of the collection ofindividual muscle set surfaces.

Note that muscle force is not an invariant.  The muscle set for a shoulder remainsthe same irrespective of the orientation of the shoulder, but the forces neededto hold shoulder in a configuration may be very different, depending upon theorientation of the shoulder. Theset of muscle forces required to abduct one’s shoulder 90° in standing is quitedifferent from set of muscle forces needed to perform the same movement whenlying on one’s side or back. Theset of muscle forces is clearly not an invariant for the geometry of ajoint/muscle system.

The Calculation of Muscle Set Surfaces

We will now consider a small number of examples of thecalculation of a muscle set surface for a single muscle, plotted againstrelative bone placement in a joint that the muscle crosses.  We will consider two simple muscleconfigurations and then a more complex muscle configuration, with multiplecomponents that act differently, but in a coordinated fashion.

In each case the calculation is essentially the same.  The location of the muscle insertionfor neutral placement, O_I , relativeto the functional joint, J, iscomputed.

The muscle insertion is rotated about the longitudinal axisof bone B, through an angle , and then about a transverse axis perpendicular to thelongitudinal axis that will cause bone Bto flex or extend upon bone A,through an angle . Inkinesiological terminology, the moving bone spins about its longitudinal axisthrough an angle of and then swingsabout a transverse axis that moves with the bone, though an angle of . The initialtransverse axis is perpendicular to the plane of the muscle and the bone, the axis of the frame of reference for the bone relative to themuscle attachment.

The muscle vector, , is the difference between the new location of the muscleinsertion and the location of the muscle origin. It is the length of that is plottedversus the placement of bone B.

Rotating bone Babout its longitudinal axis once more, after the computed movement, will alterthe placement of bone B by changingits orientation. However, thatoption will not be used here. Therefore, the placement will have its orientation determined by thelocation and the orientation will have null spin relative to neutralplacement. The advantage forpresent purposes is that placement has only two dimensions, allowing us to plotthe muscle set surfaces as two-dimensional surfaces in a three-dimensionalspace.

There are other options for creating an array of placementsof bone B. The one sketched herewill give an array that is like the lines of longitude and latitude on aglobe. In fact all of the surfacesplotted below are for a hemisphere of movement. This system seems to be a natural array for a universaljoint. Other, more restricted,joints might warrant a different type of array. In which case, the rotation quaternions might be constructeddifferently.

A Long Muscle With Its Insertion Near the Joint

The muscle extends from an origin near the proximal endof bone A to an insertion near thejoint on bone B.  Both offsets are aligned in neutralposition.

For a first example, consider a muscle like that illustratedabove to illustrate the concept of a generalized universal joint, where theorigin is far from the joint and the insertion is near it.  In particular, let the origin be 0.2units from the proximal end of bone A,on an offset of 0.1 units off the axis of the bone, and let the insertion be0.1 units distal to the universal joint in bone Bwith an offset of 0.1 units in the same direction as the offset on bone A. The joint is constructed to have a radius of curvature of 0.1units. Consequently, in neutralconfiguration the muscle has a length of 1.0 units.

The muscle set surface is fairly simple.  There is small concavity in the surfacecentered upon neutral position. Asbone B is laterally rotated through90° in either direction, there is a subtle lengthening of the muscle, so musclecontraction will have a modest tendency to bring the two offsets intoalignment. However, the much greatertendency with shortening of the muscle will be to flex the joint until it isbent about 130°, beyond which the muscle will become longer with furtherflexion and the moving bone will tend to roll laterally.  This behavior accords with ourintuitive impression of what such muscles do.

This arrangement seems to be well designed for situationswhere a large joint excursion into flexion or extension is required.  It gives large movements with modestamounts of muscle contraction. Allthe flexion movements converge on a common placement, just as the lines oflongitude converge upon the poles.

A Short Muscle With Its Origin and Insertion Near the Joint

In a second example, the muscle origin is moved distallyuntil it lies just proximal to the universal joint and the muscle insertionlies just distal to the joint and rotated laterally through 90°.  The proximal offset is 0.1 units fromthe axis of the bone and the distal offset is 0.2 units.  In words, the muscle wraps about aquarter of the way around the joint.

The muscle set surface for a muscle that crosses theuniversal joint diagonally between an origin and insertion where both are closeto the joint.

As one might expect, the muscle becomes longer as the jointis rotated so as to increase the angle between the offsets and it shortens whenthe angle is reduced until they are aligned. Flexing the joint also reduced the length of the muscle, butgenerally not as quickly. Musclecontraction that produces a rotation that brings the offsets into alignment andflexes the joint is the movement that causes the greatest shortening of themuscle. However, the extent towhich the muscle can bring about that movement is restricted by the gap betweenthe insertions when the offsets are aligned being smaller than the muscle canachieve. The maximal contractionof a muscle from greatest to shortest length is probably 50%.  That means that the muscle cannot moveinto the nearest corner of the surface in the illustration, because the gap ison the order of 40% of that in neutral position and the muscle must be able tobecome longer than it is in neutral configuration if the joint is able to turnlaterally in the direction that opens the angle between the offsets.  Consequently, only that part of thesurface that lies above 0.6 is likely to occur in a real system and the rangemay be substantially less.

As a result of these considerations, such muscles will tendto be important for laterally rotating a joint. They are most stretched and shortened by such movements andthey are comparatively insensitive to flexion and extension.  In this particular geometry, the amountof flexion is comparable to the amount of lateral rotation with muscleshortening.

A Deltoid-like Muscle

Next, consider a muscle that is in many ways like thedeltoid muscle of the shoulder. Wewill consider the muscle in terms of three component muscle descriptions thatrepresent different aspects of the muscle. The first component, the middle component, extends from anoffset that directly overhangs the joint to an insertion some distance down theshaft of bone B.  The joint is considered to be inneutral position when bone B isextended 90° relative to bone A.  The other two components of the musclediffer in having their origins anterior and posterior to the joint as well as proximalto the joint. One might imaginethe offset from bone A to have theshape of a horseshoe lying in a horizontal plane above the joint.  The insertion for all three componentsof the muscle will be at the same point on bone B.

A deltoid-like muscle takes its origin from an offsetring above the joint and it inserts into the shaft of the moving bone.  Three muscle components are drawn: oneat the apex of the offset that runs directly down, one that takes originanteriorly and one that takes origin posteriorly. They have a common insertion.

Bone B is rotatedabout its long axis through a series of angular excursions from -90° to +90°and then about an axis perpendicular to the plane that contains the shaft ofthe bone and the offset, again through a series of angular excursions from -90°to +90°. In each location bone B could be again rotated about its shaft,to give a variety of orientations, but that movement tends to have only a smalleffect on muscle length, so it is not been explored here.

In the following calculations, if the functional joint istaken to be the origin of the coordinate system, then the origins and insertionof the three components are taken to be at the following locations.

The valuesare approximations from actual shoulder joints, where the radius of curvaturefor the spherical facet is set equal to 0.1 units. One usually obtains the best results when using the valuesapproximately equal to actual anatomical values, because they usually give the bestcompromise of all the possible values. By choosing values that differ from the anatomical values, one can oftendiscover why the anatomical values are what they are.

The following figure shows the geometrical relations of thefirst component of the muscle. Itresembles the first illustration of this section, that for a long muscle thatjust crosses the joint, but it is different in a number of interesting ways.

Muscle set surface for the middle component of adeltoid-like muscle where the neutral point is chosen with the arm pendant.

Rotation about the long axis of bone B in a pendant position leads to minor lengthening of themuscle so contraction of the muscle will tend to move the bones towards neutralconfiguration. That effect is morepronounced as bone B is abducted(moving towards the left in the illustrated surface). The trend is reversed when there is more that 150° ofextension, however, anatomical joints would not normally support that muchextension. In actualglenohumeral joints, the range ofabduction is on the order of 60° to 90° of abduction and the range of medialand lateral rotation are usually less than 90° from neutral placement(Kapandji).

The more pronounced geometrical relationship is the changein muscle length as the bone B iselevated. The return oncontraction becomes less as the joint approaches 150° of elevation, but itremains the dominant consequence of muscle contraction throughout the entirephysiological range. Consequently,the contraction of the central component of this deltoid-like muscle, workingalone, tends to lift the arm directly laterally. All the contraction vectors are directed towards the centralmeridian through neutral placement and elevation.

That raises the question of what one gains by having theanterior and posterior components of the muscle. Because their arrangement is symmetrical with respect to thebones their muscle set surfaces are also symmetrical.

Muscle set surfaces for the anterior and posteriorcomponents of the deltoid-like muscle constrain the ability of a bone to movein the opposite direction and pull it towards the same direction at the origin.These two surfaces are mirror reflections of each other in the coronal planethrough neutral placement.

The anterior and posterior components are more directed atbringing bone B forwards and backwards. Rotation of the bone about its long axis will moderately lengthen amuscle for rotation in one direction and shorten it for rotation in theopposite direction. When the boneis rotated so as to lengthen the muscle, the muscle is not very effective inelevating the bone until it is already elevated about 50°.  On the other hand, when the bone isrotated so as to shorten the muscle length, the further shortening of themuscle will work to further rotate it about its axis in the same direction andto elevate it. Since portions ofthe muscle take origin anteriorly and portions take origin posteriorly, thereis an effective aid to elevation in all directions. The portion of the muscle that is contracting will act topull the moving bone further in the same direction. By resisting lengthening, an eccentric part of the musclemay prevent movements from moving into a substantial portion of the potentialmovement range. Consequently, thesecomponents may act as brakes upon movements in the direction away from theiroriginal placement.

The three component muscle set surfaces consideredabove are plotted together and viewed from a different viewpoint.

The muscle set surface for a deltoid like muscle iscomplex. In the above figure thethree surfaces that were considered individually are plotted in the samecoordinate system to illustrate their differences and how they might workcooperatively in the control of the joint. In fact, the complete muscle set surface for thedeltoid-like muscle is a stack of such surfaces, a sheave, as it were.  In the figure, we see two end sheetsand the median sheet in the sheave. In addition, as mentioned above, there are other options for orientationat each location of the bone, so each of these surfaces would extend into athird placement dimension, rotation about the axis of the bone.  So the true surfaces are actuallyvolumes in a four-dimensional space. Those volumes would have a three-dimensional mesh, like the lines drawnon the surfaces that are illustrated here, so that one would follow definitepaths through the volume surface as the placement changed along meridians oflocation and orientation.

Such surfaces are complex. In most instances, three or more dimensions of placement aswell as the muscle lengths for several muscles or muscle components.  They are usually beyond our ability toreadily visualize or comprehend in their entirety. However, with judicious simplification, one can often learninteresting things about how a muscle functions, how its geometrical anatomyinfluences its functioning, and why the muscle takes the form that itdoes. There is not space here todelve deeply into these ideas, but they warrant a separate consideration atgreater length, elsewhere.