How Ligaments RestrictMovement

Tethering may simply prevent bones from moving more than acertain distance apart, that is, to restrict active or passivetranslation. For instance, if ajoint is distracted by pulling on one of its bones while stabilizing the other,then the separation is ultimately limited by the ligaments.  The cruciate ligaments in the kneejoint operate largely by restricting anterior posterior translation of thefemur upon the tibial plateau, but the consequence is mostly to restrict rotationin the joint. The role of mostligaments is to restrain rotation, because that is largely what joint areabout.

If rotation occurs, then the ends of a ligament linking thebones will move relative to each other. The gap between the ligament attachments may increase or decrease in amanner that depends upon the geometry of the joint. When the bones move so as to stretch the ligament, then themaximal separation is the length of the ligament. That is the ligament’s principal constraint upon movement.   When the ligaments are maximallystretched there is usually an abutment between the two bones that fixes thejoint in the sense that it cannot move any further in that direction.  In this way, tethering and abutmentsoften operate together. For instance,in the following figure, bone 1 and bone 2 have rotated about axesperpendicular to the page until they are stopped by their abutment.

A tethering ligament and abutment is abstracted to atriangle with three pivot points. The joint is locked by the rotations indicated by the purple-headedarrows.

There are three loci critical to the fixation of the joint,the two attachments of the ligament (p₁and p₂) and theabutment (p₃).  This arrangement may be abstracted asthree linked points. Each point isa potential pivot point. Theattached bone can pivot about its ligamentous attachment and the two bones canpivot about the point at their mutual contact. We will say that such a joint is locked in that it can moveno further in the direction that it followed into the lock.  It can still move in many otherdirections.

The fact that the three points form a triangle is important,because triangles with sides of fixed length are rigid.  Such triangles are uniquelydetermined. There is no othertriangle that has the same three sides. That would not be true of a rectangle,which can be sheared without changing the lengths of the sides.  In everyday life, the rigidity oftriangles is the principle of structural trusses, which are used to buildstructures that need to be rigid, like roofs and bridges.

The triangular arrangement of the three critical loci isalso important because it determines what movements can occur in thejoint. For instance, the thirdpanel in the above figure indicates the directions of rotation that lock thejoint. At point 1 the axis ofrotation is into the page and point 2 it is out of the page.  The two torques at the abutment areequal and opposite to those at the ligament attachments.  Nothing is moving, so the joint islocked, but only for movements that are in the indicated directions.  Obviously, rotation in the oppositedirection will unlock the joint and it turns out that many other rotations arealso permitted. For instance,rotation about the axis of the ligament is permitted and rotation about ahorizontal axis through a ligament attachment is also permitted.  In this instance, with a littlereflection, one can see that the set of rotation axes for the permittedrotations occupy a hemisphere in the direction opposite to the direction of theaxis of rotation for the locking rotation. Consequently, there are a great many rotations that willunlock the joint.

The Formal Constraints Upon Movement

To discuss thepermitted axes of rotation formally, it is convenient to generalize thedescription of a joint limited by a ligament and an abutment.  Let us start with two bones that arelinked by a ligament. The ligamentattaches at points p₁ andp₂ and it hasa maximal length of k.  The points i₁ and i₂lie in the articular surfaces of bones 1 and 2, respectively.  If a rotation is indicated by thesymbol , then a rotation is permitted if the following conditionsare satisfied.

The first two expressions simply say that a ligament cannotbe longer than its maximal length and bones cannot overlap (there is no pointthat lies in both bones). The nextfour lines say that there can be no part of the moving surface that lies withinany other bone after the rotation.

The local surface geometry determines the allowed rotations.That is partially because the local geometry determines the center of rotationfor the surfaces. Commonly, atleast one of the surfaces is convex as with a sphere, an ellipse, or a cylinderand the local curvature has a center of rotation that is the effective centerof rotation for that surface. Inthe knee joint, the condyles are each like an ellipsoid and lying side-by-sideas they do, they enforce an axis of rotation that lies some distance superiorto the condylar surfaces, within the femur. The menisci reinforce and amplify shallow depressions in thetibial facet that help to guide the movements of the femur upon the tibia, butthey would not be very effective without the guidance of the ligaments the lie aboutand within the joint. Let usconsider a situation similar to the medial and lateral collateral ligaments.

The joint is limited by a ligament with a maximallength of k and attachment sites p₁ and p₂. The two points i₁and i₂ are in thesurfaces of bone 1 and bone 2, respectively, which occupy the volumes V₁ and V₂.

A single ligament is not particularly restrictive, becauseof the many ways in which the joint may move without violating the conditionslaid out above. Two ligamentsforce the permitted movements to satisfy two sets of constraints, which maygreatly reduce the options.

Consider the following situation. Two ligaments are set to ether side of a joint so that theyare parallel and they have a common abutment. Let them be designated as L1and L2 and let them have the common abutment . Thereis a fixed relationship between the ligamentous attachments to a bone and theabutment point for that bone. Letthe vector from the attachment of L1 to the attachment of L2 be . We write thelocation of the abutment in a common coordinate system centered upon theattachment for L1.   For an arbitrary rotation, , the new location of the abutment point () is given by two expressions.

The only rotation that satisfies both conditions is about anaxis () in the direction of the vector that joins the attachmentpoints (). Clearly, theligaments have a substantial role in restricting movement.  If the ligaments are not maximallystretched, then there is more room for variation in the axis of therotation. However, most ligamentsare situated so that the gap that they span varies little in length. And thegap is normally close to the maximal length of the ligament.

Ligament Arcs and Center of Rotation Arcs

Ligaments restrict movement in joints by only allowingmovements that do not produce a gap between their attachments that exceeds themaximal length of the ligament. That mean that that, if we fix one end to the ligament, then allpermissible movements keep the other end of the ligament within the arcgenerated by swinging the maximal gap about the fixed attachment.  That will be called the ligament arc.

The movements of the mobile end of a ligament areexamined when the center of rotation of the moving bone is collinear with theligament and when it is eccentrically placed. Only cases where the center of rotation lies on the sameside of the joint as the moving bone are illustrated. P₁ andP₂ are the attachmentsites for the ligament. Theligament arc is the arc that the moving attachment traces when the ligament ismaximally stretched. Allpermissible movements leave the moving attachment with that arc. C₁and C₂ are possiblecenters of rotation. S₁ and S₂ are the arcs of the surfaces that would havethe same center of rotation. Movements about each center of rotation will cause the mobile attachmentto follow another arc, which is its center of rotation arc (COR arc). The COR arcs aredrawn for a maximally stretched ligament.

As the bones move by rotating about a center of rotation,the mobile ligament attachment is also moved. If the center of rotation lies between the two ligamentattachments (C₂ in thefollowing figure), then the ligament is swung on an arc that that has a shorterradius of curvature than the ligament arc. That means the there is no restriction on the movement dueto the ligament. The arc followedby the ligament attachment will be called the center of rotation arc (CORarc).

As the center of rotation approaches the stable attachment,the COR arc will approach the radius of curvature of the ligament arc.  If the axis of rotation passes throughthe stable attachment and it is perpendicular to the long axis of the ligament,then the COR arc will be coincident with the ligament arc for a fully stretchedligament. If there is slack in the ligament, then the two arcs will beconcentric.

In the above figure, the center of rotation is constrainedto lie in the moving bone, because the supporting bone has a flat surface.  Consequently, the case of a center ofrotation in the stable bone is not illustrated.

The permissible movement is the excursion between theactual gap (Gap) and the maximal gap(Maximal Ligament Length).  The location of the mobile ligamentinsertion in neutral position is the green dot, but the maximal ligament lengthis the red dot and its arc. Thecenter of rotation is the blue dot. The permissible angular excursion is the blue arc. Bone1 is drawn in neutral position and the two extreme positions.

If the center of rotation lies beyond the stable attachmentof the ligament, then the radius of curvature of the arc of the mobileattachment is greater than that of the ligament arc and the arc isflatter. If the ligament is notstretched to its maximal length, then movement is possible until the center ofrotation arc intersects the ligament arc. That excursion may be substantial since the arc is going to becomparatively flat and it takes an almost tangential approach to the ligamentarc.

In general, ligaments allow the most movement in directionsperpendicular to the long axis of the ligament. So, one can often deduce the permitted or favored movementsfor a ligament by constructing the spherical shell with its center at thestable attachment site.

If the center of rotation of the bone carrying the mobileligament attachment is beyond the mobile attachment, then the center ofrotation arc is curved in the opposite direction. That means that unless there is some slack in the ligament,the bone will not be able to rotate. If there is slack, then the rotation will occur until the center ofrotation arc intersects the ligament arc.

If the center of rotation is not in the plane of theligament attachments, then the situation changes in an interesting way.  There is a direction of rotation thatallows the joint to open and a direction of rotation that will lock thejoint. The direction of rotationthat allows the joint to open depends on which side of the ligament the centerof rotation lies. In theillustration, rotation about an axis that comes perpendicularly out of the pagewill tighten the joint. However,if we move C₂ to theother side of the ligament, then the joint will open with the same rotation.

An Example: The Collateral Ligaments of the Knee

The knee is a complex joint with many interacting componentsthat constrain it to move in particular ways. We will not consider all of the features of the knee at thispoint, but it does provide an interesting context in which to explore some ofthe ideas that have just been introduced.

Drawing of the knee joint viewed from laterally andmedially, showing the bones and the collateral ligaments and the approximatetrajectories of the centers of rotation.

The two bony elements of the knee are the distal end of thefemur and the proximal end of the tibia. The tibial condyles are relatively flat, but with a slight ellipticaldepression for the medial facet and a saddle-shaped surface for the lateralfacet. The lateral facet isconcave in the medial-lateral axis and slightly convex in theanterior-posterior axis. Themedial condyle is more elliptical and longer. The flatness of the tibial facets is partially compensatedfor by the menisci that fit between the femoral and tibial surfaces,providing moveable sockets for thefemoral condyles to sit in.

The femoral condyles are most relevant to our purposes,because they are strongly curved and their shape reflects the mechanics of thejoint. The joint surface on thedistal femur takes the form of two ellipsoidal condyles with the long axesdirected anterior-posterior and convergent anteriorly.

The articular surface is actually two joint surfaces.  On the anterior face of the femoralcondyles is a single surface with medial and lateral elevations thatarticulates with the posterior surface of the patella.  That part of the joint will not berelevant to what follows.

The posterior half of the condyles is split by a deep cleft,so that there are two separate articular surfaces. The two condyles are similar, but not identical.  For instance, the medial condyle islarger. The condyles areellipsoidal, but with a curvature that flattens as one approaches the anteriorpart of the tibial articular surface. Posteriorly, the articular surface wraps around to cover the posterioraspect of the condyle, so that the knee can flex through more than 90°.  Passive flexion is estimated to beabout 160°, active flexion more like 120-140°, depending on the orientation ofthe hip joint and the amount of muscle that comes between the femur andtibia.

Because of the variation in curvature of the femoralcondyles, the center of rotation changes as the joint flexes and extends.  In the following figure, the tracks ofthe centers of rotation are sketched against the anatomy.  For flexed postures the centers ofrotation lie posteriorly and near the articular surface, but as the joint nearsfull extension the center of rotation shifts anteriorly and sharply superior.

The figure also shows the locations and distributions of theattachments for the medial and lateral collateral ligaments.  The green lines indicate the locationsof the ligaments. Both ligaments are nearly vertical and attached posteriorlyon the femoral condyles. Thetrajectory of centers of rotation passes across both proximal attachment sites.

In the above figure the pertinent features of the femoralcondyles have been abstracted. Thecenters of rotation are approximated by drawing a series of perpendiculars tothe articular surfaces. Whereadjacent lines meet is a reasonable estimate of the center of rotation for thepart of the articular surface that lies between the perpendicular lines.  The centers of rotation cluster nearthe proximal attachment sites for both collateral ligaments when consideringthe posterior part of the surface, the part that comes into play with largeflexion movements. As the knee isextended, the centers of rotation move anteriorly and up the shaft of thefemur.

The red curves are segments of a circle centered upon theligament attachment site, with a radius equal to the distance to the mostposterior aspect of the condyle. Both surfaces have an approximately circular sagittalcross-section. The medial condyleis relatively deeper than the lateral condyle.

If we superimpose the two profiles, the two condyles can beseen to be remarkably similar. Theattachment sites for the collateral ligaments are approximately aligned and thecenters of rotation are clustered along a common trajectory.  There are differences and they arerelvant for the locking of the knee when it enters terminal extension and rotatesslightly on a longitudinal axis, but, to the first approximation, we have tworockers that rotate about a common, approximately horizontal, axis.  The axis of rotation is set by thecenters of rotation, which all cluster around the proximal ligamentattachments. In flexion, the relative placements of the centers of rotation andthe ligament attachments indicates that the ligaments are unlikely to restrictmovement, but they may guide it, by preventing rotations that deviatesignificantly from an axis through the proximal ligament attachments.  As the joint approaches full extension,the centers of rotation move away from the ligament attachment sites andanteriorly. As you can easily see,rotations about those centers of rotation into extension (axis of rotationperpendicularly out of the page) will cause the attachment site to moveproximally, that is, stretch the ligaments. Consequently, the collateral ligaments restrain movementinto extension and they do so as the knee approaches full extension.  Also, because the point of contact liesanterior to the ligament attachments, there is an abutment due to the ligamentsbeing maximally stretched and the femur being unable to move any further intoextension.

It should be noted that there are a number of otherligaments that reinforce the restriction. One way to get around the restriction might be to slip the femurposteriorly, so that the abutment lies in the same vertical plane as theligaments. That is prevented by the need to make the ligaments longer to doso. First, moving the femur backwould make the ligaments oblique, therefore longer. Secondly, the socket on the medial side is concave so movingthe medial condyle posterior will move it up that slope and further from thedistal ligament attachment. However, both of those restrictions may be overcome by stretching thecollateral ligaments. Anotherrestriction is the anterior cruciate ligament, which extends from the anteriormargin of the tibial plateau to the posterior medial aspect of the lateralfemoral condyle. Moving the femurposteriorly will strain that ligament and so the anterior cruciate ligamentrestrains extension. In a fullyextended knee the femur cannot move further posterior, because, to do so wouldstrain the cruciate ligament in the direction where they have the least give,along the longitudinal axis of the ligament. We will briefly consider the geometry of the cruciateligaments in the next section. Finally, the posterior part of the joint capsule that overlays theposterior parts of the femoral condyles is thickened so that at full extensionit is pulled taut and will not allow the femoral condyles to move moresuperiorly. The cruciate ligamentsand the posterior joint capsule are different than the collateral ligament in thatthey do not guide the movement through its full excursion, but check it at itsextreme.

The Cruciate Ligaments

The cruciate ligaments lie in the center of the knee jointand as their names imply they cross each other. The anterior cruciate is attached to the anterior end of theelevation between the tibial condyles and it reaches posteriorly and superiorlyto attach to the medial aspect of the lateral condyle, just above the articularcartilage. The posterior cruciateligament attaches to the posterior rim of the tiba, between the tibial condylesand stretches anteriorly and superiorly to attach to the lateral aspect of themedial condyle, just posterior to the cartilage of the patellar facet.  Their approximate locations have beendrawn on the medial view of the knee joint.

When the knee is flexed, the center of rotation is near theposterior attachment of the anterior cruciate ligament, probably a bit closerto the tibial surface than the center of rotation. That might cause problems because the COR arc is orthogonalto the ligament arc, but the movement into flexion makes the gap between theattachments of the anterior cruciate ligament much shorter than the maximalligament length, so the anterior cruciate ligament does not constrainmovement. Flexion carries theposterior cruciate ligament into a more vertical orientation, so it may bestretched, but, since greater than 90° of knee flexion is routine, the ligamentis normally long enough and so placed that its attachments do not separate bymore than the maximal length of the ligament. However, the ligament may contribute to the posterior shiftof the femur upon the tibial plateau as the knee goes into flexion.  A posterior shift would bring the twoattachments closer together. Sincethe centers of rotation in flexion are between the two attachments, the COR arcwill have a smaller radius of curvature and therefore the ligament will not besignificantly stretched by either flexion or extension.

Extension is more interesting because the cruciate ligamentsare important elements in the fixation of the knee. The situation in endrange extension is illustratedabove. The gap between theattachments of the anterior cruciate ligament is near or at its maximum.  Consequently, the ligament will pullthe femur anteriorly and inferiorly. Further extension will further stretch the ligament, so the anteriorcruciate ligament restricts extension. The posterior cruciate ligament is probably initially less than fullystretched and extension will tend to decrease the gap between itsattachments. As a result the femurcan be pulled anteriorly on the tibial plateau until the tension in the twocruciate ligaments is equal and oppositely directed in the horizontal plane,while both ligaments will draw the femur closer to the tibia, compressing thejoint.

In endrange extension the anterior cruciate ligament isstrained so that it draws the femur anteriorly and inferiorly on the tibialplateau. Extension tends todecrease the gap between the attachments of the posterior cruciate ligament soit is able to accommodate the anterior translation. Between the two cruciate ligaments, the femur is translatedanteriorly and inferiorly as the joint enters terminal extension.

If both ligaments are tense, then the femoral attachments ofboth ligaments can be brought slightly closer to their tibial attachments byrotating the tibia laterally a few degrees. That is observed to happen as the knee joint enters theclose-packed position in endrange extension.

Wrapping Up

The intent in this chapter has been to briefly introduce anumber of ideas related to the manner in which ligaments shape movement injoints. This is a large subject tofully explore and this is not the place for an exhaustive examination.  We tend to be impressed by the mass andforce of the muscles and the wide excursions of the bones and to pay lessattention to the many ligaments that do not actively move and, when they move,it is often for short distances. They are much less voluminous than either muscles or bones.  And yet, they are central to the controlof anatomical movement. Asdemonstrated here, they often set the direction and magnitude of themovement. In another forum, itmight be interesting to systematically consider the many forms that ligamentstake and how they perform their roles in different joints.  Here we have considered only a coupleof pairs of ligaments in the knee. To fully understand the knee, it is necessary to examine each ligamentalone and in company with all the others.