Friday, April 18, 2008

Euler and the Deltoid Ligament

I haven't dabbled in calculus in quite a while, but remember it fondly. I was a bit of a math dunce, but real math dudes are fiercely intelligent. One of the greatest math dudes of all time is Leonhard Euler. If you took calculus, you will remember Euler's identity, perhaps fondly, and perhaps with a shudder:

This little gem was called "the most remarkable formula in mathematics" by the physicist Richard Feynman, no math slouch himself.

OK, by now you should be asking, "What in the world has this got to do with musculoskeletal MRI". Ah, but there is an odd but delightful connection.

Euler conceived of the mathematical entity, the deltoid, which can be defined as the trace of a point on a circle, rolling inside another circle 3 or 3/2 times as large in radius:


Note the triangular shape of the deltoid, which comes from the Greek letters delta (Greek letter D) and eidos ("resemblance").

Two well known structures of the body are the deltoid muscle of the shoulder and the deltoid ligament of the medial ankle, whose shape resemble the mathematical deltoid, triangular in configuration.

Menfiardi et al. wrote an excellent article (Radiology 242:817-824, 2007) describing the anatomy of the deltoid ligament. Here is a drawing from that article:

© RSNA, 2007

Medial aspect of the ankle displays the deep and superficial layer of the MCL complex. The superficial layer is semitransparently displayed to visualize the covered portions of the deep layer. Deep layer consists of anterior and posterior tibiotalar (aTTL and pTTL) ligaments. Superficial layer consists of tibionavicular ligament (TNL), tibiospring ligament (TSL), and the tibiocalcaneal ligament (TCL). Three portions of the spring ligament complex are displayed: superomedial calcaneonavicular ligament (smCNL), medioplantar oblique calcaneonavicular ligament (1), and inferoplantar longitudinal calcaneonavicular ligament (2). aColl = anterior colliculus, pColl = posterior colliculus.

Twenty-eight year old male with history of eversion injury three months ago:
Coronal T2-weighted image with fat saturation shows a grade 2-3 sprain of the deltoid, particularly the deep fibers (red arrow). There are multiple bone bruises as well (green asterisks).

Repeat scan at 5 months, this time without fat saturation, shows improvement in edema, but persistent abnormal thickening and abnormal signal associated with the deep (red arrow) and superficial fibers (yellow arrow) of the deltoid.

Sagittal images at 5 months:

(click on image to enlarge)

(A) T1-weighted and (B) STIR images show persistent bone marrow edema subjacent to the anterior colliculus, intercollicular groove, and the posterior colliculus, the origin of the deep fibers of the deltoid ligament. Persistent edema 5 months after injury suggests that the deltoid has not healed properly, and is the cause of the patient's medial ankle pain.


Vic David MD
Orthoradiology.com

1 comment:

Felasfa Wodajo, MD said...

way cool. I loved this post.

phil