The Longitudinal Muscle in Esophageal Disease - A Boolean Model of the Esophague

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BOOLEAN MODEL OF SWALLOWING LIQUIDS
The state of the esophagus moves along the edges of a cube defined by the 8 possible combinations of relaxation/contraction of its constituent elements: the circular, longitudinal and sphincter muscles

The status of the esophagus at any instant in time can be specified by giving the state of each muscle. These specifications are achieved by "and-ing" the three symbols or their negations. Thus, the normal condition of the esophagus is ~C&~L&S - that is, the longitudinal and circular muscles are relaxed and the sphincter is contracted.

Detecting the formula for the resting esophagus is easy because it stands still while we are doing so. We can tell that the sphincter is closed because there is no reflux, that the LM is relaxed because there is no shortening, hiatal herniation or tenting of the diaphragm or PEL. The circular muscle state is less obvious, but with the fluoroscope we can see that there is no peristalsis going on, a swallow of barium initially meets no resistance when discharged from the hypopharynx. Manometrically the pressure is zero or negative.

Even at this point, the insight provided by a model enables one to ask some questions, the answers to which, if they could be found, would not be trivial. During deglutition the state formula must change from the resting formula. Knowing the exact sequence in which the state formulas changed during this function of the esophagus would be interesting. Obviously, there is no reason to exclude L, the state of the longitudinal muscle, from consideration.

Before attempting to trace the changes in the state formula, a further simplification will be employed as it will give direction to the search. We can assume they are there, drop the ampersands and write the resting formula as ~C~LS. In addition, because its order in the formula already identifies the muscle, we can use the binary numbers 0 and 1 to stand for the relaxed and contracted states of the components. Thus simplified, the resting state formula becomes 001.

If these binary numbers are then treated as the x, y and z coordinates of a point in space,⁽¹⁾ they define the 8 vertices of a unit cube. This produces a major simplification at once. We can require that only one of the three muscles changes its state at a time. This is equivalent to a rule restricting the "state paths" to those from one vertex to an adjacent vertex. Thus, instead of having 7 possible ways things could change from one state to another, the possibilities are reduced three. Because the only normal state node of the organ is 001, a second rule is that all paths on the 3-D cube must be closed.

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BOOLEAN MODEL OF SWALLOWING AGAINST RESISTANCE

The problem is now reduced to deciding what happens first. That done, one is ready to look for the next change, one component at a time. Tracing the entire path in a given subject is not even necessary. It is enough to work out one transition. The next can be worked out at liesure. A directed observation is far easier and more likely to be accurate than sizing up the esophageal gestalt.

In addition, the longitudinal muscle (L) now becomes an essential part of the picture at least as important as the other components. Simply because it does not affect a manometer or a balloon is no reason for ignoring it.

Another consequence of the model is to emphasize that the esophagus is not a single-purpose organ designed only for swallowing. This is because there are a great many paths from vertex to vertex that can be traced on a 3-dimensional cube. Here is a clue to the ability of the organ to swallow liquids via a different path than it uses for solids or when swallowing against resistance or in the upside down position. We can now see that esophageal speech might have a still different path. Deploying its resources in varied sequences is the means by which the esophagus carries out its multiple functions - swallowing liquids and solids, belching, gagging, vomiting.

In principal, it would be also be anticipated that various malfunctions could cause interruptions of the smooth transitions along normal pathways or that arrests could occur - possibly at different vertex of the cube. When these very specific questions have been answered, we should have a much more detailed knowledge of the diverse functions of the organ.

Fluoroscopic observation is generally sufficient to map the paths of changing state patterns with the aid of this search algorithm. In principle, however, it should be possible to develop instrumentation to detect all three types of muscle contraction simultaneously. A computer could constantly monitor the changing patterns with a high degree of accuracy.

I will trace the state pattern in several modes of esophageal activity. By treating the 3-bit binary coordinates of the vertices as octal numbers and using them as labels for the vertices they can be referred to more conveniently. Thus, 111 becomes 7, 001 translates to 1, etc.

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BOOLEAN MODEL OF BELCHING

The contraction of L changes the formula to ~CLS or 011 = 3 and the subsequent sphincter relaxation to ~L (010) = 2. A rapid alternation among nodes 1, 2 and 3 then occurs with each subsequent swallow. The graph never gets off the not-C plane.

Swallowing against resistance, whether it be with solid food, in the upside down position or during a Valsalva effort causes the path to leave the left (not-C) plane and involve the circular muscle. The first event is active peristalsis (001 to 101). Peristalsis gradually "latches" the longitudinal muscle (101 to 111) causing the sphincter to yield (110). Just before the peristaltic wave reaches the sphincter, the longitudinal muscle relaxes (110 to 100), the ring of circular muscle contraction becomes the sphincter (100 to 101) and so vanishes (101 to 001). The path traced is 1, 5, 7, 6, 4, 5, 1.

As there are three possible transitions from a given vertex, and swallowing involves 6 transitions, there are 3³ or 729 possible sequences of which only one is appropriate to swallowing against resistance.

Comparison of the graphs of the two swallowing modes shows strikingly different pathways. Note that the state formula changes one bit at a time, and that it is difficult to leave the designated path without the swallow aborting. Both activities are highly mechanical acts that the model represents quite faithfully. One might say that LMC is essentially concurrent with the peristaltic wave and ought to be represented as a path on the diagonal from vertex 1 to vertex 7. However, it seems more helpful to think of paths confined to the edges of the cube between vertices even if close temporal relationships or overlapping activity warp the picture.

For belching the following schema is easily detected:

Formula	Node	Esophageal state
~C&~L&S	001	Resting state
~C&L&S	011	Longitudinal muscle contracts
~C&L&~S	010	Sphincter opens, gas escapes
~C&~L&~S	000	Longitudinal muscle relaxes
~C&~L&S	001	Sphincter closes
C&&S	101	Circular muscle evacuates gas
&&S	001	CM relaxes to resting state

It is evident from inspection of the various state-path diagrams that they are not restricted to a single plane but move in all 3 "dimensions." This highlights a remarkable aspect of esophageal physiology to date: it has been largely restricted to the CS plane. That is, it is fundamentally 2-dimensional.

If this handicap is accepted, we are as limited in our comprehension as were the Flatlanders of Edward Abbott,⁽²⁾ who could be imprisoned by circumscribing them with a line because they could not conceive of a third dimension. The available balloons, manometers and transducers are Flatlander instrumentsas they see only the circular and sphincter muscles.

This has serious consequences even for careful laboratory research. No matter how meticulously LES pressure and peristaltic wave pressure are measured or sequenced under various experimental conditions, the interactions between stimulus and response may be happening in a plane that is invisible to the experimenter who is not also tracking LM function. Unfortunately, instrumentation has not been developed to measure LM contraction conveniently in the intact subject.

Another aspect of esophageal physiology that can be read from the model is the fact that, with one exception, the organ normally never remains indefinitely at a node. It occupies them only in transit. The exception, of course, is the resting state - (001). Permanent occupation of another vertex is pathologic. Myotonia dystrophica, for example, seems to arrest the organ at 010 as does scleroderma.

The various formulas that occur can be listed and "or-ed" together and so manipulated with operations of the sentential calculus of mathematical logic⁽³⁾ to yield a result that I will subsequently show to be supported by radiological and clinical observations. It might be called the Fundamental Law of the Esophagus:

Because it ignores the function of the longitudinal muscle, the prevailing esophageal paradigm is 2-dimensional. A 3-D model reveals several rules of esophageal physiology, the most important of which is the rule that the longitudinal muscle and the sphincter are never contracted simultaneously. This is equivalent to the assertion that the LM opens the sphincter. The vastly increased number of possible state sequences together with alternative peristaltic and en masse modes of contraction account for the ability of the esophagus to carry out a variety of functions.

1. . Flegg, H.G., Boolean Algebra and its Applications, John Wiley & Sons, New York, 1964.

3. . Hilbert, D. and Ackerman, W., Principles of Mathematical Logic, Ed. Luce, Robert E., Chelsea Publishing Company, New York, 1950.

A Boolean model of the esophagus