eyeprolog

Leg length discrepancy measurement

Are this patient’s legs measurably different in length?

Four landmarks are marked on an X-ray. From their coordinates, a chain of geometric rules projects the knee points, computes each leg’s length, and raises an alarm when the difference exceeds a clinical threshold. The proof shows every arithmetic step from coordinates to alarm.

Run it in the playground · Source

The question

A leg length discrepancy (LLD) is a difference between the lengths of the two legs. Small differences are normal; larger ones can call for treatment. Given measured landmark positions for measurement meas47, are the legs different by more than 1.25 cm, and if so, by how much?

The measurement

measurement(meas47).

% measured landmark coordinates, in centimetres
val(meas47, p1xCm, 10.1).
val(meas47, p1yCm, 7.8).
val(meas47, p2xCm, 45.1).
val(meas47, p2yCm, 5.6).
val(meas47, p3xCm, 3.6).
val(meas47, p3yCm, 29.8).
val(meas47, p4xCm, 54.7).
val(meas47, p4yCm, 28.5).

% threshold used by the alarm rule, in centimetres
threshold(meas47, lld_alarm_threshold_cm, 1.25).

All raw values and every derived quantity live in one relation, val(Measurement, Name, Value).

The geometry

val(M, d53Cm, Z) :- measurement(M), val(M, ssd53Cm2, X), (Z is X ** 0.5).
val(M, d64Cm, Z) :- measurement(M), val(M, ssd64Cm2, X), (Z is X ** 0.5).
val(M, dCm, Z) :- measurement(M), val(M, d53Cm, X), val(M, d64Cm, Y), (Z is X - Y).

type(M, lld_alarm) :- measurement(M), val(M, dCm, D), threshold(M, lld_alarm_threshold_cm, T), (Negt is 0 - T), (D < Negt).
type(M, lld_alarm) :- measurement(M), val(M, dCm, D), threshold(M, lld_alarm_threshold_cm, T), (D > T).

About thirty val/3 rules build up from coordinate differences to projected knee points (p5, p6) and then to the two leg lengths as Euclidean distances. The alarm fires when the difference is below minus the threshold or above it.

The answer

eyeprolog examples/lldm.pl

type(meas47, lld_alarm).
lld_left_length_cm(meas47, 21.548900464617255).
lld_right_length_cm(meas47, 23.45713444515475).
lld_discrepancy_cm(meas47, -1.9082339805374957).
lld_threshold_cm(meas47, 1.25).
lld_reason(meas47, "discrepancy below negative threshold").

The left leg measures about 21.55 cm and the right about 23.46 cm in this image. The difference, -1.91 cm, is beyond the 1.25 cm threshold, so the alarm fires.

The reason

eyeprolog --proof examples/lldm.pl

step(type(meas47, lld_alarm),
     rule(49),
     ['M' = meas47, 'D' = -1.9082339805374957, 'T' = 1.25, 'Negt' = -1.25],
     [measurement(meas47),
      val(meas47, dCm, -1.9082339805374957),
      threshold(meas47, lld_alarm_threshold_cm, 1.25),
      -1.25 is 0 - 1.25,
      -1.9082339805374957 < -1.25]).

The alarm holds by the first alarm rule: the discrepancy, -1.908…, is less than the negated threshold, -1.25. The step for dCm in turn cites the two leg lengths and their subtraction, and so on down to the measured coordinates.

The check

condition('C1', resolution, ok, 53).
condition('C5', re_decision, ok, 39).
condition('C7', relevance, ok, 98).
recomputed(39).
trusted(0).
verdict(checked).

The proof has 92 steps. Fifty-three rule and fact steps replay against the program, and all 39 arithmetic steps (subtractions, products, squares, square roots, comparisons) are recomputed by the checker. Nothing is trusted: every number in the alarm can be traced back to a measured coordinate.

Try it

Raise the threshold to 2.0. The alarm no longer fires, and because every output relation depends on the alarm, the run prints nothing at all.
Swap the left and right landmarks (p1↔p2, p3↔p4, p5↔p6). The discrepancy becomes +1.91 cm, the alarm fires through the second rule, and the reason changes to “discrepancy above positive threshold”.
Move one landmark by a millimetre and see how far the discrepancy moves. How sensitive is the alarm to measurement error?

Learn more in The Art of EyeProlog, Chapters 14 and 29.

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