THE SECOND LETTER FROM ABROAD
In the previous letter that I wrote to you from abroad,
I told about my visit to Professor K. with whom I had sought
an interview in order that the expenses for my vacation in
this quaint, historic valley might become deductible on my
Federal income tax return. The interview with the professor
had begun auspiciously enough. He had seated me in the exam-
ining chair in his office and, in the company of a spider
busily weaving its web, we had chatted amiably about possible
progression and regression of optic disc excavation in glau-
coma. The pleasure of that conference, however, had begun to
fade, for me at least, when the professor began to regale me
with the details of a mathematical model, that I infer to be
his brainchild, describing the manner in which elevated in-
traocular pressure brings about the excavation of the optic
disc. I have never been much of a mathematician. Indeed,
aside from a love of affluence and, of course of humanity,
it was a distinct disinclination from the intellectual rigors
of mathematics that led me to dedicate myself to medicine. I
had listened politely enough to the professor's exposition,
but I made no pretence of being able to understand it. Time
was my ally and released me, but not before I had contracted
a pounding headache in the left temporal-parietal region,
which would not cease, and continued to torment me until I
had quenched its fire with the fourth glass of Moselbluem-
chen in the Wirtshaus later that evening. At that time I had
made the firm resolve to eschew further biophysics instruct-
ion and to leave the mathematical modeling of optic nerve
excavation to more cerebral ophthalmologists than myself.
My mind has changed. I am reluctant to leave this quaint
little city, all the more so, since Lotte, the young lady who
cashes my travelers checks for me at the Hotel, has agreed
to be my guide and companion on a hike, Sunday next, to the
ruins of the old Koenigswarte, where the Emperor Frederick II
is said to have stayed overnight just before setting out on
his third expedition to Lombardy. To make a long story
short, it is impossible for me to leave at this time, and,
given the intransigence of the Internal Revenue Service fa-
miliar to me from previous audits, I had no choice but to
place a telephone call, almost a week after my original meet-
ing with him, to Professor K's office. The secretary, whose
mellifluous voice immediately recalled to my memory the en-
tire scene, oboist, professor, spider, exponential equation
and all, was obviously pleased to hear from me.
"Of course I remember your visit. We have been expecting
your call. How would you like to come this afternoon at three
o'clock."
I had no expected so early an appointment. I was taken
by surprise. Did he have nothing better to do than to teach
me mathematics? The excuse which I thought of later did not
find its way to my tongue in time. At three o'clock, there-
fore, I found myself, without further introductory cere-
monies, seated once more in the examining chair in the inner-
most room of Professor K's office. "Guten Nachmittag, Herr
Kollege," the professors buoyant baritone greeted me. He
seemed much less somber than at our last meeting, and I was
reminded how exuberant I had felt as a child, when my friends
came to play with me and one of my model trains. How charm-
ing it is of you to come here all the way from St. Louis.
"From Cambridge," I corrected him.
"A suburb of St. Louis," he chuckled ingratiatingly. I
let it pass, and thought to myself, I hope his mathematics is
better than his geography.
"When we ran out of time last week," he began, "if I re-
member correctly, I had just finished describing to you the
mathematical model of the excavation of the optic disc."
"I couldn't understand it, Herr Professor, and I had a
very bad headache."
"Is your headache better now?" he asked solicitously.
"Oh yes, much better, it went away that same evening."
"That is good," he said, "then we can start from the be-
ginning."
A twitch of anguish must have contorted my face. What
will not a man endure for the sake of a tax deduction, I
thought, or for the opportunity to be shown by a pretty young
woman where the Emperor Frederick II spent the night before
his third expedition to Lombardy.
My facial expression may have given Professor K. pause
to think. There was a moment of silence. Then he said, "It
is not really necessary for you to be able to derive the mod-
el. Let us be practical. Let us begin at the end and end at
the beginning. I will show you how it works."
With that he turned in his chair to a computer terminal
that stood on the further side of his desk, and upon the
click of a switch, the room resonated to the dull monotone of
a 50 cycle per second power supply.
"In our hospital," the professor began, his voice rising
above the hum, "we have glaucoma conferences every Wednesday
morning at 8 a.m. At last week's conference, one of our as-
sistants presented a twenty-six year old patient with previ-
ously healthy eyes who had sustained a corneal laceration
with presumably flat anterior chamber which he neglected to
have treated. He comes to us with 75% of his angle closed,
and with an intraocular pressure fixed at 30 mm.Hg. How long,
the assistant asked me, do I think that this pressure can be
sustained before total cupping of the disc supervenes. What
is your estimate?"
"I can only guess," I said.
"None of us can do more."
"I tell my patients that I am only a doctor. I am not a
prophet. I am not even a professor."
"Well said," Professor K. nodded, "And yet, if he were
perfectly healthy with a life expectancy of sixty years you
would surely treat him, while if you knew that he had a ma-
lignancy, and only a year or two to live, you might spare him
the rigors of even medical therapy."
"I will bet on ten years," I said, but don't hold me to
it, because I don't know." Professor K. ignored my dis-
claimer and proceeded, "And what if the pressure were 36."
"Then I would give him four years to total cupping if
his disc was previously healthy," I said recklessly. "Does
your model require such shots in the dark?"
Professor K. was obviously hurt. "You are reluctant to
commit yourself to specific values," Professor K. continued,
"Because you wish to preserve the appearance that you know
what you are doing. You do not wish to be wrong. I fault you
for seeking to avoid error by such evasion. Implicit in the
clinical decisions that you make day after day are numerous
judgments of this sort, and there is some virtue in making
them explicit. Of course you cannot know how long this par-
ticular disc will sustain a pressure of 30 or 36, but it is
obligatory that you should have some idea of the approximate
survival time of the disc under these circumstances. How else
will you justify your clinical decisions? On what other
grounds can you give your patient advice?"
"I am sorry," I said.
"I happen to agree with your ball-park estimates. Let's
see what the model thinks of them."
With these words, Professor K. lifted the receiver of
the telephone on his desk from its cradle and inverted it in-
to a small electronic device positioned immediately adjacent.
He dialed a number. The terminal consulted briefly with its
superior in the hierarchy of machines. Then it began to im-
portune its operator with questions: "Pressure #1:" it de-
manded, and Professor K. dutifully typed: "30". "Estimated
survival time #1:" "10", the exchange continued. "Pressure
#2:" "36", "Estimated survival time #2:" "4". That was all.
Once its questions had been answered, the terminal paused, as
if to swallow the newly acquired information; then, with re-
lentless rhythm, it disgorged the tables which I have repro-
duced below. Professor K. leaned back in his chair with evi-
dent satisfaction. The performance of the machine seemed to
compensate him for my sceptical recalcitrance. The spider
that we had watched the week before, or another one just like
it, had reappeared on the moulding, and having anchored its
thread to the woodwork, prepared for another descent.
Equations
R1 = R0*exp(S*k)
(ul-u0)/t = R1
S = p*r/(2*(w-u0))
(u1-u0)/t = R0*(exp(p*r*k/(2*w-u0)))
Variables
p = intraocular pressure
r = radius of globe = 11 mm
R1 = rate of excavation
S = tangential stress in disc
t = survival time
u0 = prior excavation = 0 mm
u1 = depth of excavation = .9 mm
w = thickness of disc = 1 mm
Assumed Data
At 30 mm Hg survival time = 10 yrs.
At 36 mm Hg survival time = 4 yrs.
Calculated Constants
R0 = .9216e-3
k = .277664e-1
Predicted Survival Time of Normal Disc
tension time tension time
11 mm 182.03 yr | 41 mm 1.86 yr
12 mm 156.25 yr | 42 mm 1.60 yr
13 mm 134.12 yr | 43 mm 1.37 yr
14 mm 115.13 yr | 44 mm 1.18 yr
15 mm 98.82 yr | 45 mm 1.01 yr
16 mm 84.83 yr | 46 mm 317.04 da
17 mm 72.81 yr | 47 mm 272.14 da
18 mm 62.50 yr | 48 mm 233.60 da
19 mm 53.65 yr | 49 mm 200.52 da
20 mm 46.05 yr | 50 mm 172.12 da
21 mm 39.53 yr | 51 mm 147.74 da
22 mm 33.93 yr | 52 mm 126.82 da
23 mm 29.12 yr | 53 mm 108.86 da
24 mm 25.00 yr | 54 mm 93.44 da
25 mm 21.46 yr | 55 mm 80.21 da
26 mm 18.42 yr | 56 mm 68.85 da
27 mm 15.81 yr | 57 mm 59.10 da
28 mm 13.57 yr | 58 mm 50.73 da
29 mm 11.65 yr | 59 mm 43.54 da
30 mm 10.00 yr | 60 mm 37.38 da
31 mm 8.58 yr | 61 mm 32.08 da
32 mm 7.37 yr | 62 mm 27.54 da
33 mm 6.32 yr | 63 mm 23.64 da
34 mm 5.43 yr | 64 mm 20.29 da
35 mm 4.66 yr | 65 mm 17.42 da
36 mm 4.00 yr | 66 mm 14.95 da
37 mm 3.43 yr | 67 mm 12.83 da
38 mm 2.95 yr | 68 mm 11.02 da
39 mm 2.53 yr | 69 mm 9.46 da
40 mm 2.17 yr | 70 mm 8.12 da
"Please explain to me what all this means," I demanded
impatiently. "182.03 years seems a rather protracted life
span. In Cambridge, patients don't live that long."
"Very good," Professor K. answererd. "You have already
learned the most important lesson that the model has to
teach."
"What is that?"
"Not to take it too seriously, to be sceptical."
"But to learn to be sceptical, one doesn't need an elab-
orate model, just common sense," I retorted.
Professor K. was annoyed. "It is true, that the model
makes no provision for the span of human life, and is pre-
pared to calculate survival times for the optic disc far be-
yond any possible survival of the human body. At the other
end of the pressure range, the model is also patently defi-
cient, inasmuch as it describes total cupping, for example,
produced by a pressure of 70 mm. Hg. as occuring over a pe-
riod of only 8.12 days. Since we would always operate to low-
er the pressure in such a situation, we have only rare occa-
sion to observe the developemnt of cupping at such very high
pressures, but it is my impression that a healthy disc does
not develop cupping in less than several weeks, perhaps even
a few months. The eye, of course, usually goes blind from
optic atrophy much sooner, the result apparently of a differ-
ent pathophysiologic mechanism."
"But what about the other 'survival times' as you call
them?"
The term 'survival time' refers to the period required
for a given pressure to produce a given amount of cupping.
The table above is calculated on the assumption that 90% of
the disc thickness is eroded by a pressure of 30 mm.Hg. in
10 years, and by a pressure of 36 mm.Hg. in 4 years. These
assumptions serve to determine the other values in the table.
The calculations also assume a radius of the globe of 11 mm.
and and initial thickness of the disc of 1 mm. These last
assumptions are not at all critical, they must of course re-
main the same if survival time calculations are to be com-
pared one with the other. The model can also be used to ca-
clculate the effect of variation of the radius of the eye, as
in axial refractive errors. For example:
Ametropia and Survival Time
refrctve survival refrctve survival
error time | error time
Tension = 18 mm. Hg.
-18 D 33.81 yr | -2 D 58.38 yr
-16 D 36.20 yr | 0 D 62.50 yr
-14 D 38.76 yr | 2 D 66.92 yr
-12 D 41.49 yr | 4 D 71.64 yr
-10 D 44.43 yr | 6 D 76.71 yr
-8 D 47.56 yr | 8 D 82.13 yr
-6 D 50.92 yr | 10 D 87.93 yr
-4 D 54.52 yr | 12 D 94.14 yr
Tension = 21 mm. Hg.
-18 D 19.30 yr | -2 D 36.50 yr
-16 D 20.90 yr | 0 D 39.53 yr
-14 D 22.63 yr | 2 D 42.81 yr
-12 D 24.51 yr | 4 D 46.35 yr
-10 D 26.54 yr | 6 D 50.20 yr
-8 D 28.74 yr | 8 D 54.36 yr
-6 D 31.13 yr | 10 D 58.87 yr
-4 D 33.71 yr | 12 D 63.75 yr
Tension = 24 mm. Hg.
-18 D 11.02 yr | -2 D 22.82 yr
-16 D 12.07 yr | 0 D 25.00 yr
-14 D 13.22 yr | 2 D 27.38 yr
-12 D 14.48 yr | 4 D 29.99 yr
-10 D 15.86 yr | 6 D 32.85 yr
-8 D 17.37 yr | 8 D 35.98 yr
-6 D 19.03 yr | 10 D 39.41 yr
-4 D 20.84 yr | 12 D 43.17 yr
Tension = 27 mm. Hg.
-18 D 6.29 yr | -2 D 14.27 yr
-16 D 6.97 yr | 0 D 15.81 yr
-14 D 7.72 yr | 2 D 17.52 yr
-12 D 8.55 yr | 4 D 19.41 yr
-10 D 9.48 yr | 6 D 21.50 yr
-8 D 10.50 yr | 8 D 23.82 yr
-6 D 11.63 yr | 10 D 26.38 yr
-4 D 12.88 yr | 12 D 29.23 yr
Tension = 30 mm. Hg.
-18 D 3.59 yr | -2 D 8.92 yr
-16 D 4.02 yr | 0 D 10.00 yr
-14 D 4.51 yr | 2 D 11.21 yr
-12 D 5.05 yr | 4 D 12.56 yr
-10 D 5.66 yr | 6 D 14.07 yr
-8 D 6.34 yr | 8 D 15.76 yr
-6 D 7.11 yr | 10 D 17.66 yr
-4 D 7.96 yr | 12 D 19.79 yr
Tension = 33 mm. Hg.
-18 D 2.05 yr | -2 D 5.58 yr
-16 D 2.32 yr | 0 D 6.32 yr
-14 D 2.63 yr | 2 D 7.17 yr
-12 D 2.98 yr | 4 D 8.12 yr
-10 D 3.38 yr | 6 D 9.21 yr
-8 D 3.83 yr | 8 D 10.43 yr
-6 D 4.34 yr | 10 D 11.83 yr
-4 D 4.92 yr | 12 D 13.40 yr
"Thus the model appears to corroborate what has long
been reported anecdotally, that the myopic disc is relatively
more sensitive and the hyperopic disc is relatively more re-
sistant to the destructive effects of elevated intraocular
pressure."
"It is interesting also to look at the effect on the
survival time of prior excavation of the disc:"
Prior Excavation and Survival Time
prior survival prior survival
cup time | cup time
Tension = 6 mm. Hg.
0 mm 390.63 yr | .40 mm 117.81 yr
.10 mm 313.61 yr | .50 mm 69.44 yr
.20 mm 241.62 yr | .60 mm 32.94 yr
.30 mm 175.84 yr | .70 mm 10.23 yr
Tension = 9 mm. Hg.
0 mm 247.05 yr | .40 mm 54.90 yr
.10 mm 188.50 yr | .50 mm 27.78 yr
.20 mm 136.28 yr | .60 mm 10.48 yr
.30 mm 91.39 yr | .70 mm 2.22 yr
Tension = 12 mm. Hg.
0 mm 156.25 yr | .40 mm 25.58 yr
.10 mm 113.30 yr | .50 mm 11.11 yr
.20 mm 76.86 yr | .60 mm 3.33 yr
.30 mm 47.49 yr | .70 mm 176.14 da
Tension = 15 mm. Hg.
0 mm 98.82 yr | .40 mm 11.92 yr
.10 mm 68.10 yr | .50 mm 4.44 yr
.20 mm 43.35 yr | .60 mm 1.06 yr
.30 mm 24.68 yr | .70 mm 38.25 da
Tension = 18 mm. Hg.
0 mm 62.50 yr | .40 mm 5.56 yr
.10 mm 40.93 yr | .50 mm 1.78 yr
.20 mm 24.45 yr | .60 mm 123.12 da
.30 mm 12.83 yr | .70 mm 8.31 da
Tension = 21 mm. Hg.
0 mm 39.53 yr | .40 mm 2.59 yr
.10 mm 24.60 yr | .50 mm 259.56 da
.20 mm 13.79 yr | .60 mm 39.16 da
.30 mm 6.67 yr | .70 mm 1.80 da
Tension = 24 mm. Hg.
0 mm 25.00 yr | .40 mm 1.21 yr
.10 mm 14.79 yr | .50 mm 103.82 da
.20 mm 7.78 yr | .60 mm 12.46 da
.30 mm 3.46 yr | .70 mm 0.39 da
Tension = 27 mm. Hg.
0 mm 15.81 yr | .40 mm 205.20 da
.10 mm 8.89 yr | .50 mm 41.53 da
.20 mm 4.39 yr | .60 mm 3.96 da
.30 mm 1.80 yr | .70 mm 0.09 da
Tension = 30 mm. Hg.
0 mm 10.00 yr | .40 mm 95.62 da
.10 mm 5.34 yr | .50 mm 16.61 da
.20 mm 2.47 yr | .60 mm 1.26 da
.30 mm 341.56 da | .70 mm 0.02 da
Tension = 33 mm. Hg.
0 mm 6.32 yr | .40 mm 44.56 da
.10 mm 3.21 yr | .50 mm 6.64 da
.20 mm 1.40 yr | .60 mm 0.40 da
.30 mm 177.51 da |
Tension = 36 mm. Hg.
0 mm 4.00 yr | .40 mm 20.76 da
.10 mm 1.93 yr | .50 mm 2.66 da
.20 mm 287.28 da | .60 mm 0.13 da
.30 mm 92.25 da |
Tension = 39 mm. Hg.
0 mm 2.53 yr | .40 mm 9.68 da
.10 mm 1.16 yr | .50 mm 1.06 da
.20 mm 162.03 da | .60 mm 0.04 da
.30 mm 47.95 da |
"This table illustrates two important clinical impres-
sions. It makes explicit how prior excavation increases the
susceptibility of the disc to yet further excavation. Specif-
ically, a flat disc might tolerate a pressure of 18 for 62.5
years. But a disc already excavated to .4 mm would survive a
pressure of 18 for only 5.56 years. Hence a tension adequate
to preserve vision in a healthy eye might be entirely inade-
quate to protect from further destruction a disc already dis-
eased. It is also apparent from this table that for any given
pressure the rate of disc excavation is initially very small
and then becomes ever greater as the excavation deepens.
This characteristic of optic nerve cupping, which I believe
the model exaggerates, explains the origin of the disagree-
ment about the nosology of glaucoma. Those optometrists and
ophthalmologists who believe that ocular hypertension can be
distinguished from glaucoma, look at the early section of the
curve, where disc excavation is so slight as to be impercep-
tible. Those, on the other hand, who believe all ocular
hypertension should be considered glaucomatous, look to the
end of the process, no matter how remote, where excavation
and consequent field loss must always be expected."
"If I understand you correctly, all the figures in the
tables you have shown me are extrapolated from my two guess-
es," I said.
"You guessed well."
"But I do not really believe that a disc subject to a
pressure of 30 mm.Hg. will survive for exactly 10 years, or
that subjected to a pressure of 36, it would survive exactly
four. Besides, no patient maintains a constant pressure for
four years, not to speak of ten."
"But that is not what we said. We said only that if the
pressure remained constant, the average survival might be 10
and 4 years respectively. Nothing was said about the scatter
of actual measurements about the hypothetical means. Perhaps
that scatter is so great that the means are relatively mean-
ingless."
"We agree that we cannot, with any certainty, predict
the course of glaucomatous excavation in a given patient.
Nonetheless we advise him, on the basis of pressure measure-
ments, to take medication or to undergo surgery. I can't
reconcile these facts unless I assume that we have some no-
tion of the average rate of disc destruction induced by any
given pressure."
"That, exactly, is what makes the model compelling.
This model may not be optimal. It is certainly not the only
possible model. One can immediately suggest important emen-
dations. The point is that we advise our patients on the ba-
sis of our intuitive conceptions of the disease and its
course. It is unavoidable that our judgment should be guided
by a model. The mathematical analysis serves to make this
model explicit. The point is that there must be a model of
some kind to guide our judgment."
"Just as we must have in mind some model of the anatomy
before we begin a surgical operation."
"Yes. The anatomic drawing which guides us is always
more or less schematic. It does not include all variations.
It shows us our problem in only one of many possible perspec-
tives.
* * * * *
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Copyright 2006, Ernst Jochen Meyer