An electrical model of the human eye
I. The basic model
W. K. McEwen, Marvin Shepherd, and Earle H. McBain
An electrical model which simulates the action of the human eye is developed from experimental data and current concepts. Three fundamental parameters of the eye (volume, pressure,
and time) are directly translated into analogous electrical elements. All other parameters are
derived from these fundamental ones. The classical elastic Friedemoald model is compared
to the viscoelastic model.
M.
also because supplemental electronic circuitry (to be given in later papers) can
substitute for many manual procedures,
reiterative matching or reproducing techniques are facilitated.
.any physical systems, hydrodynamic,
mechanical, acoustical, and electrical, are
analogous.1 This article is concerned with
the development of an electrical model
which is analogous to the hydrodynamic
and mechanical properties of the human
eye.
In general, changes in these properties
of the eye, occasioned by any test, procedure, or altered physiology should be
matched or reproduced by this electrical
model within the limits of its error. If there
is no correspondence, either our present
concepts of the mechanism of the action
of the eye or the experimental data regarding its fundamental properties are faulty.
This model then allows us to test the
validity of concepts and data by comparing the model's action with the action
of the eye during clinical or experimental
tests.
Because the model performs in a time
continum 60 times faster than the eye, and
Analogous quantities and units
In the system of the eye there are three
fundamental quantities, volume, pressure
and time, from which all other quantities
are derived. These quantities are translated
into the electrical system by changing the
units without changing the symbols. Volume (V) in microliters (fiL) of the eye
system is translated into the electrical system as microcoulombs (ju-Q). Similarly,
pressure (P) in mm. Hg becomes volts,
and time (t) in minutes becomes seconds.
This last translation allows the model to
act sixty times faster than the eye. For instance, a 4 min. tonogram of the eye will
be followed by the model in 4 sec.
Other parameters of the eye are easily
derived from the above quantities and
translated into electrical units. Flow (F)
in /xL per minute translates into juQ per
second, which by definition is microamperes
(jiamp), and facility of outflow (C) is
the ratio of flow to pressure (/JL per minute per mm. Hg) which becomes ^amp.
From the Department of Ophthalmology and
the Francis I. Proctor Foundation of the University of California-San Francisco Medical
Center, San Francisco, Calif.
Supported in part by National Institutes of Health
Grant No. NB-05451.
155
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156 McEioen, Shepherd, and McBain
per volts. By Ohm's law, the ratio of microamperes to volts is the reciprocal of resistance in megohms. The outflow resistance in megohms (abbreviated Rout) is
thus seen as the reciprocal of facility of
outflow. Other terms will be developed
later, and all symbols will be found in
Table I.
The model and the eye under steady
state conditions
The eye maintains a steady pressure
because the rate of inflow (secretion) is
equal to the rate of outflow.
The rate of inflow is probably determined mostly by the active process of
secretion of aqueous by the ciliary body.
The inflow rate is probably not influenced
by pressures within the physiological range,
but is considered to be constant under
steady state conditions. The electrical analogue of constant secretion is constant
current. This is obtained by a source of
electrical energy with a high internal
resistance which tends to maintain a constant current independent of external circuit changes. The actual component in
the model is a high-voltage source (battery) in series with a large resistance. The
series resistance is kept at least ten times
greater than the value of any external
resistance. This means that fluctuations of
current, due to the external resistance
changes, will be less than 10 per cent.
Investigative Ophthalmology
April 1967
The rate of outflow is related to the
pressure drop from inside the eye to the
episcleral venous pressure (Po - P v ) and
the resistance to flow (reciprocal of facility).
The episcleral venous pressure has been
measured2'3 and found to be constant
within a few mm. Hg. This indicates that
the episcleral venous plexus represents a
constant pressure pool uninfluenced to
any large extent by the outflow or the
resistance to outflow. The electrical component analogous in this situation is a voltage source (battery) which is variable
within small limits.
The resistance to aqueous outflow from
the eye is represented electrically by a resistance to current flow, which is a resistor. The resistance from inside the eye to
the episcleral venous system is probably
not uniform—that is, the resistance may be
greatest across the inner wall of Schlemm's
canal and less in the corneoscleral trabecular mesh work or in the intrascleral plexus.
Since we are not yet in a position to measure the individual resistance at intermediate points, we will call the total resistance to outflow (1/C) and represent it with
a single electrical resistance (ROut)The constant current device may now
be associated in a circuit with the outflow
resistance and the voltage source representing episcleral venous pressure. A diagram of the steady state condition of the
Table I. Conversion of units between eye and model
Units
Quantity
Fundamental
Volume
Pressure
Time
Derived
Flow
Facility of outflow
Resistance to outflow
Ocular rigidity dP/dV
Reciprocal of ocular rigidity dV/dP
V
P
T
F
C
Rout
K
CAP
nQ
mm. Hg
min.
/tL/min.
/tL/min./mm. Hg
mm. Hg/|iL/min.
)
d (mm. Hg)/d (ML
d(/iL)/d(mm. Hg)
Note that Rout is the reciprocal of C, and CAP the reciprocal of K.
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Electrical
Eye
Symbol
volts
sec.
/iQ/sec. = /xamp
Ma/volts
volts/juamp = megohms
d (volts )/d(MQ)
d(fiQ)/d (volts) = /xfarads
Volume 6
Number 2
An electrical model of the human eye. I
eye is compared with the analogous electrical circuit in Fig. 1. It will be noticed
that the well-known formula4 for the eye
under steady state conditions (F = C [Po
- Pv] ) describes both the eye and the electrical model when the appropriate units
are used.
The model of the eye under
dynamic conditions
Any change in pressure in an eye must
be caused by a change in the stress in the
walls of the ocular coats. Thus, to represent
dynamic conditions with changing pressure we must take into account the stressstrain relation of the ocular coats. The
stress-strain relation may be divided into
the immediate stress-strain relation5 commonly known as ocular rigidity and the
time-dependent stress-strain relation known
as stress-relaxation or change of ocular
rigidity with time.0
The immediate stress-strain relation or
ocular rigidity has been determined by
many investigators, culminating in a unifying formulation.5 In developing our model,
we are using an average of the data of
McBain (0.023P + 0.25) and Prijot (0.025P
+ 0.20) on enucleated eyes. Both authors
used modern rapid methods for determining
the immediate stress-strain relation, thus
OUTER COATS
C(Po-Pv)
Po
R or
I
J
Fig. 1. Comparison of eye and model under steady
state conditions.
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avoiding complications of the time-dependent variation, and the data of both were
in close agreement. We will use, also,
Friedenwald's data for comparative purposes; the data of other investigators may
be used by substituting those particular
numerical data in the translation from eye
quantities to electrical quantities.
The stress-strain relation for the eye is
dP per dV. We will use5 finite quantities
rather than infinitesimals:
dP/dV ~ AP/AV = 0.050 P Friedemvald
= 0.024 P + 0.23 average of
McBain and Prijot
From Table I it can be seen that ocular
rigidity is K in units of
d (mm. Hg)
To convert K to capacitance (CAP), we
need to take the reciprocal, i.e.,
d (mm. Hg)
and express the value in microfarads, i.e.,
d (volts)
This results in the following:
CAP =
1
0.050P
Friedenwald
average of
0.024P + 0.23
McBain and Prijot.
INTRAOCULAR
CONSTANT
CURRENT
DEVICE
I
1 F
157
It may be noticed that the capacitance
value changes with pressure. Since variable capacitors in the microfarad range are
difficult to obtain, we have employed the
device of using a fixed capacitor over a
pressure range which would be covered
by allowing an error of ± 20 per cent in
the capacitance value. The capacitance values obtained from the above equations
and the pressure ranges used are given
in Table II.
The actual value used will therefore
depend upon: (1) which investigator's
values are used, (2) the pressure range
over which the model is operating, and
Investigative Ophthalmology
April 1967
158 McEwen, Shepherd, and McBain
Table II
McBain-Prijot
Friedenwald
Pressure
Capacitance
value
Pressure
range
Pressure
Capacitance
value
Pressure
range
15
25
40
1.70
1.21
0.84
10-20
20-30
30-50
15
25
35
1.33
0.80
0.57
12.5-20
20-30
30-40
(3) changes which might be made to
match the ocular rigidity of a specific eye
if it is known that that eye has an ocular
rigidity much greater or less than the
average value.
The time dependent stress-relation has
been determined quantitatively by St.
Helen and McEwen,0 and Itoi and associates,7 but we shall use the former data.
St. Helen and McEwen stressed human
scleral segments and analyzed the resultant
decay of the stress (stress relaxation) into
three components: an equilibrium elastic
component, and two pressure decay components with half-lives of 0.09 and 1.7 min.
The decay components are composed of
an elastic part and a loss part. Heretofore
we have considered ocular rigidity as a
single elastic component. The above authors' analysis equates ocular rigidity to
the sum of three separate parts: the equilibrium elastic part (K E Q), and the two
elastic parts (KQ* and K s ) of the decay
components. The values for KEQ, KQ, and
Ks were given as 1.5, 0.5, and 0.9 respectively, yielding a total K of 2.9. It will be
noticed that this total K of 2.9 when converted in CAP, by taking the reciprocal
(1/2.9 = 0.345), is 3V2 times smaller than
the value previously given for McBain and
Prijot (1.21) in the pressure range of 20
to 30 mm. Hg. This discrepancy caused
by the geometry used in stressing scleral
segments: the periphery of the segments
clamped, rather than free to move. To
correct this, we divide each individual K
by 3.5, and take the reciprocal, obtaining
Table III
Pressure
ranges
CAPc
CAPKQ
CAP*
10-20
20-30
30-50
9.8
7.0
4.9
3.3
2.3
1.6
5.5
3.9
2.7
the CAP values (i.e. — = CAPEQ = 2.3;
l.o
± | = CAPQ = 7.0; £ | = CAPS - 3.9)
and the total equals 1.21, since capacitances in series add as reciprocals (i.e.
— + — + — = — ). Therefore the ocu2.3 7.0 3.9
1.2
lar rigidity from McBain and Prijot may
be considered as the sum of three separate
capacitances in series, and have values in
the different ranges given in Table III.
Similar calculations may be made for any
other ocular rigidity data. Because the
capacitances remain in a fixed ratio, the
designation of CAPQ, for example, as 7.0
indicates the values for the older capacitances.
We are now in a position to convert
the loss elements of the decay components
into resistances. The decay components
are recorded as half-life values, the time
necessary to decay to V2 its value in minutes. This is a time constant which is composed of an elastic element
d (mm. H g )
K or
"The original notation was Kr, "f" standing for "fast",
but because of confusion of "f" with flow, the subscript
"Q" for "quick" is used.
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that is combined with a loss component
An electrical model of the human eye. I 159
Volume 6
Number 2
STRESS-RELAXA TION :
C or
min. x mm. Hg
• )
ROUT = 3.6
in order to give units in reciprocal minutes
(
d ((mm. Hg)
d (j"L)
:O28
/*L
min. x mm. Hg
min. /
and half-life in minutes
FRIEDENWALD :
A capacitor in parallel to a resistance also
gives a time constant
(
d (nQ)
d (volts)
x
volts x sec.
nQ
"OUT
1.33 =F
ROUT
= 0.28
•IS
= sec. 1
in units of seconds, but this time constant
is the time necessary for it to decay to l/e th
of its value, which is 1.47 times longer
than a half-life. Thus, we may establish
the following relation:
= 3.6
C=
—
T
PV=1O
C (PO-Pv)
0.28(15-10)
Fig. 2. Viscoelastic and Friedenwald electrical
models of the eye compared. Note that the capacitors have the value for the range of pressures from
10 to 20 mm. Hg.
R x CAP (in sec.) = 1.47 x half-life (in min.).
Since we know the half-life and the
capacitance for each decay component, we
can solve for R:
1.47 x 0.09
R« =
= 0.019 megohms,"
and
Rs = -
1.47 x 1.7
= 0.64 megohms.
These values are in the ratio of approximately 1:35.
The complete basic electrical model
of the eye
We have translated data obtained from
laboratory conditions into analogous electrical components and have calculated average values for these components from
two sets of data: (1) the classical data of
Friedenwald, and (2) the newer data of
an average of McBain-Prijot for immediate ocular rigidity, combined with that
of St. Helen and McEwen for time dependent ocular rigidity. The two models
are shown in Fig. 2. It will be noticed that
"Usually recorded as 19 kilohms.
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under steady state conditions the models
are indistinguishable and both obey the
formula: F = C (P o - P v ). In a subsequent
paper it will be specifically shown that
tonography can distinguish between the
two models and that the electrical model
incorporating stress relaxation is a more
faithful analog of the action of the eye.
REFERENCES
1. Olson, Harry F.: Dynamical analogies, ed. 2,
New York, 1958, van Nostrand Company, Inc.
2. Goldmann, H.: Der Druck in Schlaemmschen
Kanal bei Normalen und bei glaucoma simplex,
Experientia 6: 110, 1950.
3. Linner, E.: Further studies of the episcleral
venous pressure in glaucoma, Am. J. Ophth.
41: 646, 1956.
4. Becker, B.: Glaucoma (1955-1956), Arch.
Ophth. 56: 898, 1956.
5. McEwen, W. K., and St. Helen, R.: Rheology
of the human sclera. II. Unifying formulation
of ocular rigidity, Ophthalmologica 150: 321,
1965.
6. St. Helen, R., and McEwen, W. K.: Rheology
of the human sclera. I. Anelastic behavior, Am.
J. Ophth. 52: 539, 1961.
7. Itoi, M., Kaneko, H., and Sugimachi, T.:
Anelasticity of eyeball and errors in tonometry,
Jap. J. Ophth. 9: 61, 1965.