- Email: [email protected]
A New Single-Parameter Ocular Rigidity Function
A N E W S I N G L E - P A R A M E T E R O C U L A R RIGIDITY F U N C T I O N T E R R Y J. VAN D E R W E R F F , D.
PHIL.
Cape Town, South Africa
A new empirical one-parameter ocular rigidity function accurately represents the pressure-volume relationship of the corneoscleral shell over a wide range: AV = i (P2* - P,*) where Pi is the initial intraocular pressure and P2 the pressure after an alteration AV in the intraocular volume. Unlike Friedenwald's coefficient of ocular rigidity, the proportionality constant k in the cube root ocular rigidity function is insensitive to changes in intraocular pressure. Its value for enucleated human eyes is approximately k = 0.03 mm Hg14 μΐ - 1 . The cube root ocular rigidity function is as faithful to experimental data as more complicated two- or threeparameter models.
The ocular rigidity function relates pressure changes to volume changes within the eye and is a measure of the elasticity of the corneoscleral shell. A precise formulation for the ocular rigidity function is important to characterize this elasticity, especially in clinical situations. Unfortunately, the mechanical properties of the eye are nonlinear and viscoelastic. Thus, an empirical formulation for the ocular rigidity function must be used. The ocular rigidity function is deter mined experimentally by injecting vol ume increments into the anterior chamber and measuring the resultant intraocular pressure changes for several initial pressures. It is determined clini cally with a standard tonometer. Friedenwald's ocular rigidity func tion—The most commonly used pressureFrom the Department of Biomédical Engineering, University of Cape Town, and Groote Schuur Hospi tal, Cape Town, South Africa. Reprint requests to Terry J. van der Werff, D. Phil., School of Science and Engineering, Seattle University, Seattle, WA 98122.
volume relationship is the ocular rigidity function developed by Friedenwald. 1 He noticed that the slope of the pressurevolume curve in his experiments and in those of previous investigators seemed to be proportional to the intraocular pres sure, that is
which when integrated becomes AV = -ln(P2/Pi)
=^log(P 2 /P 1 )
(2)
where Pi is the initial intraocular pres sure and P2 the intraocular pressure after an alteration AV in the intraocular vol ume. M = 2.303 is the conversion factor between common logarithms to the base 10 and natural logarithms to the base e. The proportionality factor K is called Friedenwald's coefficient of ocular rigidi-
AMERICAN JOURNAL OF OPHTHALMOLOGY 92:391-395, 1981
391
392
AMERICAN JOURNAL OF OPHTHALMOLOGY
ty and is always calculated using loga rithms to the base 10. Equation 2 is an empirical relation based upon thousands of experimental observations. Although Gloster 2 reported a mean value for K in humans of 0.025 μΐ -1 and Ytteborg 3 reported a mean value for K of 0.0232, Friedenwald gave K a value of 0.0215, the figure most com monly used. Unfortunately, Friedenwald's equation does not represent the pressure-volume relation of the eye with complete accura cy. K is not truly constant, but decreases as the intraocular pressure increases. This has been observed both in enucleat ed human eyes4"6 and in living human eyes. 3 Ytteborg 3 also found that the greater the volume of the eye, the lower the rigidity coefficient. Other empirical ocular rigidity func tions—Because Friedenwald's equation is not valid over the full range of intraocular pressures normally encountered experi mentally or clinically, several authors have proposed different empirical formu lations of the ocular rigidity. McBain 7 successfully fitted his data on enucleated eyes by introducing a power law (twoparameter) relation rather than the sim ple proportional relation used by Frie denwald: dP aP n dV which when integrated becomes ΔΥ
1 1 a ( l - n ) [P2 -
P11-
(3)
(4)
McBain's mean values for the constants are {a(l-n)}" 1 = 29.38, and b = 1 - n = 0.356. (Units of pressure and volume here and elsewhere in this paper are measured in millimeters of mercury and microliters, respectively.) The constant b varies less from eye to eye than a, so that for a fixed value of b , one coefficient would suffice to describe a particular eye. (This thought resulted in the present paper.)
SEPTEMBER, 1981
Holland, Madison, and Bean 8 proposed a formula similar to McBain's though it introduced a third parameter:
§ = a(P - of
(5)
which when integrated becomes AV
= a(î^) KP« - <
!-n — IV.~ ( P i - c^Ι-η' )1-]).
(6)
The constants they found for enucleated cat eyes are {a(l-n)}" 1 = 2.779, b = 1 - n = 0.611, a n d c = 10.71. McEwen and St. Helen 9 attempted to draw together all published data for human, cat, and rabbit eyes, both in vivo and enucleated, into a two-parameter form which they called the unifying for mulation of ocular rigidity: d P
dV
r> +j . bu
=
a P
(7)
which when integrated becomes Av - *i
Pa + b/a
- ä
Pi + b/a
AV
ln
(8)
Friedenwald's formulation is a special case of this equation, that is, when b = 0 and a = MK. McEwen and St. Helen successfully applied equation 8 to much of the available data. For human eyes, they gave values for a ranging from 0.015 to 0.027 μΐ' 1 and for b from 0.03 to 0.31 mm Hg/μΐ, with mean values for enucleated or post mortem in situ eyes of a = 0.022 μΐ"1 and b = 0.21 mm Hg/μΐ. A relationship between pressure and volume of the form shown in equation 7 implies that stress within the corneoscleral shell is exponentially related to strain. In uniaxial stress-strain experi ments on isolated strips of human sciera, Graebel and van Alphen 10 confirmed this exponential behavior, a nonlinear feature that the corneoscleral shell shares with most biological soft tissues. 11
VOL. 92, NO. 3
CUBE ROOT OCULAR RIGIDITY FUNCTION
Using a flying spot scanner and a finite element mathematical method to deter mine the elastic properties of the sciera, stroma, and optic disk separately, Woo and associates 12 also found a linear rela tionship between the stress and the de rivative of stress with respect to strain. In a companion paper, 13 they expressed their calculated intraocular pressurevolume relation as
393
The advantage of Friedenwald's formu lation is that it contains only one parame ter. Its disadvantage, however, is that it is not accurate over the entire range of intraocular pressures normally encoun tered. Although the other formulations are more accurate, they have the disad vantage of containing two or three parameters. RESULTS
dP = 0.016 P + 0.13. dV
A simple formulation for the ocular rigidity function that is accurate and which contains only one parameter is
(9)
Note that the constants fall within the range of values determined by McEwen and St. Helen. 9 Similarly, Hibbard and associates 14 found that for enucleated human eyes dP = 0.020 P + 0.24. dV
1 ΔΥ = £(P 2 * - P ^
(ID
which I call the cube root ocular rigidity function. Equation 11 corresponds to a rigidity of form:
(10)
This also falls within McEwen and St. Helen's range. Note that for each of the empirical relationships presented, the eye's rigidity E = dP/dV increases monotonically with increasing intraocular pressure.
dV
J k F
'
(12)
The proportionality factor k is a constant of approximate value 0.03 for enucleated human eyes. The values for k as deter mined at P = 15.5 mm Hg from the
TABLE 1 O C U L A R RIGIDITY FUNCTIONS W I T H PARAMETERS ADJUSTED*
Study
Rigidity Function
Equation No.
Friedenwald 1
AV = 20.2 ln(P2/Pi)
(13)
McBain7
ΔΥ = 21.4 (Pz356 - Pr 356 )
(14)
Holland and associates 8
AV = 3.92{(P2 - 10.71)· β11 -(Ρι - 10.71) 611 } P 2 + 5.8 ΔΥ = 27.8 In Pi + 5.8
(15)
McEwen and St. Helen 9
(16)
Woo and associates 12,13
ΔΥ = 24.3 In
P 2 + 3.2 Pi + 3.2
(17)
Hibbard and associates 14
ΔΥ = 29.4 In
P 2 + 7.1 Pi + 7.1
(18)
Cube root
ΔΥ = 24.3 (P2:* - Pi 1 4 )
(19)
*All parameters have a rigidity at (ΔΥ, P) = (0, 15.5) of 0.767 mm Hg/μΐ. Pi = 15.5 mm Hg.
394
AMERICAN JOURNAL OF OPHTHALMOLOGY
Fig. 1 (van der Werft). Comparison of ocular rigidity functions with coefficients adjusted to give the same rigidity of 0.767 mm Hg/μΐ at an intraocular pressure of 15.5 mm Hg. The curves correspond to the equations given in Table 1. F refers to Frieden wald,1 M to McBain,7 HO to Holland and associates,8 MS to McEwen and St. Helen,9 W to Woo and associates,12,13 HI to Hibbard and associates,14 C to the cube root ocular rigidity function, and 106 to McBain's experimental result for eye No. 106.
ocular rigidity functions of Ytteborg, 4 McBain, 7 and McEwen and St. Helen 9 are 0.0296, 0.0299, and 0.0295, respectively. I cannot present a sound theoretical basis for this formulation. It must, there fore, be classified at this time as a purely
SEPTEMBER, 1981
empirical formulation. (There is some reason to believe from similitude argu ments that the exponent should be 3/8 rather than 1/3, but this needs to be investigated further.) Comparison of ocular rigidity func tions—It is difficult to make a simple comparison of all the ocular rigidity func tions considered so far, because the para metric values given by different authors are necessarily derived from different ex periments and, in some cases, from dif ferent species. The primary obstacle to direct comparison, however, lies in the functional form associated with each ocu lar rigidity function. The cube root and Friedenwald's formulations have one pa rameter; McBain's, McEwen and St. Helen's, Woo and associates', and Hib bard and associates' have two; and Hol land, Madison, and Bean's has three. Clearly, the more parameters in an ex pression, the more accurately it should fit any experimental data. One comparative approach is to re quire all formulations to have the same rigidity or pressure-volume slope at
TABLE 2 OCULAR RIGIDITY FUNCTIONS WITH PARAMETERS GIVEN BY THE AUTHORS*
Rigidity Function
Study 1
(13)
Friedenwald
AV = 20.2 In (P2/P1)
McBain7
AV = 29.38{P2·366 - Pi356} 18
Holland and associates
McEwen and St. Helen9
611
(20) 611
AV = 2.779{(P2 - 10.7)- -(Pi - 10.7) } „7 Ρ2 + 9.55 ir. „κ , - *,.*, ,π ρ ι + g 5 5
Woo and associates 12,13
P + 8.1 AV = 6 2 5 ΐ η 2 ^ · Ρ;+8.ι
Hibbard and associates1'
AV = 5 0 0 ■ ·
Cube root
AV == 33.3 (P2% - Pi%)
*Pi = 15.5 mm Hg.
Equation No.
ln
P2 + 12.0 P ! +12.0
(21) (22) (23) (24) (25)
VOL. 92, NO. 3
CUBE ROOT OCULAR RIGIDITY FUNCTION
395
most commonly used, gives results for humans that are considerably different from the others. The formula of Woo and associates 12 ' 13 also gives considerably different results and may indicate the difficulty of deriving whole-eye rigidity coefficients from data obtained from seg ments alone. 16 REFERENCES 0
,o
20
30
40
50
av(pi) 60
Fig. 2 (van der Werfï). Comparison of ocular rigidity functions with all coefficients as given by the authors. The curves correspond to the equations given in Table 2. F refers to Friedenwald, 1 M to McBain, 7 HO to Holland and associates, 8 MS to McEwen and St. Helen, 9 W to Woo and associ ates, 1213 HI to Hibbard and associates, 14 C to the cube root ocular rigidity function, and 106 to Mc Bain's experimental results for eye No. 106.
(AV, P) = (0, 15.5) as Friedenwald's form ulation: rigidity = E = dP/dV = 0.767 mm Hg/μΐ. For formulations with more than one parameter, I have adjust ed only the initial parameter a in the equations and left the other parameters as given by the authors. The formulas obtained in this manner are shown in Table 1. The values for a of 0.036, 0.041, and 0.034 in equations 16, 17, and 18, respectively, are considerably outside the range of values found normally. These formulas in Table 1 are plotted in Figure 1 along with the experimental curve for McBain's enucleated human eye No. 106, whose parameters differ somewhat from those given in equation 14. Another comparative approach is to use the authors' own parameters, given in Table 2. These formulas are plotted in Figure 2. The experimental curve for McBain's eye No. 106 is again given for comparison. All curves except for that of Holland, Madison, and Bean, 8 which is for cats, refer to enucleated human eyes. Note that Friedenwald's formula, the
1. Friedenwald, J. S.: Contribution to the theory and practice of tonometry. Am. J. Ophthalmol. 20:985, 1937. 2. Gloster, J.: Tonometry and Tonography. Lon don, J. & A. Churchill, 1966. 3. Ytteborg, J.: Further investigations of factors influencing size of rigidity coefficient. Acta Ophthal mol. 38:643, 1960. 4. : The effect of intraocular pressure on rigidity coefficient in the human eye. Acta Ophthal mol. 38:548, 1960. 5. Macri, F. J., Wanko, T., and Grimes, P. A.: The elastic properties of the human eye. Arch. Ophthalmol. 60:1021, 1958. 6. Gloster, J., and Perkins, E. S.: Distensibility of the human eye. Br. J. Ophthalmol. 43:97, 1959. 7. McBain, E. H.: Tonometer calibration. II. Oc ular rigidity. Arch. Ophthalmol. 60:1080, 1958. 8. Holland, M. G., Madison, J., and Bean, W.: The ocular rigidity function. Am. J. Ophthalmol. 50:958, 1960. 9. McEwen, W. K., and St. Helen, R.: Rheology of the human sciera. Unifying formulation of ocular rigidity. Ophthalmologica 150:321, 1965. 10. Graebel, W. P., and van Alphen, G. W. H. M.: The elasticity of sciera and choroid of the human eye, and its implication on scierai rigidity and accommodation. J. Biomech. Eng. 99:203, 1977. 11. Fung, Y. C. : Elasticity of soft tissues in simple elongation. Am. J. Physiol. 213:1532, 1967. 12. Woo, S. L-Y., Kobayashi, A. S., Schlegel, W. A., and Lawrence, C : Nonlinear material prop erties of intact cornea and sciera. Exp. Eye Res. 14:29, 1972. 13. Woo, S. L-Y., Kobayashi, A. S., Lawrence, C , and Schlegel, W. A.: Mathematical model of the corneo-scleral shell as applied to intraocular pressure-volume relations and applanation tonome try. Ann. Biomed. Eng. 1:87, 1972. 14. Hibbard, R. R., Lyon, C. S., Shepard, M. D., McBain, E. H., and McEwen, W. K.: Im mediate rigidity of an eye. I. Whole, segments and strips. Exp. Eye Res. 9:137, 1970. 15. Collins, R., and van der Werff, T. J.: Mathe matical Models of the Dynamics of the Human Eye. Berlin, Springer-Verlag, 1980, pp. 36-38. 16. van der Werff, T. J. : Whole eye rigidity coeffi cients from segment experiments. Exp. Eye Res. 13:181, 1972.
PHIL.
Cape Town, South Africa
A new empirical one-parameter ocular rigidity function accurately represents the pressure-volume relationship of the corneoscleral shell over a wide range: AV = i (P2* - P,*) where Pi is the initial intraocular pressure and P2 the pressure after an alteration AV in the intraocular volume. Unlike Friedenwald's coefficient of ocular rigidity, the proportionality constant k in the cube root ocular rigidity function is insensitive to changes in intraocular pressure. Its value for enucleated human eyes is approximately k = 0.03 mm Hg14 μΐ - 1 . The cube root ocular rigidity function is as faithful to experimental data as more complicated two- or threeparameter models.
The ocular rigidity function relates pressure changes to volume changes within the eye and is a measure of the elasticity of the corneoscleral shell. A precise formulation for the ocular rigidity function is important to characterize this elasticity, especially in clinical situations. Unfortunately, the mechanical properties of the eye are nonlinear and viscoelastic. Thus, an empirical formulation for the ocular rigidity function must be used. The ocular rigidity function is deter mined experimentally by injecting vol ume increments into the anterior chamber and measuring the resultant intraocular pressure changes for several initial pressures. It is determined clini cally with a standard tonometer. Friedenwald's ocular rigidity func tion—The most commonly used pressureFrom the Department of Biomédical Engineering, University of Cape Town, and Groote Schuur Hospi tal, Cape Town, South Africa. Reprint requests to Terry J. van der Werff, D. Phil., School of Science and Engineering, Seattle University, Seattle, WA 98122.
volume relationship is the ocular rigidity function developed by Friedenwald. 1 He noticed that the slope of the pressurevolume curve in his experiments and in those of previous investigators seemed to be proportional to the intraocular pres sure, that is
which when integrated becomes AV = -ln(P2/Pi)
=^log(P 2 /P 1 )
(2)
where Pi is the initial intraocular pres sure and P2 the intraocular pressure after an alteration AV in the intraocular vol ume. M = 2.303 is the conversion factor between common logarithms to the base 10 and natural logarithms to the base e. The proportionality factor K is called Friedenwald's coefficient of ocular rigidi-
AMERICAN JOURNAL OF OPHTHALMOLOGY 92:391-395, 1981
391
392
AMERICAN JOURNAL OF OPHTHALMOLOGY
ty and is always calculated using loga rithms to the base 10. Equation 2 is an empirical relation based upon thousands of experimental observations. Although Gloster 2 reported a mean value for K in humans of 0.025 μΐ -1 and Ytteborg 3 reported a mean value for K of 0.0232, Friedenwald gave K a value of 0.0215, the figure most com monly used. Unfortunately, Friedenwald's equation does not represent the pressure-volume relation of the eye with complete accura cy. K is not truly constant, but decreases as the intraocular pressure increases. This has been observed both in enucleat ed human eyes4"6 and in living human eyes. 3 Ytteborg 3 also found that the greater the volume of the eye, the lower the rigidity coefficient. Other empirical ocular rigidity func tions—Because Friedenwald's equation is not valid over the full range of intraocular pressures normally encountered experi mentally or clinically, several authors have proposed different empirical formu lations of the ocular rigidity. McBain 7 successfully fitted his data on enucleated eyes by introducing a power law (twoparameter) relation rather than the sim ple proportional relation used by Frie denwald: dP aP n dV which when integrated becomes ΔΥ
1 1 a ( l - n ) [P2 -
P11-
(3)
(4)
McBain's mean values for the constants are {a(l-n)}" 1 = 29.38, and b = 1 - n = 0.356. (Units of pressure and volume here and elsewhere in this paper are measured in millimeters of mercury and microliters, respectively.) The constant b varies less from eye to eye than a, so that for a fixed value of b , one coefficient would suffice to describe a particular eye. (This thought resulted in the present paper.)
SEPTEMBER, 1981
Holland, Madison, and Bean 8 proposed a formula similar to McBain's though it introduced a third parameter:
§ = a(P - of
(5)
which when integrated becomes AV
= a(î^) KP« - <
!-n — IV.~ ( P i - c^Ι-η' )1-]).
(6)
The constants they found for enucleated cat eyes are {a(l-n)}" 1 = 2.779, b = 1 - n = 0.611, a n d c = 10.71. McEwen and St. Helen 9 attempted to draw together all published data for human, cat, and rabbit eyes, both in vivo and enucleated, into a two-parameter form which they called the unifying for mulation of ocular rigidity: d P
dV
r> +j . bu
=
a P
(7)
which when integrated becomes Av - *i
Pa + b/a
- ä
Pi + b/a
AV
ln
(8)
Friedenwald's formulation is a special case of this equation, that is, when b = 0 and a = MK. McEwen and St. Helen successfully applied equation 8 to much of the available data. For human eyes, they gave values for a ranging from 0.015 to 0.027 μΐ' 1 and for b from 0.03 to 0.31 mm Hg/μΐ, with mean values for enucleated or post mortem in situ eyes of a = 0.022 μΐ"1 and b = 0.21 mm Hg/μΐ. A relationship between pressure and volume of the form shown in equation 7 implies that stress within the corneoscleral shell is exponentially related to strain. In uniaxial stress-strain experi ments on isolated strips of human sciera, Graebel and van Alphen 10 confirmed this exponential behavior, a nonlinear feature that the corneoscleral shell shares with most biological soft tissues. 11
VOL. 92, NO. 3
CUBE ROOT OCULAR RIGIDITY FUNCTION
Using a flying spot scanner and a finite element mathematical method to deter mine the elastic properties of the sciera, stroma, and optic disk separately, Woo and associates 12 also found a linear rela tionship between the stress and the de rivative of stress with respect to strain. In a companion paper, 13 they expressed their calculated intraocular pressurevolume relation as
393
The advantage of Friedenwald's formu lation is that it contains only one parame ter. Its disadvantage, however, is that it is not accurate over the entire range of intraocular pressures normally encoun tered. Although the other formulations are more accurate, they have the disad vantage of containing two or three parameters. RESULTS
dP = 0.016 P + 0.13. dV
A simple formulation for the ocular rigidity function that is accurate and which contains only one parameter is
(9)
Note that the constants fall within the range of values determined by McEwen and St. Helen. 9 Similarly, Hibbard and associates 14 found that for enucleated human eyes dP = 0.020 P + 0.24. dV
1 ΔΥ = £(P 2 * - P ^
(ID
which I call the cube root ocular rigidity function. Equation 11 corresponds to a rigidity of form:
(10)
This also falls within McEwen and St. Helen's range. Note that for each of the empirical relationships presented, the eye's rigidity E = dP/dV increases monotonically with increasing intraocular pressure.
dV
J k F
'
(12)
The proportionality factor k is a constant of approximate value 0.03 for enucleated human eyes. The values for k as deter mined at P = 15.5 mm Hg from the
TABLE 1 O C U L A R RIGIDITY FUNCTIONS W I T H PARAMETERS ADJUSTED*
Study
Rigidity Function
Equation No.
Friedenwald 1
AV = 20.2 ln(P2/Pi)
(13)
McBain7
ΔΥ = 21.4 (Pz356 - Pr 356 )
(14)
Holland and associates 8
AV = 3.92{(P2 - 10.71)· β11 -(Ρι - 10.71) 611 } P 2 + 5.8 ΔΥ = 27.8 In Pi + 5.8
(15)
McEwen and St. Helen 9
(16)
Woo and associates 12,13
ΔΥ = 24.3 In
P 2 + 3.2 Pi + 3.2
(17)
Hibbard and associates 14
ΔΥ = 29.4 In
P 2 + 7.1 Pi + 7.1
(18)
Cube root
ΔΥ = 24.3 (P2:* - Pi 1 4 )
(19)
*All parameters have a rigidity at (ΔΥ, P) = (0, 15.5) of 0.767 mm Hg/μΐ. Pi = 15.5 mm Hg.
394
AMERICAN JOURNAL OF OPHTHALMOLOGY
Fig. 1 (van der Werft). Comparison of ocular rigidity functions with coefficients adjusted to give the same rigidity of 0.767 mm Hg/μΐ at an intraocular pressure of 15.5 mm Hg. The curves correspond to the equations given in Table 1. F refers to Frieden wald,1 M to McBain,7 HO to Holland and associates,8 MS to McEwen and St. Helen,9 W to Woo and associates,12,13 HI to Hibbard and associates,14 C to the cube root ocular rigidity function, and 106 to McBain's experimental result for eye No. 106.
ocular rigidity functions of Ytteborg, 4 McBain, 7 and McEwen and St. Helen 9 are 0.0296, 0.0299, and 0.0295, respectively. I cannot present a sound theoretical basis for this formulation. It must, there fore, be classified at this time as a purely
SEPTEMBER, 1981
empirical formulation. (There is some reason to believe from similitude argu ments that the exponent should be 3/8 rather than 1/3, but this needs to be investigated further.) Comparison of ocular rigidity func tions—It is difficult to make a simple comparison of all the ocular rigidity func tions considered so far, because the para metric values given by different authors are necessarily derived from different ex periments and, in some cases, from dif ferent species. The primary obstacle to direct comparison, however, lies in the functional form associated with each ocu lar rigidity function. The cube root and Friedenwald's formulations have one pa rameter; McBain's, McEwen and St. Helen's, Woo and associates', and Hib bard and associates' have two; and Hol land, Madison, and Bean's has three. Clearly, the more parameters in an ex pression, the more accurately it should fit any experimental data. One comparative approach is to re quire all formulations to have the same rigidity or pressure-volume slope at
TABLE 2 OCULAR RIGIDITY FUNCTIONS WITH PARAMETERS GIVEN BY THE AUTHORS*
Rigidity Function
Study 1
(13)
Friedenwald
AV = 20.2 In (P2/P1)
McBain7
AV = 29.38{P2·366 - Pi356} 18
Holland and associates
McEwen and St. Helen9
611
(20) 611
AV = 2.779{(P2 - 10.7)- -(Pi - 10.7) } „7 Ρ2 + 9.55 ir. „κ , - *,.*, ,π ρ ι + g 5 5
Woo and associates 12,13
P + 8.1 AV = 6 2 5 ΐ η 2 ^ · Ρ;+8.ι
Hibbard and associates1'
AV = 5 0 0 ■ ·
Cube root
AV == 33.3 (P2% - Pi%)
*Pi = 15.5 mm Hg.
Equation No.
ln
P2 + 12.0 P ! +12.0
(21) (22) (23) (24) (25)
VOL. 92, NO. 3
CUBE ROOT OCULAR RIGIDITY FUNCTION
395
most commonly used, gives results for humans that are considerably different from the others. The formula of Woo and associates 12 ' 13 also gives considerably different results and may indicate the difficulty of deriving whole-eye rigidity coefficients from data obtained from seg ments alone. 16 REFERENCES 0
,o
20
30
40
50
av(pi) 60
Fig. 2 (van der Werfï). Comparison of ocular rigidity functions with all coefficients as given by the authors. The curves correspond to the equations given in Table 2. F refers to Friedenwald, 1 M to McBain, 7 HO to Holland and associates, 8 MS to McEwen and St. Helen, 9 W to Woo and associ ates, 1213 HI to Hibbard and associates, 14 C to the cube root ocular rigidity function, and 106 to Mc Bain's experimental results for eye No. 106.
(AV, P) = (0, 15.5) as Friedenwald's form ulation: rigidity = E = dP/dV = 0.767 mm Hg/μΐ. For formulations with more than one parameter, I have adjust ed only the initial parameter a in the equations and left the other parameters as given by the authors. The formulas obtained in this manner are shown in Table 1. The values for a of 0.036, 0.041, and 0.034 in equations 16, 17, and 18, respectively, are considerably outside the range of values found normally. These formulas in Table 1 are plotted in Figure 1 along with the experimental curve for McBain's enucleated human eye No. 106, whose parameters differ somewhat from those given in equation 14. Another comparative approach is to use the authors' own parameters, given in Table 2. These formulas are plotted in Figure 2. The experimental curve for McBain's eye No. 106 is again given for comparison. All curves except for that of Holland, Madison, and Bean, 8 which is for cats, refer to enucleated human eyes. Note that Friedenwald's formula, the
1. Friedenwald, J. S.: Contribution to the theory and practice of tonometry. Am. J. Ophthalmol. 20:985, 1937. 2. Gloster, J.: Tonometry and Tonography. Lon don, J. & A. Churchill, 1966. 3. Ytteborg, J.: Further investigations of factors influencing size of rigidity coefficient. Acta Ophthal mol. 38:643, 1960. 4. : The effect of intraocular pressure on rigidity coefficient in the human eye. Acta Ophthal mol. 38:548, 1960. 5. Macri, F. J., Wanko, T., and Grimes, P. A.: The elastic properties of the human eye. Arch. Ophthalmol. 60:1021, 1958. 6. Gloster, J., and Perkins, E. S.: Distensibility of the human eye. Br. J. Ophthalmol. 43:97, 1959. 7. McBain, E. H.: Tonometer calibration. II. Oc ular rigidity. Arch. Ophthalmol. 60:1080, 1958. 8. Holland, M. G., Madison, J., and Bean, W.: The ocular rigidity function. Am. J. Ophthalmol. 50:958, 1960. 9. McEwen, W. K., and St. Helen, R.: Rheology of the human sciera. Unifying formulation of ocular rigidity. Ophthalmologica 150:321, 1965. 10. Graebel, W. P., and van Alphen, G. W. H. M.: The elasticity of sciera and choroid of the human eye, and its implication on scierai rigidity and accommodation. J. Biomech. Eng. 99:203, 1977. 11. Fung, Y. C. : Elasticity of soft tissues in simple elongation. Am. J. Physiol. 213:1532, 1967. 12. Woo, S. L-Y., Kobayashi, A. S., Schlegel, W. A., and Lawrence, C : Nonlinear material prop erties of intact cornea and sciera. Exp. Eye Res. 14:29, 1972. 13. Woo, S. L-Y., Kobayashi, A. S., Lawrence, C , and Schlegel, W. A.: Mathematical model of the corneo-scleral shell as applied to intraocular pressure-volume relations and applanation tonome try. Ann. Biomed. Eng. 1:87, 1972. 14. Hibbard, R. R., Lyon, C. S., Shepard, M. D., McBain, E. H., and McEwen, W. K.: Im mediate rigidity of an eye. I. Whole, segments and strips. Exp. Eye Res. 9:137, 1970. 15. Collins, R., and van der Werff, T. J.: Mathe matical Models of the Dynamics of the Human Eye. Berlin, Springer-Verlag, 1980, pp. 36-38. 16. van der Werff, T. J. : Whole eye rigidity coeffi cients from segment experiments. Exp. Eye Res. 13:181, 1972.