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Von Frey's method of measuring pressure sensibility in the hand: An engineering analysis of the Weinstein-Semmes pressure aesthesiometer
Von Frey's method of measuring pressure sensibility in the hand: An engineering analysis of the Weinstein-Semmes pressure aesthesiometer The development of microsurgical techniques has generated a resurgence of interest in estimating local pressure sensibility as a measure of sensory improvement. Because our experience with Weinstein's modification of Von Frey's probes yielded variable and poorly understood results, we measured two sets of probes and examined them in the light of the engineering principles on which their behavior is based. The mechanical behavior of the nylon monofilaments can be described as buckling with one end built in and the other end pinned. The probes are relatively uniform and consistent. However, no loss in sensitivity would accompany division of the set into two or three equivalent sets. Variations in the buckling stress as high as a factor of eight are difficult to avoid. Gross errors arise from careless application, variations in the elastic modulus due to changes in temperature and humidity, and variations in the attachment of fibers to handles and differences in the ends of the filaments. Interpreting results for this instrument requires an understanding of the factors which can influence those results. The probes are simple to use but easy to misinterpret.
Scott Levin, B.S.,* George Pearsall, Sc.D.,** and Robert J. Ruderman, M.D.,* Durham, N. C.
In 1898 Von Frey introduced a clinical methodusing what have been referred to subsequently as Von Frey hairs or Von Frey filaments-for evaluating the skin's sensitivity to touch.! By pressing on the skin with a thorn glued on the end of a hair until the hair started to bow out, he obtained a measure of the pressure sensibility of nerve fibers in the skin. Von Frey calibrated the hairs on a balance, varying stiffness by changing length or by using hairs of different "hardness." He recorded the pressure sensibility by noting whether a given hair pressed on the skin produced any sensation. In subsequent papers, Von Frey 2-4 elaborated on this method of measuring pressure sensibility, but his technique essentially remained unchanged. Recently Weinstein 5 reintroduced Von Frey's method, using nylon monofilaments mounted in plastic handles as substitutes for the more variable natural hairs. These probes now are known as the WeinsteinSemmes pressure aesthesiometer and frequently are From the Division of Orthopaedic Surgery, Duke University Medical Center, * and the Department of Engineering, Duke University, ** Durham, N. C. Received for publication June 2, 1977. Reprint requests: Robert J. Ruderman, M.D., P. O. Box 3023, Duke University Medical Center, Durltam, NC 27710.
employed in the diagnosis and postoperative monitoring of nerve regeneration. The advantages of this method of recording pressure sensibility are ease of application, discrimination according to location, and measurement of threshold sensitivity. The development of new surgical techniques has created more interest in the reliability of assessing local pressure sensibility as a measure of sensory improvement and a need to monitor patients with more accurate clinical testing methods. A number of methods are used currently: the two-point discrimination test utilizing a paper clip, the sweat test, vibratory stimulus response, moving touch, and constant touch 6 ; but the Von Frey type of filaments provide one of the most common means of monitoring local sensory improvement. Sensory return, as measured by constant touch, is thought to be the result of regeneration of slowadapting group A beta fibers and their proposed end organ, the Merkel disc, which is said to be pressure sensitive. 7 • 8 According to this view, Von Frey filaments measure the threshold sensitivity of individual or groups of Merkel nerve endings, whereas the twopoint discrimination test measures the density of these endings. Our experiences at Duke University Medical Center have indicated that the probes in the Weinstein-
0363-5023/78/0303-0211 $00.60/0 © 1978 American Society for Surgery of the Hand
THE JOURNAL OF HAND SURGERY
211
212
The Joumal of HAND SURGERY
Levin, Pearsall, and Rudennan
Materials Two sets of probes were used-one for obtaining force and geometry data, the other for measuring elastic modulus and evaluating consistency between sets. The monofilaments were taken from Weinstein-Semmes pressure aesthesiometers manufactured by Research Media, Inc. (currently available from Stoelting Co., 1350 S. Kostner St., Chicago, IL 60623). A Mettler top-loading balance, Model P-163, was used to record the force exerted on the monofilaments. The balance capacity was 160 gm. Monofilament diameters (D := 2R) were measured to approximately ±2% in a microscope with a filarmeasuring eyepiece and calibrated objectives, part of a Tukon Microhardness Tester. Lengths (L) were measured to approximately ±0.5% by vernier calipers. We determined the elastic modulus of the nylon monofilaments by sacrificing 11 filaments from one Weinstein-Semmes aesthesiometer and stretching them in tension in an Instron mechanical testing machine, Model TT-C.
Methods
Fig. 1. Experimental arrangement for measuring buckling forces.
Semmes pressure aesthesiometer are simple to use but easy to misinterpret. For example, each probe is marked with a number (M) which represents the logarithm of 10 times the force in milligrams required to bow the monofilament: M = log(lO Fmg)
(1)
But these numbers (M) often are misinterpreted and quoted erroneously in the medical literature as the actual force in grams, resulting in quoted force values from 67 times too small (for the largest probe) to 365 times too large (for the smallest probe). In addition, the probes are calibrated according to force rather than stress. Stress (defined as force per unit area) is the more appropriate variable for measuring pressure sensibility. The data obtained may not only be somewhat inconsistent, but also less accurate than the user might assume. As a result ot these experiences, we measured two sets of probes and examined them from the standpoint of the engineering principles on which their behavior is based.
CaUbration. One set of probes was calibrated by applying each monofilament, in the manner suggested by Werner and Orner!' for clinical testing, to the pan of the Mettler balance. A piece of rubber glove was placed on the pan to avoid slippage (Fig. I). The probe was maintained perpendicular to the balance, with the plastic handle horizontal, and the mass in grams which produced sufficient force to initiate a bow in the monofilament was recorded as the "grams force" for that monofilament. At least 10 tests were performed on each monofilament, and a standard deviation was calculated for each set of values; the temperature was 74° ± 1° F and relative humidity was typical of a hospital laboratory environment, with no pronounced changes occurring during the testing period. The monofilaments were tested at random to avoid any systematic errors. Mechanical properties. The elastic modulus was determined for 11 nylon monofilaments cut from a Weinstein-Semmes pressure aesthesiometer. The test procedure involved clamping a 1.4 in length of monofilament between two sets of rubber-faced grips, initially 1.0 in apart, in an Instron testing machine and moving the grips apart, at a relative velocity (v) of 0.2 in/min. The results were recorded as a force-time (F-t) curve, which was converted to a stress-strain (S-e) curve when the force axis was divided by the crosssectional area (A) of the fiber,
Vol. 3, No.3
Von Frey's method of measuring pressure sensibility
May, 1978
213
Table I. Stress and force calculations and measurements for a Weinstein-Semmes pressure aesthesiometer
2L1R
Measured force F (gm)
Standard deviation uF (gm)
Measured stress S (gmlsq mm)
Standard deviation uS (gmlsq mm)
130 141 180 200 251 280 312 341 453 473 520 593 650 689 851 1,050 1,090 1,420 1,930 2,300
86.5 73.2 22 .3 18.6 17.0 10.6 3.14 2.81 1.85 1.58 0.977 0.562 0.213 0.112 0.091 0.034 0.0094 0.0040
4.3 1.9 1.5 0.8 1.5 0.5 0.07 0.06 0.06 0.06 0.030 0.011 0.004 0.007 0.005 0.002 0.0001 0.0001
171 175 82.0 84.9 94.9 76.1 38.9 36.6 29.5 33.7 23.9 15.7 9.29 7.50 6.52 4.02 2.14 1.29
16 13 9.6 7.9 13 7.4 2.8 2.6 2.4 2.9 1.9 1.1 0.64 0.84 0.68 0.43 0.13 0.09
Manufacturer's marking
Calculated force F' (gm)
Diameter D (mm)
Area A (sq m)
Calculated stress S' (gmlsq mm)
Slenderness
6.65 6.45 6.10 5.88 5.46 5.18 5.07 4.93 4.74 4.56 4.31 4 . 17 4.08 3.84 3.61 3.22 2.83 2.44 2.36 1.65
448 283 126 76.0 28.9 15.2 11.8 8.53 5.51 3.64 2.05 1.48 1.20 0.693 0.408 0.166 0.068 0.0276 0.0229 0.0045
1.\42 1.033 0.805 0.732 0.582 0.525 0.475 0.423 0.322 0.313 0.284 0.244 0.228 0.214 0.171 0. 137 0.132 0. 104 0.075 0.063
1.02 0.84 0.51 0.42 0.27 0.22 0. 18 0.14 0.081 0.077 0.063 0.047 0.041 0.036 0.023 0.015 0.014 0.0085 0.0044 0.0031
439 337 243 181 107 69.1 65.6 60.9 68 .0 47 .3 33.1 31.5 29.3 19.3 17.7 11.1 4 .86 3.25 5.20 1.45
S=
f. A
(2)
and the time axis was multiplied by the ratio of separation velocity (v) to initial length between grips (£0): €=
M vt to to
-=-
(3)
The elastic modulus was calculated as the slope of the stress-strain curve: E = dS de
(4)
during the initial (elastic or recoverable) stage of deformation . Results The elastic modulus appeared to exhibit a slight increase for finer diameter monofilaments, but the data can be represented quite well by a mean elastic modulus, E = 3.56 X lOB dynes/sq mm (516,000 psi) with a standard deviation of ±O.22 x lOB dynes/sq mm (±32,220 psi), The calibration data for the probes are presented in Table I and plotted in Fig. 2. The data are plotted in terms of stress because the sensibility that these probes measure is a sensibility to pressure rather than to force . The units of gram per square millimeter were chosen for stress because the user can be expected to have a
better physical feeling for grams and millimeters on this scale of clinical testing than for the more correct engineering units of dynes per square millimeter, Newtons per square meter, or pounds per square inch (psi) . The stress data are plotted vs. the slenderness ratio (2LiR) for each monofilament because this ratio describes a principal geometric factor in the theory of buckling slender columns .
Discussion In engineering terms, a Von Frey filament works by buckling elastically at a critical force or stress. Qualitatively, buckling is the lateral bending, or bowing out, of a beam or column that is compressed axially. Fig. 3 illustrates how the lateral displacement increases markedly in the vicinity of the critical buckling stress, providing an easily observed signal that the critical stress has been reached. Historically, Euler10 was the first investigator to apply the calculus to the analysis of column buckling. In 1736 he described the derivation of the equation: C1T 2
F"r= 4U
(5)
for the critical force to buckle a column loaded, as shown in Fig. 4, A . Long before Von Frey's experiments in physiology, other investigators in mechanics showed that Euler's constant, C = EI, where E is the
214
The Journal of HAND SURGERY
Levin, Pearsall, and Ruderman
1.2r------------.
o Stresses calclJlated from manlJfactlJrer's mar/cings and measlJred diafMt.rs. Stresses calclJlated from measlJred bvclcling forces and diameters.
•
100
~8 70
"i)
50
:.: :~
"0 .!:!
E
-=
20
"'E
-!: E o a .S
Vl Vl W
......
II)
10
9.0 8.0 7.0 6.0 5.0 4.0 3.0
2.0
'" ~
Vl
< 0.6
'"
Vl
II)
w
'"
\
0.4
~
Vl
\ \
\\ \\ \\
0.2
OL---------------__~ LATERAL DEFLECTION -
\\
Fig. 3. The variation of lateral deflection with stress for a typical case of buckling. Note that the deflection increases markedly when the stress reaches the critical buckling stress. (After Higdon et a1. 12 )
\\
\\
\\
\\ \\ \\ \\
\\
1.0
\\ \\ \\ \\
0.9 0.8 0.7 0.6 0.5 0.4 0.3
\\
\\ \\
0.2
O.IL-_--L._L-L.J._ _.L.-_-l...~'___ __
100
0.8
Q
\\
.~
g>
Vl
-- 5 TT2E db ./ . - - cr= 4{2lIR}2 ' one en IJllt-m, ontl tlndfrH.
\
60
t---------
u
5 2.05TT2E db ·z . cr = (2l1R12 ,one en IJI tom, t one end pmntld.
-
200
1.0 ~
200 300 400 500
1000
2000 3000
2L1R (logarithmic scale)
Fig. 2. Buckling stress VS. slenderness ratio (logarithmic scales) for one set of probes from a Weinstein-Semmes pressure aesthesiometer, as compared with theoretical buckling curves for two end conditions. The error bars on the data points and the double curves for each theoretical case represent expected limits of error when they are greater than the size of a data point.
elastic modulus of the material comprising the column and I is the second moment of area for the crosssection.lJ For a circular cross-section I = 7TR4/4 and A = 7TR2, so: Fer Srr = A
7T 2E
= 4(2L/R)2
(6)
for a column loaded as shown in Fig. 4, A. Generalizing equation (6) to account for different end conditions and the possibility of higher orders of buckling, (7)
equation (7) provides a reasonably accurate description of the buckling stress for slenderness ratios greater than approximately 150. 12 As can be seen from Table I, the lowest slenderness ratio for a probe in the WeinsteinSemmes pressure aesthesiometer is 130, so equation (7) is appropriate for our analysis. Among the buckling end constraints 13. 14 that have been analyzed in the literature of mechanics, the two which seem most appropriate to consider in analyzing the behavior of the filaments in a Weinstein-Semmes pressure aesthesiometer are presented in Fig. 4 and 5. In these figures the top of the filament is considered to be "built-in" to the handle. The lower end of the filament in Fig. 4 is described as "free," and this mode of buckling would be expected if care were not exercised in maintaining the top of the filament directly above the bottom during a test. For the buckling illustrated in Fig. 4, A. n = ':4 in equation (7). The second-order buckling mode for the same end constraints is illustrated in Fig. 4, B. It is almost impossible to obtain the second-order mode without additional constraints on the filament; but if it could be forced, it would lead to n = 9/4. Fig. 5 describes the buckling that a carefully applied Weinstein-Semmes probe would be expected to experience. The upper end is built in to the handle as before, but the lower end constraint in this case is described as "pinned." The pinned condition is achieved on the lower end of the filament in contact with the skin when one applies whatever lateral forces are necessary to keep the top end directly above the lower end. For the firstorder buckling mode described in Fig. 5, A, n = 2.05 in equation (7). For the second-order buck-
Vol. 3, No.3 Von Frey's method of measuring pressure sensibility
May, 1978
u=O at y = l
u =-h at y=O
CD
I
t,L • F
u=O at
y~L
T
,L
l
u~-, j
at y=O -
CD
u=O at y=l
F
u=O at y =l
215
T l
u=O aty=O
~
CD
u= 0 at y= 0
CD
1
~
Fig. 4. A slender column with one end (top) built in and the other end (bottom) free to move laterally under the influence of an axial compression force F. A, First-order buckling mode. D, Second-order buckling mode, which requires nine times the force in A.. The deflection of the column in the horizontal (x) direction is denoted by the variable u, and the maximum absolute value of u is denoted by a.
Fig. 5. A slender column with one end (top) built in and the other end (bottom) pinned to prevent lateral motion under the influence of the axial compression force F. A, First-order buckling mode. D, Second-order buckling mode, which requires approximately three times the force in A. The deflection of the column in the horizontal (x) direction is denoted by the variable u.
ling mode described in Fig. 5, B, n = 6.05. Generally, higher order modes are not observed in buckling, because the primary mode occurs at a much lower stress. But it is conceivable, in the case of a Von Frey filament, that the primary mode could be suppressed somewhat by finger and wrist action on the handle, leading to slightly higher buckling stress. In Fig. 2, we have superposed the theoretical curves for the two types of buckling illustrated in Fig. 4, A. and 5, A, on the stress data points calculated from the manufacturer's information and our experimental measurements. To convert stress from dynes per square millimeter to grams per square millimeter, equation (7) was divided by the acceleration of gravity, g = 980.7 cm/ sq sec. The double curves for each case represent our estimate of the probable limit of error due to variations in measuring E and (L/R). As can be seen from Fig. 2, the experimental results agree quite well with the buckling model for which one end is built in and the other is pinned. Nevertheless, some scatter is apparent, even though extreme care was taken to apply the probes consistently. The probes from a second Weinstein-Semmes pressure aesthesiometer broadened the scatter of Fig. 2 by a factor of 1.5 to 2; but within these limits, the data exhibited reasonable agreement with the theory. When the manufacturer's force data are converted to stress (open circles in Fig. 2), some filaments change places in the sequence, which suggests that the Weinstein-Semmes pressure aesthesiometer is not the finely tuned device that one might assume from the number of different probes which comprise each set. Note, for example, that two monofilaments (4.74 and
2.36) are out of sequence, as marked, if rank ordered by buckling stress calculated from the manufacturer's equation for buckling force and the measured crosssectional area; however, when rank ordered by buckling stress calculated from the Euler buckling equation and measured values for elastic modulus and crosssectional area, all monofilaments are in proper sequence as packaged.
Summary and conclusions 1. When the probes in a Weinstein-Semmes pressure aesthesiometer are carefully applied, the mechanical behavior of the nylon monofilaments can be described as buckling, with one end built in and the other end pinned. But variations in the monofilaments or in their use can lead to considerable scatter. The statistics of our results suggest that no loss in sensitivity would accompany the use of every other probe, or perhaps only every third one. One set actually could be divided into two or three approximately equivalent sets.
2. Markings on probes are not in grams and should not be reported in grams. Since nerve response is dependent on local stress (pressure) rather than force, relative measures of sensitivity should be described by the stress at which sensation is recorded. For example, monofilament 5.88 buckles at a force about nine times that required by monofilament 4.93, yet the stress differs by only a factor of three. In the extreme case, monofilament 6.65 requires about 100,000 times the force required by monofilament 1.65; yet the difference in stress (pressure) is only a factor of about 300.
216
The Journal of HAND SURGERY
Levin, Pearsall, and Rudemum
3. Correctly interpreting the results from a Weinstein-Semmes pressure aesthesiometer clearly requires an understanding of the factors which can influence those results. According to this engineering
iations observed in length produce a negligible effect on buckling stress.
analysis, the principal factors that can lead to variations in the stress required to buckle a Von Frey filament are as follows: The method of application. As described above, the difference between applying a probe as shown in Fig. 4, A, and applying it as shown in Fig. 5, A, is roughly a factor of eight in buckling stress or force. The more carelessly a filament is applied, the closer one can expect to come to the lower curves in Fig. 2. Variations in end conditions. We observed that not all fibers are glued into their handles equally well. A filament that is not built in to its handle securely will buckle at a stress lower than the calculated stress because its end constraint is less. We also observed a variety of surfaces on the ends of the filaments that contact the skin: some were sharp and angular, others were smooth, and still others had fibrillated to some degree. An end that can slip easily on the skin could lead to a lower buckling stress, as it might tend to approach the case illustrated in Fig. 4, A. An end that does not slip will produce the geometry of Fig. 5, A, which agrees most closely with the manufacturer's markings. If the skin is sufficiently compliant that it depresses noticeably under the probe, the dimple produced will help create the desired pinned end condition. But note that the stress produced at the surface of the skin when the filament buckles otherwise is independent of the degree of skin depression. Variations in elastic modulus. These variations should not be great if the fibers are maintained at a reasonably constant temperature and moderate humidity. However, elevated temperatures or high humidity can decrease the elastic modulus of nylon by at least a factor of two or three, 15. 16 thereby decreasing the stress required for buckling by the same factor. Variations in geometry. These appear to be minor. Microscopic examination of the nylon monofilaments used in the probes shows them to be quite uniform and approximately circular in cross-section. The slight var-
I. Von Frey M: Verspatete Schmerzempfindungen. Z Gesamte Neurol Psychiat 79:324-33, 1922 2. Von Frey M: Zur Physiologic Der Juckempfindung. Arch Neerl Physiologie 7:142-145, 1922 3. Von Frey M: Physiologie der Sinnesorgane der menschlichen Haut. Ergebn Physiol 9:351-368, 1910 4. Von Frey M: Gibt es tiefe Druckempfindungen? Dtsch Med Wochenschr 51:113-124, 1925 5. Weinstein S: Tactile sensitivity of the phalanges. Percept Mot Skills 14:351-354, 1962 6. Moberg E: Objective methods for determining the functional value of sensibility in the hand. J Bone Joint Surg 40:454-475, 1958 7. Dellon, AL, Witebsky FG, Terrill RE: The denervated Meissner corpuscle: A sequential histological study after nerve division in the rhesus monkey. Plast Reconstr Surg 56:182-193, 1975 8. Boudreau JC, Tsuchitani C: Sensory neurophysiology, New York, 1973, D Van Nostrand Co, Inc, pp 201-268 9. Werner, JL, Orner GE Jr: Evaluating cutaneous pressure sensation of the hand. Am J Occup Ther 24:347-356, 1970 10. Euler, L: Mechanica sive motus scientia analytice exposita, St. Petersburg, 1736, discussed by Timoshenko SP: History of strength of materials, New York, 1953, McGraw-Hill Book Co, Inc 11. Wainwright, SA, Biggs WD, Currey JD, et al.: Mechanical design in organisms, New York, 1976, WileyHalsted 12. Higdon A, Ohlsen EG, Stiles WB, et al.: Mechanics of Materials, ed 3, New York, 1976, John Wiley & Sons, Inc 13. Douglas RA: Introduction to solid mechanics, Belmont, Calif., 1963, Wadsworth 14. Faupel JH: Engineering design, New York, 1964, John Wiley & Sons, Inc 15. Kohan MI, editor: Nylon plastics, New York, 1973, Interscience & Publishers Inc 16. Ogorkiewicz RM, editor: Engineering properties of thermoplastics, New York, 1970, Interscience Publishers Inc
REFERENCES
Scott Levin, B.S.,* George Pearsall, Sc.D.,** and Robert J. Ruderman, M.D.,* Durham, N. C.
In 1898 Von Frey introduced a clinical methodusing what have been referred to subsequently as Von Frey hairs or Von Frey filaments-for evaluating the skin's sensitivity to touch.! By pressing on the skin with a thorn glued on the end of a hair until the hair started to bow out, he obtained a measure of the pressure sensibility of nerve fibers in the skin. Von Frey calibrated the hairs on a balance, varying stiffness by changing length or by using hairs of different "hardness." He recorded the pressure sensibility by noting whether a given hair pressed on the skin produced any sensation. In subsequent papers, Von Frey 2-4 elaborated on this method of measuring pressure sensibility, but his technique essentially remained unchanged. Recently Weinstein 5 reintroduced Von Frey's method, using nylon monofilaments mounted in plastic handles as substitutes for the more variable natural hairs. These probes now are known as the WeinsteinSemmes pressure aesthesiometer and frequently are From the Division of Orthopaedic Surgery, Duke University Medical Center, * and the Department of Engineering, Duke University, ** Durham, N. C. Received for publication June 2, 1977. Reprint requests: Robert J. Ruderman, M.D., P. O. Box 3023, Duke University Medical Center, Durltam, NC 27710.
employed in the diagnosis and postoperative monitoring of nerve regeneration. The advantages of this method of recording pressure sensibility are ease of application, discrimination according to location, and measurement of threshold sensitivity. The development of new surgical techniques has created more interest in the reliability of assessing local pressure sensibility as a measure of sensory improvement and a need to monitor patients with more accurate clinical testing methods. A number of methods are used currently: the two-point discrimination test utilizing a paper clip, the sweat test, vibratory stimulus response, moving touch, and constant touch 6 ; but the Von Frey type of filaments provide one of the most common means of monitoring local sensory improvement. Sensory return, as measured by constant touch, is thought to be the result of regeneration of slowadapting group A beta fibers and their proposed end organ, the Merkel disc, which is said to be pressure sensitive. 7 • 8 According to this view, Von Frey filaments measure the threshold sensitivity of individual or groups of Merkel nerve endings, whereas the twopoint discrimination test measures the density of these endings. Our experiences at Duke University Medical Center have indicated that the probes in the Weinstein-
0363-5023/78/0303-0211 $00.60/0 © 1978 American Society for Surgery of the Hand
THE JOURNAL OF HAND SURGERY
211
212
The Joumal of HAND SURGERY
Levin, Pearsall, and Rudennan
Materials Two sets of probes were used-one for obtaining force and geometry data, the other for measuring elastic modulus and evaluating consistency between sets. The monofilaments were taken from Weinstein-Semmes pressure aesthesiometers manufactured by Research Media, Inc. (currently available from Stoelting Co., 1350 S. Kostner St., Chicago, IL 60623). A Mettler top-loading balance, Model P-163, was used to record the force exerted on the monofilaments. The balance capacity was 160 gm. Monofilament diameters (D := 2R) were measured to approximately ±2% in a microscope with a filarmeasuring eyepiece and calibrated objectives, part of a Tukon Microhardness Tester. Lengths (L) were measured to approximately ±0.5% by vernier calipers. We determined the elastic modulus of the nylon monofilaments by sacrificing 11 filaments from one Weinstein-Semmes aesthesiometer and stretching them in tension in an Instron mechanical testing machine, Model TT-C.
Methods
Fig. 1. Experimental arrangement for measuring buckling forces.
Semmes pressure aesthesiometer are simple to use but easy to misinterpret. For example, each probe is marked with a number (M) which represents the logarithm of 10 times the force in milligrams required to bow the monofilament: M = log(lO Fmg)
(1)
But these numbers (M) often are misinterpreted and quoted erroneously in the medical literature as the actual force in grams, resulting in quoted force values from 67 times too small (for the largest probe) to 365 times too large (for the smallest probe). In addition, the probes are calibrated according to force rather than stress. Stress (defined as force per unit area) is the more appropriate variable for measuring pressure sensibility. The data obtained may not only be somewhat inconsistent, but also less accurate than the user might assume. As a result ot these experiences, we measured two sets of probes and examined them from the standpoint of the engineering principles on which their behavior is based.
CaUbration. One set of probes was calibrated by applying each monofilament, in the manner suggested by Werner and Orner!' for clinical testing, to the pan of the Mettler balance. A piece of rubber glove was placed on the pan to avoid slippage (Fig. I). The probe was maintained perpendicular to the balance, with the plastic handle horizontal, and the mass in grams which produced sufficient force to initiate a bow in the monofilament was recorded as the "grams force" for that monofilament. At least 10 tests were performed on each monofilament, and a standard deviation was calculated for each set of values; the temperature was 74° ± 1° F and relative humidity was typical of a hospital laboratory environment, with no pronounced changes occurring during the testing period. The monofilaments were tested at random to avoid any systematic errors. Mechanical properties. The elastic modulus was determined for 11 nylon monofilaments cut from a Weinstein-Semmes pressure aesthesiometer. The test procedure involved clamping a 1.4 in length of monofilament between two sets of rubber-faced grips, initially 1.0 in apart, in an Instron testing machine and moving the grips apart, at a relative velocity (v) of 0.2 in/min. The results were recorded as a force-time (F-t) curve, which was converted to a stress-strain (S-e) curve when the force axis was divided by the crosssectional area (A) of the fiber,
Vol. 3, No.3
Von Frey's method of measuring pressure sensibility
May, 1978
213
Table I. Stress and force calculations and measurements for a Weinstein-Semmes pressure aesthesiometer
2L1R
Measured force F (gm)
Standard deviation uF (gm)
Measured stress S (gmlsq mm)
Standard deviation uS (gmlsq mm)
130 141 180 200 251 280 312 341 453 473 520 593 650 689 851 1,050 1,090 1,420 1,930 2,300
86.5 73.2 22 .3 18.6 17.0 10.6 3.14 2.81 1.85 1.58 0.977 0.562 0.213 0.112 0.091 0.034 0.0094 0.0040
4.3 1.9 1.5 0.8 1.5 0.5 0.07 0.06 0.06 0.06 0.030 0.011 0.004 0.007 0.005 0.002 0.0001 0.0001
171 175 82.0 84.9 94.9 76.1 38.9 36.6 29.5 33.7 23.9 15.7 9.29 7.50 6.52 4.02 2.14 1.29
16 13 9.6 7.9 13 7.4 2.8 2.6 2.4 2.9 1.9 1.1 0.64 0.84 0.68 0.43 0.13 0.09
Manufacturer's marking
Calculated force F' (gm)
Diameter D (mm)
Area A (sq m)
Calculated stress S' (gmlsq mm)
Slenderness
6.65 6.45 6.10 5.88 5.46 5.18 5.07 4.93 4.74 4.56 4.31 4 . 17 4.08 3.84 3.61 3.22 2.83 2.44 2.36 1.65
448 283 126 76.0 28.9 15.2 11.8 8.53 5.51 3.64 2.05 1.48 1.20 0.693 0.408 0.166 0.068 0.0276 0.0229 0.0045
1.\42 1.033 0.805 0.732 0.582 0.525 0.475 0.423 0.322 0.313 0.284 0.244 0.228 0.214 0.171 0. 137 0.132 0. 104 0.075 0.063
1.02 0.84 0.51 0.42 0.27 0.22 0. 18 0.14 0.081 0.077 0.063 0.047 0.041 0.036 0.023 0.015 0.014 0.0085 0.0044 0.0031
439 337 243 181 107 69.1 65.6 60.9 68 .0 47 .3 33.1 31.5 29.3 19.3 17.7 11.1 4 .86 3.25 5.20 1.45
S=
f. A
(2)
and the time axis was multiplied by the ratio of separation velocity (v) to initial length between grips (£0): €=
M vt to to
-=-
(3)
The elastic modulus was calculated as the slope of the stress-strain curve: E = dS de
(4)
during the initial (elastic or recoverable) stage of deformation . Results The elastic modulus appeared to exhibit a slight increase for finer diameter monofilaments, but the data can be represented quite well by a mean elastic modulus, E = 3.56 X lOB dynes/sq mm (516,000 psi) with a standard deviation of ±O.22 x lOB dynes/sq mm (±32,220 psi), The calibration data for the probes are presented in Table I and plotted in Fig. 2. The data are plotted in terms of stress because the sensibility that these probes measure is a sensibility to pressure rather than to force . The units of gram per square millimeter were chosen for stress because the user can be expected to have a
better physical feeling for grams and millimeters on this scale of clinical testing than for the more correct engineering units of dynes per square millimeter, Newtons per square meter, or pounds per square inch (psi) . The stress data are plotted vs. the slenderness ratio (2LiR) for each monofilament because this ratio describes a principal geometric factor in the theory of buckling slender columns .
Discussion In engineering terms, a Von Frey filament works by buckling elastically at a critical force or stress. Qualitatively, buckling is the lateral bending, or bowing out, of a beam or column that is compressed axially. Fig. 3 illustrates how the lateral displacement increases markedly in the vicinity of the critical buckling stress, providing an easily observed signal that the critical stress has been reached. Historically, Euler10 was the first investigator to apply the calculus to the analysis of column buckling. In 1736 he described the derivation of the equation: C1T 2
F"r= 4U
(5)
for the critical force to buckle a column loaded, as shown in Fig. 4, A . Long before Von Frey's experiments in physiology, other investigators in mechanics showed that Euler's constant, C = EI, where E is the
214
The Journal of HAND SURGERY
Levin, Pearsall, and Ruderman
1.2r------------.
o Stresses calclJlated from manlJfactlJrer's mar/cings and measlJred diafMt.rs. Stresses calclJlated from measlJred bvclcling forces and diameters.
•
100
~8 70
"i)
50
:.: :~
"0 .!:!
E
-=
20
"'E
-!: E o a .S
Vl Vl W
......
II)
10
9.0 8.0 7.0 6.0 5.0 4.0 3.0
2.0
'" ~
Vl
< 0.6
'"
Vl
II)
w
'"
\
0.4
~
Vl
\ \
\\ \\ \\
0.2
OL---------------__~ LATERAL DEFLECTION -
\\
Fig. 3. The variation of lateral deflection with stress for a typical case of buckling. Note that the deflection increases markedly when the stress reaches the critical buckling stress. (After Higdon et a1. 12 )
\\
\\
\\
\\ \\ \\ \\
\\
1.0
\\ \\ \\ \\
0.9 0.8 0.7 0.6 0.5 0.4 0.3
\\
\\ \\
0.2
O.IL-_--L._L-L.J._ _.L.-_-l...~'___ __
100
0.8
Q
\\
.~
g>
Vl
-- 5 TT2E db ./ . - - cr= 4{2lIR}2 ' one en IJllt-m, ontl tlndfrH.
\
60
t---------
u
5 2.05TT2E db ·z . cr = (2l1R12 ,one en IJI tom, t one end pmntld.
-
200
1.0 ~
200 300 400 500
1000
2000 3000
2L1R (logarithmic scale)
Fig. 2. Buckling stress VS. slenderness ratio (logarithmic scales) for one set of probes from a Weinstein-Semmes pressure aesthesiometer, as compared with theoretical buckling curves for two end conditions. The error bars on the data points and the double curves for each theoretical case represent expected limits of error when they are greater than the size of a data point.
elastic modulus of the material comprising the column and I is the second moment of area for the crosssection.lJ For a circular cross-section I = 7TR4/4 and A = 7TR2, so: Fer Srr = A
7T 2E
= 4(2L/R)2
(6)
for a column loaded as shown in Fig. 4, A. Generalizing equation (6) to account for different end conditions and the possibility of higher orders of buckling, (7)
equation (7) provides a reasonably accurate description of the buckling stress for slenderness ratios greater than approximately 150. 12 As can be seen from Table I, the lowest slenderness ratio for a probe in the WeinsteinSemmes pressure aesthesiometer is 130, so equation (7) is appropriate for our analysis. Among the buckling end constraints 13. 14 that have been analyzed in the literature of mechanics, the two which seem most appropriate to consider in analyzing the behavior of the filaments in a Weinstein-Semmes pressure aesthesiometer are presented in Fig. 4 and 5. In these figures the top of the filament is considered to be "built-in" to the handle. The lower end of the filament in Fig. 4 is described as "free," and this mode of buckling would be expected if care were not exercised in maintaining the top of the filament directly above the bottom during a test. For the buckling illustrated in Fig. 4, A. n = ':4 in equation (7). The second-order buckling mode for the same end constraints is illustrated in Fig. 4, B. It is almost impossible to obtain the second-order mode without additional constraints on the filament; but if it could be forced, it would lead to n = 9/4. Fig. 5 describes the buckling that a carefully applied Weinstein-Semmes probe would be expected to experience. The upper end is built in to the handle as before, but the lower end constraint in this case is described as "pinned." The pinned condition is achieved on the lower end of the filament in contact with the skin when one applies whatever lateral forces are necessary to keep the top end directly above the lower end. For the firstorder buckling mode described in Fig. 5, A, n = 2.05 in equation (7). For the second-order buck-
Vol. 3, No.3 Von Frey's method of measuring pressure sensibility
May, 1978
u=O at y = l
u =-h at y=O
CD
I
t,L • F
u=O at
y~L
T
,L
l
u~-, j
at y=O -
CD
u=O at y=l
F
u=O at y =l
215
T l
u=O aty=O
~
CD
u= 0 at y= 0
CD
1
~
Fig. 4. A slender column with one end (top) built in and the other end (bottom) free to move laterally under the influence of an axial compression force F. A, First-order buckling mode. D, Second-order buckling mode, which requires nine times the force in A.. The deflection of the column in the horizontal (x) direction is denoted by the variable u, and the maximum absolute value of u is denoted by a.
Fig. 5. A slender column with one end (top) built in and the other end (bottom) pinned to prevent lateral motion under the influence of the axial compression force F. A, First-order buckling mode. D, Second-order buckling mode, which requires approximately three times the force in A. The deflection of the column in the horizontal (x) direction is denoted by the variable u.
ling mode described in Fig. 5, B, n = 6.05. Generally, higher order modes are not observed in buckling, because the primary mode occurs at a much lower stress. But it is conceivable, in the case of a Von Frey filament, that the primary mode could be suppressed somewhat by finger and wrist action on the handle, leading to slightly higher buckling stress. In Fig. 2, we have superposed the theoretical curves for the two types of buckling illustrated in Fig. 4, A. and 5, A, on the stress data points calculated from the manufacturer's information and our experimental measurements. To convert stress from dynes per square millimeter to grams per square millimeter, equation (7) was divided by the acceleration of gravity, g = 980.7 cm/ sq sec. The double curves for each case represent our estimate of the probable limit of error due to variations in measuring E and (L/R). As can be seen from Fig. 2, the experimental results agree quite well with the buckling model for which one end is built in and the other is pinned. Nevertheless, some scatter is apparent, even though extreme care was taken to apply the probes consistently. The probes from a second Weinstein-Semmes pressure aesthesiometer broadened the scatter of Fig. 2 by a factor of 1.5 to 2; but within these limits, the data exhibited reasonable agreement with the theory. When the manufacturer's force data are converted to stress (open circles in Fig. 2), some filaments change places in the sequence, which suggests that the Weinstein-Semmes pressure aesthesiometer is not the finely tuned device that one might assume from the number of different probes which comprise each set. Note, for example, that two monofilaments (4.74 and
2.36) are out of sequence, as marked, if rank ordered by buckling stress calculated from the manufacturer's equation for buckling force and the measured crosssectional area; however, when rank ordered by buckling stress calculated from the Euler buckling equation and measured values for elastic modulus and crosssectional area, all monofilaments are in proper sequence as packaged.
Summary and conclusions 1. When the probes in a Weinstein-Semmes pressure aesthesiometer are carefully applied, the mechanical behavior of the nylon monofilaments can be described as buckling, with one end built in and the other end pinned. But variations in the monofilaments or in their use can lead to considerable scatter. The statistics of our results suggest that no loss in sensitivity would accompany the use of every other probe, or perhaps only every third one. One set actually could be divided into two or three approximately equivalent sets.
2. Markings on probes are not in grams and should not be reported in grams. Since nerve response is dependent on local stress (pressure) rather than force, relative measures of sensitivity should be described by the stress at which sensation is recorded. For example, monofilament 5.88 buckles at a force about nine times that required by monofilament 4.93, yet the stress differs by only a factor of three. In the extreme case, monofilament 6.65 requires about 100,000 times the force required by monofilament 1.65; yet the difference in stress (pressure) is only a factor of about 300.
216
The Journal of HAND SURGERY
Levin, Pearsall, and Rudemum
3. Correctly interpreting the results from a Weinstein-Semmes pressure aesthesiometer clearly requires an understanding of the factors which can influence those results. According to this engineering
iations observed in length produce a negligible effect on buckling stress.
analysis, the principal factors that can lead to variations in the stress required to buckle a Von Frey filament are as follows: The method of application. As described above, the difference between applying a probe as shown in Fig. 4, A, and applying it as shown in Fig. 5, A, is roughly a factor of eight in buckling stress or force. The more carelessly a filament is applied, the closer one can expect to come to the lower curves in Fig. 2. Variations in end conditions. We observed that not all fibers are glued into their handles equally well. A filament that is not built in to its handle securely will buckle at a stress lower than the calculated stress because its end constraint is less. We also observed a variety of surfaces on the ends of the filaments that contact the skin: some were sharp and angular, others were smooth, and still others had fibrillated to some degree. An end that can slip easily on the skin could lead to a lower buckling stress, as it might tend to approach the case illustrated in Fig. 4, A. An end that does not slip will produce the geometry of Fig. 5, A, which agrees most closely with the manufacturer's markings. If the skin is sufficiently compliant that it depresses noticeably under the probe, the dimple produced will help create the desired pinned end condition. But note that the stress produced at the surface of the skin when the filament buckles otherwise is independent of the degree of skin depression. Variations in elastic modulus. These variations should not be great if the fibers are maintained at a reasonably constant temperature and moderate humidity. However, elevated temperatures or high humidity can decrease the elastic modulus of nylon by at least a factor of two or three, 15. 16 thereby decreasing the stress required for buckling by the same factor. Variations in geometry. These appear to be minor. Microscopic examination of the nylon monofilaments used in the probes shows them to be quite uniform and approximately circular in cross-section. The slight var-
I. Von Frey M: Verspatete Schmerzempfindungen. Z Gesamte Neurol Psychiat 79:324-33, 1922 2. Von Frey M: Zur Physiologic Der Juckempfindung. Arch Neerl Physiologie 7:142-145, 1922 3. Von Frey M: Physiologie der Sinnesorgane der menschlichen Haut. Ergebn Physiol 9:351-368, 1910 4. Von Frey M: Gibt es tiefe Druckempfindungen? Dtsch Med Wochenschr 51:113-124, 1925 5. Weinstein S: Tactile sensitivity of the phalanges. Percept Mot Skills 14:351-354, 1962 6. Moberg E: Objective methods for determining the functional value of sensibility in the hand. J Bone Joint Surg 40:454-475, 1958 7. Dellon, AL, Witebsky FG, Terrill RE: The denervated Meissner corpuscle: A sequential histological study after nerve division in the rhesus monkey. Plast Reconstr Surg 56:182-193, 1975 8. Boudreau JC, Tsuchitani C: Sensory neurophysiology, New York, 1973, D Van Nostrand Co, Inc, pp 201-268 9. Werner, JL, Orner GE Jr: Evaluating cutaneous pressure sensation of the hand. Am J Occup Ther 24:347-356, 1970 10. Euler, L: Mechanica sive motus scientia analytice exposita, St. Petersburg, 1736, discussed by Timoshenko SP: History of strength of materials, New York, 1953, McGraw-Hill Book Co, Inc 11. Wainwright, SA, Biggs WD, Currey JD, et al.: Mechanical design in organisms, New York, 1976, WileyHalsted 12. Higdon A, Ohlsen EG, Stiles WB, et al.: Mechanics of Materials, ed 3, New York, 1976, John Wiley & Sons, Inc 13. Douglas RA: Introduction to solid mechanics, Belmont, Calif., 1963, Wadsworth 14. Faupel JH: Engineering design, New York, 1964, John Wiley & Sons, Inc 15. Kohan MI, editor: Nylon plastics, New York, 1973, Interscience & Publishers Inc 16. Ogorkiewicz RM, editor: Engineering properties of thermoplastics, New York, 1970, Interscience Publishers Inc
REFERENCES