The Journal of Pain, Vol 13, No 6 (June), 2012: pp 519-523Available online at www.jpain.org and www.sciencedirect.com

Focus Article

Estimating Efficacy and Drug ED50’s Using von Frey Thresholds:

Impact of Weber’s Law and Log Transformation

Charles Mills, David LeBlond, Shailen Joshi, Chang Zhu, Gin Hsieh, Peer Jacobson,Michael Meyer, and Michael DeckerAbbott Laboratories, Abbott Park, Illinois.

All authoAddressR4DK, B6115. E-m

1526-590

ª 2012 b

doi:10.10

Abstract: The use of von Frey filaments, originally developed by Maximilian von Frey, has become

the cornerstone for assaying mechanical sensitivity in animal models and is widely used for human

assessment. While there are certain limitations associated with their use that make comparisons be-

tween studies not straightforward at times, such as stimulus duration and testing frequency, von Frey

filaments provide a good measurement of mechanosensation. Here we describe the application of

von Frey filaments to testing in animal models, specifically with respect to determining changes in

sensory thresholds in a pain state using the Dixon up-down method. In a literature survey, we found

that up to 75% of reports using this method analyze the data with parametric statistical analysis and

of those that used nonparametric analysis, none took into account that mechanical sensation is

perceived on a logarithmic scale (Weber’s Law) when calculating efficacy. Here we outline a more rig-

orous analysis for calculating efficacy and ED50’s from von Frey data that incorporates Weber’s Law.

We show that this analysis makes statistical and biological sense and provide a specific example of

how this change affects data analysis that brings results from animal models more in line with clinical

observations.

Perspective: This focus article argues that analyzing von Frey paw withdrawal threshold data

obtained by using the Dixon up-down method without considering Weber’s Law is inappropriate.

An analysis method that incorporates how mechanical sensation is perceived and how its application

brings results from animal models more in line with clinical data is presented.

ª 2012 by the American Pain Society

Key words: Pain, von Frey filament, paw withdrawal threshold, ED50.

Assaying Mechanical Sensitivity

Original work by Ernst Weber and Gustav Fechnerdemonstrated that our senses, including mecha-nosensation, are interpreted on a logarithmic

scale. While the absolute relationships and the formulasused to describe these relationships have been thesubject of debate, the principal is easy to understand.Consider the ability to discriminate the weight differ-ence between 2 objects. The ability to tell the differencebetween them is proportional to the difference in their

rs are employees of Abbott Laboratories.reprint requests to Charles Mills, Abbott Laboratories, Dept.ldg AP9A-LL, 100 Abbott Park Road, Abbott Park, IL 60064-ail: charles.mills@abbott.com

0/$36.00

y the American Pain Society

16/j.jpain.2012.02.009

weights, which is known as Weber’s law. For example,if you hold a 1-gram weight in your palm, it would beeasier to detect a 1-gram weight being added than ifyou were holding a 100-gram weight and a 1-gramweight were added. This also illustrates that the lowerend of the detection range is more sensitive, in an abso-lute sense, than the higher end. One consequence ofWe-ber’s Law is that the biological response to a tactilestimulus (say the force exerted by a von Frey filament)is linearly related to the logarithm of the stimulus.This has important implications for assayingmechanical

sensitivity using von Frey filaments. First, the up-downmethod1 is widely used in the pain field for calculating50% paw withdrawal thresholds (PWTs). A literaturesurvey of 200 publications4 indicated that over 60% of re-ports that use von Frey filaments to determinemechanicalthresholds use theDixonup-downor amodifiedup-downmethod.1,2 This method uses the sequential testing

519

-1.0 -0.5 0.0 0.5 1.0

0

25

50

75

100

Force [log(g)]

Percen

t W

ith

draw

als

Nerve injury

Baseline

Figure 1. Logarithmic nature of mechanical sensation. Graph-ical representation of mouse ipsilateral paw withdrawal fre-quencies to von Frey filament stimulation before (baseline)and at 7 days after a peripheral nerve injury (spared nerve in-jury). Y-axis plotted as percent of withdrawals to 10 applicationsof a given von Frey filament. X-axis is the force (log g) of the vonFrey filaments. Peripheral nerve injury induces mechanical hy-persensitivity indicated by a leftward, linear shift in responsefrequencies. Data are plotted as mean 6 SEM. N = 10 group.

520 The Journal of Pain Estimating Efficacy Using von Frey Filaments

approach of Dixon3 along with the equation below1 togive a value (in grams) that will likely give a response50% of the time.

PWT ¼ 50% g threshold ¼ �10½xf1kd���10; 000 [1]

where Xf = value of the final filament (in log units), K =tabular value, d = mean difference between stimuli (inlog units). While the actual biological perception cannotbe measured directly, the PWT may be taken as a surro-gate measure of pain perception.For example, a PWTof 4 grams means that 50% of the

time a 4-gram filament will elicit a response. The selec-tion of the filaments used in this method is rooted inthe fact that tactile stimuli are sensed as a logarithmicfunction. The output of equation [1] is an absolute num-ber and the basis of the formula is that the filaments arelog units apart. Due to the increased sensitivity at thelower end of the scale, it is easier to detect subtlechanges at the lower range of sensitivity, opposed tothe higher end of the scale. However, notice from equa-tion [1] that the relationship between PWT and stimulusstrength is nonlinear. While the von Frey stimuli areevenly separated on a log-gram scale, the PWT responsefrom equation [1] has units of grams. When percent re-sponse is plotted versus the log of stimulus strength (ie,Force), a region of linear response frequencies can beidentified (see Fig 1; Figs 1 and 2 of Chaplan1) as ex-pected by equation [1]. This illustrates that the responseto mechanical stimuli follows Weber’s Law. After injury,sensitivity is increased and the percentage responsecurve shifts to the left, while a region of linearity is pre-served between percent response and log [Force] (Fig 1).Dixon’s method presumes equal stimuli intervals on anappropriate scale. Fig 1 shows that for von Frey testing,the appropriate scale is logarithmic. While the von Freystimulus strength is recorded properly on a log-gramscale, equation [1] produces a response metric (PWT)which is on a gram scale that ignores the significanceof Weber’s Law.Since the behavioral data obtained from Dixon’s

method (ie, theXf1 kd) reflect the vonFreyfilamenthan-dle indices that are spaced logarithmically, the antilogtransformation in equation [1] is actually unnecessary.Taking the log of both sides of equation [1] produces:

logðPWTÞ ¼ Xf1kd� 4 [2]

where log(PWT) is the appropriate response that takesWeber’s Law into account. Thus log(PWT) can be ob-tained simply and directly from the behavioral datawith-out an intermediate step of calculating PWT.

Calculating %MPE – Linear vs. Log ThinkingDose-response profiles are often nonlinear with

shapes that depend on pharmacological mechanisms un-related to Weber’s Law. However, the statistical analysis,comparisons, and interpretation of the underlyingmodel parameters (eg, ED50) may be based on assump-tions (such as normality, variance homogeneity, andlinearity of the response) for which Weber’s Law has

significance. Since the biological endpoint (response tomechanical stimulus) is linearly related to the log of thestimulus, it seems appropriate to report PWT measure-ments on a log scale. However, only 10% of reportsgraphing von Frey PWTs use a log scale.4 When evaluat-ing candidate compounds for pain relief, drug effecthas traditionally been defined as the percent change inPWT from vehicle (placebo treatment) relative to themaximumpossible change. The drug dose correspondingto 50% of the maximum possible drug effect (MPE) isdefined as a drug’s ED50. The ED50 is an important charac-teristic that may determine a candidate drug’s suitabilityand dose selection. However,whatmay not bewidely ap-preciated is the impact ofWeber’s Law on the estimationof ED50. For the same set of drug testing data, the ED50

estimate will differ depending on whether Weber’slaw, and the corresponding log transformation of PWT,is applied. This impact is illustrated below.

Linear ThinkingTraditionally, %MPE is taken as a linear function of

PWT. In this case, 50% MPE corresponds to a PWT halfway between vehicle and maximum possible effect.This was our (and the field’s) way of determining 50%MPE from the PWT calculated by the up-down formula.For example, 15 g (arbitrarily defined as the maximumpossible value) will always be a 100% drug effect. If thevehicle response is 1 g (0% drug effect), then halfwaybetween the 2 (50%MPE) is (15 1 1)/2 = 8 g. For thisexample, the %MPE formula would be:

%MPE ¼ 100�ðPWT� 1Þ=ð15� 1Þ [3]

However, given that the biological response is logarith-mically related to PWT (by Weber’s law), it seems

%MPE = 100*(observed grams – 1)/(15-1) %MPE = 100*(log[observed grams] – log[1])/(log[15]-log[1])

0

2

4

6

8

10

12

14

16

Linear

50%

PW

T (g

)

0%

100%

1

10

Log

50%

PW

T [lo

g (g

)]

15

0%

100%

4

6

8

2

3

Figure 2. Effect of using linear or log values to calculate%MPE. Plotting and using linear analysis (A) to calculate a 50% effect (50%MPE), assuming 1g = 0%and 15 g = 100% (maximumvalue using the up-downmethod), gives a 50%MPE of 8.0 g. Using log values foranalysis (B), with the same assumptions of 1 g = 0% and 15 g = 100%, gives a 50%MPE of 3.8 g. Formulas for calculating%MPE witheach condition are given below their respective graphs.

Mills et al The Journal of Pain 521

inappropriate to take %MPE (regarded as a measure ofbiological response) to be a linear function of PWT.Recall that PWT’s are derived from equations [1] and[2] that presume sensations follow a logarithmicrelationship. This makes the use of a linear scale inequation [3] inappropriate.

0 10 20 30 40 50 60 70 80 90 100

0

10

20

30

40

50

60

70

80

90

100

%MPE (g)

%M

PE

(lo

g [g

])

1 gram

5 gram

2 gram

3 gram

4 gram

Figure 3. Comparison of %MPEs using linear and log analysis.Graphical representation of %MPE using linear versus log calcu-lations with values 1 to 5 g defining the zero percent effect. A20% MPE using linear analysis translates into a 50% MPE usinglog analysis, when assuming 1 g = 0%. As the number defining0% increases, the difference between the 2 analyses becomessmaller. For example, using 5 g as 0%, 37% MPE (linear) trans-lates to a 50% MPE (log).

Log ThinkingSince the biological response follows Weber’s Law and

is therefore linearly related to the log of PWT measure-ments, it is appropriate to use a log transformation ofPWT (or derive directly from equation [2]) in estimating%MPE. When the force quantities in equation [3] arelog transformed, the %MPE formula becomes:

%MPE ¼ 100�ðlog½PWT� � log½1�Þ=ðlog½15�� log½1�Þ [4]

The resulting 50%MPE = 3.8 g, very different from the 8 gon the linear scale. The difference between the linearand log methods for the above example is illustratedgraphically in Fig 2.

Another way to visualize the effects of using the logscale for %MPE calculation is shown by Fig 3. The %MPE, as calculated with equation [3], is plotted on theabscissa. On the ordinate is the corresponding measurecalculated on a log scale (equation [4]). Using vehicle =1g (the minimum value) as in the example above, ourtraditional ED50, defined as 50% on the abscissa, corre-sponds to about 77% when calculated on the log scale(ordinate) and that 50%on the log scale (ordinate) corre-sponds to about 20% on our traditional linear scale. Thisrelationship depends on the assumed values of 1 g and 15g for the minimum and maximum response, respectively.The nature of the relationship between linear and logvalues can vary widely depending on theminimum value(due to the nature of the log scale). Effect values canchange up to 20% depending on the minimum PWTvalue. The effect of changing minimum PWT values be-tween 1 and 5 grams is illustrated in Fig 3.Since the strength of the von Frey filaments are in log

units and force responses are best represented on a logscale, ED50 and %MPE are best calculated both mathe-matically and biologically (see below) as a ½ log step

(half the distance on the log scale). Importantly, logtransformation of the data helps equalize the variance,allowing for parametric analysis. The original Chaplanreport1 recommended nonparametric tests because theresponses are discrete. While nonparametric tests are of-ten less powerful, they avoid assumptions of ANOVA andt-tests such as homogeneity of variance. This guidance islargely ignored in the literature; less than 30% of reportswith von Frey data used nonparametric analysis.4 Whilediscrete, the PWTresponse reflects an underlying contin-uous measure. The use of a log transformation on PWTsrenders the response more normal and its variancemore constant, reducing the need for less powerful non-parametric tests. Fig 4 is a randomly selected, typical dataset from a gabapentin dose response study in the ratBennett model of neuropathic pain. Each plot shows in-dividual animal PWTs versus the treatment group. Notice

Untransformed PWT

Gabapentin (mg/kg)

50%

PW

T (

g)

Vehicle 10 30 100 150

0

5

10

15

20

Transformed PWT

Gabapentin (mg/kg)

Lo

g P

WT

(g

)

Vehicle 10 30 100 150

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

1.25

Figure 4. The impact of log transformation of PWT measurements on variance homogeneity. Untransformed measurements on an-imals are plotted against their treatment group (N = 9 or 10). (A) Raw PWT measurements. (B) log[PWT] values. Note that log trans-formation makes the variance (scatter) across the different treatment groups more uniform. Data are from a typical, randomlyselected dose response curve for gabapentin in the rat Bennett model of neuropathic pain. Gabapentin was given orally at 2 m/kgin water 1 hour prior to testing.

522 The Journal of Pain Estimating Efficacy Using von Frey Filaments

in Fig 4A (untransformed) that with the low dose groupsthe direct gram measure (50%PWT) is very tight relativeto the higher dose (higher mean) treatments. However,with Log[PWT] as shown in Fig 4B, the vertical scatter ismore or less constant across the range of PWT measure-ments. The effect of the log transformation on variancehomogeneity is confirmed by Bartlett’s variancehomogeneity test (P < .0001 and P = .75 for untrans-formed and log transformed, respectively).It should be noted that the log transformation must

be applied to the individual PWT measurements, notto their averages. Taking the log of a group averageis not the same as the average of the individual logs.Consider 3 measurements (2, 4, and 8 grams) wherewe take log to the base 2 (the base used for log trans-

Linear Scale

0

2

4

6

8

10

12

14

16

Gabapentin ( mol/kg p.o.)

PW

T (g

)

Veh 100 250 500

Linear Scale

Gabapentin ( mol/kg)

%M

PE

Vehicle 100 250 500

0

20

40

60

80

100

13%

27%

48%

Figure 5. Effects of the log scale on a dose response curve for gabadrawal thresholds plotted on a linear scale (A) and log scale (B) of acalculated on the linear scale (A) and log scale (B). Changes in %MPData are plotted as means and SEMs with 95% confidence intervals aequals 50%MPE (9.25 g in A and .85 (7.1 g) in B). Data are from a typirat Bennett model of neuropathic pain. Gabapentin was given orall

formation is not important as long as the same base isused consistently):

log of group average ¼ log2 ð14=3Þ¼ LNð14=3Þ=LNð2Þ ¼ 2:22

average of individual logs

¼ ðlog2ð2Þ 1 log2ð4Þ1 log2ð8ÞÞ=3¼ ð11213Þ=3 ¼ 2

The average of the individual logs is equal to the log ofthe geometric mean. The geometric mean will be lessthan the group average (unless all the individual values

Gabapentin ( mol/kg p.o.)

Log Scale

0.4

0.6

0.8

1.0

1.2

PW

T (lo

g (g

))

Veh 100 250 500

Log Scale

Gabapentin ( mol/kg)

%M

PE

Vehicle 100 250 500

0

20

40

60

80

100

24%

39%

63%

pentin in the rat Bennett model of neuropathic pain. Paw with-dose response for gabapentin 2 weeks after nerve injury. %MPEE range from 11 to 15% when converting to the log scale (C, D).fter linear regression analysis in A and B. Dotted line in A and Bcal, randomly selected dose response curve for gabapentin in they at 2 m/kg in water 1 hour prior to testing.

Mills et al The Journal of Pain 523

are equal). Therefore, for any set of PWT measurementswith variability, the average of the individual logs = log(geometric mean) < log(group average).

What Makes Sense - ImplicationsSince tactile sense follows Weber’s law and is detected

in a log fashion, it is more discriminating on the lowerend of the scale. Referring back to the hand held weightexample, it is easier to detect absolute changes at thelower end of the scale. What this means for our assaysis that we can detect changes more easily andmore accu-rately at the lower end of our measures. When using lin-ear calculations, the significance of an effect at the highend of the scale is exaggerated. Furthermore, if you con-sider the number of von Frey filaments at the high end ofthe scale, it accentuates the lack of significance of rela-tively large differences at the top end derived by linearcalculations. Correspondingly, an observed biological re-sponse associated with a fixed stimulus change (eg, 1gram) is more likely to represent a bona fide responseto the stimulus (rather than random experimental noise)when the stimuli applied are at the lower end of thescale. The minimum biologically detectable differencebetween 2 tactile stimuli (eg, as with the Von Frey fila-ment series) is constant over the stimulus range whenthe stimulus measure is expressed on a log scale (ie, log[force in grams]). Thus, stimuli are sensed on a log scaleand it is easier to produce (and detect) biological effectsat the lower end of our measures. Therefore, 50%MPEshould correspond to where it would be felt/sensed asbeing halfway to a maximum stimuli. For the exampleabove, it would be a ½ log step up from 1 g, which corre-sponds to 3.8 g.The use of log PWT for %MPE shifts ED50 values to the

left compared with linear anlaysis. To illustrate, consider

another typical, randomly chosen data set from a doseresponse study for gabapentin in the rat Bennett modelof neuropathic pain. Calculating PWTs on a linear scale(Fig 5) gives an ED50 of �500 mmol/kg (actually cannotbe calculated from the data set). Analyzing the data us-ing log PWT gives an ED50 of 300 mmol/kg. Despitesome unavoidable methodological constraints compar-ing animal to human data (eg, acute dose effects in ani-mals versus maintenance dosing levels in patients;different endpoints, etc.), data generated in animalmodels of pain generally yield efficacy ranges that areright shifted compared with available human data.5

The leftward shift in efficacy seen when incorporatingWeber’s law and log transformation brings animalmodelresults closer to clinical reports.In summary, tactile sensation is perceived in a logarith-

mic fashion; therefore, the use of linear analysis on vonFrey PWT thresholds to determine efficacy is incorrect.Log transformation (or direct derivation with equation[2]) of PWTs makes mathematical sense (normalizingthe variance, allowing the use of more powerful para-metric analysis), and more importantly, makes biologicalsense (taking into account stimulus perception). Usinglog values of PWTs to generate ranges of efficacy bringspreclinical animal painmodel data more in line with clin-ical observations. The basis of the formula wewere usingwas correct (equation [3]), but we were neglecting theappropriate transformation to account for Weber’s law(equation [4]).

AcknowledgmentsThe authors wish to thank the reviewer for noting that

log values can be obtained directly with equation [2], by-passing the need for a log transformation of PWTs.

References

1. Chaplan SR, Back FW, Pogrel JW, Chung JM, Yaksh TL:Quantitative assessment of tactile allodynia in the rat paw.J Neurosci Methods 53:55-63, 1994

2. Dixon WJ: The up-and-down method for small samples.J Am Statist Assoc 60:967-978, 1965

3. DixonWJ: Efficient analysis of experimental observations.Ann Rev Pharmacol Toxicol 20:441-462, 1980

4. Mills CD; PubMed search: First 100 articles with keyword search for spinal nerve ligation, run 9.12.11,and first 100 articles with key word chronic constric-tion injury, run 9.13.11. Limitations: English, free fulltext

5. Whiteside GT, Adedoyin A, Leventhal L: Predictivevalidity of animal pain models? A comparison ofthe pharmacokinetic-pharmacodynamic relationship forpain drugs in rats and humans. Neuropharm 54:767-775,2008