JBJS | Multiplier Method for Predicting Limb-Length Discrepancy*

The Journal of Bone & Joint Surgery, Volume 82, Issue 10

Articles | October 01, 2000

Multiplier Method for Predicting Limb-Length Discrepancy*

Dror Paley, M.D., F.R.C.S.(C)†; Anil Bhave, P.T.†; John E. Herzenberg, M.D., F.R.C.S.(C)†; J. Richard Bowen, M.D‡

Investigation performed at the Maryland Center for Limb Lengthening and Reconstruction, Baltimore, Maryland
*No benefits in any form have been received or will be received from a commercial party related directly or indirectly to the subject of this article. No funds were received in support of this study.
†Maryland Center for Limb Lengthening and Reconstruction, The James Lawrence Kernan Hospital, 2200 Kernan Drive, Baltimore, Maryland 21207. E-mail address for D. Paley: drorpaley@ hotmail.com. E-mail address for A. Bhave: anilbhave@yahoo.com. E-mail address for J. E. Herzenberg: frscs@aol.com.
‡The Alfred I. duPont Institute, 1600 Rockland Road, Wilmington, Delaware 19899. E-mail address: jrbowen@nemours.org.

The Journal of Bone & Joint Surgery. 2000; 82:1432-1432

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Abstract

Background: In patients with a congenital or developmental limb-length discrepancy, the short limb grows at a rate proportional to that of the normal, long limb. This is the basis of predicting limb-length discrepancy with existing methods, which are complicated and require multiple data points. The purpose of our study was to derive a simple arithmetic formula that can easily and accurately predict limb-length discrepancy at skeletal maturity.

Methods: Using available databases, we divided the femoral and tibial lengths at skeletal maturity by the femoral and tibial lengths at each age for each percentile group. The resultant number was called the multiplier. Using the multiplier, we derived formulae to predict the limb-length discrepancy and the amount of growth remaining. We verified the accuracy of these formulae by evaluating two groups of patients with congenital shortening who were managed with epiphysiodesis or limb-lengthening. We also calculated and compared the multipliers for other databases according to radiographic, clinical, and anthropological lower-limb measurements.

Results: The multipliers for the femur and tibia were equivalent in all percentile groups, varying only by age and gender. Because congenital limb-length discrepancy increases at a rate proportional to growth, the discrepancy at maturity can be calculated as the current discrepancy times the multiplier for the current age and the gender. This calculation can be performed with use of a single measurement of limb-length discrepancy. For progressive developmental (noncongenital) discrepancies, the discrepancy at skeletal maturity can be calculated as the current discrepancy plus the growth inhibition times the amount of growth remaining. The timing of the epiphysiodesis can also be calculated with the multiplier. The predictions made with use of the multiplier method correlated well with those made with use of the Moseley method as well as with the actual limb-length discrepancy in both the limb-lengthening and epiphysiodesis groups. The multipliers derived from the radiographic, clinical, and anthropological measurements of femora and tibiae were all similar to each other despite differences in race, ethnicity, and generation.

Conclusions: The multiplier method allows for a quick calculation of the predicted limb-length discrepancy at skeletal maturity, without the need to plot graphs, and is based on as few as one or two measurements. This method is independent of percentile groups and is the same for the prediction of femoral, tibial, and total-limb lengths. The multiplier values are also independent of generation, height, socioeconomic class, ethnicity, and race. We verified the accuracy of this method clinically by evaluating patients who had been managed with limb-lengthening or epiphysiodesis. The method was also comparable with or more accurate than the Moseley method of limb-length prediction.

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Introduction

Limb-length discrepancy in children is generally progressive until skeletal maturity. Treatment decisions depend on the predicted limb-length discrepancy at skeletal maturity. Accurate prediction of the discrepancy is therefore important. We present a quick, convenient, accurate method for predicting limb-length discrepancy at skeletal maturity.

Shapiro³⁴ identified five patterns of progression of limb-length discrepancy in children. The current methods of predicting limb-length discrepancy at skeletal maturity are applicable only to the Shapiro type-I proportionate progression pattern. Limb-length discrepancies associated with other types of progression (Shapiro types II through V) have periods of acceleration or deceleration and therefore cannot be predicted accurately. Because most lower-limb-length discrepancies with congenital causes (such as congenital short femur, fibular hemimelia, hemiatrophy, and hemihypertrophy) and developmental causes (such as enchondromatosis [Ollier disease], poliomyelitis, and growth arrest) follow a type-I proportionate progression pattern, they can be predicted.

The current methods of predicting limb-length discrepancy at skeletal maturity for patients who have the Shapiro type-I progression pattern are based on the longitudinal data presented by Anderson et al.⁵. These data include the lengths of the femur and tibia from the age of one year to skeletal maturity for boys and girls, within one or two standard deviations from the mean.

For lower-limb-length discrepancies that are present at birth, the predicted length of the short lower limb can be determined on the basis of the observation that the percentage of growth inhibition remains the same until skeletal maturity^{1,2,15,18,29,31,42}. To predict the limb-length discrepancy at skeletal maturity, Amstutz^1,2 and Hootnick et al.¹⁵ recommended multiplying the ratio of the current length of the short limb to the current length of the long limb by the predicted length of the long limb at skeletal maturity to calculate the predicted length of the short limb at skeletal maturity. Subtracting the predicted length of the short limb from the predicted length of the long limb yields the predicted limb-length discrepancy at skeletal maturity. To determine the predicted length of the long limb at skeletal maturity, the current femoral and tibial lengths of the long limb are compared with the measurements collected by Anderson et al.⁵ for the current age and the gender of the child to determine the correct percentile group^1,2. The predicted length of the normal femur and tibia at skeletal maturity for that percentile group is recorded from the Anderson table or graph⁵.

Perhaps the most popular tool for predicting limb-length discrepancy at skeletal maturity is the Moseley straight-line graph^26,27. Moseley converted the Anderson growth curve⁵ of the normal limb into a straight line with a 45-degree slope by shifting the data points along the x axis and altering the distance between the age scale on the x axis by a comparable amount. This is why, in the Moseley straight-line graph, the age scale is not linear but is similar to a semilogarithmic scale. The Moseley straight-line method requires serial follow-up (with at least three data points) to accurately predict limb-length discrepancy. It allows for refinement of the prediction because the growth percentile of the patient is based on more than one measurement. The Moseley graph can also be used to predict limb-length discrepancy with only one data point by incorporating the Amstutz method^1,2 graphically. The Amstutz method requires the availability of tables, and the Moseley method requires the availability of graphs. Neither method can be used for children younger than one year, so prediction during the first year of life is not possible with these methods. The greatest difficulty associated with these methods is determining the percentile of the patient within the skeletal age range. This determination is improved somewhat by taking skeletal rather than chronological age into account in children who are older than nine years^3,4,12,13.

All of these methods of prediction can be cumbersome, confusing, and time-consuming. Therefore, the purpose of the present study was to simplify the method of predicting limb-length discrepancy at skeletal maturity.

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Fig. 1:: Comparison of mean femoral and tibial multipliers for boys and girls.

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Fig. 2-A:: Figs. 2-A through 2-G: Graphs showing the actual limb-length discrepancies and the predictions made with use of the Moseley^26,27 and multiplier methods for patients who underwent lengthening or epiphysiodesis for the treatment of congenital limb-length discrepancy. The solid line represents exact prediction between the two data groups being compared.
Fig. 2-A: Graph comparing the predictions made with the Moseley method and those made with the multiplier method for patients in the epiphysiodesis group.

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Fig. 2-B:: Graph comparing the actual discrepancies and the predictions made with the multiplier method for patients in the epiphysiodesis group.

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Fig. 2-C:: Graph comparing the actual discrepancies and the predictions made with the Moseley method for patients in the epiphysiodesis group.

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Fig. 2-D:: Graph comparing the predictions made with the Moseley method and those made with the multiplier method for congenital discrepancies for patients in the lengthening group.

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Fig. 2-E:: Graph comparing the actual discrepancies and the predictions made with the multiplier method for congenital discrepancies for patients in the lengthening group.

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Fig. 2-F:: Graph comparing the actual discrepancies and the predictions made with the Moseley method for patients in the lengthening group.

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Fig. 2-G:: Graph comparing the actual discrepancies and the predictions made with use of the multiplier method for developmental discrepancies for patients in the lengthening group.

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Fig. 3:: Graph comparing multipliers for boys from different radiographic and clinical databases.

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Fig. 4:: Graph comparing multipliers for girls from different radiographic and clinical databases.

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Fig. 5:: Graph comparing multipliers for boys from different anthropological databases.

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Fig. 6:: Graph comparing multipliers for girls from different anthropological databases.

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Table I: Femoral Multipliers for Boys by Percentile Group

*SD = standard deviation.

Age (yrs.)	Multiplier*					Avg.	Variability
Mean	+2 SD	+1 SD	-1 SD	-2 SD
0	5.09	5.06	5.14	5.14	5.22	5.13	±0.08
1	3.26	3.25	3.30	3.27	3.28	3.27	±0.03
2	2.60	2.57	2.62	2.62	2.64	2.61	±0.04
3	2.24	2.21	2.26	2.26	2.28	2.25	±0.03
4	2.00	1.96	2.01	2.02	2.04	2.00	±0.04
5	1.82	1.79	1.83	1.84	1.86	1.83	±0.03
6	1.68	1.64	1.69	1.70	1.73	1.69	±0.04
7	1.56	1.52	1.56	1.58	1.61	1.57	±0.05
8	1.46	1.43	1.46	1.49	1.51	1.47	±0.04
9	1.37	1.34	1.38	1.40	1.42	1.38	±0.04
10	1.30	1.27	1.30	1.32	1.35	1.31	±0.04
11	1.24	1.20	1.24	1.26	1.29	1.24	±0.05
12	1.18	1.14	1.17	1.20	1.23	1.18	±0.05
13	1.12	1.07	1.11	1.15	1.18	1.13	±0.06
14	1.07	1.03	1.06	1.09	1.12	1.08	±0.03
15	1.03	1.01	1.04	1.05	1.06	1.04	±0.03
16	1.01	1.00	1.02	1.02	1.03	1.02	±0.02
17	1.00	1.00	1.02	1.01	1.01	1.01	±0.01
18	1.00	1.00	1.01	1.00	1.00	1.00	±0.00

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Table I: Femoral Multipliers for Boys by Percentile Group

*SD = standard deviation.

Age (yrs.)	Multiplier*					Avg.	Variability
Mean	+2 SD	+1 SD	-1 SD	-2 SD
0	5.09	5.06	5.14	5.14	5.22	5.13	±0.08
1	3.26	3.25	3.30	3.27	3.28	3.27	±0.03
2	2.60	2.57	2.62	2.62	2.64	2.61	±0.04
3	2.24	2.21	2.26	2.26	2.28	2.25	±0.03
4	2.00	1.96	2.01	2.02	2.04	2.00	±0.04
5	1.82	1.79	1.83	1.84	1.86	1.83	±0.03
6	1.68	1.64	1.69	1.70	1.73	1.69	±0.04
7	1.56	1.52	1.56	1.58	1.61	1.57	±0.05
8	1.46	1.43	1.46	1.49	1.51	1.47	±0.04
9	1.37	1.34	1.38	1.40	1.42	1.38	±0.04
10	1.30	1.27	1.30	1.32	1.35	1.31	±0.04
11	1.24	1.20	1.24	1.26	1.29	1.24	±0.05
12	1.18	1.14	1.17	1.20	1.23	1.18	±0.05
13	1.12	1.07	1.11	1.15	1.18	1.13	±0.06
14	1.07	1.03	1.06	1.09	1.12	1.08	±0.03
15	1.03	1.01	1.04	1.05	1.06	1.04	±0.03
16	1.01	1.00	1.02	1.02	1.03	1.02	±0.02
17	1.00	1.00	1.02	1.01	1.01	1.01	±0.01
18	1.00	1.00	1.01	1.00	1.00	1.00	±0.00

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Table II: Tibial Multipliers for Boys by Percentile Group

*SD = standard deviation.

Age (yrs.)	Multiplier*					Avg.	Variability
Mean	+2 SD	+1 SD	-1 SD	-2 SD
0	5.04	4.98	5.01	5.07	5.08	5.04	±0.05
1	3.21	3.26	3.24	3.19	3.16	3.21	±0.05
2	2.56	2.59	2.58	2.55	2.54	2.56	±0.03
3	2.22	2.24	2.23	2.21	2.20	2.22	±0.02
4	2.00	2.01	2.00	1.99	1.99	2.00	±0.01
5	1.82	1.82	1.82	1.82	1.82	1.82	±0.00
6	1.69	1.68	1.68	1.68	1.68	1.68	±0.01
7	1.57	1.55	1.56	1.58	1.60	1.57	±0.02
8	1.47	1.45	1.46	1.48	1.50	1.47	±0.03
9	1.38	1.35	1.37	1.40	1.42	1.38	±0.04
10	1.31	1.28	1.29	1.33	1.35	1.31	±0.04
11	1.24	1.21	1.22	1.26	1.29	1.24	±0.04
12	1.17	1.14	1.15	1.20	1.23	1.18	±0.03
13	1.11	1.07	1.09	1.14	1.18	1.12	±0.06
14	1.06	1.02	1.04	1.08	1.11	1.06	±0.04
15	1.03	1.01	1.01	1.04	1.05	1.03	±0.02
16	1.01	1.00	1.00	1.01	1.02	1.01	±0.01
17	1.00	1.00	1.00	1.00	1.01	1.00	±0.00
18	1.00	1.00	1.00	1.00	1.00	1.00	±0.00

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Table II: Tibial Multipliers for Boys by Percentile Group

*SD = standard deviation.

Age (yrs.)	Multiplier*					Avg.	Variability
Mean	+2 SD	+1 SD	-1 SD	-2 SD
0	5.04	4.98	5.01	5.07	5.08	5.04	±0.05
1	3.21	3.26	3.24	3.19	3.16	3.21	±0.05
2	2.56	2.59	2.58	2.55	2.54	2.56	±0.03
3	2.22	2.24	2.23	2.21	2.20	2.22	±0.02
4	2.00	2.01	2.00	1.99	1.99	2.00	±0.01
5	1.82	1.82	1.82	1.82	1.82	1.82	±0.00
6	1.69	1.68	1.68	1.68	1.68	1.68	±0.01
7	1.57	1.55	1.56	1.58	1.60	1.57	±0.02
8	1.47	1.45	1.46	1.48	1.50	1.47	±0.03
9	1.38	1.35	1.37	1.40	1.42	1.38	±0.04
10	1.31	1.28	1.29	1.33	1.35	1.31	±0.04
11	1.24	1.21	1.22	1.26	1.29	1.24	±0.04
12	1.17	1.14	1.15	1.20	1.23	1.18	±0.03
13	1.11	1.07	1.09	1.14	1.18	1.12	±0.06
14	1.06	1.02	1.04	1.08	1.11	1.06	±0.04
15	1.03	1.01	1.01	1.04	1.05	1.03	±0.02
16	1.01	1.00	1.00	1.01	1.02	1.01	±0.01
17	1.00	1.00	1.00	1.00	1.01	1.00	±0.00
18	1.00	1.00	1.00	1.00	1.00	1.00	±0.00

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Table III: Femoral Multipliers for Girls by Percentile Group

*SD = standard deviation.

Age (yrs.)	Multiplier*					Avg.	Variability
Mean	+2 SD	+1 SD	-1 SD	-2 SD
0	4.64	4.71	4.63	4.60	4.56	4.63	±0.04
1	2.94	2.97	2.96	2.93	2.91	2.94	±0.03
2	2.39	2.40	2.40	2.39	2.38	2.39	±0.01
3	2.05	2.04	2.05	2.05	2.05	2.05	±0.01
4	1.82	1.81	1.81	1.83	1.85	1.82	±0.02
5	1.66	1.65	1.65	1.66	1.67	1.66	±0.01
6	1.53	1.51	1.52	1.54	1.55	1.53	±0.02
7	1.42	1.40	1.41	1.44	1.45	1.43	±0.03
8	1.33	1.31	1.32	1.35	1.36	1.33	±0.03
9	1.26	1.23	1.24	1.27	1.29	1.26	±0.03
10	1.19	1.16	1.17	1.20	1.22	1.19	±0.03
11	1.12	1.10	1.11	1.14	1.16	1.13	±0.03
12	1.07	1.05	1.06	1.08	1.10	1.07	±0.03
13	1.03	1.02	1.02	1.04	1.05	1.03	±0.02
14	1.00	1.00	1.00	1.00	1.00	1.00	±0.00
15	1.00	1.00	1.00	1.00	1.00	1.00	±0.00
16	1.00	1.00	1.00	1.00	1.00	1.00	±0.00

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Table III: Femoral Multipliers for Girls by Percentile Group

*SD = standard deviation.

Age (yrs.)	Multiplier*					Avg.	Variability
Mean	+2 SD	+1 SD	-1 SD	-2 SD
0	4.64	4.71	4.63	4.60	4.56	4.63	±0.04
1	2.94	2.97	2.96	2.93	2.91	2.94	±0.03
2	2.39	2.40	2.40	2.39	2.38	2.39	±0.01
3	2.05	2.04	2.05	2.05	2.05	2.05	±0.01
4	1.82	1.81	1.81	1.83	1.85	1.82	±0.02
5	1.66	1.65	1.65	1.66	1.67	1.66	±0.01
6	1.53	1.51	1.52	1.54	1.55	1.53	±0.02
7	1.42	1.40	1.41	1.44	1.45	1.43	±0.03
8	1.33	1.31	1.32	1.35	1.36	1.33	±0.03
9	1.26	1.23	1.24	1.27	1.29	1.26	±0.03
10	1.19	1.16	1.17	1.20	1.22	1.19	±0.03
11	1.12	1.10	1.11	1.14	1.16	1.13	±0.03
12	1.07	1.05	1.06	1.08	1.10	1.07	±0.03
13	1.03	1.02	1.02	1.04	1.05	1.03	±0.02
14	1.00	1.00	1.00	1.00	1.00	1.00	±0.00
15	1.00	1.00	1.00	1.00	1.00	1.00	±0.00
16	1.00	1.00	1.00	1.00	1.00	1.00	±0.00

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Table IV: Tibial Multipliers for Girls by Percentile Group

*SD = standard deviation.

Age (yrs.)	Multiplier*					Avg.	Variability
Mean	+2 SD	+1 SD	-1 SD	-2 SD
0	4.76	4.58	4.54	4.58	4.67	4.63	±0.11
1	2.99	3.03	3.01	2.98	2.95	2.99	±0.04
2	2.39	2.44	2.41	2.36	2.33	2.39	±0.06
3	2.06	2.10	2.08	2.04	2.02	2.06	±0.04
4	1.84	1.84	1.84	1.83	1.83	1.84	±0.01
5	1.67	1.67	1.67	1.67	1.67	1.67	±0.00
6	1.54	1.53	1.53	1.54	1.55	1.54	±0.01
7	1.43	1.42	1.42	1.44	1.45	1.43	±0.02
8	1.34	1.32	1.33	1.35	1.36	1.34	±0.02
9	1.26	1.24	1.25	1.27	1.29	1.26	±0.03
10	1.18	1.16	1.17	1.20	1.22	1.19	±0.03
11	1.12	1.09	1.10	1.14	1.16	1.12	±0.04
12	1.06	1.04	1.05	1.08	1.09	1.06	±0.03
13	1.02	1.01	1.02	1.03	1.04	1.03	±0.02
14	1.00	1.00	1.00	1.00	1.00	1.00	±0.00
15	1.00	1.00	1.00	1.00	1.00	1.00	±0.00
16	1.00	1.00	1.00	1.00	1.00	1.00	±0.00

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Table IV: Tibial Multipliers for Girls by Percentile Group

*SD = standard deviation.

Age (yrs.)	Multiplier*					Avg.	Variability
Mean	+2 SD	+1 SD	-1 SD	-2 SD
0	4.76	4.58	4.54	4.58	4.67	4.63	±0.11
1	2.99	3.03	3.01	2.98	2.95	2.99	±0.04
2	2.39	2.44	2.41	2.36	2.33	2.39	±0.06
3	2.06	2.10	2.08	2.04	2.02	2.06	±0.04
4	1.84	1.84	1.84	1.83	1.83	1.84	±0.01
5	1.67	1.67	1.67	1.67	1.67	1.67	±0.00
6	1.54	1.53	1.53	1.54	1.55	1.54	±0.01
7	1.43	1.42	1.42	1.44	1.45	1.43	±0.02
8	1.34	1.32	1.33	1.35	1.36	1.34	±0.02
9	1.26	1.24	1.25	1.27	1.29	1.26	±0.03
10	1.18	1.16	1.17	1.20	1.22	1.19	±0.03
11	1.12	1.09	1.10	1.14	1.16	1.12	±0.04
12	1.06	1.04	1.05	1.08	1.09	1.06	±0.03
13	1.02	1.01	1.02	1.03	1.04	1.03	±0.02
14	1.00	1.00	1.00	1.00	1.00	1.00	±0.00
15	1.00	1.00	1.00	1.00	1.00	1.00	±0.00
16	1.00	1.00	1.00	1.00	1.00	1.00	±0.00

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Table V: Lower-Limb Multipliers for Boys and Girls

*NA = not applicable.

Age (yrs. + mos.)	Multiplier
Boys	Girls*
Birth	5.080	4.630
0 + 3	4.550	4.155
0 + 6	4.050	3.725
0 + 9	3.600	3.300
1 + 0	3.240	2.970
1 + 3	2.975	2.750
1 + 6	2.825	2.600
1 + 9	2.700	2.490
2 + 0	2.590	2.390
2 + 3	2.480	2.295
2 + 6	2.385	2.200
2 + 9	2.300	2.125
3 + 0	2.230	2.050
3 + 6	2.110	1.925
4 + 0	2.000	1.830
4 + 6	1.890	1.740
5 + 0	1.820	1.660
5 + 6	1.740	1.580
6 + 0	1.670	1.510
6 + 6	1.620	1.460
7 + 0	1.570	1.430
7 + 6	1.520	1.370
8 + 0	1.470	1.330
8 + 6	1.420	1.290
9 + 0	1.380	1.260
9 + 6	1.340	1.220
10 + 0	1.310	1.190
10 + 6	1.280	1.160
11 + 0	1.240	1.130
11 + 6	1.220	1.100
12 + 0	1.180	1.070
12 + 6	1.160	1.050
13 + 0	1.130	1.030
13 + 6	1.100	1.010
14 + 0	1.080	1.000
14 + 6	1.060	NA
15 + 0	1.040	NA
15 + 6	1.020	NA
16 + 0	1.010	NA
16 + 6	1.010	NA
17 + 0	1.000	NA

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Table V: Lower-Limb Multipliers for Boys and Girls

*NA = not applicable.

Age (yrs. + mos.)	Multiplier
Boys	Girls*
Birth	5.080	4.630
0 + 3	4.550	4.155
0 + 6	4.050	3.725
0 + 9	3.600	3.300
1 + 0	3.240	2.970
1 + 3	2.975	2.750
1 + 6	2.825	2.600
1 + 9	2.700	2.490
2 + 0	2.590	2.390
2 + 3	2.480	2.295
2 + 6	2.385	2.200
2 + 9	2.300	2.125
3 + 0	2.230	2.050
3 + 6	2.110	1.925
4 + 0	2.000	1.830
4 + 6	1.890	1.740
5 + 0	1.820	1.660
5 + 6	1.740	1.580
6 + 0	1.670	1.510
6 + 6	1.620	1.460
7 + 0	1.570	1.430
7 + 6	1.520	1.370
8 + 0	1.470	1.330
8 + 6	1.420	1.290
9 + 0	1.380	1.260
9 + 6	1.340	1.220
10 + 0	1.310	1.190
10 + 6	1.280	1.160
11 + 0	1.240	1.130
11 + 6	1.220	1.100
12 + 0	1.180	1.070
12 + 6	1.160	1.050
13 + 0	1.130	1.030
13 + 6	1.100	1.010
14 + 0	1.080	1.000
14 + 6	1.060	NA
15 + 0	1.040	NA
15 + 6	1.020	NA
16 + 0	1.010	NA
16 + 6	1.010	NA
17 + 0	1.000	NA

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Table VI: Comparison of Databases of Limb-Length Measurements

Reference	No. of Patients or Skeletons Studied	Ethnic or Racial Origin	Comments
Radiographic data
Anderson et al.⁴ (1963)	100	American	50 percent of patients had poliomyelitis involving one lower limb; Boston, Massachusetts, USA
Anderson et al.⁵ (1964)	134	American (predominantly Irish ancestry)	Longitudinal; Boston, Massachusetts, USA
Beumer et al.⁷ (1997)	182	Dutch	Partially longitudinal
Maresh²¹ (1955)	113	American (predominantly Northern European ancestry)	Longitudinal; Denver, Colorado, USA
Maresh²² (1970)	265	American (predominantly Northern European ancestry)	Longitudinal; Denver, Colorado, USA
Clinical data
Cheng et al.⁸ (1996)	2193	Chinese	Tibia only
Meredith²³ (1939)	100	American	Measurements of standing minus sitting height; Iowa City, Iowa, USA
Meredith and Goldstein²⁴ (1952)	400	Mexican-American	Measurements of standing minus sitting height
Low and Kung²⁰ (1985)	24,978	Chinese	Tibia only
Snyder et al.³⁷ (1977)	4000	American	Includes foot height
Steyn and Henneberg³⁸ (1996)	61	African Negro	Femur only
Anthropological data
Armelagos et al.⁶ (1972)	68	Negro (350 b.c.e.-1400 c.e.)	Sudan
Hoppa¹⁶ (1992)	69	Anglo-Saxon (900 c.e.)	England
Johnston¹⁷ (1962)	165	American Indian (3000 b.c.e.)	Kentucky, USA
Miles and Bulman²⁵ (1994)	120	Scottish (1500-1850 c.e.)	Ensay, Scotland
Saunders and Hoppa³³ (1993)	241	Canadian (1800-1900 c.e.)	Ontario, Canada
Steyn and Henneberg³⁸ (1996)	106	Negro (1000-1200 c.e.)	South Africa
Stloukal and Hanáková³⁹ (1978)	200	Slavic (800 c.e.)	Eastern Europe
Sundick⁴⁰ (1978)	128	German (500-600 c.e.)	Germany
Y'Edynak⁴³ (1976)	100	Eskimo and Aleut (1700-1900 c.e.)	Kodiak Island, Alaska, USA

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Table VI: Comparison of Databases of Limb-Length Measurements

Reference	No. of Patients or Skeletons Studied	Ethnic or Racial Origin	Comments
Radiographic data
Anderson et al.⁴ (1963)	100	American	50 percent of patients had poliomyelitis involving one lower limb; Boston, Massachusetts, USA
Anderson et al.⁵ (1964)	134	American (predominantly Irish ancestry)	Longitudinal; Boston, Massachusetts, USA
Beumer et al.⁷ (1997)	182	Dutch	Partially longitudinal
Maresh²¹ (1955)	113	American (predominantly Northern European ancestry)	Longitudinal; Denver, Colorado, USA
Maresh²² (1970)	265	American (predominantly Northern European ancestry)	Longitudinal; Denver, Colorado, USA
Clinical data
Cheng et al.⁸ (1996)	2193	Chinese	Tibia only
Meredith²³ (1939)	100	American	Measurements of standing minus sitting height; Iowa City, Iowa, USA
Meredith and Goldstein²⁴ (1952)	400	Mexican-American	Measurements of standing minus sitting height
Low and Kung²⁰ (1985)	24,978	Chinese	Tibia only
Snyder et al.³⁷ (1977)	4000	American	Includes foot height
Steyn and Henneberg³⁸ (1996)	61	African Negro	Femur only
Anthropological data
Armelagos et al.⁶ (1972)	68	Negro (350 b.c.e.-1400 c.e.)	Sudan
Hoppa¹⁶ (1992)	69	Anglo-Saxon (900 c.e.)	England
Johnston¹⁷ (1962)	165	American Indian (3000 b.c.e.)	Kentucky, USA
Miles and Bulman²⁵ (1994)	120	Scottish (1500-1850 c.e.)	Ensay, Scotland
Saunders and Hoppa³³ (1993)	241	Canadian (1800-1900 c.e.)	Ontario, Canada
Steyn and Henneberg³⁸ (1996)	106	Negro (1000-1200 c.e.)	South Africa
Stloukal and Hanáková³⁹ (1978)	200	Slavic (800 c.e.)	Eastern Europe
Sundick⁴⁰ (1978)	128	German (500-600 c.e.)	Germany
Y'Edynak⁴³ (1976)	100	Eskimo and Aleut (1700-1900 c.e.)	Kodiak Island, Alaska, USA

Materials and Methods

Development of the Multiplier

The data collected by Anderson et al.⁵ are divided into the mean, mean plus one standard deviation, mean plus two standard deviations, mean minus one standard deviation, and mean minus two standard deviations of the femoral and tibial lengths at different chronological ages for boys and girls. These values correspond to the fifth, thirty-third, fiftieth, sixty-seventh, and ninety-fifth percentiles, respectively. For each percentile group, we divided the length of the femur and tibia for boys and girls at skeletal maturity (Lm) by the corresponding length of the femur or tibia at each year of age from one year to skeletal maturity (L). This converted every data point from the Anderson tables to a multiplier for length at skeletal maturity (M): M = Lm/L. Conversely, the current length of the femur or tibia can be multiplied by the age-specific multiplier to calculate the length of that bone at skeletal maturity: LM = Lm.

The data of Anderson et al.⁵ begin at the age of one year. Maresh^21,22 presented data on femoral and tibial lengths that were measured radiographically between birth and skeletal maturity. We incorporated the data presented by Maresh to include the period between birth and the age of one year.

Development and Clinical Testing of the Formulae

To predict the limb-length discrepancy and the amount of growth remaining, we developed formulae using the multipliers that had been calculated.

We chose the Moseley method^26,27 as the so-called gold standard for limb-length prediction. We compared the predictions that were made with use of the Moseley method with the predictions that were made with use of the multiplier formulae for two groups of patients who had reached skeletal maturity: a group managed with epiphysiodesis and a group managed with limb-lengthening.

The epiphysiodesis group consisted of sixteen patients who were managed and followed at the Alfred I. duPont Institute in Wilmington, Delaware, by one of the authors (J. R. B.). The only treatment administered to fifteen of the sixteen patients was epiphysiodesis. The procedure was performed in the distal aspect of the femur in ten of these fifteen patients, in the proximal aspect of the tibia in one, and in both the distal aspect of the femur and the proximal aspect of the tibia in four. In the remaining patient, distal femoral and proximal tibial epiphysiodesis (6.5 centimeters) was performed with simultaneous femoral and tibial lengthening (7.5 and five centimeters, respectively). The predictions were made preoperatively with use of the Moseley and multiplier methods and were compared with use of the system presented by Little et al.¹⁹. The accuracy of the predictions made with use of each method was then checked postoperatively. The effect of epiphysiodesis was factored in with use of Moseley's method for the Moseley predictions and with use of the calculation of the amount of growth remaining for the multiplier predictions. With both the Moseley method and the multiplier method, we used the Anderson approximation that 71 percent of the total amount of femoral growth occurs at the distal aspect of the femur and 57 percent of the total amount of tibial growth occurs at the proximal aspect of the tibia⁴.

The limb-lengthening group consisted of fourteen patients who were managed with equalization of limb length by means of femoral and/or tibial lengthening at the Maryland Center for Limb Lengthening and Reconstruction in Baltimore, Maryland. For this group, the prediction of final limb-length discrepancy included the contribution of the lengthening process itself, assuming that no inhibition or stimulation of growth occurred as a result of the lengthening. The total actual limb-length difference was compared with the predicted limb-length difference with use of both the Moseley method and the multiplier method.

Comparison of Available Growth Databases

In addition to the two databases presented by Anderson et al.^4,5, eighteen other databases of femoral, tibial, and/or limb-length measurements in children were identified^{6-8,16,17,20-25,33,37-40,43}. We used the same methods to calculate the age and gender-related multipliers from these databases. In total, we analyzed and compared twenty databases: eleven based on radiographic or clinical measurements of the lower limbs of living children and nine based on anthropological measurements of the skeletal remains of children.

Results

The multiplier for each percentile group (mean, mean plus one standard deviation, mean plus two standard deviations, mean minus one standard deviation, and mean minus two standard deviations) at each age was approximately the same (mean variability, 0.03; maximum variability, ±0.08) for the data on both the femur and the tibia (Table I, Table II, Table III, and Table IV). This variability was highest at birth and decreased with increasing age. The multipliers for the femur for each age-group and percentile of boys and girls were approximately the same as the respective multipliers for the tibia (mean variability, 0.008; maximum variability, ±0.05) (Fig. 1). Because the multipliers for the tibia and femur were nearly identical, we took the mean multipliers for both bones and averaged them to obtain overall lower-limb multipliers for boys and girls (Table V). These multipliers can be used to calculate clinically relevant information, such as the predicted limb-length discrepancy and the amount of growth remaining, as described in the following two sections.

Multiplier Method for Congenital Discrepancies

The length of the long limb at skeletal maturity (Lm) can be predicted by multiplying the current length of the long limb (L) by the multiplier (M) for the current (chronological) age. Because the multiplier is the same for each percentile group, there is no need to consult the Anderson tables⁵ or to determine the percentile group of the patient.

By definition,

M = Lm/L

and

Lm = ML.

As Amstutz^1,2 noted, the ratio of the length of the short limb to the length of the long limb stays the same throughout growth in patients with congenital deficiencies. With the Amstutz method, the length of the short limb at skeletal maturity (Sm) can be calculated by multiplying the ratio of the current length of the short limb (S) to the current length of the long limb (L) by the predicted length of the long limb at skeletal maturity (Lm).

For congenital deformities,

Sm/Lm = S/L.

Therefore,

Sm = Lm × S/L.

Substituting ML for Lm,

Sm = ML × S/L = MS.

The limb-length discrepancy at skeletal maturity (Dm) is then calculated by subtracting the short-limb length from the long-limb length at skeletal maturity:

Dm = Lm - Sm.

Substituting ML for Lm and MS for Sm,

Dm = ML - MS = M(L - S).

Because D = L - S,

Dm = D × M.

The congenital limb-length discrepancy at skeletal maturity can be predicted even when only the currentage-specific multiplier (M) and the current limb-length difference (D) are known.

For example, a seven-year-old boy has a limb-length discrepancy of 6.3 centimeters due to congenital short femur. The multiplier for seven-year-old boys is 1.57. Therefore, the predicted discrepancy at skeletal maturity is Dm = MD = 1.57 × 6.3 = 9.9 centimeters.

Multiplier Method for Developmental Discrepancies

With developmental discrepancies (for example, Ollier disease, poliomyelitis, and growth arrest), the rate of growth of the short limb relative to the rate of growth of the long limb is fixed. The limb-length discrepancy increases because the short limb is growing slower than the long limb is. The rate of growth of both limbs changes proportionally with age according to the normal pattern of growth. The inhibition of growth of the short limb in comparison with the growth of the long limb remains fixed. To predict the limb-length discrepancy at skeletal maturity, the inhibition (I), the amount of growth remaining (G), and the current limb-length discrepancy (D) must be known. The inhibition can be calculated from the ratio of growth of the short limb to growth of the long limb during the same time-interval. Two separate measurements of limb length made since the beginning of the growth disturbance are needed to calculate inhibition. Inhibition is defined as the amount of growth of the short limb (S - SÂ´) divided by the amount of growth of the long limb (L - LÂ´) during the same time-interval, subtracted from 1:

I = 1 - (S - SÂ´)/(L - LÂ´).

SÂ´ and LÂ´ are the lengths of the short and long limbs, respectively, measured on previous radiographs that preferably were made at least six or twelve months before the current radiographs. The radiographs must have been made after the date of the growth disturbance, with use of the same radiographic method and the same magnification. S and L are the lengths of the short and long limbs, respectively, measured on the current radiographs.

The next parameter that needs to be calculated is the amount of growth remaining from the current time until skeletal maturity. We know that

G = Lm - L.

Substituting LM for Lm,

G = LM - L = L(M - 1).

Therefore, the amount of growth remaining in the normal long limb is

G = L(M - 1).

The amount of additional limb-length discrepancy that will occur from the present time until skeletal maturity (Dg = discrepancy in growth remaining) is the inhibition multiplied by the amount of growth remaining:

Dg = I × G.

We can substitute the individual formulae that we derived above for I and G:

Dg = [1 - (S - SÂ´)/(L - LÂ´)] × L(M - 1).

The predicted discrepancy at skeletal maturity in cases of developmental discrepancy is the current discrepancy (D) added to the discrepancy in growth remaining:

Dm = D + Dg.

Substituting for Dg from the formulae above, we obtain the formula for predicting limb-length discrepancy at skeletal maturity:

Dm = D + (I × G).

For example, an eight-year-old girl has a limb-length discrepancy of 2.7 centimeters due to distal femoral growth arrest following trauma three years earlier. The current length of the normal femur (L) is twenty-eight centimeters, and the length of the normal femur one year ago (LÂ´) was twenty-six centimeters. The current length of the short femur (S) is 25.3 centimeters, and the length of the short femur one year ago (SÂ´) was 24.4 centimeters. The multiplier for eight-year-old girls is 1.33. The amount of growth remaining in the normal femur is G = L(M - 1) = 28(1.33 - 1) = 9.2 centimeters. The amount of growth in the normal femur during the previous year was (L - LÂ´) = 28 - 26 = 2 centimeters, and the amount of growth in the short femur during the previous year was (S - SÂ´) = 25.3 - 24.4 = 0.9 centimeter. Therefore, the inhibition can be calculated as I = 1 - (S - SÂ´)/(L - LÂ´) = 1 - (0.9/2) = 0.55. The discrepancy in the amount of growth remaining is Dg = I × G = 0.55 × 9.2 = 5.1 centimeters. Therefore, the total predicted limb-length discrepancy is Dm = D + Dg = 2.7 + 5.1 = 7.8 centimeters.

Clinical Testing of the Formulae

The limb-length discrepancies in both the epiphysiodesis group and the limb-lengthening group were of congenital origin.

Epiphysiodesis Group

The sixteen patients in the epiphysiodesis group ranged in age from two to ten years at the time that the first radiographs were made. Two patients had substantial differences between their skeletal and chronological ages. Of the sixteen patients, four had two preoperative radiographs, three had three, and nine had four. All radiographs were made with three-level radiography, with the patient supine. The multiplier predictions were based on each single measurement, and the Moseley predictions were based on all of the radiographs. Both the Moseley and the multiplier predictions were based on chronological, not skeletal, age because the latter was available only for patients who had radiographs that were made just before epiphysiodesis was performed.

The correlation coefficient that was derived by comparing the Moseley and multiplier predictions was 0.93 (Fig. 2-A). Comparing the predicted discrepancy with the actual discrepancy revealed a correlation coefficient of 0.94 for the multiplier method (Fig. 2-B) and 0.90 for the Moseley method (Fig. 2-C). When the threshold of acceptable accuracy was set at ±1 centimeter, five Moseley predictions and two multiplier predictions fell outside the zone. When the threshold was set at ±1.5 centimeters, five Moseley predictions and one multiplier prediction fell outside the zone. The five Moseley predictions that were outside the zone had errors of prediction of 1.8, 2.1, 2.2, 2.8, and 3.5 centimeters. The one multiplier prediction that was outside the zone had an error of 1.6 centimeters.

Limb-Lengthening Group

The fourteen patients in the limb-lengthening group ranged in age from eight to thirteen years at the time that the first radiographs were made. Ten patients had one lengthening each, and four had two lengthenings each. Five patients had lengthening of the femur, seven had lengthening of the tibia, and two had lengthening of both bones. All limb-length radiographs were made with use of a long x-ray cassette, with the patient standing. This type of radiograph shows both lower limbs in the frontal plane from the ankles to the hips. We had previously determined that the magnification factor is approximately 5 percent for these long radiographs when the x-ray tube is positioned ten feet away and is centered on the knee joints. Therefore, all of the measurements were corrected by a magnification factor of 5 percent. The amount of lengthening was measured on the radiographs and was factored into the prediction as described by Moseley^26,27. The final actual discrepancy was based on the sum of the lengthening amount or amounts and any residual limb-length discrepancy at skeletal maturity. This value was compared with the values determined with the Moseley and multiplier methods. We calculated the multiplier prediction using both the congenital formula and the developmental formula. The congenital formula required a single radiograph and a single calculation. This calculation was performed for each limb-length discrepancy that was measured. Although all of these discrepancies were congenital, the developmental formula can be applied when two interval radiographs are used. This calculation was performed with one pair of preoperative measurements for each patient. The correlation coefficient that was derived by comparing the multiplier method for congenital discrepancies with the Moseley method was 0.98 (Fig. 2-D). The correlation coefficients that were derived by comparing the multiplier method for congenital discrepancies and the Moseley method with the actual difference were both 0.99 (Fig. 2-E and Fig. 2-F). Finally, the correlation coefficient that was derived by comparing the multiplier method for developmental discrepancies with the actual difference was 0.98 (Fig. 2-G).

Percentage of Total Bone Growth from the Distal Aspect of the Femur and the Proximal Aspect of the Tibia

By using the formula G = L(M - 1) to calculate the amount of growth remaining in an entire bone, it is possible to calculate the amount of growth remaining from the proximal or distal physis, provided that the relative contributions of the proximal and distal physes are known. In 1963, Anderson et al.⁴ determined that the relative growth rates of the distal femoral and proximal tibial physes were 71 and 57 percent, respectively, relative to the entire length of the bones. In accordance with the method presented by Anderson et al., multiplying the amount of femoral growth remaining by 0.71 indicates the amount of growth remaining from the distal femoral physis and multiplying the amount of tibial growth remaining by 0.57 indicates the amount of growth remaining from the proximal tibial physis.

Calculation of Timing of Epiphysiodesis with the Multiplier Method

To calculate the timing of epiphysiodesis of the femur or tibia, the desired amount of correction (e) must be determined for each bone. Because the entire correction will result from epiphysiodesis of only one of the two physes in each bone, the amount of growth remaining at the time of epiphysiodesis (Ge) can be written as Ge = e/0.71 for the femur and as Ge = e/0.57 for the tibia. We therefore want to calculate the length of the long femur or tibia at skeletal maturity (Lm = LM, where L is the current length of the long femur or tibia and M is the current age multiplier) and subtract from that length e/0.71 for the femur and e/0.57 for the tibia. This difference is the length of the femur or tibia at the desired age of epiphysiodesis (Lm - Ge = Le). Substituting Le for L, we can calculate the multiplier at the age of epiphysiodesis (Me): Lm = LeMe, with Me = Lm/Le.We can now look in the multiplier table for the value of Me and determine what age corresponds to this multiplier value. The calculation of the timing of epiphysiodesis is based on the desired correction. For a particular desired correction, this calculation can be performed for all Shapiro types because the epiphysiodesis is performed on the long limb, which grows according to the data presented by Anderson et al.

For example, a seven-year-old girl who has a predicted femoral length discrepancy at maturity (Dm) of five centimeters has a current femoral length (L) of twenty-eight centimeters. The multiplier (M) for seven-year-old girls is 1.43. The desired correction following epiphysiodesis (e) is five centimeters. The amount of femoral growth remaining at the time of epiphysiodesis is seven centimeters (Ge = e/0.71 = 5/0.71). The length of the femur at skeletal maturity is forty centimeters (Lm = L × M = 28 × 1.43). The length of the femur at the time of epiphysiodesis is thirty-three centimeters (Le = Lm - Ge = 40 - 7). The multiplier at the age of epiphysiodesis is 1.21 (Me = Lm/Le = 40/33). The multiplier for girls is 1.19 at the age of ten years and 1.22 at the age of nine years and six months. Therefore, M = 1.21 corresponds best to the age of nine years and eight months. The length of the femur should be followed radiographically at six-month intervals until the age of nine years and eight months. If the length of the femur reaches thirty-three centimeters before this age, recalculate to fine-tune the accuracy. If the length of the femur is less than thirty-three centimeters at the age of nine years and eight months, epiphysiodesis should be delayed until that length is achieved to avoid overcorrection of the limb-length discrepancy. This method provides a check mechanism for epiphysiodesis by providing the predicted length of the bone at the age of epiphysiodesis.

Relationship of Multipliers for Boys to Multipliers for Girls

The multiplier values for boys and girls are clearly different from each other (Fig. 1). The shape of the curved-line graph representing multiplier versus age is similar for boys and girls. When the multiplier for boys is divided by the multiplier for girls for each age from birth through thirteen years, the ratio is always approximately 1.09. The multiplier for boys is 1.09 at the age of thirteen years and nine months. If the femoral and tibial lengths for boys at the age of thirteen years and nine months instead of seventeen years are used to calculate the multiplier (by dividing the femoral and tibial lengths for boys at the age of thirteen years and nine months by the femoral and tibial lengths, respectively, for boys at birth through the age of thirteen years and nine months), then the multiplier for boys is equivalent to the multiplier for girls at each age. Therefore, the only reason that multipliers differ for boys and girls is that boys grow for approximately two years and three months longer than girls do. This also indicates that the growth patterns of boys and girls are actually identical until the age of thirteen years and nine months. This relationship between multipliers for boys and girls is very useful for analyzing the multipliers of femoral and tibial lengths from pooled measurements of boys' and girls' bone lengths.

Most anthropological measurements of children's bone lengths cannot distinguish boys' bones from girls' bones. We used anthropological data to calculate the multipliers from tibial and femoral-length measurements from as far back as 3000 b.c.e. to learn whether the multipliers have changed over time. For pooled gender data, we calculated the multipliers for girls with use of the femoral or tibial length at the age of thirteen years and six months and the multipliers for boys with use of the femoral or tibial length at the age of sixteen years.

Comparison with Other Growth Databases

Radiographic Data

The data of Anderson et al.⁵ are the best known measurements of femoral and tibial lengths. Anderson et al. conducted a longitudinal study of sixty-seven normal boys and sixty-seven normal girls from the age of one year to skeletal maturity. Radiographs were made as closely as possible to the anniversary of the date of birth in order to simplify the recording of the patient's bone length relative to age. The length measurements included the epiphyses at both ends of the bone. Although this is the most clinically utilized database of femoral and tibial lengths in children, there have been other studies of femoral and tibial length measurements in children. Anderson et al.⁴ presented a separate, earlier series of femoral-length and tibial-length measurements for 100 patients (fifty boys and fifty girls), half of whom had poliomyelitis involving one lower limb. The multipliers from the earlier study⁴ were equivalent to the multipliers from the later study⁵ (Fig. 3 and Fig. 4, Table VI).

Maresh²² presented a longitudinal series of radiographic measurements of femoral and tibial lengths for 123 boys and 121 girls who were evaluated in the Denver area from 1935 until 1967. The data were gathered by the Denver Child Research Council. Radiographs were made from birth to skeletal maturity on patients' birthdays and at six-month intervals between birthdays. Measurements for children less than ten years old were made without the epiphyses, measurements for children between the ages of ten and twelve years were made both with and without the epiphyses, and measurements for children more than twelve years old were made with the epiphyses. We normalized all of the data for children older than twelve years by subtracting the height of the epiphyses on the basis of the percentages that the epiphyses contributed to the length of the bone between the ages of ten and twelve years. We then calculated the multipliers for these data.

The multipliers that were calculated on the basis of the data presented by Maresh were equivalent to those that were calculated on the basis of the data presented by Anderson et al.⁵ (Fig. 3 and Fig. 4, Table VI). Maresh's data were published twice, once in 1955²¹ and once in 1970²². The values for femoral and tibial lengths were slightly different in the two publications, and the more recent publication included more complete longitudinal data. The multipliers derived from the two publications were virtually identical. The data presented by Maresh were gathered between 1935 and 1967 in Denver, Colorado, and the data presented by Anderson et al. were gathered during a similar time-period in the area surrounding Boston, Massachusetts. Most of the participants in the Denver study were of Northwest European ancestry, whereas a predominance of the participants in the Boston study were of Irish ancestry. The participants from Denver were from a more homogeneous socioeconomic group (middle and upper-middle class), and they were substantially taller than the participants from Boston³⁰. Despite these ethnic, socioeconomic, and height differences, there was no significant difference between the multipliers for the two study groups (p = 0.9039).

Beumer et al.⁷ recently studied the radiographs of 182 children from Rotterdam, The Netherlands, and reported femoral-length and tibial-length measurements that were significantly longer than those reported by Anderson et al.⁵ (p = 0.0326). The Dutch children were evaluated between 1979 and 1994 and were taller than the American children studied by Anderson et al.⁵. The Rotterdam study was partially longitudinal, with an average of 3.3 radiographic measurements (range, two to fourteen measurements) per child. On the basis of their data, Beumer et al. created a new straight-line graph for Dutch children (the Rotterdam straight-line graph). They reported that the Rotterdam straight-line graph was more accurate than the Moseley straight-line graph for predicting limb-length discrepancy and the amount of growth remaining and thus for determining the timing of epiphysiodesis. We calculated femoral and tibial multipliers from the data reported by Beumer et al. and found that they were approximately the same as the multipliers that were calculated from the data reported by Anderson et al.⁵ (Fig. 3 and Fig. 4, Table VI).

Clinical Data

Clinical measurements of lower-limb length and of femoral and tibial length are not as accurate as radiographic measurements are. Clinical measurements are less costly, are more readily available, and are not associated with the risks related to exposure to radiation. Meredith²³ measured lower-limb length as the difference between standing height and sitting height in children from the age of seven years to skeletal maturity^11,24. He studied children from Iowa City²³ and, in a second study, North American children of Mexican ancestry²⁴. The multipliers derived from these two studies were similar to those calculated with the data of Anderson et al.⁵ (Fig. 3 and Fig. 4, Table VI).

Snyder et al.³⁷, in a publication commissioned by the automotive industry, presented clinical measurements of "iliocristale height" (the length from the iliac crest to the floor), "buttock-knee length," and "tibiale length" (the length from the knee to the floor) in American children in 1977. These measurements were made between well defined surface-anatomy landmarks. We performed the multiplier calculation for all three clinical measurements, which corresponded to total limb length including foot height ("sphyrion"), femoral length, and tibial length including foot height. Because there were no substantial differences among the multipliers for these three measurements, we represented the data of Snyder et al. with one average multiplier for the femur and the tibia, similar to what was done with the data of Anderson et al. The multipliers derived from the data of Snyder et al. were almost identical to the multipliers derived from the 1964 data of Anderson et al.⁵. The study by Snyder et al. represents perhaps the largest and most detailed study involving clinical measurements. Because foot height is included in both tibiale and iliocristale-length measurements, the present study demonstrates that the multiplier method can be used to predict limb length and limb-length discrepancy inclusive of foot height. The Moseley and Anderson techniques of predicting limb-length discrepancy do not account for discrepancies related to decreased foot height. This becomes especially important when assessing congenital discrepancies, which often involve a hypoplastic foot. The multiplier method can therefore be applied to total limb-length discrepancy, which includes femoral, tibial, and foot-height differences.

All of the previously mentioned radiographic and clinical studies were of Caucasian children of North American or European descent. We found two clinical studies of Chinese children^8,20. The multipliers derived from these two sources were very similar to those derived from the study by Anderson et al.⁵ (Fig. 3 and Fig. 4, Table VI).

Steyn and Henneberg³⁸ reported upper leg-length growth (symphysion-tibiale) in "Cape Coloured" children from rural South Africa. The multipliers calculated from this study of Negro children were also equivalent to those derived from Anderson et al.'s study of Caucasian children⁵ (Fig. 3 and Fig. 4, Table VI).

Anthropological Data

Femoral and tibial lengths have been measured from children's skeletal remains recovered from archaeological sites and cemeteries^{6,16,17,25,33,38-40,43}. The ages of the children were determined from the dental remains. The measurements reflect the length of the bone, not including the epiphyses. In only one⁴³ of these studies was it possible to differentiate between the skeletal remains of boys and those of girls. We calculated the multipliers from nine anthropological study groups (Fig. 5 and Fig. 6, Table VI) dating from as far back as 3000 b.c.e. to as recently as the nineteenth century. The groups studied represent a wide cross section of ethnic and racial origins: Eskimos, Slavs, Anglo-Saxons, Scots, Canadians, Germans, Nubian Negroes, Southern African Negroes, and North American Indians. As described above, the bone length at the age of sixteen years was used as the skeletal-maturity length for boys and the length at the age of thirteen years and six months was used as the skeletal-maturity length for girls. When data were not available beyond fourteen years of age, only the multiplier for girls was calculated. The multipliers for all of these groups were very similar to those calculated from the data of Anderson et al.⁵ when compared with regard to age and gender (Fig. 5 and Fig. 6, Table VI).

Discussion

Prediction of limb-length discrepancy with the Anderson, Amstutz, and Moseley methods assumes that both lower limbs grow in the same pattern, albeit at different rates, without periods of acceleration or deceleration of one limb relative to the other. This corresponds to a Shapiro type-I growth pattern. We subclassify the type-I growth pattern according to whether the limb-length discrepancy originated in utero and was present at birth (type IA) (as in patients with congenital short femur, fibular hemimelia, hemiatrophy, and hemihypertrophy) or developed after birth (type IB) (as in patients with Ollier disease, poliomyelitis, and growth arrest). In both subtypes, the discrepancy develops proportionately after the date of origin because the short limb is growing slower than the long limb at a constant rate of growth inhibition. The ratio of the growth rate of the short limb to the growth rate of the long limb does not change in patients with a Shapiro type-I growth pattern. When this ratio is subtracted from 1 and expressed as a percentage, the resulting value is called the growth inhibition. Shapiro type-I growth patterns are characterized by a constant growth inhibition of the short limb compared with the long limb. Only when there is constant growth inhibition can the discrepancy be predicted with use of the Amstutz, Moseley, and Anderson methods. This is also the case for the multiplier formulae.

Intuitively, multiple measurements of limb length lead to a more accurate prediction than a single measurement does. We do not suggest that prediction be based on only a single radiograph. Nevertheless, in the clinical setting, one is often faced with a patient without previous radiographs or a very young patient with few previous radiographs. Families and physicians want to know the predicted discrepancy at maturity.

At the very least, the multiplier method provides a simple, rapid method of early prediction. Because both the multiplier method and the Moseley method are based on the data of Anderson et al.⁵, both methods should be equally accurate when the predictions are based on the same number of measurements. As more limb-length measurements become available, the accuracy of both methods increases. The clinical data in our study confirmed that the multiplier method correlates closely with the Moseley method. In the limb-lengthening group there was no difference in the accuracy of the two methods, whereas in the epiphysiodesis group the multiplier method was more accurate than the Moseley method.

In addition to speed and simplicity, another major advantage of the multiplier method is that the percentile group does not have to be taken into account because the multiplier for each age and gender group is the same across percentile groups. This is perhaps the most surprising and interesting finding of our study. The consistency of this finding lends credence to the data of Anderson et al.⁵. Such a relationship could not have occurred by chance and therefore must be a basic biological-design feature of the normal pattern of human growth and development.

The accuracy of the Moseley method in predicting equalization after epiphysiodesis has been verified by comparing actual discrepancies with predicted discrepancies. We verified our method in the same way and by comparing it with the Moseley method. The ideal patient group with which to verify the accuracy of prediction is a natural-history (nonoperatively managed) group. Such a group is difficult to find because most patients with limb-length discrepancy undergo either epiphysiodesis or limb-lengthening before skeletal maturity. For this reason, we compared our method with the Moseley method with respect to the epiphysiodesis group. Although this group did have a limb-length equalization procedure, we know that the measurements made before the epiphysiodesis were not affected by any intervention. Therefore, the pre-epiphysiodesis radiographs represent the closest that we can get to a natural-history group. Lengthening procedures can lead to inhibition or stimulation of femoral and tibial growth^32,35,36. Despite this, we found no difference between the actual discrepancies and the discrepancies that were predicted with either the Moseley method or the multiplier method. If inhibition or stimulation was present, its effect was limited. As more children are managed with lengthening procedures at younger ages^14,28,32, there are also fewer preoperative measurements on which to base a prediction of limb-length discrepancy at skeletal maturity. Therefore, a method that relies on only one or two measurements is advantageous.

The simpler that a method is, the more likely that people are to use it. The Moseley graph frequently is not used because it is perceived to be complicated and time-consuming⁴¹. Table V provides a simple chart of the multipliers, which can be kept in the outpatient clinic. Because the data are so compact, a card-sized version can be carried in a wallet for handy reference. The Appendix provides a list of the formulae.

Anderson et al.⁴ recommended using skeletal age instead of chronological age for girls from the age of nine years to skeletal maturity and for boys from the age of twelve years to skeletal maturity. The recent Rotterdam study⁷ showed that skeletal age and chronological age do not diverge until even later (at the age of thirteen years in girls and at the age of fourteen years in boys). Little et al.¹⁹ compared eight different methods of predicting limb-length discrepancy, including the methods of Anderson et al. and Moseley. They found that the methods that used skeletal age were no more accurate than those that used chronological age. Cundy et al.⁹ reported interobserver variability of more than two years in 10 percent of patients whose skeletal age was graded by four radiologists. The multiplier described in the present study was derived from the data on chronological age as reported by Anderson et al.⁵. The Moseley straight-line graph was derived from the same chronological data, despite the fact that the age axis on the Moseley graph is labeled as skeletal age. The Moseley straight-line graph was not derived from the data of Anderson et al.⁴ on the amount of growth remaining, which includes skeletal age. Anderson et al.^3,4 recommended using skeletal age for children older than nine years to reduce the standard deviation in the data. To comply with the accepted standard, skeletal age should be used for patients who are ten years or older. However, in most cases, we prefer to use chronological age.

One of the criticisms that is often leveled against the data of Anderson et al.⁵ is related to differences in femoral and tibial lengths between people of different ethnicities, races, and generations. The femoral and tibial lengths reported by Maresh²² were different from those reported by Anderson et al. Pritchett³⁰ suggested that these differences were related to differences in ethnic origin, height, and socioeconomic status. The femoral and tibial lengths reported by Beumer et al.⁷ were significantly different from those reported by Anderson et al. (p = 0.0326), and the differences may have been related to ethnic origin, height, and generation. Differences in femoral and tibial lengths were present in all three of these studies^5,7,22, even though all of the patients were Caucasian. Despite these differences, the multipliers calculated from all three studies were equivalent. The clinical-measurement studies allowed us to calculate multipliers for non-Caucasians. The multipliers for Negroes and Chinese were not substantially different from those for Caucasians.

The anthropological data (Fig. 5 and Fig. 6) allowed us to calculate multipliers for many more ethnic and racial groups and to determine whether they have changed over time. The anthropological multipliers demonstrated greater variability than the radiographically and clinically calculated multipliers did. This increased variablity may have been related to the fact that, in the anthropological studies, age was determined retrospectively with use of dental data. Nevertheless, the amount of variability that was observed was smaller than what would be expected for such uncontrolled, retrospectively grouped data. In comparison, the data presented by Anderson et al. and Maresh were derived from longitudinal studies in which chronological age was determined accurately. The variability between these two independent longitudinal studies was almost negligible. The anthropological studies suggest that the multipliers for the lower limb have remained essentially unchanged by generation, ethnicity, and race. This may not be surprising in that anthropologists have failed to find major differences between the pattern of growth when modern and prehistoric groups have been compared, even though prehistoric groups seem to have been shorter than their modern counterparts³⁸. It has been proposed that we all reach a certain proportion of our lower-limb growth by a certain age¹⁰. The multiplier is therefore a measure of the growth-proportion pattern. Although some of us are destined to be tall and others to be short, it stands to reason that we all reach a quarter, a third, a half, three-quarters, and so on, of our lower-limb growth by certain ages. Such consistency among the multipliers in different studies suggests that this pattern may be genetically programmed.

Because the multipliers remain the same across percentiles, ethnic groups, generations, and races, the multiplier method may prove to be more accurate than other methods of prediction that are not immune to ethnic, generational, and racial differences. Multipliers also may be applicable in predicting skeletal-maturity and growth-remaining values for height, spine length, foot length, and upper-limb length.

In conclusion, we recommend the multiplier method of predicting limb-length discrepancy, amount of growth remaining until skeletal maturity, and timing of epiphysiodesis as an alternative, simple, and quick method of assessing lower-limb-length discrepancy.

Appendix

Multiplier Method for Predicting Lower-Limb-Length Discrepancies

Congenital Limb-Length Discrepancy

Dm = D × M.

This formula can be used to determine limb-length discrepancy in patients with congenital short femur, fibular hemimelia, hemihypertrophy, or hemiatrophy.

Developmental Limb-Length Discrepancy

Dm = D + (I × G),

where I = 1 - (S - SÂ´)/(L - LÂ´) and G = L(M - 1). This formula can be used to determine the limb-length discrepancy in patients with Ollier disease, poliomyelitis, or growth arrest. It can also be used to determine the discrepancy in patients with a congenital discrepancy. It is also useful in predicting the growth-remaining discrepancy in patients who have already undergone one or more limb-lengthening procedures.

Length at Skeletal Maturity

Lm = L × M.

This formula can be used to determine the length of the femur, tibia, femur and tibia, or entire lower limb, including the foot height. It applies equally to the short and long limbs.

Timing of Epiphysiodesis

Le = Lm - Ge

and

Me = Lm/Le.

Look in the multiplier table for the value of Me and determine which age corresponds to this multiplier value. This is the age of the patient at the time of epiphysiodesis.

Key

G = amount of growth remaining.

I = amount of growth inhibition.

L = current length of long limb.

LÂ´ = length of long limb as measured on previous radiographs (preferably made at least six or twelve months before current radiographs).

Lm = length of femur or tibia at skeletal maturity.

M = multiplier.

S = current length of short limb.

SÂ´ = length of short limb as measured on previous radiographs (preferably made at least six or twelve months before current radiographs).

D = current limb-length discrepancy.

Dm = limb-length discrepancy at skeletal maturity.

e = desired correction following epiphysiodesis.

Ge = amount of femoral or tibial growth remaining at age of epiphysiodesis (Ge = e/0.71 for femur and e/0.57 for tibia).

Le = desired length of bone to undergo epiphysiodesis at time of epiphysiodesis.

Me = multiplier at age of epiphysiodesis.

References

Amstutz, H. C.: The morphology, natural history, and treatment of proximal femoral focal deficiency. In Proximal Femoral Focal Deficiency. A Congenital Anomaly, pp. 50-76. Edited by G. T. Aitken. Washington, D.C., National Academy of Sciences, 1969.

Amstutz, H. C.: Natural history and treatment of congenital absence of the fibula. In Proceedings of the American Academy of Orthopaedic Surgeons. J. Bone and Joint Surg., 54-A: 1349, Sept. 1972.

Anderson, M., and Green, W. T.: Lengths of the femur and the tibia. Norms derived from orthoroentgenograms of children from five years of age until epiphyseal closure. Am. J. Dis. Child., 75: 279-290, 1948.

Anderson, M.; Green, W. T.; and Messner, M. B.: Growth and predictions of growth in the lower extremities. J. Bone and Joint Surg., 45-A: 1-14, Jan. 1963.

Anderson, M.; Messner, M. B.; and Green, W. T.: Distribution of lengths of the normal femur and tibia in children from one to eighteen years of age. J. Bone and Joint Surg., 46-A: 1197-1202, Sept. 1964.

Armelagos, G. J.; Mielke, J. H.; Owen, K. H.; Van Gerven, D. P.; Dewey, J. R.; and Mahler, P. E.: Bone growth and development in prehistoric populations from Sundanese Nubia. J. Hum. Evol., 1: 89-119, 1972.

Beumer, A.; Lampe, H. I.; Swierstra, B. A.; Diepstraten, A. F.; and Mulder, P. G.: The straight line graph in limb length inequality. A new design based on 182 Dutch children. Acta Orthop. Scandinavica, 68: 355-360, 1997.

Cheng, J. C. Y.; Leung, S. S. F.; and Lau, J.: Anthropometric measurements and body proportions among Chinese children. Clin. Orthop., 323: 22-30, 1996.

Cundy, P.; Paterson, D.; Morris, L.; and Foster, B.: Skeletal age estimation in leg length discrepancy. J. Pediat. Orthop., 8: 513-515, 1988.

Dimeglio, A.: Notions pratiques. Ces chiffres qu'il faut connaitre. In Les Inegalites de Longueur des Membres. Collection de Pathologie Locomotrice 28, pp. 273-280. Edited by A. Dimeglio, J. Caton, C. Herisson, and L. Simon. Paris, Masson, 1994.

Gill, G. G., and Abbott, L. C.: Practical method of predicting the growth of the femur and tibia in the child. Arch. Surg., 45: 286-315, 1942.

Green, W. T., and Anderson, M.: Experiences with epiphyseal arrest in correcting discrepancies in length of the lower extremities in infantile paralysis. A method of predicting the effect. J. Bone and Joint Surg., 29-A: 659-675, July 1947.

Green, W. T., and Anderson, M.: Epiphyseal arrest for the correction of discrepancies in length of the lower extremities. J. Bone and Joint Surg., 39-A: 853-872, July 1957.

Herzenberg, J. E.; Preston, D.; and Paley, D.: Leg lengthening in toddlers. Orthop. Trans., 18: 996, 1994-1995.

Hootnick, D.; Boyd, N. A.; Fixsen, J. A.; and Lloyd-Roberts, G. C.: The natural history and management of congenital short tibia with dysplasia or absence of the fibula. J. Bone and Joint Surg., 59-B(3): 267-271, 1977.

Hoppa, R. D.: Evaluating human skeletal growth: an Anglo-Saxon example. Internat. J. Osteoarchaeol., 2: 275-288, 1992.

Johnston, F. E.: Growth of the long bones of infants and young children at Indian Knoll. Am. J. Phys. Anthropol., 20: 249-254, 1962.

Koman, L. A.; Meyer, L. C.; and Warren, F. H.: Proximal femoral focal deficiency. Natural history and treatment. Clin. Orthop., 162: 135-143, 1982.

Little, D. G.; Nigo, L.; and Aiona, M. D.: Deficiencies of current methods for the timing of epiphysiodesis. J. Pediat. Orthop., 16: 173-179, 1996.

Low, W. D., and Kung, L. S.: Linear growth of the tibia in Chinese children. Zeitschr. Morphol. und Anthropol., 75: 327-330, 1985.

Maresh, M. M.: Linear growth of long bones of extremities from infancy through adolescence. Am. J. Dis. Child., 89: 725-742, 1955.

Maresh, M. M.: Measurements from roentgenograms. In Human Growth and Development, pp. 157-181. Edited by R. W. McCammon. Springfield, Illinois, Charles C Thomas, 1970.

Meredith, H. V.: Length of head and neck, trunk and lower extremities on Iowa City children aged seven to seventeen years. Child Devel., 10: 129-144, 1939.

Meredith, H. V., and Goldstein, M. S.: Studies on the body size of North American children of Mexican ancestry. Child Devel., 23: 91-110, 1952.

Miles, A. E. W., and Bulman, J. S.: Growth curves of immature bones from a Scottish island population of sixteenth to mid-nineteenth century: limb-bone diaphyses and some bones of the hand and foot. Internat. J. Osteoarchaeol., 4: 121-136, 1994.

Moseley, C. F.: A straight-line graph for leg-length discrepancies. J. Bone and Joint Surg., 59-A: 174-179, March 1977.

Moseley, C. F.: A straight line graph for leg length discrepancies. Clin. Orthop., 136: 33-40, 1978.

Naudie, D.; Hamdy, R. C.; Fassier, F.; Morin, B.; and Duhaime, M.: Management of fibular hemimelia: amputation or limb lengthening. J. Bone and Joint Surg., 79-B(1): 58-65, 1997.

Pappas, A. M.: Congenital abnormalities of the femur and related lower extremity malformations: classification and treatment. J. Pediat. Orthop., 3: 45-60, 1983.

Pritchett, J. W.: Longitudinal growth and growth-plate activity in the lower extremity. Clin. Orthop., 275: 274-279, 1992.

Ring, P. A.: Congenital short femur. Simple femoral hypoplasia. J. Bone and Joint Surg., 41-B(1): 73-79, 1959.

Sabharwal, S.; Paley, D.; Bhave, A.; and Herzenberg, J. E.: Growth patterns after lengthening of congenitally short lower limbs in young children. J. Pediat. Orthop., 20: 137-145, 2000.

Saunders, S. R., and Hoppa, R. D.: Growth deficit in survivors and non-survivors: biological mortality bias in subadult skeletal samples. Yearbook Phys. Anthropol., 36: 127-151, 1993.

Shapiro, F.: Developmental patterns in lower-extremity length discrepancies. J. Bone and Joint Surg., 64-A: 639-651, June 1982.

Shapiro, F.: Longitudinal growth of the femur and tibia after diaphyseal lengthening. J. Bone and Joint Surg., 69-A: 684-690, June 1987.

Sharma, M.; MacKenzie, W. G.; and Bowen, J. R.: Severe tibial growth retardation in total fibular hemimelia after limb lengthening. J. Pediat. Orthop., 16: 438-444, 1996.

Snyder, R. G.; Schneider, L. W.; Owings, C. L.; Reynolds, H. M.; Golomb, D. H.; and Schork, M. A.: Anthropometry in Infants, Children, and Youths to Age 18 for Product Safety Design, SP-450. Warrendale, Pennsylvania, Society of Automotive Engineers, 1977.

Steyn, M., and Henneberg, M.: Skeletal growth of children from the Iron Age site at K2 (South Africa). Am. J. Phys. Anthropol., 100: 389-396, 1996.

Stloukal, M., and Hanáková, H.: Die Lünge der Lüngsknochen altslawischer Bevölkerungen. Unter besonderer Berücksichtigung von Wachstumsfragen. Homo, 29: 53-69, 1978.

Sundick, R. I.: Human skeletal growth and age determination. Homo, 29: 228-249, 1978.

Westh, R. N., and Menelaus, M. B.: A simple calculation for the timing of epiphyseal arrest. A further report. J. Bone and Joint Surg., 63-B(1): 117-119, 1981.

Wood, W. L.; Zlotsky, N.; and Westin, G. W.: Congenital absence of the fibula. Treatment by Syme amputation - indications and technique. J. Bone and Joint Surg., 47-A: 1159-1169, Sept. 1965.

Y'Edynak, G.: Long bone growth in Western Eskimo and Aleut skeletons. Am. J. Phys. Anthropol., 45: 569-574, 1976.

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Fig. 1:: Comparison of mean femoral and tibial multipliers for boys and girls.

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Fig. 2-B:: Graph comparing the actual discrepancies and the predictions made with the multiplier method for patients in the epiphysiodesis group.

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Fig. 2-C:: Graph comparing the actual discrepancies and the predictions made with the Moseley method for patients in the epiphysiodesis group.

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Fig. 2-D:: Graph comparing the predictions made with the Moseley method and those made with the multiplier method for congenital discrepancies for patients in the lengthening group.

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Fig. 2-E:: Graph comparing the actual discrepancies and the predictions made with the multiplier method for congenital discrepancies for patients in the lengthening group.

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Fig. 2-F:: Graph comparing the actual discrepancies and the predictions made with the Moseley method for patients in the lengthening group.

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Fig. 2-G:: Graph comparing the actual discrepancies and the predictions made with use of the multiplier method for developmental discrepancies for patients in the lengthening group.

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Fig. 3:: Graph comparing multipliers for boys from different radiographic and clinical databases.

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Fig. 4:: Graph comparing multipliers for girls from different radiographic and clinical databases.

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Fig. 5:: Graph comparing multipliers for boys from different anthropological databases.

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Fig. 6:: Graph comparing multipliers for girls from different anthropological databases.

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Table I: Femoral Multipliers for Boys by Percentile Group

*SD = standard deviation.

Age (yrs.)	Multiplier*					Avg.	Variability
Mean	+2 SD	+1 SD	-1 SD	-2 SD
0	5.09	5.06	5.14	5.14	5.22	5.13	±0.08
1	3.26	3.25	3.30	3.27	3.28	3.27	±0.03
2	2.60	2.57	2.62	2.62	2.64	2.61	±0.04
3	2.24	2.21	2.26	2.26	2.28	2.25	±0.03
4	2.00	1.96	2.01	2.02	2.04	2.00	±0.04
5	1.82	1.79	1.83	1.84	1.86	1.83	±0.03
6	1.68	1.64	1.69	1.70	1.73	1.69	±0.04
7	1.56	1.52	1.56	1.58	1.61	1.57	±0.05
8	1.46	1.43	1.46	1.49	1.51	1.47	±0.04
9	1.37	1.34	1.38	1.40	1.42	1.38	±0.04
10	1.30	1.27	1.30	1.32	1.35	1.31	±0.04
11	1.24	1.20	1.24	1.26	1.29	1.24	±0.05
12	1.18	1.14	1.17	1.20	1.23	1.18	±0.05
13	1.12	1.07	1.11	1.15	1.18	1.13	±0.06
14	1.07	1.03	1.06	1.09	1.12	1.08	±0.03
15	1.03	1.01	1.04	1.05	1.06	1.04	±0.03
16	1.01	1.00	1.02	1.02	1.03	1.02	±0.02
17	1.00	1.00	1.02	1.01	1.01	1.01	±0.01
18	1.00	1.00	1.01	1.00	1.00	1.00	±0.00

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Table I: Femoral Multipliers for Boys by Percentile Group

*SD = standard deviation.

Age (yrs.)	Multiplier*					Avg.	Variability
Mean	+2 SD	+1 SD	-1 SD	-2 SD
0	5.09	5.06	5.14	5.14	5.22	5.13	±0.08
1	3.26	3.25	3.30	3.27	3.28	3.27	±0.03
2	2.60	2.57	2.62	2.62	2.64	2.61	±0.04
3	2.24	2.21	2.26	2.26	2.28	2.25	±0.03
4	2.00	1.96	2.01	2.02	2.04	2.00	±0.04
5	1.82	1.79	1.83	1.84	1.86	1.83	±0.03
6	1.68	1.64	1.69	1.70	1.73	1.69	±0.04
7	1.56	1.52	1.56	1.58	1.61	1.57	±0.05
8	1.46	1.43	1.46	1.49	1.51	1.47	±0.04
9	1.37	1.34	1.38	1.40	1.42	1.38	±0.04
10	1.30	1.27	1.30	1.32	1.35	1.31	±0.04
11	1.24	1.20	1.24	1.26	1.29	1.24	±0.05
12	1.18	1.14	1.17	1.20	1.23	1.18	±0.05
13	1.12	1.07	1.11	1.15	1.18	1.13	±0.06
14	1.07	1.03	1.06	1.09	1.12	1.08	±0.03
15	1.03	1.01	1.04	1.05	1.06	1.04	±0.03
16	1.01	1.00	1.02	1.02	1.03	1.02	±0.02
17	1.00	1.00	1.02	1.01	1.01	1.01	±0.01
18	1.00	1.00	1.01	1.00	1.00	1.00	±0.00

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Table II: Tibial Multipliers for Boys by Percentile Group

*SD = standard deviation.

Age (yrs.)	Multiplier*					Avg.	Variability
Mean	+2 SD	+1 SD	-1 SD	-2 SD
0	5.04	4.98	5.01	5.07	5.08	5.04	±0.05
1	3.21	3.26	3.24	3.19	3.16	3.21	±0.05
2	2.56	2.59	2.58	2.55	2.54	2.56	±0.03
3	2.22	2.24	2.23	2.21	2.20	2.22	±0.02
4	2.00	2.01	2.00	1.99	1.99	2.00	±0.01
5	1.82	1.82	1.82	1.82	1.82	1.82	±0.00
6	1.69	1.68	1.68	1.68	1.68	1.68	±0.01
7	1.57	1.55	1.56	1.58	1.60	1.57	±0.02
8	1.47	1.45	1.46	1.48	1.50	1.47	±0.03
9	1.38	1.35	1.37	1.40	1.42	1.38	±0.04
10	1.31	1.28	1.29	1.33	1.35	1.31	±0.04
11	1.24	1.21	1.22	1.26	1.29	1.24	±0.04
12	1.17	1.14	1.15	1.20	1.23	1.18	±0.03
13	1.11	1.07	1.09	1.14	1.18	1.12	±0.06
14	1.06	1.02	1.04	1.08	1.11	1.06	±0.04
15	1.03	1.01	1.01	1.04	1.05	1.03	±0.02
16	1.01	1.00	1.00	1.01	1.02	1.01	±0.01
17	1.00	1.00	1.00	1.00	1.01	1.00	±0.00
18	1.00	1.00	1.00	1.00	1.00	1.00	±0.00

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Table II: Tibial Multipliers for Boys by Percentile Group

*SD = standard deviation.

Age (yrs.)	Multiplier*					Avg.	Variability
Mean	+2 SD	+1 SD	-1 SD	-2 SD
0	5.04	4.98	5.01	5.07	5.08	5.04	±0.05
1	3.21	3.26	3.24	3.19	3.16	3.21	±0.05
2	2.56	2.59	2.58	2.55	2.54	2.56	±0.03
3	2.22	2.24	2.23	2.21	2.20	2.22	±0.02
4	2.00	2.01	2.00	1.99	1.99	2.00	±0.01
5	1.82	1.82	1.82	1.82	1.82	1.82	±0.00
6	1.69	1.68	1.68	1.68	1.68	1.68	±0.01
7	1.57	1.55	1.56	1.58	1.60	1.57	±0.02
8	1.47	1.45	1.46	1.48	1.50	1.47	±0.03
9	1.38	1.35	1.37	1.40	1.42	1.38	±0.04
10	1.31	1.28	1.29	1.33	1.35	1.31	±0.04
11	1.24	1.21	1.22	1.26	1.29	1.24	±0.04
12	1.17	1.14	1.15	1.20	1.23	1.18	±0.03
13	1.11	1.07	1.09	1.14	1.18	1.12	±0.06
14	1.06	1.02	1.04	1.08	1.11	1.06	±0.04
15	1.03	1.01	1.01	1.04	1.05	1.03	±0.02
16	1.01	1.00	1.00	1.01	1.02	1.01	±0.01
17	1.00	1.00	1.00	1.00	1.01	1.00	±0.00
18	1.00	1.00	1.00	1.00	1.00	1.00	±0.00

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Table III: Femoral Multipliers for Girls by Percentile Group

*SD = standard deviation.

Age (yrs.)	Multiplier*					Avg.	Variability
Mean	+2 SD	+1 SD	-1 SD	-2 SD
0	4.64	4.71	4.63	4.60	4.56	4.63	±0.04
1	2.94	2.97	2.96	2.93	2.91	2.94	±0.03
2	2.39	2.40	2.40	2.39	2.38	2.39	±0.01
3	2.05	2.04	2.05	2.05	2.05	2.05	±0.01
4	1.82	1.81	1.81	1.83	1.85	1.82	±0.02
5	1.66	1.65	1.65	1.66	1.67	1.66	±0.01
6	1.53	1.51	1.52	1.54	1.55	1.53	±0.02
7	1.42	1.40	1.41	1.44	1.45	1.43	±0.03
8	1.33	1.31	1.32	1.35	1.36	1.33	±0.03
9	1.26	1.23	1.24	1.27	1.29	1.26	±0.03
10	1.19	1.16	1.17	1.20	1.22	1.19	±0.03
11	1.12	1.10	1.11	1.14	1.16	1.13	±0.03
12	1.07	1.05	1.06	1.08	1.10	1.07	±0.03
13	1.03	1.02	1.02	1.04	1.05	1.03	±0.02
14	1.00	1.00	1.00	1.00	1.00	1.00	±0.00
15	1.00	1.00	1.00	1.00	1.00	1.00	±0.00
16	1.00	1.00	1.00	1.00	1.00	1.00	±0.00

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Table III: Femoral Multipliers for Girls by Percentile Group

*SD = standard deviation.

Age (yrs.)	Multiplier*					Avg.	Variability
Mean	+2 SD	+1 SD	-1 SD	-2 SD
0	4.64	4.71	4.63	4.60	4.56	4.63	±0.04
1	2.94	2.97	2.96	2.93	2.91	2.94	±0.03
2	2.39	2.40	2.40	2.39	2.38	2.39	±0.01
3	2.05	2.04	2.05	2.05	2.05	2.05	±0.01
4	1.82	1.81	1.81	1.83	1.85	1.82	±0.02
5	1.66	1.65	1.65	1.66	1.67	1.66	±0.01
6	1.53	1.51	1.52	1.54	1.55	1.53	±0.02
7	1.42	1.40	1.41	1.44	1.45	1.43	±0.03
8	1.33	1.31	1.32	1.35	1.36	1.33	±0.03
9	1.26	1.23	1.24	1.27	1.29	1.26	±0.03
10	1.19	1.16	1.17	1.20	1.22	1.19	±0.03
11	1.12	1.10	1.11	1.14	1.16	1.13	±0.03
12	1.07	1.05	1.06	1.08	1.10	1.07	±0.03
13	1.03	1.02	1.02	1.04	1.05	1.03	±0.02
14	1.00	1.00	1.00	1.00	1.00	1.00	±0.00
15	1.00	1.00	1.00	1.00	1.00	1.00	±0.00
16	1.00	1.00	1.00	1.00	1.00	1.00	±0.00

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Table IV: Tibial Multipliers for Girls by Percentile Group

*SD = standard deviation.

Age (yrs.)	Multiplier*					Avg.	Variability
Mean	+2 SD	+1 SD	-1 SD	-2 SD
0	4.76	4.58	4.54	4.58	4.67	4.63	±0.11
1	2.99	3.03	3.01	2.98	2.95	2.99	±0.04
2	2.39	2.44	2.41	2.36	2.33	2.39	±0.06
3	2.06	2.10	2.08	2.04	2.02	2.06	±0.04
4	1.84	1.84	1.84	1.83	1.83	1.84	±0.01
5	1.67	1.67	1.67	1.67	1.67	1.67	±0.00
6	1.54	1.53	1.53	1.54	1.55	1.54	±0.01
7	1.43	1.42	1.42	1.44	1.45	1.43	±0.02
8	1.34	1.32	1.33	1.35	1.36	1.34	±0.02
9	1.26	1.24	1.25	1.27	1.29	1.26	±0.03
10	1.18	1.16	1.17	1.20	1.22	1.19	±0.03
11	1.12	1.09	1.10	1.14	1.16	1.12	±0.04
12	1.06	1.04	1.05	1.08	1.09	1.06	±0.03
13	1.02	1.01	1.02	1.03	1.04	1.03	±0.02
14	1.00	1.00	1.00	1.00	1.00	1.00	±0.00
15	1.00	1.00	1.00	1.00	1.00	1.00	±0.00
16	1.00	1.00	1.00	1.00	1.00	1.00	±0.00

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Table IV: Tibial Multipliers for Girls by Percentile Group

*SD = standard deviation.

Age (yrs.)	Multiplier*					Avg.	Variability
Mean	+2 SD	+1 SD	-1 SD	-2 SD
0	4.76	4.58	4.54	4.58	4.67	4.63	±0.11
1	2.99	3.03	3.01	2.98	2.95	2.99	±0.04
2	2.39	2.44	2.41	2.36	2.33	2.39	±0.06
3	2.06	2.10	2.08	2.04	2.02	2.06	±0.04
4	1.84	1.84	1.84	1.83	1.83	1.84	±0.01
5	1.67	1.67	1.67	1.67	1.67	1.67	±0.00
6	1.54	1.53	1.53	1.54	1.55	1.54	±0.01
7	1.43	1.42	1.42	1.44	1.45	1.43	±0.02
8	1.34	1.32	1.33	1.35	1.36	1.34	±0.02
9	1.26	1.24	1.25	1.27	1.29	1.26	±0.03
10	1.18	1.16	1.17	1.20	1.22	1.19	±0.03
11	1.12	1.09	1.10	1.14	1.16	1.12	±0.04
12	1.06	1.04	1.05	1.08	1.09	1.06	±0.03
13	1.02	1.01	1.02	1.03	1.04	1.03	±0.02
14	1.00	1.00	1.00	1.00	1.00	1.00	±0.00
15	1.00	1.00	1.00	1.00	1.00	1.00	±0.00
16	1.00	1.00	1.00	1.00	1.00	1.00	±0.00

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Table V: Lower-Limb Multipliers for Boys and Girls

*NA = not applicable.

Age (yrs. + mos.)	Multiplier
Boys	Girls*
Birth	5.080	4.630
0 + 3	4.550	4.155
0 + 6	4.050	3.725
0 + 9	3.600	3.300
1 + 0	3.240	2.970
1 + 3	2.975	2.750
1 + 6	2.825	2.600
1 + 9	2.700	2.490
2 + 0	2.590	2.390
2 + 3	2.480	2.295
2 + 6	2.385	2.200
2 + 9	2.300	2.125
3 + 0	2.230	2.050
3 + 6	2.110	1.925
4 + 0	2.000	1.830
4 + 6	1.890	1.740
5 + 0	1.820	1.660
5 + 6	1.740	1.580
6 + 0	1.670	1.510
6 + 6	1.620	1.460
7 + 0	1.570	1.430
7 + 6	1.520	1.370
8 + 0	1.470	1.330
8 + 6	1.420	1.290
9 + 0	1.380	1.260
9 + 6	1.340	1.220
10 + 0	1.310	1.190
10 + 6	1.280	1.160
11 + 0	1.240	1.130
11 + 6	1.220	1.100
12 + 0	1.180	1.070
12 + 6	1.160	1.050
13 + 0	1.130	1.030
13 + 6	1.100	1.010
14 + 0	1.080	1.000
14 + 6	1.060	NA
15 + 0	1.040	NA
15 + 6	1.020	NA
16 + 0	1.010	NA
16 + 6	1.010	NA
17 + 0	1.000	NA

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Table V: Lower-Limb Multipliers for Boys and Girls

*NA = not applicable.

Age (yrs. + mos.)	Multiplier
Boys	Girls*
Birth	5.080	4.630
0 + 3	4.550	4.155
0 + 6	4.050	3.725
0 + 9	3.600	3.300
1 + 0	3.240	2.970
1 + 3	2.975	2.750
1 + 6	2.825	2.600
1 + 9	2.700	2.490
2 + 0	2.590	2.390
2 + 3	2.480	2.295
2 + 6	2.385	2.200
2 + 9	2.300	2.125
3 + 0	2.230	2.050
3 + 6	2.110	1.925
4 + 0	2.000	1.830
4 + 6	1.890	1.740
5 + 0	1.820	1.660
5 + 6	1.740	1.580
6 + 0	1.670	1.510
6 + 6	1.620	1.460
7 + 0	1.570	1.430
7 + 6	1.520	1.370
8 + 0	1.470	1.330
8 + 6	1.420	1.290
9 + 0	1.380	1.260
9 + 6	1.340	1.220
10 + 0	1.310	1.190
10 + 6	1.280	1.160
11 + 0	1.240	1.130
11 + 6	1.220	1.100
12 + 0	1.180	1.070
12 + 6	1.160	1.050
13 + 0	1.130	1.030
13 + 6	1.100	1.010
14 + 0	1.080	1.000
14 + 6	1.060	NA
15 + 0	1.040	NA
15 + 6	1.020	NA
16 + 0	1.010	NA
16 + 6	1.010	NA
17 + 0	1.000	NA

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Table VI: Comparison of Databases of Limb-Length Measurements

Reference	No. of Patients or Skeletons Studied	Ethnic or Racial Origin	Comments
Radiographic data
Anderson et al.⁴ (1963)	100	American	50 percent of patients had poliomyelitis involving one lower limb; Boston, Massachusetts, USA
Anderson et al.⁵ (1964)	134	American (predominantly Irish ancestry)	Longitudinal; Boston, Massachusetts, USA
Beumer et al.⁷ (1997)	182	Dutch	Partially longitudinal
Maresh²¹ (1955)	113	American (predominantly Northern European ancestry)	Longitudinal; Denver, Colorado, USA
Maresh²² (1970)	265	American (predominantly Northern European ancestry)	Longitudinal; Denver, Colorado, USA
Clinical data
Cheng et al.⁸ (1996)	2193	Chinese	Tibia only
Meredith²³ (1939)	100	American	Measurements of standing minus sitting height; Iowa City, Iowa, USA
Meredith and Goldstein²⁴ (1952)	400	Mexican-American	Measurements of standing minus sitting height
Low and Kung²⁰ (1985)	24,978	Chinese	Tibia only
Snyder et al.³⁷ (1977)	4000	American	Includes foot height
Steyn and Henneberg³⁸ (1996)	61	African Negro	Femur only
Anthropological data
Armelagos et al.⁶ (1972)	68	Negro (350 b.c.e.-1400 c.e.)	Sudan
Hoppa¹⁶ (1992)	69	Anglo-Saxon (900 c.e.)	England
Johnston¹⁷ (1962)	165	American Indian (3000 b.c.e.)	Kentucky, USA
Miles and Bulman²⁵ (1994)	120	Scottish (1500-1850 c.e.)	Ensay, Scotland
Saunders and Hoppa³³ (1993)	241	Canadian (1800-1900 c.e.)	Ontario, Canada
Steyn and Henneberg³⁸ (1996)	106	Negro (1000-1200 c.e.)	South Africa
Stloukal and Hanáková³⁹ (1978)	200	Slavic (800 c.e.)	Eastern Europe
Sundick⁴⁰ (1978)	128	German (500-600 c.e.)	Germany
Y'Edynak⁴³ (1976)	100	Eskimo and Aleut (1700-1900 c.e.)	Kodiak Island, Alaska, USA

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Table VI: Comparison of Databases of Limb-Length Measurements

Reference	No. of Patients or Skeletons Studied	Ethnic or Racial Origin	Comments
Radiographic data
Anderson et al.⁴ (1963)	100	American	50 percent of patients had poliomyelitis involving one lower limb; Boston, Massachusetts, USA
Anderson et al.⁵ (1964)	134	American (predominantly Irish ancestry)	Longitudinal; Boston, Massachusetts, USA
Beumer et al.⁷ (1997)	182	Dutch	Partially longitudinal
Maresh²¹ (1955)	113	American (predominantly Northern European ancestry)	Longitudinal; Denver, Colorado, USA
Maresh²² (1970)	265	American (predominantly Northern European ancestry)	Longitudinal; Denver, Colorado, USA
Clinical data
Cheng et al.⁸ (1996)	2193	Chinese	Tibia only
Meredith²³ (1939)	100	American	Measurements of standing minus sitting height; Iowa City, Iowa, USA
Meredith and Goldstein²⁴ (1952)	400	Mexican-American	Measurements of standing minus sitting height
Low and Kung²⁰ (1985)	24,978	Chinese	Tibia only
Snyder et al.³⁷ (1977)	4000	American	Includes foot height
Steyn and Henneberg³⁸ (1996)	61	African Negro	Femur only
Anthropological data
Armelagos et al.⁶ (1972)	68	Negro (350 b.c.e.-1400 c.e.)	Sudan
Hoppa¹⁶ (1992)	69	Anglo-Saxon (900 c.e.)	England
Johnston¹⁷ (1962)	165	American Indian (3000 b.c.e.)	Kentucky, USA
Miles and Bulman²⁵ (1994)	120	Scottish (1500-1850 c.e.)	Ensay, Scotland
Saunders and Hoppa³³ (1993)	241	Canadian (1800-1900 c.e.)	Ontario, Canada
Steyn and Henneberg³⁸ (1996)	106	Negro (1000-1200 c.e.)	South Africa
Stloukal and Hanáková³⁹ (1978)	200	Slavic (800 c.e.)	Eastern Europe
Sundick⁴⁰ (1978)	128	German (500-600 c.e.)	Germany
Y'Edynak⁴³ (1976)	100	Eskimo and Aleut (1700-1900 c.e.)	Kodiak Island, Alaska, USA