our mathematical analysis was based on the following assumptions ; 1 ) the unstapled side of the operated physis grows as fast as that of the contralateral limb , and 2 ) the hinge point of angular correction is located at the perichondrium of the stapled physis .
the limb length distal to the stapled physis is represented by l , the width of the stapled physis by d , and the amount of angular deformity to be corrected by . the gain of effective leg length resulting from angular correction can be written as and the limb shortening effect due to hemiepiphyseal stapling can be written as from the above two trigonometric formulae , the resultant change in effective leg length discrepancy ( elld ) is when elld = 0 or when lengthening effect equals shortening effect , equation 1 reduces to ; which can be rearranged to ; in an attempt to validate this equation , a patient database was reviewed from january 1997 to may 2003 , and data on those who fulfilled the following inclusion criteria was retrieved : 1 ) a unilateral angular deformity around the knee joint , 2 ) an angular deformity only in the coronal plane , 3 ) no obvious partial physeal arrest , 4 ) treated by unilateral hemiepiphyseal stapling , and 5 ) those who were followed up regularly and whose angular deformity had been corrected completely by hemiepiphyseal stapling . among the patients satisfying the inclusion criteria , seven patients had complete medical records and radiographs taken before and after the angular correction .
mean age at the time of surgery was 9.7 years ( range , 5.8 to 12.8 years ) .
the amount of angular deformity to be corrected (  ) was the angle formed between the mechanical axes of the femur and tibia , which averaged 15.3. the width of the stapled physis ( d ) and the limb length distal to the stapled physis ( l ) were measured on a preoperative standing teleradiographs of the lower extremity .
change in effective leg length discrepancy was predicted using equation 1 , and denoted as elldpredicted .
changes in effective leg length discrepancy were also measured on preoperative and post - correction radiographs , and denoted as elldmeasured .
first we created a line along the mechanical axis of the femur , and we created second line , that was perpendicular to the first line , from the ankle joint center to the first line .
the distance from femoral head center to the cross point of the first and second line was defined as the effective leg length ( fig .
angular correction was considered to be achieved successfully when the mechanical axes of the femur and tibia were co - linear , which took 17.8 months at average ( range , 10 to 28 months ) .
significance was set at p < 0.05 . in order to determine the range of l / d ratio in growing children , the width of the stapled physis ( d ) and the limb length distal to the stapled physis ( l ) were measured on standing teleradiographs of the lower extremities in twenty seven children , who underwent hemiepiphyseal stapling or percutaneous hemiepiphysiodesis to correct angular deformity , including the six cases above .
the mean and standard deviation of l / d ratio were calculated in these twenty - seven patients .
our mathematical analysis was based on the following assumptions ; 1 ) the unstapled side of the operated physis grows as fast as that of the contralateral limb , and 2 ) the hinge point of angular correction is located at the perichondrium of the stapled physis .
the limb length distal to the stapled physis is represented by l , the width of the stapled physis by d , and the amount of angular deformity to be corrected by . the gain of effective leg length resulting from angular correction can be written as and the limb shortening effect due to hemiepiphyseal stapling can be written as from the above two trigonometric formulae , the resultant change in effective leg length discrepancy ( elld ) is when elld = 0 or when lengthening effect equals shortening effect , equation 1 reduces to ; which can be rearranged to ;
in an attempt to validate this equation , a patient database was reviewed from january 1997 to may 2003 , and data on those who fulfilled the following inclusion criteria was retrieved : 1 ) a unilateral angular deformity around the knee joint , 2 ) an angular deformity only in the coronal plane , 3 ) no obvious partial physeal arrest , 4 ) treated by unilateral hemiepiphyseal stapling , and 5 ) those who were followed up regularly and whose angular deformity had been corrected completely by hemiepiphyseal stapling . among the patients satisfying the inclusion criteria , seven patients had complete medical records and radiographs taken before and after the angular correction .
mean age at the time of surgery was 9.7 years ( range , 5.8 to 12.8 years ) .
the amount of angular deformity to be corrected (  ) was the angle formed between the mechanical axes of the femur and tibia , which averaged 15.3. the width of the stapled physis ( d ) and the limb length distal to the stapled physis ( l ) were measured on a preoperative standing teleradiographs of the lower extremity .
change in effective leg length discrepancy was predicted using equation 1 , and denoted as elldpredicted .
changes in effective leg length discrepancy were also measured on preoperative and post - correction radiographs , and denoted as elldmeasured .
first we created a line along the mechanical axis of the femur , and we created second line , that was perpendicular to the first line , from the ankle joint center to the first line .
the distance from femoral head center to the cross point of the first and second line was defined as the effective leg length ( fig .
angular correction was considered to be achieved successfully when the mechanical axes of the femur and tibia were co - linear , which took 17.8 months at average ( range , 10 to 28 months ) .
in order to determine the range of l / d ratio in growing children , the width of the stapled physis ( d ) and the limb length distal to the stapled physis ( l ) were measured on standing teleradiographs of the lower extremities in twenty seven children , who underwent hemiepiphyseal stapling or percutaneous hemiepiphysiodesis to correct angular deformity , including the six cases above .
the mean and standard deviation of l / d ratio were calculated in these twenty - seven patients .
equation 2 indicates that effective leg length of the operated limb may increase , remain the same , or decrease after angular correction by hemiepiphyseal stapling , according to l / d ratio and  value ( fig .
2 ) . if a case is plotted above and right to the curve of fig .
if it is plotted below or left to the curve , it will lose effective leg length . with a constant l / d ratio ,
the larger angular deformity is to be corrected , the more likely is it to increase effective leg length .
table 2 shows the preoperative anatomical parameters and changes in predicted and measured effective leg length discrepancies in 6 cases .
changes in effective leg length discrepancy which were predicted by equation 1 ( elldpredicted ) coincided with the actual changes measured on preoperative and post - correction radiographs ( elldmeasured ) , although the amount of changes was not exactly the same . a significant correlation was found between predicted and measured elld 's with correlation coefficient of 0.823 ( p = 0.044 ) .
the mean of l / d ratio in twenty - seven children was 4.82 with a standard deviation of 0.51 .
range of l / d ratio within 2 standard deviations was from 5.85 to 3.78 .
range of value that fulfilled equation 2 with this range of l / d ratio was from 9.9 to 15.6 ( fig .
equation 2 indicates that effective leg length of the operated limb may increase , remain the same , or decrease after angular correction by hemiepiphyseal stapling , according to l / d ratio and  value ( fig .
2 ) . if a case is plotted above and right to the curve of fig .
if it is plotted below or left to the curve , it will lose effective leg length . with a constant l / d ratio ,
the larger angular deformity is to be corrected , the more likely is it to increase effective leg length .
table 2 shows the preoperative anatomical parameters and changes in predicted and measured effective leg length discrepancies in 6 cases .
changes in effective leg length discrepancy which were predicted by equation 1 ( elldpredicted ) coincided with the actual changes measured on preoperative and post - correction radiographs ( elldmeasured ) , although the amount of changes was not exactly the same . a significant correlation was found between predicted and measured elld 's with correlation coefficient of 0.823 ( p = 0.044 ) .
the mean of l / d ratio in twenty - seven children was 4.82 with a standard deviation of 0.51 .
range of l / d ratio within 2 standard deviations was from 5.85 to 3.78 .
range of value that fulfilled equation 2 with this range of l / d ratio was from 9.9 to 15.6 ( fig .
epiphyseal stapling is well established as a simple and effective means of controlling physeal growth.7 - 10 ) by applying staple(s ) to one side of the physis angular deformities of a long bone can be corrected .
it appears reasonable to conclude that hemiepiphyseal stapling inevitably causes limb shortening because it inhibits physeal growth .
this could lead to adopt distraction osteogenesis or open wedge osteotomy instead of hemiepiphyseal stapling in cases with pre - existing limb shortening on the affected side .
the inhibition of physeal growth on the stapled side does decrease long bone longitudinal growth to some extent . on the other hand ,
a significant portion of the length can be restored by correction of an angular deformity .
gain of effective leg length by acute angular correction has already been well demonstrated.6,11 - 14 ) paley explained the lengthening effect of angular correction in terms of the difference between mechanical and anatomical lengths.15 ) however , angular correction by hemiepiphyseal stapling is different from acute angular correction in some aspects . as it takes time to obtain the attempted correction , physeal growth during that time should be considered .
depending on the amounts of each of these contradictory effects , hemiepiphyseal stapling can either increase or decrease effective leg length compared with the contralateral side .
however , to the best of our knowledge , no systematic analysis of these contradictory effects has been reported that considered physeal growth and its asymmetrical inhibition by hemiepiphyseal stapling . in the current study , we investigated mathematically the positive and negative effects of hemiepiphyseal stapling on effective leg length .
the formula indicates that ; a smaller width of the operated physis , a longer distance from the operated physis to the ankle joint , and a larger angular deformity will increase effective leg length of the operated limb .
for example , in case 3 , which had the largest angular correction , a relatively small physeal width , but a substantial distance between the operated physis and the ankle joint , effective leg length discrepancy improved to a maximal extent in terms of both predicted and measured values ( fig .
3 ) . predicted changes in effective leg length discrepancy in the 6 patients concurred with clinical results .
however , the amount of changes varied between the predicted and measured values as the assumptions upon which the mathematical model was based are not always satisfied in reality . the unstapled side of the operated physis , which was assumed not to be affected by any pathologic process , may have some functional ( though not structural ) defect .
moreover , the hinge point of angular correction by hemiepiphyseal stapling is not necessarily at the perichondrium but possibly nearer to the midline of the limb segment .
another source of error for this model is that physeal growth of the unstapled side of the operated physis may also be retarded by hemiepiphyseal stapling to some extent.7,16 ) if this occurs , it would interfere with the length gain effect and subsequently aggravate the effective leg length discrepancy .
nevertheless , this study revealed that hemiepiphyseal stapling could improve effective leg length in a certain circumstance .
as l / d ratio was found to remain in a relatively narrow range , it is largely influenced by the amount of angular correction whether the effective leg length discrepancy will improve or aggravate .
our data suggest that the hemiepiphyseal stapling is likely to lose effective leg length if the amount of angular correction is less than 10 , whereas it is to gain effective leg length if it is larger than 16. this should be taken into consideration when determining the surgical method for angular deformity correction in skeletally immature patients .
 backgroundthe hemiepiphyseal stapling has both positive and negative effects on effective leg length . the purpose of this study was to analyze change in effective leg length after angular correction by hemiepiphyseal stapling , and to validate in clinical cases.methodsmathematical analysis of a hemiepiphyseal stapling model 
 was conducted . 
 the induced formula was validated in 6 cases fulfilling the assumptions of the model . 
 anatomical parameters involved in this formula were measured in additional 21 cases undergoing hemiepiphyseal stapling or hemiepiphysiodesis.resultseffective leg length increased or decreased according to three parameters in this model : 1 ) limb length distal to the operated physis ( l ) , 2 ) width of the operated physis ( d ) , and 3 ) the amount of angular deformity to be corrected (  ) . 
 actual change in effective leg length of 6 cases similar to this model coincided with the predicted change at least in its direction . 
 l / d ratio was 4.82  0.51.conclusionsconsidering the narrow range of the l / d ratio , hemiepiphyseal stapling is likely to decrease effective leg length if the amount of angular correction is less than 10 , whereas to increase it if the amount of angular correction is larger than 16. this should be taken into consideration when selecting the surgical method for angular deformity correction in skeletally immature patients . 
METHODS
Mathematical Analysis
Validation of the Equation in Clinical Cases 
Range of 
RESULTS
Mathematical Analysis Predicting Change in Effective Leg Length Discrepancy
Validation of the Equation in Clinical Cases
Range of 
DISCUSSION