Abstract
Purpose
The purpose of this study is to generate validated prediction rules for metacarpal lengths that can be applied without the need for computation tools to assist with restoration of anatomic length after fracture and utilizes only ipsilateral metacarpals.
Methods
The anatomic lengths of all hand bones in 50 hands (25 men, 25 women) were used along with linear regression subset analysis to determine which metacarpals are the most predictive of each other. The most predictive metacarpals were then used to generate simple addition and subtraction prediction rules via simplifying the linear equation generated with linear regression analysis. Those rules were then applied to subsequent test cases, and percent accuracy within various cutoffs were analyzed and compared to the accuracy when using the contralateral side.
Results
The prediction rules were generated and were found to be identical for both men and women. When applied to the test cases, the estimated metacarpal lengths were within 3 mm of the actual value in 97.5% of the cases for women and 90% of the cases for men compared to 95% when using the contralateral side.
Conclusion
The simple additional and subtraction rules generated in this analysis were as good as or superior to using the contralateral side in all cases for women and were as good as or superior to using the contralateral side in for metacarpals 3–5 for men.
Clinical Relevance
Using these simple estimating rules may be superior to using the contralateral side in most cases and provides a secondary method for determining anatomic lengths when contralateral radiographs are not available or when contralateral radiographs were obtained in different enough conditions such that the lengths may not be representative of the hand of interest.
Key words: Biomechanics, Fracture, Length, Metacarpal, Prediction
Metacarpal fractures are one of the most treated hand fractures in orthopedics accounting for between 18% and 44% of all hand fractures.1 Restoration of anatomic alignment is crucial in reserving function after fracture.2 However, it has been shown that some amount of metacarpal length may be lost without affecting overall function. Previous studies have determined patients can tolerate anywhere from 3 to 10 mm of shortening before clinically significant effects are likely. One study evaluating extension found that for every 2 mm of shortening, 7º of extension lag are lost.3 Given that the average patient had 20º of hyperextension, they concluded patients could tolerate up to 6 mm of shortening.3 Further studies investigating flexion found that significant effects on flexion force at least 5 mm of shortening.4 Additional studies evaluating the theoretic effect on intrinsic muscles of the hand concluded that 3 mm was enough to likely cause significant effects.5 The most recent study on the effect of metacarpal shortening evaluated both effects on range of motion and the force generated by the finger of the fractured metacarpal and found that up to 10.0 mm of shortening would likely not affect a patient’s ability to reach terminal flexion.6 Additionally, it was determined that while a significant effect on flexion force may be found with shortening lengths as low as 2.5 mm in the context of overall grip strength, accounting for the fact that no finger contributes more than 30% to grip strength, those significant effects on finger strength are unlikely to cause global grip strength defects.6 In fact, in clinical studies evaluating conservative nonsurgical management of metacarpal fractures where up to 5 mm shortening was noted, no patient experienced significant loss of grip strength.7 Given these studies, it appears that while restoration of metacarpal length is important, there is some room for error without affecting clinical outcomes.
These results rely on the ability to determine the amount of metacarpal length discrepancy. In vitro experiments rely on precise changes applied by the researchers; thus, the amount of shortening is controlled. In patients, however, determining anatomic length after fracture is more challenging as there is rarely an unfractured reference for comparison. The gold standard has always been to use the contralateral side as a reference in these situations; however, that does not come without issues. In some contexts, contralateral radiographs may not be available because of polytrauma involving both hands or simply because contralateral radiographs were not obtained preoperatively and obtaining them intraoperatively would be infeasible. Additionally, if the contralateral radiographs obtained are not at the same magnification or angle, the length measurements may be affected and not representative. Further, while using the contralateral side is the gold standard, it has been shown to not be entirely accurate. Studies evaluating the symmetry of human metacarpal lengths have found that in fact there are significant differences in lengths in size for metacarpals 1, 2, 3, and 5.8 Therefore, alternate techniques for determining the anatomic length are needed for when the contralateral side is unavailable or to act as a secondary reference that the anatomic length is correct. Previous studies investigating methods to determine anatomic length have been conducted. One study reported the ratio of all metacarpal lengths in 100 patients.9 While the ratios determined may be accurate, the study did not provide validation of the ratios. In addition, using the ratios determined required mathematical manipulation involving multiplication to the 100th digit, which is not likely to be performed without a computational tool. However, this paper did include the first metacarpal, whereas this paper does not because this paper relies on the predetermined acceptable amount of shortening, which has not been validated as extensively in the first metacarpal. Additionally, recent studies have determined bivariate models for metacarpal length prediction but again require a computational tool for use.8 For surgeons in the operating room, simple estimates that can be done in their head without a computer may provide far more utility even if they are slightly less accurate given the fact it has been concluded that some amount of inaccuracy is likely acceptable. Therefore, the intention of this study was twofold. First, the study sought to evaluate the gold standard of using the contralateral side as a length predictor. Second, the study sought to produce a set of simple addition and subtraction rules for metacarpal length prediction that can be implemented without access to a computer using only alternate ipsilateral metacarpals as a reference. Additionally, given that metacarpal length and volume can be used as a gender-identifying characteristic, attention to gender specificity will be included.10,11
Materials and Methods
In total, 50 bilateral hand radiographs (25 men, 25 women) were used to generate the models and estimates. Demographics of all patients included in the study are provided in Table 1.
Table 1.
Demographics of All Patients
| Demographics | Mean | SD | |
|---|---|---|---|
| Age (years) | 54.7 | 13.0 | |
| BMI | 32.5 | 7.31 | |
| Race | |||
| White | 39.1% | ||
| Black or African American | 58.7% | ||
| Native American/Native Alaskan | 2.2% | ||
| Ethnicity | |||
| Hispanic | 25% | ||
| Non-Hispanic | 75% |
BMI, body mass index; SD, standard deviation.
Inclusion criteria included any patient over the age of 18 years seen in the orthopedic hand clinical with bilateral anteroposterior hand radiographs. Patients with a history of any fracture, hand surgery, congenital anomaly, or severe arthritis were excluded from the study. Two separate observers of differing experience recorded the lengths of all metacarpals and phalanges according to Figure 1. The averages and standard deviations of the metacarpals of interest are listed in Table 2.
Figure 1.
Example of measurements recorded for all bones of the fingers.
Table 2.
Averages and Standard Deviations for all Recorded Metacarpals of Interest for Both Men and Women
| Metacarpal 2 | Metacarpal 3 | Metacarpal 4 | Metacarpal 5 | |
|---|---|---|---|---|
| Men | ||||
| Mean (cm) | 7.27 | 7.04 | 6.28 | 5.75 |
| SD | 0.56 | 0.56 | 0.5 | 0.4 |
| Women | ||||
| Mean (cm) | 6.77 | 6.54 | 5.82 | 5.27 |
| SD | 0.36 | 0.36 | 0.34 | 0.31 |
SD, standard deviation.
All measurements were obtained using the electronic health record native measurement tool. All data were collected with institutional review board approval.
Linear regression
The initial stage of analysis involved regression subset analysis using the R package Leaps and allowing for up to 10 inputs.12,13 Leaps provides the best overall linear regression model for a given number of inputs based on the adjusted R2 value. The model determined to have the best correlation with the metacarpal of interest was then selected. The models determined using these methods while highly accurate based on the R2 value require significant computational access to implement. To determine which single input (metacarpal or phalanges) alone was the most predictive for each metacarpal of interest, the Leaps package was rerun allowing for only a single input. The model with the highest adjusted R2 again was selected. If two or more models had similar R2 values, the model with the coefficient closest to one was selected. The coefficient and constant of that model were used to generate simple addition and subtraction rules.
Determining simple prediction rules
Any linear equation in the form y = mx + b can be rewritten as y = x + x (m − 1) + b. This is the starting point for our analysis. For the purpose of these methods, y represents the metacarpal of interest, and x represents the input variable determined by the Leaps subset analysis with only one input. The slope and constant of the equation were taken from the model determined using Leaps. Assuming a narrow distribution of input lengths within our population, the average value of the input for the entire population was then substituted for x in x (m − 1), resulting in a constant Xavg (m − 1) such that the final equation is y = x + Xavg (m − 1) + b. Given that Xavg, m and b are known, (Xavg (m − 1) + b) is a constant (c). Then, c was rounded to the nearest 0.05 cm to simplify the equation. Thus, the final form is y = x + c, yielding a simple equation involving only addition where y is the metacarpal of interest, x is the input determined by the subset analysis and varies by patient, and c is a constant calculated as described above. Using this simple formulation, a simple estimate can be determined for all metacarpals. All calculations and rules were generated separately for both genders to ensure no gender biases were introduced.
Contralateral side
The rules determined using the strategy above were then compared to the contralateral side using two methods. First, the output of the simple estimate and the contralateral were used as an input for a secondary linear regression model using the metacarpal of interest as the independent variable. The adjusted R2 values of these models were then compared. Additionally, categorical analysis was performed by subdividing the results into error bins by 1 mm, 2 mm, and 3 mm, ie, predictions within 1 mm, 2 mm, and 3 mm or greater of the actual value, respectively. Using this method, a percentage accuracy can be obtained based on the threshold of interest.
Test set
Up to this point, all validation and accuracy analyses were performed on the same set of data used to generate the model, thus potentially biasing the results. Therefore, a third observer recorded 10 more bilateral radiographs (20 hands) from 5 men and 5 women. The rules were then applied to the tests set and again compared to the contralateral side via categorical accuracy based on threshold of 1 mm, 2 mm, and 3 mm or greater error, respectively. The thresholds were used as reported in previous studies and described in the Introduction section. These studies found that 3 mm of error was highly unlikely to cause any functional issues in patients. Thus, less than 3 mm of error in measurements will be sufficient in almost all patients. However, previous studies focused on the biomechanics of metacarpals 2–5, and their recommendations are based on the motion of metacarpal 2–5, which significantly differ from metacarpal 1. Furthermore, 88% of metacarpal fractures involve metacarpals 2–5. Thus, this analysis was restricted only to those metacarpals. All analysis was performed separately by gender to ensure no gender bias was introduced.
Results
Linear regression models with up to 10 inputs produced highly accurate models using R2 values. The resulting adjusted R2 values for each metacarpal's best model are detailed in Table 3. All models were statically significant at P <.001.
Table 3.
R2 Values of High Input Linear Regression Models
| Metacarpal 2 | Metacarpal 3 | Metacarpal 4 | Metacarpal 5 | |
|---|---|---|---|---|
| Men | 0.969 | 0.962 | 0.98 | 0.904 |
| Women | 0.922 | 0.95 | 0.96 | 0.948 |
When restricting the input to only one variable, it was found that the same adjacent metacarpal was the most predictive for both men and women. The input for each respective metacarpal is presented in Table 4.
Table 4.
Most Predictive Metacarpal for Each Metacarpal of Interest
| Metacarpal of Interest | M2 | M3 | M4 | M5 |
|---|---|---|---|---|
| Predictor Metacarpal | M3 | M4 | M3 | M4 |
Using these models, both the coefficient and constant, and the strategy described in the Methods section, the final simple equations were determined as described in Table 5 and redemonstrated in graphical form (Fig. 2). Note that despite being determined separately, the rules for both men and women were the same.
Table 5.
Final Prediction Rules for Each Metacarpal
| M2 | M3 | M4 | M5 | |
|---|---|---|---|---|
| Men | M3 + 0.2 cm | M4 + 0.75 cm | M3 − 0.75 cm | M4 − 0.5 cm |
| Women | M3 + 0.2 cm | M4 + 0.75 cm | M3 − 0.75 cm | M4 − 0.5 cm |
Figure 2.
A graphical representation of the final prediction rules depicted on a standard hand radiograph.
A representative radiograph demonstrating that the differences in metacarpal length are all within 0.05 of the predicted differences is presented in Figure 3.
Figure 3.
Representative radiographs with metacarpal length differences highlighted. All differences are within 0.05 cm of the predicted difference, which is well within the acceptable error of 3 mm to allow for functional recovery.
When these rules were applied to the data set and the resulting prediction was used as the input for a secondary linear regression compared to use of the contralateral side, the adjusted R2 values presented in Table 6 were obtained.
Table 6.
R2 Values of Linear Regression Models Using Predicted Metacarpal Lengths and the Contralateral Side
| M2 | M3 | M4 | M5 | |
|---|---|---|---|---|
| Men | 0.936 | 0.937 | 0.938 | 0.854 |
| Women | 0.835 | 0.891 | 0.891 | 0.838 |
| Contralateral | 0.954 | 0.938 | 0.941 | 0.884 |
When the simple estimate was calculated and categorized by the amount of error compared to the contralateral side, the percentage within the respective error (Table 7) was obtained. Given that there were 25 men and 25 women with bilateral radiographs, there were 50 total hands in the male and female groups. The contralateral analysis combined the men and women, thus resulting in 50 contralateral comparisons with an even amount of data in each category.
Table 7.
Percent Within Respective Error for the Predicted and Contralateral Side
| Within 3 mm | M2 | M3 | M4 | M5 |
|---|---|---|---|---|
| Men | 96.2 | 96.2 | 96.2 | 90.4 |
| Women | 91.7 | 97.9 | 97.9 | 91.7 |
| Contralateral | 98 | 96 | 96 | 98 |
| Within 2 mm | ||||
| Men | 82.7 | 84.6 | 86.5 | 76.9 |
| Women | 82.7 | 89.6 | 89.6 | 81.3 |
| Contralateral | 88 | 86 | 90 | 90 |
| Within 1 mm | ||||
| Men | 53.9 | 51.9 | 53.9 | 46.2 |
| Women | 54.2 | 56.3 | 56.3 | 68.8 |
| Contralateral | 66 | 64 | 64 | 66 |
Test set
After the 10 additional patients’ data were recorded and analyzed, the percentage within the respective errors were obtained as detailed in Table 8. In Table 8, the highest percent accuracy is noted in blue for men and green for women. If there was a tie between the contralateral side and the estimate, the estimate was favored given its simplicity.
Table 8.
Percent Accuracy of Estimated Metacarpal Lengths of the Test Set Using the Simple Rules Determined Based on the Training Set Compared to the Contralateral Side
|
Within 3 mm |
Percent Within mm Range |
|||
|---|---|---|---|---|
| M2 | M3 | M4 | M5 | |
| Estimate for Men | 80 | 100 | 100 | 80 |
| Estimate for Women | 100 | 100 | 100 | 90 |
| Contralateral | 100 | 100 | 100 | 80 |
| Within 2 mm | ||||
| Estimate for Men | 80 | 90 | 80 | 70 |
| Estimate for Women | 100 | 90 | 90 | 80 |
| Contralateral | 100 | 80 | 80 | 40 |
| Within 1 mm | ||||
| Estimate for Men | 30 | 40 | 40 | 50 |
| Estimate for Women | 70 | 70 | 70 | 70 |
| Contralateral | 50 | 60 | 40 | 40 |
Discussion
The goal of this study was to produce simple addition and subtraction rules using ipsilateral adjacent metacarpals that could predict anatomic length within an acceptable amount of error as determined by previous studies, which is reported to be at least 3 mm. The resulting rules generated were in fact as good as or superior to use of the contralateral side for women in all the metacarpals and threshold of interest in the test set. When applied to an external data set, the rules again were confirmed to be highly predictive. A study that evaluated metacarpal length parameters of 57 patients using computed tomography found that the average lengths of metacarpals 2–5 were 67.6 mm, 65.6 mm, 58.0 mm, and 52.5 mm, respectively.14 The resulting differences are 2 mm, 7.6 mm, and 5.5 mm, which matching well with our predicted estimates of 2 mm, 7.5 mm, and 5 mm. For men, the estimate was as good as or superior to the contralateral side at predicting within 3 mm and 2 mm of error for metacarpals 3–5. For metacarpal 2, there was one patient whose metacarpal was greater than 2 standard deviations above the average, resulting in an estimate that was off by 0.38 cm and 0.36 cm for the patient’s right and left hand, respectively. However, given that previous studies found 3 mm to be the lowest possible threshold for functional effects and these values were only off by 0.6 mm from that threshold, it is likely that this estimate would still be acceptable for the patient’s functional outcomes. If 4 mm of discrepancy is acceptable, then the estimates are in fact as good as or superior to use of the contralateral side in all cases tested for both men and women. Given this finding and the fact the contralateral radiographs may have been obtained at different angles or magnifications such that it is no longer representative of the hand of interest, it may be preferable to use these estimates over the contralateral side in most cases. Caution may be needed, however, for patients whose metacarpal lengths are greater than 2 standard deviations from the average, as detailed in Table 2.
Limitations
This study is not without limitations. It relies on the assumption that the discrepancy in length can be up to 3 mm in both directions, including shortened and lengthened. However, even use of the contralateral side in 37 of 50 cases when comparing right to left yielded overestimates. Therefore, even when using the contralateral side, this method cannot be relied upon to constantly over- or underestimate the metacarpal of interest. Additionally, the metacarpal length measurement technique is crucial when applying these rules. The technique for measuring length involved the use of the apex of the metacarpal head distally, and measurements were obtained parallel to the shaft until the most proximal aspect of the bone was reached. The use of 2 separate observers to record the training data along with a third observer to obtain the test data set helps account for measurement variability and adds external validity to the results. However, these rules may not be applicable should a different measurement technique be used.
Footnotes
Declaration of interests: No benefits in any form have been received or will be received related directly to this article.
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