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00005768-200706000-0001700005768_2007_39_1013_zoeller_allometric_6miscellaneous-article< 101_0_21_6 >Medicine & Science in Sports & Exercise©2007The American College of Sports MedicineVolume 39(6)June 2007pp 1013-1019Allometric Scaling of Biceps Strength before and after Resistance Training in Men[APPLIED SCIENCES: Physical Fitness and Performance]ZOELLER, ROBERT F.1; RYAN, ERIC D.2; GORDISH-DRESSMAN, HEATHER3; PRICE, THOMAS B.4,5; SEIP, RICHARD L.4; ANGELOPOULOS, THEODORE J.6; MOYNA, NIALL M.7; GORDON, PAUL M.8; THOMPSON, PAUL D.4; HOFFMAN, ERIC P.31Department of Exercise Science and Health Promotion, Florida Atlantic University, Davie, FL; 2Department of Health and Exercise Science, University of Oklahoma, Norman, OK; 3Research Center for Genetic Medicine, Children's National Medical Center, Washington, DC; 4Division of Cardiology, Hartford Hospital, Hartford, CT; 5Department of Diagnostic Radiology, Yale University School of Medicine, New Haven, CT; 6Center for Lifestyle Medicine and Department of Health Professions, University of Central Florida, Orlando, FL; 7Department of Sport Science and Health, Dublin City University, Dublin, IRELAND; and 8Division of Exercise Physiology, West Virginia University, Morgantown, WVAddress for correspondence: Robert F. Zoeller, Ph.D., Florida Atlantic University, Dept. of Exercise Science and Health Promotion, 2912 College Ave., Davie, FL 33314; E-mail: rzoeller@fau.edu.Submitted for publication August 2006.Accepted for publication January 2007.ABSTRACTPurpose: The purposes of this study were 1) derive allometric scaling models of isometric biceps muscle strength using pretraining body mass (BM) and muscle cross-sectional area (CSA) as scaling variables in adult males, 2) test model appropriateness using regression diagnostics, and 3) cross-validate the models before and after 12 wk of resistance training.Methods: A subset of FAMuSS (Functional SNP Associated with Muscle Size and Strength) study data (N = 136) were randomly split into two groups (A and B). Allometric scaling models using pretraining BM and CSA were derived and tested for group A. The scaling exponents determined from these models were then applied to and tested on group B pretraining data. Finally, these scaling exponents were applied to and tested on group A and B posttraining data.Results: BM and CSA models produced scaling exponents of 0.64 and 0.71, respectively. Regression diagnostics determined both models to be appropriate. Cross-validation of the models to group B showed that the BM model, but not the CSA model, was appropriate. Removal of the largest six subjects (CSA > 30 cm2) from group B resulted in an appropriate fit for the CSA model. Application of the models to group A posttraining data showed that both models were appropriate, but only the body mass model was successful for group B.Conclusion: These data suggest that the application of scaling exponents of 0.64 and 0.71, using BM and CSA, respectively, are appropriate for scaling isometric biceps strength in adult males. However, the scaling exponent using CSA may not be appropriate for individuals with biceps CSA > 30 cm2. Finally, 12 wk of resistance training does not alter the relationship between BM, CSA, and muscular strength as assessed by allometric scaling.Muscle strength testing is a common modality used to evaluate, assess, and articulate data regarding muscle function within groups. Results from strength testing are commonly confounded by body size, and inconsistencies can arise when strength data are nonnormalized for body size or normalized using inappropriate methods (9,10). Absolute methods of strength testing tend to yield a bias toward larger individuals (23). Ratio methods (i.e., strength relative to body mass) tend to yield a bias toward smaller individuals (2). Normalizing strength relative to muscle cross-sectional area (CSA) has been proposed as the gold standard within fusiform muscle groups (i.e., biceps) (14). However, the ability to measure CSA is often limited to expensive technologies such as computerized tomography (CT) or magnetic resonance imaging (MRI). Allometric scaling models, typically using body mass, have been proposed as an alternative approach to control for differences in body size and/or muscle mass.Allometric scaling is based on the theory of geometric similarity, which holds that all humans have the same shape and differ only in size (1,13). More specifically, limb lengths are proportional to body height (L), and all areas (e.g., CSA) are proportional to L2, whereas all volumes and volume-associated indices, such as body mass, are proportional to L3. As such, muscle CSA is proportional to body mass2/3. Alternatively, any muscle CSA is directly proportional to the strength/force produced by that muscle. However, these assumptions may be confounded by other factors such as body composition, fiber type distribution, distances of the tendon insertion from the center of rotation of the joint, and limitations inherent in measures of CSA compared with muscle volume. For an excellent review of these issues, see Bruce et al. (4).Allometric scaling models use linear, log-linear, or other regression models to remove the potential confounders of differences in body size or muscle mass/CSA. However, allometric scaling models, whether using CSA or body mass, must be carefully evaluated for appropriateness of fit (3,18). This fit is assessed using regression diagnostics, a group of statistical techniques designed to check the underlying assumptions of a model (i.e., the normality and distribution of residual errors).The data for this project were derived from a subset of participants from the Functional Single Nucleotide Polymorphisms Associated with Muscle Size and Strength (FAMuSS) study (22). The FAMuSS study is a multisite, controlled, unilateral biceps resistance exercise study assessing four specific variables: 1) baseline biceps muscle strength, 2) baseline biceps muscle CSA, 3) posttraining biceps muscle strength, and 4) posttraining muscle CSA. The goal of the study is to search for relationships between these muscle traits and specific genetic markers (single-nucleotide polymorphisms (SNP)).The well-controlled design of the FAMuSS study also provided a unique opportunity to develop and evaluate allometric scaling models in a large cohort of previously untrained adult males, both before and after 12 wk of progressive unilateral biceps resistance exercise training. We are unaware of any studies that have examined the effects of resistance training on the relation between body size (i.e., body mass or CSA) and muscle strength as determined by allometric scaling. It is well known that, especially in the first 8 wk of resistance training, neurological adaptations can contribute significantly to increases in strength independently of morphological changes in the muscle (8,15,17,21). We hypothesized that these training-induced neurological adaptations could potentially alter the relation between CSA, body mass, and isometric strength. As such, the purposes of this study were to 1) derive allometric scaling models of biceps isometric strength using pretraining CSA and body mass as the independent or scaling variables; 2) test model appropriateness and fit, using regression diagnostics; and 3) assess the models by cross-validation within the FAMuSS subject cohort, both before and after 12 wk of unilateral biceps resistance training.METHODSThe methods for the FAMuSS project have been detailed previously by Thompson et al. (22). However, a brief description is presented below.ParticipantsThis study used archival data from five of the eight FAMuSS study institutions: Florida Atlantic University, University of Central Florida, Hartford Hospital, West Virginia University, and Dublin City University (Ireland). Data from 136 men who completed the FAMuSS study were analyzed. Before initiating the study, all participants were informed of all procedures and risks associated with the study and signed an informed consent in accordance with each institution's institutional review board for human subject experimentation. Further approval to use these archived data was obtained from the Florida Atlantic University institutional review board for human subjects experimentation.Experimental MeasurementsBody mass.A body mass measurement (kg) was taken before the initiation of all strength testing and training for all FAMuSS study participants. At the conclusion of all posttraining strength testing, all participants' body masses were measured again. All subjects chosen for this study showed no appreciable change in body mass during their participation in the study. The average of the pre- and posttraining body mass measurements served as the criterion measure of body mass.Magnetic resonance imaging.Magnetic resonance imaging (MRI) was performed before and after exercise training to assess biceps brachialis CSA, as previously described (22). Because of concerns that postexertion swelling can spuriously increase MRI measurements, pretraining MRI were performed before or 24-48 h after the isometric strength tests. Posttraining MRI were performed 48-96 h after the final training session. This ensured that temporary exercise effects, such as water shifts, were avoided, while also avoiding any reduction of muscle size from detraining.Isometric strength.Isometric strength of the elbow flexor muscles of the nondominant arm was measured before and after 12 wk of strength training using a specially constructed, modified preacher bench and strain gauge (model 32628CTL, Lafayette Instrument Company, Lafayette, IN). Baseline measures of isometric strength were assessed on three separate days spaced no more than 2 d apart to control for learning effect. Posttraining measures of isometric strength were performed immediately before the last training session and 48 h after the last training session. On each of the testing days, three maximal isometric contractions were performed with each arm. Each contraction lasted 2 s, with 1 min of recovery allowed between contractions. The average of the peak force produced during the three contractions was used as the criterion score. To obtain three consistent peak force values, up to two more contractions were performed if a peak value deviated by more than 2.25 kg from the other two peak values. The average of the results obtained on the second and third pretraining testing days was used as the baseline criterion measurement (intraclass correlation = 0.90), and the results obtained 48 h after the last training session were used for the posttraining criterion measurement.Construction and Evaluation of Allometric Scaling ModelsPretraining isometric strength of the nondominant arm served as the dependent variable. Muscle CSA (cm2) and body mass (kg) served as the independent variables to construct two separate regression models and to identify scaling exponents. The following steps outline the procedures used to construct and then evaluate the appropriateness of each model: 1. Subjects were randomly sorted and split into two groups (A and B) for cross-validation. Independent t-tests were used to identify any between-group differences with regard to age, height, weight, and pretraining isometric strength and nondominant-arm CSA. Alpha was adjusted using Bonferroni correction. Allometric scaling models were first constructed and tested for appropriateness of fit for group A and then cross-validated on group B (step 6). 2. Normality of the dependent variable (pretraining isometric strength of the nondominant arm) was assessed in the entire cohort, as well as for groups A and B. 3. A log-linear regression analysis was performed on the independent and dependent variables for group A. The slope of the regression line was used as the allometric scaling exponent. 4. Distribution of residuals and the assumption of homoscedasticity were tested by the Anderson-Darling normality test, Breusch-Pagan/Cook-Weisberg test, and visual inspection of the residuals. The residual errors should demonstrate a constant variance (homoscedasticity) and a normal distribution, indicating that the model fits all individuals across the entire range (18). 5. Independence of the power ratio (allometrically scaled strength) and independent variables were assessed. For an allometric model to be deemed appropriate, there should be no significant correlation between the allometrically scaled strength measurement and the independent variable. 6. Step 5 was repeated on group B, using the same criterion of appropriateness of fit. The newly derived scaling exponents were applied to group B and were evaluated as an internal test of cross-validation. Again, there should be no significant correlation between the allometrically scaled strength and the independent variable. 7. Step 5 was repeated for posttraining strength measures of both groups A and B for each model. The newly derived scaling exponents were applied to posttraining strength measures to examine the influence of training-induced changes and its influence on the application and/or cross-validation of such models.RESULTSThe pre- and posttraining descriptive characteristics for groups A and B are presented in Table 1. Briefly, group A was older (mean ± SEM = 26.7 ± 0.74 yr for group A vs 24.2 ± 0.66 yr for group B (P < 0.05)), whereas group B was stronger (mean ± SEM for isometric strength = 60.28 ± 2.27 kg for group A vs 66.66 ± 2.05 kg for group B (P < 0.05)). Anderson-Darling normality tests showed that the raw pretraining isometric strength values for the entire cohort, as well as groups A and B, were normally distributed (P > 0.05 for all three).TABLE 1. Descriptive characteristics for groups A (N = 68) and B (N = 68).Body mass as a scaling variable.Log-linear regression analysis using body mass as the independent variable was applied to group A, resulting in a scaling exponent of 0.64 (Fig. 1). The Breusch-Pagan/Cook-Weisberg test for heteroscedasticity was performed, and the residual errors from the body mass model in group A were found to be randomly distributed or homoscedastic (P > 0.05). The independence test of the power ratio (Fig. 2) was nonsignificant (P > 0.05). Visual inspection of residuals showed no apparent systematic variation. Additionally, the Anderson-Darling normality test of the residuals revealed a normal distribution (P > 0.05).FIGURE 1-Log-linear regression analysis of the group A baseline isometric biceps strength using baseline body mass (top panel, y = 0.637x + 1.2956; r2 = 0.158; P = 0.001) and CSA (bottom panel, y = 0.7095x + 1.963; r2 = 0.331; P < 0.001) as independent variables.FIGURE 2-Test of the independence of the power ratio for group A baseline body mass (top panel, r2 = 0.004, P = 0.613) and CSA (bottom panel, r2 < 0.001, P = 0.877) models.CSA as a scaling variable.The independent variable CSA yielded a scaling coefficient of 0.71 (Fig. 1). The group A CSA model was also nonsignificant (P > 0.05) for the Breusch-Pagan/Cook-Weisberg test for heteroscedasticity and for the independence test of the power ratio (Fig. 2). Visual inspection of residuals showed no systematic variation, and the Anderson-Darling normality test indicated that the residuals were normally distributed (P > 0.05).Cross-validation of the models.The newly derived scaling exponents were applied to group B pretraining data and evaluated using the independence of the power ratio as the criterion. Linear regression analysis revealed no correlation (P > 0.05) between group B's body mass and the newly scaled isometric strength (Fig. 3). Group B's CSA was significantly correlated (P < 0.05) with its newly scaled isometric strength (Fig. 3), suggesting that the model did not entirely remove the influence of CSA on the newly scaled isometric strength variable.FIGURE 3-Test of the independence of the power ratio for group B baseline body mass (top panel, r2 = 0.014, P = 0.335) and CSA (center panel, r2 = 0.090, P = 0.013) after applying scaling exponents derived from group A. The bottom panel is the same as the center panel but with the six largest (CSA > 30 cm2) subjects removed (r2 = 0.004, P = 0.614).Application to posttraining data.The newly derived scaling exponents from group A were then applied to the posttraining isometric strength data from groups A and B and again evaluated, using the independence of the power ratio as the criterion. The linear regression analysis revealed no correlation (P > 0.05) between the independent variables (both body mass and CSA) and the scaled posttraining isometric biceps strength for group A (Fig. 4). For group B, on the other hand, there was no correlation (P > 0.05) between body mass and the newly scaled posttraining isometric strength (Fig. 4), but there was a significant relationship between CSA and the scaled posttraining isometric strength (P = 0.011) (Fig. 4). It is important to note that the CSA model did not meet all criteria of appropriateness, because of the lack of independence of the power ratio when applied to group B, either before or after 12 wk of resistance training.FIGURE 4-Test of the independence of the power ratio on posttraining measures for group A body mass (panel A, r2 = 0.002, P = 0.719), group A CSA (panel B, r2 = 0.003, P = 0.654), group B body mass (panel C, r2= 0.044, P= 0.085), and group B CSA (panel D, r2 = 0.093, P = 0.011) after applying scaling exponents derived from group A baseline measurements.DISCUSSIONThe purpose of this investigation was to construct, apply, and evaluate allometric scaling models, using data from the FAMuSS study (22). Allometric scaling attempts to remove the influence or confounder of body size on physiological measurements, in this case isometric strength of the biceps brachii. A number of studies have now demonstrated the validity of allometric scaling as long as specific statistical criteria are satisfied (3,13). The FAMuSS study provided a unique opportunity to evaluate allometric scaling when applied to a large, relatively heterogeneous (in terms of body size) cohort of previously untrained adult males. Another unique aspect of the present investigation was the availability of muscle CSA, which was determined by MRI. Further, and to the best of our knowledge, we are the first to derive allometric models and test their applicability in scaling muscular strength before and after 12 wk of progressive resistance training.Body mass scaling exponents (pretraining).Body mass is the most commonly used allometric scaling variable (9,10). The body mass-derived scaling exponent for group A was 0.64. Similar body mass allometric parameters for muscle force with adult males were found by Atkins (2) (b = 0.62 in rugby players), Challis (5) (b = 0.64 for Olympic lifting and b = 0.65 for power lifting), Dooman and Vanderburgh (6) (b = 0.57 for bench press and b = 0.60 for squat), Jaric, Ugarkovic, and Kukolj (12) (b = 0.61 in elite athletes), and Jaric, Radosavljevic-Jaric, and Johansson (11) (average slope of six muscles b = 0.67, however, for the elbow flexors resulted in b = 0.97). Additionally, the application of this scaling exponent to group B's pretraining strength data was found to meet all statistical criteria, further suggesting that the model derived from group A was appropriate and was unaffected by 12 wk of resistance training.Of equal importance, the body mass exponent derived presently was also comparable with that of the theory of geometric similarity (1,13). Briefly, the theory of geometric similarity holds that all humans have the same shape and differ only in size. More specifically, limb lengths are proportional to body height (L), and all areas (e.g., CSA) are proportional to L2, whereas all volumes and volume-associated indices, such as body mass, are proportional to L3. Alternatively, and consistent with this theory, any area is proportional to mass2/3. If we accept that muscle strength is proportional to CSA, then strength should be proportional to mass2/3 (13).CSA scaling exponents (pretraining).To the best of our knowledge, this is the first study using CSA as a scaling variable in adults; thus, direct comparisons with the existing literature cannot be made. However, it is interesting to note that the scaling exponent of 0.71 does not support the theory of geometric similarity, which holds that the strength/force produced by a muscle is proportional to its CSA. The theory of geometric similarity assumes that muscle size, as measured by CSA, is the only factor that determines differences or changes in muscular strength. However, as mentioned previously, other factors that may contribute to individual differences in muscle strength are independent of muscle size or CSA. For example, measuring CSA does not reveal the relative distribution of muscle fiber types, which is known to influence muscle force production (16). Further, even in parallel fibers, not all fibers may be included by measuring CSA, given that some may not span the entire length of the muscle. (7). Differences in the distance from the muscle insertion to the center of rotation of the elbow joint and, therefore, the moment arm, may alter the measured tension for a given force-producing unit (19). There is also some evidence that muscles may not be fully activated during a maximal voluntary contraction. For example, Rutherford et al. (20) found evidence of incomplete activation of biceps brachii in 28% of their subjects. The results of the present investigation support the contention that other factors in addition to CSA contribute to individual differences in the force produced during a maximal isometric contraction of the biceps brachii.Application of the CSA model was unsuccessful when the exponent derived from group A was applied to group B, as evidenced by the significant relation between the newly scaled strength measure and CSA (P < 0.05; Fig. 3). Although the model produced a very low r2 value (0.090), it did not completely remove the influence of CSA on muscle force. Visual inspection revealed that the six largest males (in terms of CSA) might have contributed to the inappropriate fit. To examine this premise, we removed six subjects with a baseline biceps CSA > 30 cm2, which resulted in an insignificant model (P > 0.05) with an r2 of 0.004 (Fig. 3). As such, these data suggest that a scaling exponent of 0.71 may not be appropriate for individuals with CSA > 30 cm2. However, and given the small number of subjects with CSA > 30 cm2, this conclusion must be regarded with discretion until it can be more fully investigated.Cross-validation of posttraining measures.To the best of our knowledge, this is also the first study examining the effects of resistance training on the relation between body mass and CSA and isometric strength as determined by allometric scaling models. Application of the body mass model was nonsignificant (P > 0.05) when tested for the independence of the power ratio in posttraining isometric biceps strength measures for groups A and B. However, as was the case with the pretraining data, the application of the CSA model was successful for group A but not group B posttraining data. We again removed subjects with CSA > 30 cm2 and reapplied the model. As with the pretraining data, removal of these individuals resulted in a nonsignificant relationship (P = 0.153) between posttraining CSA and the newly scaled isometric strength (Fig. 5). These data suggest that models using CSA may not be appropriate for individuals with CSA > 30 cm2. The well-established neurological adaptations associated with resistance training are independent of muscle size and represent potential confounders to the models. Further, it has been suggested that strength increases disproportionately and at a lower rate than mass (13). However, the robustness of the models seems to be unaffected by 12 wk of resistance exercise training, forcing us to reject our original hypothesis that such training would alter the fit of allometric scaling models. Further, the fact that the two subgroups studied were significantly different in age (although, given the age range of the subjects, it would not be expected to exert much of an influence) and especially muscular strength also serves to demonstrate the robustness and/or generalizability of these models.FIGURE 5-Test of the independence of the power ratio on posttraining measures for group B CSA after removing individuals with CSA > 30 cm2 (r2 = 0.039, P = 0.153).With regard to the issue of generalizability, it is important to note that the present study evaluated allometric scaling models before and after resistance training, using a small, fusiform muscle mass. This was done with the expectation that it would be easier to control for potential changes in physical activity during the course of the intervention compared with using the larger, weight-bearing muscles of the lower extremities. However, when comparing changes in CSA, it is well known that previous studies using large, pennate muscles such as the knee extensors have not reported a strong relationship between CSA and muscular strength, probably because of the inherent differences in the arrangement of the fibers (4). As such, changes in CSA as measured by MRI may not parallel changes in strength. Therefore, the results of this study cannot be generalized to larger and especially pennate muscles.In conclusion, regression diagnostics applied to allometric scaling models derived from pretraining biceps isometric strength measures determined the model based on body mass to be appropriate and consistent with the theory of geometric similarity. However, cross-validation of the models on pre- and posttraining data showed that the CSA model did not fit when applied to group B subjects. The lack of appropriate fit (P < 0.05) seen for this specific model may have been influenced by individuals with CSA > 30 cm2. 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