Rapid muscle volume prediction using anthropometric measurements and population-derived statistical models

Abstract

Knowledge of subject-specific muscle volumes may be used as surrogates for evaluating muscle strength and power generated by ‘fat-free’ muscle mass. This study presents population-based statistical learning approaches for predicting ‘fat-free’ muscle volume from known anthropometric measurements. Using computed tomography (CT) imaging data to obtain lower-limb muscle volumes from 50 men and women, this study evaluated six statistical learning methods for predicting muscle volumes from anthropometric measurements: (i) stepwise regression, (ii) linear support vector machine (SVM), (iii) 2nd-order polynomial SVM, (iv) linear partial least squares regression (PLSR), (v) quadratic PLSR, and (vi) 3rd-order spline fit PLSR. These techniques have successfully been demonstrated in bioengineering applications and freely available in open-source toolkits. Analysis revealed that separating a general population into sexes and/or cohorts based on adipose level may improve prediction accuracies. The most important measures that statistically influence muscle volume predictions were shank girth, followed by sex and finally leg length, as identified using stepwise regression. SVM learning predicted muscle volume with an accuracy of 85 ± 4% when using linear interpolation, but performed poorly with an accuracy of 59 ± 6% using polynomial interpolation. The simpler linear PLSR exhibited muscle volume prediction accuracy of 87 ± 2%, while quadratic PLSR was slightly reduced at 82 ± 3%. For the spline fit PLSR, high accuracy was observed on the training data set (~ 99%) but over-fitting (a drawback of high-interpolation methods) resulted in erroneous predictions on testing data, and hence, the model was deemed unsuitable. In conclusion, use of linear PLSR models with variables of sex, leg length, and shank girth is a useful tool for predicting ‘fat-free’ muscle volume.

Appendix

1.1 Stepwise regression

Stepwise linear regression was used to determine the optimal subset of anthropometric parameters needed to predict muscle volume. The stepwise regression model used in this study has a forward selection approach, which starts with no variables followed by adding one variable at a time to test and compare criterion for any improvements to the model. The criterion used to assess improvement in this study was the Akaike information criteria (AIC) fitness criterion (James et al. 2013) defined by

$${\text{AIC }} = 2k {-} 2\ln \left( L \right)$$

(1)

where k is the number of estimated parameters in the model and L is the maximum value of the likelihood function for the model. This process is repeated until no improvements to the model are observed and parameters are ranked according to significance.

1.2 SVM

Support vector machine is a supervised machine learning method used for classification and regression of data. With a training data set, categories are defined and a function is used to design a hyper-plane (multidimensional plane) that splits and classifies data into categorical classes, denoted C. This non-probabilistic linear classifier, C, defines minimised weight vectors, w, of the function

$$\mathop {\hbox{min} }\limits_{{w,b,\zeta ,\zeta^{*} }} \frac{1}{2}w^{T} w + C\mathop \sum \limits_{i = 1}^{n} \left( {\zeta_{i} + \zeta_{i}^{*} } \right)$$

(2)

that maximises the difference from both classes (Smola and Schölkopf 2004), where ζ_i is the distance between data point, i, and the plane (known as a support vector).Therefore, new data are mapped onto the functional space and defined as the class based on its relative position to the hyper-plane and the training data set categories. In this study, n is 50 and the classes were shank girth, sex, and leg length.

A polynomial SVM differs in that it uses a polynomial function for the internal linear kernel. For this study, we used a 2nd-order polynomial (quadratic) as the defining function, which results in a nonlinear weighting and scaling factors. The linear and polynomial SVM used in this study was from the standard python library (http://scikit-learn.org/stable/modules/svm.html).

1.3 Linear PLSR

Partial least squares regression (PLSR) (Wold et al. 1984) was used to model the relationships between total muscle volume and anthropometric measurements, thereby creating a total muscle volume predictor. The two fundamental equations in PLSR are the predictor matrix (X) and the response matrix (Y) given by

$$X_{NM} = T_{NL} P_{ML}^{T} + E_{NM} ,$$

(3)

and

$$Y_{NP} = U_{NL} Q_{PL}^{T} + F_{NP} ,$$

(4)

where N is the number of data sets (50 in this study), M is the number of predictor variables (being shank girth, sex, and leg length), P is the number of response variables (muscle volume), and L is the number of principal components. T and U are the projection matrices (also known as scores), and P and Q are the transposed orthogonal loading matrices (where the rows are created from eigenvectors or principal components), and E and F are the error or residual terms. The score vectors are related using a linear function

$$\varvec{U} = f\left( \varvec{T} \right) + \varvec{H}$$

(5)

where H is the vector of residuals. For quadratic PLSR, a nonlinear relationship can be used to relate U and T of the form

$$\varvec{U} = c_{0} + c_{1} \varvec{T} + c_{2} \varvec{T}^{2} + \varvec{H}$$

(6)

where c_i represents quadratic fitting constants. For 3rd-order spline PLSR, the relationship between the scores is given using a cubic spline function

$$\varvec{U} = S^{3} \left( \varvec{T} \right) + \varvec{H}$$

(7)