Annals of Biomedical Engineering

, Volume 36, Issue 7, pp 1242–1253 | Cite as

An Integer Programming Model for Optimizing Shoulder Rehabilitation

  • Christopher J. Gatti
  • Jason Scibek
  • Oleg Svintsitski
  • James E. Carpenter
  • Richard E. HughesEmail author


Strength restoration is one goal of shoulder rehabilitation following rotator cuff repair surgery. However, the time spent in a physical rehabilitation setting is limited. The objective of this study was to develop a novel mathematical formulation for determining the optimal shoulder rehabilitation exercise protocol to restore normal shoulder strength given a time-constrained rehabilitation session. Strength gain was modeled using a linear dose–response function and biomechanical parameters of the shoulder musculature. Two different objective functions were tested: (1) one based on a least squares support vector machine using healthy and pathologic shoulder strengths (normative objective function), and (2) one which seeks to match the strength of the contralateral shoulder (contralateral objective function). The normative objective function was subject-independent and the optimal protocol consisted of four sets each of adduction and external rotation. The contralateral objective function was subject-specific and the typical optimal protocol consisted of various set combinations of abduction and internal and external rotation. These results are only partially consistent with current practice. Improvement of the current model is dependent on a better understanding of strength training adaptation and shoulder rehabilitation.


Shoulder Rehabilitation Optimization Integer programming Computer model 

Glossary of Terms


physiological cross-sectional area of muscle m, m2

\( A^{0}_{m} \)

initial (prerehabilitation) physiological cross-sectional area of muscle m, m2


muscle hypertrophy for muscle m due to an entire exercise protocol, m2


muscle hypertrophy for muscle m due to an individual exercise l, m2


intercept of strength dose–response, 0.895 dimensionless


slope of strength dose–response, 0.355 sets−1


objective function coefficient for strength measurement n, dimensionless


time required to perform one strength exercise set including rest, 2 min


contralateral shoulder strength (goal) for strength measurement n, N m


muscle hypertrophy factor due to strength exercise l, dimensionless


strength exercise index


muscle index


strength measurement index


muscle moment arm for muscle m for strength exercise l, m


muscle moment arm for muscle m for strength measurement n, m


time budget allotted for the rehabilitation session, min


number of sets of strength exercise l, sets


strength for strength measurement n, N m

\( Y^{0}_{n} \)

prerehabilitation strength for strength measurement n, N m

\( Y^{ * }_{n} \)

postrehabilitation strength for strength measurement n, N m


increase in strength for strength exercise l, N m


increase in strength for strength measurement n, N m


specific tension of muscle, constant for all muscles and postures, N m−2


standard deviation of strength measurements from dose–response, 2.895 N m


length–tension relationship for muscle m for strength exercise l, dimensionless


length–tension relationship for muscle m for strength measurement n, dimensionless


increase in strength for x sets of strength exercise l from dose–response, N m


set of indices for muscles which contribute to strength exercise l based on muscle moment arms


set of indices for muscles which contribute to strength measurement n based on muscle moment arms



The authors would like to thank Marjorie Johnson, Kai-Nan An, and Shawn O’Driscoll of the Mayo Clinic for sharing shoulder strength data with us, and David Gabriel for identifying the necessary meta-analysis paper. We also thank the National Institutes of Health for financial support via grant AR048540.


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Copyright information

© Biomedical Engineering Society 2008

Authors and Affiliations

  • Christopher J. Gatti
    • 1
  • Jason Scibek
    • 2
  • Oleg Svintsitski
    • 1
  • James E. Carpenter
    • 1
  • Richard E. Hughes
    • 1
    Email author
  1. 1.Laboratory for Optimization and Computation in Orthopaedic SurgeryUniversity of MichiganAnn ArborUSA
  2. 2.Department of Athletic TrainingDuquesne UniversityPittsburghUSA

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