Skip to main content

Chest Wall Kinematics Using Triangular Cosserat Point Elements in Healthy and Neuromuscular Subjects


Optoelectronic plethysmography (OEP) is a noninvasive method for assessing lung volume variations and the contributions of different anatomical compartments of the chest wall (CW) through measurements of the motion of markers attached to the CW surface. The present study proposes a new method for analyzing the local CW kinematics from OEP measurements based on the kinematics of triangular Cosserat point elements (TCPEs). 52 reflective markers were placed on the anterior CW to create a mesh of 78 triangles according to an anatomical model. Each triangle was characterized by a TCPE and its kinematics was described using four time-variant scalar TCPE parameters. The total CW volume (\(V_{\text{TCW}}\)) and the contributions of its six compartments were also estimated, using the same markers. The method was evaluated using measurements of ten healthy subjects, nine patients with Pompe disease, and ten patients with Duchenne muscular dystrophy (DMD), during spontaneous breathing (SB) and vital capacity maneuvers (VC) in the supine position. TCPE parameters and compartmental volumes were compared with \(V_{\text{TCW}}\) by computing the phase angles \(\theta\) (for SB) and the correlation r (for VC) between them. Analysis of \(\theta\) and r of the outward translation parameter \(P_{\text{T}}\) of each TCPE revealed that for healthy subjects it provided similar results to those obtained by compartmental volumes, whereas for the neuromuscular patients the TCPE method was capable of detecting local asynchronous and paradoxical movements also in cases where they were undistinguished by volumes. Therefore, the TCPE approach provides additional information to OEP that may enhance its clinical evaluation capabilities.

This is a preview of subscription content, access via your institution.

Figure 1
Figure 2
Figure 3


  1. Aliverti, A., R. L. Dellacà, and A. Pedotti. Transfer impedance of the respiratory system by forced oscillation technique and optoelectronic plethysmography. Ann. Biomed. Eng. 29:71–82, 2001.

    Article  CAS  PubMed  Google Scholar 

  2. Aliverti, A., R. L. Dellacà, P. Pelosi, D. Chiumello, L. Gattinoni, and A. Pedotti. Compartmental analysis of breathing in the supine and prone positions by optoelectronic plethysmography. Ann. Biomed. Eng. 29:60–70, 2001.

    Article  CAS  PubMed  Google Scholar 

  3. Aliverti, A., R. Dellacà, P. Pelosi, D. Chiumello, A. Pedotti, and L. Gattinoni. Optoelectronic plethysmography in intensive care patients. Am. J. Respir. Crit. Care Med. 161:1546–1552, 2000.

    Article  CAS  PubMed  Google Scholar 

  4. Aliverti, A., M. Quaranta, B. Chakrabarti, A. L. P. Albuquerque, and P. M. Calverley. Paradoxical movement of the lower ribcage at rest and during exercise in COPD patients. Eur. Respir. J. 33:49–60, 2009.

    Article  CAS  PubMed  Google Scholar 

  5. Bolton, C. F., F. Grand’maison, A. Parkes, and M. Shkrum. Needle electromyography of the diaphragm. Muscle Nerve 15:678–681, 1992.

    Article  CAS  PubMed  Google Scholar 

  6. Boudarham, J., D. Pradon, H. Prigent, L. Falaize, M. C. Durand, H. Meric, M. Petitjean, and F. Lofaso. Optoelectronic plethysmography as an alternative method for the diagnosis of unilateral diaphragmatic weakness. Chest 144:887–895, 2013.

    Article  PubMed  Google Scholar 

  7. Boudarham, J., D. Pradon, H. Prigent, I. Vaugier, F. Barbot, N. Letilly, L. Falaize, D. Orlikowski, M. Petitjean, and F. Lofaso. Optoelectronic vital capacity measurement for restrictive diseases. Respir. Care 58:633–638, 2013.

    Article  PubMed  Google Scholar 

  8. Cala, S. J., C. M. Kenyon, G. Ferrigno, P. Carnevali, A. Aliverti, A. Pedotti, P. T. Macklem, and D. F. Rochester. Chest wall and lung volume estimation by optical reflectance motion analysis. J. Appl. Physiol. (Bethesda, Md. 1985) 81:2680–2689, 1996.

    CAS  Google Scholar 

  9. Charnwood Dynamics Limited. CODA cx1 User Guide., 2008.

  10. Dellacà, R. L., A. Aliverti, K. R. Lutchen, and A. Pedotti. Spatial distribution of human respiratory system transfer impedance. Ann. Biomed. Eng. 31:121–131, 2003.

    Article  PubMed  Google Scholar 

  11. Dellaca, R. L., M. L. Ventura, E. A. Zannin, M. Natile, A. Pedotti, and P. Tagliabue. Measurement of total and compartmental lung volume changes in newborns by optoelectronic plethysmography. Pediatr. Res. 67:11–16, 2010.

    Article  PubMed  Google Scholar 

  12. Ferrigno, G., P. Carnevali, A. Aliverti, F. Molteni, G. Beulcke, and A. Pedotti. Three-dimensional optical analysis of chest wall motion. J. Appl. Physiol. (Bethesda, Md. 1985) 77:1224–1231, 1994.

    CAS  Google Scholar 

  13. Fredberg, J. J., and D. Stamenovic. On the imperfect elasticity of lung tissue. J. Appl. Physiol. 67:2408–2419, 1989.

    CAS  PubMed  Google Scholar 

  14. Hahn, A., J. R. Bach, A. Delaubier, A. Renardel-Irani, C. Guillou, and Y. Rideau. Clinical implications of maximal respiratory pressure determinations for individuals with Duchenne muscular dystrophy. Arch. Phys. Med. Rehabil. 78:1–6, 1997.

    Article  CAS  PubMed  Google Scholar 

  15. Harte, J. M., C. K. Golby, J. Acosta, E. F. Nash, E. Kiraci, M. A. Williams, T. N. Arvanitis, and B. Naidu. Chest wall motion analysis in healthy volunteers and adults with cystic fibrosis using a novel Kinect-based motion tracking system. Med. Biol. Eng. Comput. 54:1631–1640, 2016.

    Article  PubMed  PubMed Central  Google Scholar 

  16. Howard, R. S., C. M. Wiles, G. T. Spencer, R. S. Howard, C. M. Wiles, N. P. Hirsch, and C. M. Wiles. Respiratory involvement in primary muscle disorders: assessment and management. Qjm 86:175–189, 1993.

    CAS  PubMed  Google Scholar 

  17. Kiernan, D., and M. Walsh. R. O’sullivan, and D. Fitzgerald. Reliability of the CODA cx1 motion analyser for 3-dimensional gait analysis. Gait Posture 39:S99–S100, 2014.

    Article  Google Scholar 

  18. Lanini, B., M. Masolini, R. Bianchi, B. Binazzi, I. Romagnoli, F. Gigliotti, and G. Scano. Chest wall kinematics during voluntary cough in neuromuscular patients. Respir. Physiol. Neurobiol. 161:62–68, 2008.

    Article  PubMed  Google Scholar 

  19. Lessard, M. R., F. Lofaso, and L. Brochard. Expiratory muscle activity increases intrinsic positive end-expiratory pressure independently of dynamic hyperinflation in mechanically ventilated patients. Am. J. Respir. Crit. Care Med. 151:562–569, 1995.

    Article  CAS  PubMed  Google Scholar 

  20. LoMauro, A., M. G. D’Angelo, M. Romei, F. Motta, D. Colombo, G. P. Comi, A. Pedotti, E. Marchi, A. C. Turconi, N. Bresolin, and A. Aliverti. Abdominal volume contribution to tidal volume as an early indicator of respiratory impairment in Duchenne muscular dystrophy. Eur. Respir. J. 35:1118–1125, 2010.

    Article  CAS  Google Scholar 

  21. LoMauro, A., S. Pochintesta, M. Romei, M. G. D’Angelo, A. Pedotti, A. C. Turconi, and A. Aliverti. Rib cage deformities alter respiratory muscle action and chest wall function in patients with severe Osteogenesis imperfecta. PLoS One 7:e35965, 2012.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  22. Malvern, L. Introduction to the Mechanics of a Continuous Medium. New Jersey: Prentice Hall, 1969.

    Google Scholar 

  23. Meric, H., L. Falaize, D. Pradon, D. Orlikowski, H. Prigent, and F. Lofaso. 3D analysis of the chest wall motion for monitoring late-onset Pompe disease patients. Neuromuscul. Disord. 26:146–152, 2016.

    Article  PubMed  Google Scholar 

  24. Meric, H., F. Lofaso, L. Falaize, and D. Pradon. Comparison of two methods to compute respiratory volumes using optoelectronic plethysmography. J. Appl. Biomech. 32:221–226, 2016.

    Article  PubMed  Google Scholar 

  25. Miller, M. R., J. Hankinson, V. Brusasco, F. Burgos, R. Casaburi, A. Coates, R. Crapo, P. Enright, C. P. M. van der Grinten, P. Gustafsson, R. Jensen, D. C. Johnson, N. MacIntyre, R. McKay, D. Navajas, O. F. Pedersen, R. Pellegrino, G. Viegi, and J. Wanger. Standardisation of spirometry. Eur. Respir. J. 26:319–338, 2005.

    Article  CAS  PubMed  Google Scholar 

  26. Pellegrini, N., P. Laforet, D. Orlikowski, M. Pellegrini, C. Caillaud, B. Eymard, J. C. Raphael, and F. Lofaso. Respiratory insufficiency and limb muscle weakness in adults with Pompe’s disease. Eur. Respir. J. 26:1024–1031, 2005.

    Article  CAS  PubMed  Google Scholar 

  27. Povšič, K., M. Jezeršek, and J. Možina. Real-time 3D visualization of the thoraco-abdominal surface during breathing with body movement and deformation extraction. Physiol. Meas. 36:1497–1516, 2015.

    Article  PubMed  Google Scholar 

  28. Redlinger, R. E., R. E. Kelly, D. Nuss, M. Goretsky, M. A. Kuhn, K. Sullivan, A. E. Wootton, A. Ebel, and R. J. Obermeyer. Regional chest wall motion dysfunction in patients with pectus excavatum demonstrated via optoelectronic plethysmography. J. Pediatr. Surg. 46:1172–1176, 2011.

    Article  PubMed  Google Scholar 

  29. Remiche, G., A. L. Mauro, P. Tarsia, D. Ronchi, A. Bordoni, F. Magri, G. P. Comi, A. Aliverti, and M. G. D’Angelo. Postural effects on lung and chest wall volumes in late onset type II glycogenosis patients. Respir. Physiol. Neurobiol. 186(3):308–314, 2013.

    Article  PubMed  Google Scholar 

  30. Romei, M., M. G. D’Angelo, A. Lomauro, S. Gandossini, S. Bonato, E. Brighina, E. Marchi, G. P. Comi, A. C. Turconi, A. Pedotti, N. Bresolin, and A. Aliverti. Low abdominal contribution to breathing as daytime predictor of nocturnal desaturation in adolescents and young adults with Duchenne Muscular Dystrophy. Respir. Med. 106:276–283, 2012.

    Article  CAS  PubMed  Google Scholar 

  31. Rubin, M. B. A simplified implicit Newmark integration scheme for finite rotations. Comput. Math. Appl. 53:219–231, 2007.

    Article  Google Scholar 

  32. Solav, D., V. Camomilla, A. Cereatti, A. Barré, K. Aminian, and A. Wolf. Bone orientation and position estimation errors using Cosserat point elements and least squares methods: application to gait. J. Biomech. 2017. doi:10.1016/j.jbiomech.2017.01.026.

    PubMed  Google Scholar 

  33. Solav, D., M. B. Rubin, A. Cereatti, V. Camomilla, and A. Wolf. Bone pose estimation in the presence of soft tissue artifact using triangular cosserat point elements. Ann. Biomed. Eng. 44:1181–1190, 2016.

    Article  PubMed  Google Scholar 

  34. Solav, D., M. B. Rubin, and A. Wolf. Soft tissue artifact compensation using triangular cosserat point elements (TCPEs). Int. J. Eng. Sci. 85:1–9, 2014.

    Article  Google Scholar 

  35. van der Ploeg, A. T. Monitoring of pulmonary function in Pompe disease: a muscle disease with new therapeutic perspectives. Eur. Respir. J. 26:984–985, 2005.

    Article  PubMed  Google Scholar 

  36. Vogiatzis, I., A. Aliverti, S. Golemati, O. Georgiadou, A. LoMauro, E. Kosmas, E. Kastanakis, and C. Roussos. Respiratory kinematics by optoelectronic plethysmography during exercise in men and women. Eur. J. Appl. Physiol. 93:581–587, 2005.

    Article  PubMed  Google Scholar 

  37. Zoumot, Z., A. LoMauro, A. Aliverti, C. Nelson, S. Ward, S. Jordan, M. I. Polkey, P. L. Shah, and N. S. Hopkinson. Lung volume reduction in emphysema improves chest wall asynchrony. Chest 148:185, 2015.

    Article  PubMed  PubMed Central  Google Scholar 

Download references


Dana Solav was partially supported by the Aharon and Ephraim Katzir Study Grant of the Batsheva de Rothschild Fund. MB Rubin was partially supported by his Gerard Swope Chair in Mechanics.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Dana Solav.

Additional information

Associate Editor Merryn Tawhai oversaw the review of this article.

Electronic supplementary material

Below is the link to the electronic supplementary material.


Supplementary material 1 (TIFF 3618 kb)


Supplementary material 2 (TIFF 3618 kb)


Supplementary material 3 (PDF 33 kb)

Supplementary material 4 (MP4 7382 kb)

supplementary material 5 (MP4 3417 kb)



The present configuration of a given TCPE is denoted by the position vectors \(\{ \varvec{x}_{1} ,\varvec{x}_{2} ,\varvec{x}_{3} \}\) of its vertices. The values of these position vectors in the reference configuration are denoted by \(\{ \varvec{X}_{1} ,\varvec{X}_{2} ,\varvec{X}_{3} \}\), with \(\varvec{X}_{i} = \varvec{x}_{i} (t = t_{0} ){ ; }i = 1,2,3\). Furthermore, the centroids of the TCPE \(\left\{ {\bar{\varvec{X}},\bar{\varvec{x}}\,} \right\}\) in its reference and present configurations, respectively, and the translation vector \(\bar{\varvec{t}}\) between these two configurations are given by

$$\bar{\varvec{X}} = \frac{1}{3}\left( {\varvec{X}_{1} + \varvec{X}_{2} + \varvec{X}_{3} } \right),\,\,\,\bar{\varvec{x}} = \frac{1}{3}\left( {\varvec{x}_{1} + \varvec{x}_{2} + \varvec{x}_{3} } \right),\,\,\,\bar{\varvec{t}} = \bar{\varvec{x}} - \bar{\varvec{X}}.$$

Moreover, the reference and present configurations are characterized by the director vectors \(\{ \varvec{D}_{1} ,\varvec{D}_{2} \varvec{,D}_{3} \}\) and \(\{ \varvec{d}_{1} ,\varvec{d}_{2} \varvec{,d}_{3} \}\), respectively, with \(\varvec{D}_{3}\) and \(\varvec{d}_{3}\) being unit vectors normal to the plane of the TCPE, and \(A\) being the TCPE’s reference area.

$$\begin{aligned} \varvec{d}_{1} = \varvec{x}_{2} - \varvec{x}_{1} , { }\,\varvec{d}_{2} = \varvec{x}_{3} - \varvec{x}_{1} ,\, \, \varvec{d}_{3} = \frac{{\varvec{d}_{1} \times \varvec{d}_{2} }}{{\left| {\varvec{d}_{1} \times \varvec{d}_{2} } \right|}} \hfill \\ \varvec{D}_{i} = \varvec{d}_{\varvec{i}} (t = t_{0} );\,\,\,i = 1,2,3 \hfill \\ \varvec{D}_{1} \times \varvec{D}_{2} \cdot \varvec{D}_{3} = 2A\,. \hfill \\ \end{aligned}$$

The vertices and director vectors of each TCPE are ordered such that for each TCPE the director vector \(\varvec{d}_{3}\) points outwards from the CW surface. Then, the reference reciprocal vectors \(\{ \varvec{D}^{1} ,\varvec{D}^{2} \varvec{,D}^{3} \}\) are defined by

$$\varvec{D}^{1} = \frac{1}{2A}(\varvec{D}_{2} \times \varvec{D}_{3} );\,\,\,\varvec{D}^{2} = \frac{1}{2A}(\varvec{D}_{3} \times \varvec{D}_{1} );\,\,\,\varvec{D}^{3} = \frac{1}{2A}(\varvec{D}_{1} \times \varvec{D}_{2} ) = \varvec{D}_{3} \,,$$

such that \(\varvec{D}^{i} \cdot \varvec{D}_{j} = \delta_{j}^{i}\), where \(\delta_{j}^{i}\) is the Kronecker delta symbol. In addition, the deformation gradient tensor \(\varvec{F}\) of the TCPE is defined by

$$\varvec{F} = \, \sum\limits_{i = 1}^{3} {} \varvec{d}_{i} \otimes \varvec{D}^{i} { ,}$$

where \(\otimes\) is the tensor product (outer product) operator.

It follows that the transformation of each TCPE from its reference configuration to a present configuration can be described by a translation vector \(\bar{\varvec{t}}\) of the TCPE’s centroid, and the deformation gradient tensor \(\varvec{F}\), such that each point in the TCPE is transformed by

$$\varvec{x}_{i} = \bar{\varvec{X}} + \bar{\varvec{t}} + \varvec{F}\left( {\varvec{X}_{i} - \bar{\varvec{X}}} \right)\,.$$

The translation vector \(\bar{\varvec{t}}\) of the TCPE’s centroid and the unit vector \(\varvec{d}_{3}\) normal to the TCPE plane are used to define a scalar parameter \(P_{T}\) describing the outward normal translation of the TCPE.

$$P_{T} = \bar{\varvec{t}} \cdot \varvec{d}_{3} \,.$$

Furthermore, the deformation gradient \(\varvec{F}\) is decomposed using the polar decomposition theorem22 to determine the unique proper orthogonal rotation tensor \(\varvec{R}\) and the unique positive definite symmetric stretch tensor \(\varvec{N}\) by

$$\begin{aligned} \varvec{F} = \varvec{NR}{ ; [}\varvec{R}_{{}}^{T} \varvec{R} = \varvec{I} , { }\det \left( \varvec{R} \right) = 1 , { }\varvec{N}_{{}}^{T} = \varvec{N}] \hfill \\ \varvec{B} = \varvec{FF}^{T} = \varvec{NR}\left( {\varvec{NR}} \right)^{T} = \varvec{NRR}^{T} \varvec{N}^{T} = \varvec{NN}^{T} \Rightarrow \varvec{N} = \varvec{B}^{1/2} \hfill \\ \varvec{R} = \varvec{N}^{ - 1} \varvec{F} = \left( {\varvec{FF}^{T} } \right)^{ - 1/2} \varvec{F}\,. \hfill \\ \end{aligned}$$

Then, \(\varvec{R}\) is used to define a scalar parameter \(P_{R}\) describing the TCPE rotation angle.31

$$P_{\text{R}} = \cos^{ - 1} \left[ {\frac{{{\text{trace}}(\varvec{R}) - 1}}{2}} \right].$$

Since the deformations of the TCPE are two-dimensional it follows that the left Cauchy-Green deformation tensor \(\varvec{B}\) has the spectral form

$$\varvec{B} = \varvec{FF}^{T} = \lambda_{1}^{2} \varvec{p}_{1} \otimes \varvec{p}_{1} + \lambda_{2}^{2} \varvec{p}_{2} \otimes \varvec{p}_{2} + \varvec{d}_{3} \otimes \varvec{d}_{3} ,$$

where \(\left\{ {\varvec{p}_{1} ,\varvec{p}_{2} } \right\}\) are orthogonal unit eigenvectors in the plane of the TCPE. Furthermore, it follows that the dilatation \(J\) is given by

$$J = \det (\varvec{F}) = \sqrt {\det (\varvec{B})} = \lambda_{1} \lambda_{2} = a/A,$$

where \(a\) is the TCPE’s present area and \(A\) is the reference area, such that a scalar parameter \(P_{\text{A}}\) describing the area change is given by

$$P_{\text{A}} = J - 1.$$

Moreover, it follows from (10) and (11) that

$${\text{trace}}(\varvec{B}) - 1 = \lambda_{1}^{2} + \lambda_{2}^{2} = \lambda_{1} \lambda_{2} \left( {\frac{{\lambda_{1} }}{{\lambda_{2} }} + \frac{{\lambda_{2} }}{{\lambda_{1} }}} \right) = J\left( {\lambda^{2} + \frac{1}{{\lambda^{2} }}} \right).$$

where \(\lambda^{2} = {{\lambda_{1} } \mathord{\left/ {\vphantom {{\lambda_{1} } {\lambda_{2} }}} \right. \kern-0pt} {\lambda_{2} }}\). Using this expression, the scalar parameter \(P_{\text{S}}\) describing the distortional deformation (shape change with no area change) is defined by

$$P_{\text{S}} = J^{ - 1} \left[ {{\text{trace}}(\varvec{B}) - 1} \right] - 2 = \lambda^{2} + \frac{1}{{\lambda^{2} }} - 2.$$

It is noted that the parameters \(\left\{ {P_{\text{R}} ,P_{\text{S}} } \right\}\) are positive definite, while \(\left\{ {P_{\text{T}} ,P_{\text{A}} } \right\}\) can have positive or negative values. Moreover, it should be noted that all four TCPE parameters are independent on the size of the TCPE.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Solav, D., Meric, H., Rubin, M.B. et al. Chest Wall Kinematics Using Triangular Cosserat Point Elements in Healthy and Neuromuscular Subjects. Ann Biomed Eng 45, 1963–1973 (2017).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Breathing pattern
  • Chest wall kinematics
  • Neuromuscular disorder
  • Optoelectronic plethysmography
  • Respiratory motion