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

Abstract

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.

Appendix

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

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

(2)

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.

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

(3)

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

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

(4)

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

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

(5)

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

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

(6)

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.

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

(7)

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

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

(8)

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

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

(9)

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

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

(10)

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

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

(11)

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

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

(12)

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

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

(13)

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

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

(14)

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.