Geometric and kinematic modelling of a human costal slice

J Biomechanicv Printed

in Great

Vol. 24. No. 314. pp. 213-221,

CCJZl-92VO/Vl $3.00+.00 Pergamon Press plc

1991

Britain

GEOMETRIC

AND KINEMATIC MODELLING HUMAN COSTAL SLICE

OF A

P. MINOTTI and C. LEXCELLENT Universite de Franche-Comte,

Laboratoire de Mecanique Appliqued, Associe au C.N.R.S., 25030 Besancon Cedex, France

A~~ct-More and more powerful c~culation methods are being used in the m~el~ation of the human thorax, and considering the progress made in the domain of numerical analysis, this modelization is naturally being oriented toward the utilization of finite element methods. However, thoracic models are usually based on extremely simple geometric hypotheses, due mostly to the lack of dependable experimental data. Hence, the exploitation of sophisticated software is far from optimal. This study is based on experimental observations which allow the capabilities of the current means of calculation to be exploited to a maximum. The objectives of the study are the geometric and kinematic representations of a typical costal slice. A precise topographical measurement, performed by a robot, allows description of the costal geometry. The exploitation of these measurements then allows the identification of the costo-vertebral articuIation. INTRODUCTION traumatisms occur more and more frequently due to the increase in the practice of sports and the marked increase in the number of automobile accidents (Cartier and Verriest, 1983). Research into protective measures and the development of thoracic surgery under artificial ventilation (excess internal pressure created in the lungs) requires the evaluation of the resistive capacities of the human thorax. Several attempts have been made in the past few years to create a thoracic model. Both mechanical models (Kaleps and Von Gierke, 1971; Lobdell, 1973; Free et al., 1976; Viano, 1978) and finite element models (Roberts, 1975,1979; Reddi and Tsai, 1977; Sundaram and Feng, 1977; Chen, 1978) can be found in the literature, and most are based on questionable mechanical and geometrical hypotheses. A critical assessment of these models shows that only the transverse geometry of the ribs (Roberts and Chen, 1971) and the mechanical properties of the materials (bone and costal cartilage) are well known (Yamada, 1973; Got ef al., 1975). The longitudinal geometry of the ribs, even though it has been modelled analytically by Cotte and Leung (1974) and Roberts (1977), is still based on an over-simplified description. The mechanical and kinematical characteristics of the costo-vertebral articulation are poorly known. A rapid appraisal of this state of affairs leads to the observation that a global modelization of the thorax requires the acquisition and processing of a multitude of data, and underscores the necessity of a wellstructured theoretical and experimental approach. The objective of this article is to give a representation of a single half-costal slice, consisting of a rib, the cartilage, the portion, of the sternum and the two corresponding vertebrae. Thoracic

Received in final form 2 October 1990. 213

The obtained experimental results lead to a numerical visualization of the 7th rib and, in particular, to the analytical calculation of the kinematical equivalent system of the costo-vertebral joint. Finally, a synthetic analysis leads to a measurement methodology which allows the identification of the pertinent parameters of the elementary costal slice. DESCRIPTION OF THE COSTAL SLICE MODEL

The proposed model is the result of precise experiments performed by a measurement robot, on a recently removed human costal slice which has been preserved in a condition as near as possible to the physiolo~cal state. It responds to the m~elling objectives which concern the complete representation of the thorax and, in particular, gives a detailed geometric description of the costo-vertebral joint which, until recently, has been little known. The analysis of the experimental measurements shows that this joint is the result of the combination of three elementary joints, each of which is obtained by the conjunction of pairs of geometrically simple surfaces. Referring to Fig. 1, the following joints will be distinguished in this article: (a) the elementary costo-vertebral joint; (b) the costo-transversal joint; (c) the compound costo-vertebral joint. DEFINITIONS OF NOTATION AND REFERENCE SYSTEMS

The

reference

systems

R&9,, x0, yO, to)

and

R$(O, x0, y,, zo) are tied to the vertebra. The refer-

ence system Rb(O’, XL,yb, zb) is tied to the rib. Its origin 0’ coincides with the center 0 of the elementary joint 4po. These reference systems are defined in Fig. 1. The reference system R*(P, x*, y*, z*) is defined in Figs 4 and 6. The point P coincides with the center of

214

P. MINO~~Iand C. LEXCELLENT

Costa1 tuberosity Costa1 head

a : Elementarycosto-vertebral joint "

b : Costo-transversal joint

b c-aub:

COMPOUNDCOSTO-VERTEBBAL JOINT 0

Fig. 1. Compound costo-vertebral joint surfaces.

/

x0

Fig. 3. Convergence of the lines 6, and 8,. Fig. 2. Geometry of the elementary joints _VO,9, and S’*. We- will note: the compound joint Yi u .YZ. The axis x* is equipollent to A,P; z* supports the unit vector N of this joint. The reference system R(0, x, y, z) tied to the rib is defined in Figs 6 and 7. Its origin coincides with the origin of Rb previously defined. The axis z supports OP

the unit vector = of the compound costo-vertebral IlOPll joint. The kinematical and geometric properties of the different elementary joints which appear in this work are summarized in Table 1.

V0=@2, V”) as a pair of rotation and translation velocity vectors associated with the movement of a solid S with respect to a reference system R. By definition: Q=O(S/R) is the angular velocity vector of S with respect to the reference system R; V”= V (0 E S/R) is the translation velocity vector of the point 0 with respect to the reference system R; p, q, r, u, v, w are the respective components of Sl and V” in the reference system R.

215

Modclling of a human costal slice

marked in Table 2. This table expresses the coordinates of the different vectors which appear in the calculations, in three appropriate reference systems: R,, R* and R.

Whatever point P which belongs to the solid S: vp=vO+fl

A OP

where A is the external vector product. The velocity vectors n and V”,i.e. P’, are defined in Table 1 for the ponctual, annular, spherical and revolt& pairs. The other notations used in this paper are sum-

IDENTIFICATION

OF THE JOINT SURFACES

Costo-transversal joint Measurement points have been taken on the vertebrae. Using the notations in Table 2, a surface identification program which solves the non-linear

Y*

Fig. 4. Optimal

reference

system

of the

combination

Fix. 5. Geometrv of the elementarv costo-vertebral

ioint

Table 1. Definition of pairs linked to the identification of the compound costo-vertebral joint

lame of pail

Geometric

form

Relative

degrees

of freedom

Components

in

the refe-

rence system R of fi(C,/C,I

3

and 'i(O E $/C,)

3 rotations

fi(p.q,r)

Ponctual 2 translations

iiO(u,v,o)

3 rctations

@p,q,r)

1 translation

iiO(u,o,o)

w-a".z=o

Annular

,=qLO w=3°.z=o

3 rotations

fi(p,q,r)

Spherical JO(o,o,o>

0 translation

1 rotation

1 ‘iO4

z-w’;

Qp,o,o)

1

I

Revolute P-6

o"(o,o,o)

0 translation

I

I

I

216

P. MINOTB and C. LEXCELLENT

system: (X:-X,0)2+(Y~-Y;)2+(Zp-Z~)2-R2=0

where i= 1, . . . , n. In a least-squares sense, this reveaied that these points describe a spherical surface with radius R at center 0 whose coordinates in the reference system R, are: 0 =(6.98, 39.4, 32.05)?

Fig. 6. Geometry

of the compound

The coordinates of 15 m~urement points which have been taken in the reference system RX of origin 0 are listed in Table 3 (in marble center, (b marble = 1 mm). The same procedure has been performed on the costal tuberosity, showing that the turberosity is also a spherical surface with radius r at center 0’. Coordinates of me~urement points in the reference system R6 tied to the rib are listed in Table 3. The

costo-vertebral joint

Y,U‘9;p,VYEp,.

distances Di and di from the contact points to the mean

Table 2. Notations of the vector coordinates in the ref~nce systems R,, R* and R Reference system

~o,x~,~o,~~

R* (P, x*9 y*, z*1

n

vo

vAi 01

02

N

0,O

O,A,

po

t40

uy b;

a; b;

xi

x;

WP

e

c:

a0 b0 co

xp

v” wo

up vp

x:

4O r”

e zp

c zp

q z;

Y9 2;

p* q* r*

u* u* w*

u: uf w:

1 0 0

a: bf c;

P 4 r

u 0 w

OOP OP

o*

XP YP

C

5

AiO

A,A,

A n2

PA2 A n2

z Lo

XP YP zp

f V0

X;

P

Y: zf

m* n*

L* M* N*

Table 3. Sampled points Costa1 tuberosity

Vertebra i

X

Y

Z

i

X

Y

Z

- 2.499 - 5.095 -6.195 - 5.098 -4.196 - 3.395 - 3.399 -4.599 -6.099 - 6.498 -6.198 -5.300 - 3.396 - 2.599 - 4.496

10.475 8.844 9.741 10.369 10.690 10.964 10.572 10.023 9.331 8.811 8.662 8.762 9.782 10.760 9.19s

4.507 5.509 1.334 1.307 1.308 2.008 3.409 3.407 3.407 3.907 4.507 5.406 5.406 3.815 5.406

(*Of): m) 1 2 3 4 2 7 1 10 11 12 13 14 15

4.034 8.558 5.603 5.121 4.446 6.370 5.394 6.116 6.502 7.528 7.283 6.965 6.215 5.117 5.693

- 6.525 -9.431 - 9.402

5.219 2.432 1.332 2.637

0.022 -0.014 0.004 -0.020

--8.131 9.372 - 8.872 -8.851 - 8.907 - 8.040 -8.121 -8.045 - 8.055 -9.135 -9.135

3.823 3.930 3.822 2.523 1.033 1.134 1.835 3.027 4.334 3.535 2.432

z:: 0:017 0.003 0.029 0.025 0.015 0.016 0.012

- 8.889

R=11.55~10-~m

: 3 4 2 : 9 10 11 12 13 14 1.5

r= 11.13 x lo-‘In

0.044 -0.031 -0.009 -0.002 -0.072 0.022 -0.014 - 0.088 0.026 -0.006 -0.065 -0.051 0.050 0.078 - 0.05s

Modelling of a human costal slice

217

Table 4. Experimental results in R, Costa1head

Vertebra

i

Y

Z

X

Y

Z

iX

14.730 14.879 14.089 13.328 13.614 14.167 14.570 14.861 15.192 15.115 14.769 14.118 13.878 14.180 14.568

31.285 31.285 28.783 27.484 27.719 28.88I 29.192 30.884 31.787 31.678 30.189 28.682 28.695 29.686 30.988

-5.100 -6.197 -6.909 -6.980 - 7.689 -7.753 - 6.665 -8.588 -8.503 -8.168 - 6.697 -5.300

15.703 15.878 15.048 14.894 15.108 14.778 14.354 15.294 15.712 15.877 15.811 15.701

21.892 21.892 23.594 23.891 23.889 24.311 24.300 23.797 23.088 22.590 22.096 22.092

1 2 3 4 5 6 7 8 9 10

X

2 3 4 5 6 7 8 9

10 11 12 13 14 15

- 5.121 - 7.208 - 7.432 - 1.525 -8.740 - 8.621 -8.610 -8.514 -8.424 -9.126 -9.818 - 9.926 -5.406 -5.309 -5.211

sphere justify the spherical approximation. The dehydration of the costal cartilage during the time required for the measurement explains the difference between the two radii. These results show that the costo-transversal joint 49, is a spherical joint, which will be denoted (S), at the center 0. The geometrical properties of this joint have allowed the position of the rib on the vertebra to be fixed at the point 0: the common center of the two mean spheres (0’ coincides with 0). Elementary costo-vertebral joint Table 4 indicates the experimental readings obtained on the two articular facets of this joint. The coordinates of the sampled points on the vertebra and on the costal head are respectively expressed in R, and in the reference system Rb, translated to the center of the mean sphere calculated on the costal turberosity. A numerical visualization of the joint surfaces allows the position of the rib on the vertebra. The result shows that the two solids are in contact with two points A, and A, whose coordinates in the reference system R, are: A, =( -8.59,

14.43, 29.75)=

A, =(-7.66,

15.13, 2X68)?

The Cartesian equations of the tangent planes z1 and n, to these points were calculated from the two sets of coordinates in Table 4. We obtained, in the reference system R,: For the plane n1 : 0.138~~ +0.968y, -0.2342,

+5.82=0

where M,(x,y,z,) is any point in xi. For the plane x2: 0.171x,+0.953y,+0.2582,+19.22=0 where M,(x,y,z,)

is any point on n,.

C,

C,

V,

VI

11 12 13 14 15 16

22.696 21.975 21.503 21.586 21.m 21.828 20.739 22.395 22.223 22.002 21.943 22.558 22.666 22.871 23.040

Y

Z

X

Y

Z

14.602 17.405 17.402 17.402 17.401 17.401 18.519 16.301 16.301 16.301 16.301 14.9cn 14.902 14.902 13.999

9.085 11.891 8.384 9.286 9.884 10.787 10.385 11.288 10.579 9.585 8.188 8.185 9.685 10.788 9.589

16.793 20.094 21.193 20.795 18.994 17.793 17.492 18.493 19.892 21.493 21.493 19.893 18.294 17.095 18.493 19.894

17.308 13.876 16.335 17.275 17.308 17.197 16.622 15.979 15.885 15.502 14.738 15.103 15.459 15.582 13.860 13.172

14.085 17.388 14.735 13.585 13.584 13.586 14.885 15.384 15.385 15.385 16.285 16.185 16.185 16.185 17.485 17.485

Table 5. Identification

n1

n2

0.138 0.968 -0.234

0.171 0.953 0.258

of the compound joint

costo-vertebral

W,

%A,

A,.%

N

-8.59 14.43 29.75

- 7.66 15.13 23.68

0.93 0.7 -6.07

0.985 -0.158 -0.071

The coordinates of the normal vectors n1 and n2 to these tangent planes are listed in Table 5. Kinematical study Global analysis of the compound costo-vertebraljoint. The angular and translation velocities associated with each of the joints Y,, Y, and Y, in Fig. 2 are given by the theory of mechanisms. In addition, the general kinematically equivalent system V” at the point 0 (see definitions of notations) can be defined for the compound costo-vertebral joint: _YOu Ip, u P2. Experience shows that the calculations are reduced if the most constraining characteristic point of the joint is chosen as the center of the reduction. It is therefore convenient to calculate the velocities at the center 0 of the joint Y,. The expression at the point 0 of Y” is: Y” = (4 V’) = (p’, q”, r”, u”, v”, w”)

(1)

where p”, q”, r”, u”, v”, and w” are the respective components of n and V” in the reference system R, tied to

the vertebra. The elementary joints Y,, Y, and Y, introduce, respectively, the following kinematical constraints: Yo(0)+VO=O Y,(A,,

n,)+V”’

*nl =0

Y,(A,,

II~)-+V”~-~~=O.

(2) (3) (4)

P. MINOI-TIand C. LEXCELUZNT

218

(A,, x*, y*, z*) of Fig. 4:

Taking into account (1) and (2), it follows that:

YAi =(Q VA’)=(p*, q*, r.*, 0, v:, w:)

u~=u~=wo~O. Furthermore, the velocities associated with the elementary joints Yi (i = 1,2) can be written at the points Ai as: Y”i = (4 VA’) (5) where: VAi=VD+ArO A f&=A,O A 0. The kinematic constraints joints satisfy the equality:

introduced

(AiO A fz)*o,=(AiO

by these two

A 4)*Q=O.

(6)

The notations in Table 2 allow equation written in scalar form in R0 as:

(6) to be

(cpyp-bpzp)p”+(upzp-cpx~)qO +(bpx;-aPyp)ro=O

(7)

This initial analysis yields the expression of the equivalent system Y ’ associated with the compound costo-vertebral joint: Y-O= (p”, 40, IO,0, 0, 0)

where: u:=O since VA’ an1 =0 with n,(l, 0,O) in the reference system R*. The kinematical constraints introduced by the joint Y, can also be written with respect to this same point: VA2~n2=VA1*n2+(A1A2

Local analysisof the elementary costo-vertebraljoint Geometricalaspects. A detailed study of the experimental results shows that the combination 2, WY* possesses special geometrical properties. In particular, the following calculations show the convergence of the lines S,(A,,n,) and 6,(A,,az). By definition, the shortest distance ~(~,,~~) between the lines 6, and 6, is: (9) W6,,J,)=lA,A,*NI

A n,)*a=O.

(13)

Let (I*, nt*, n*) be the co-ordinates of the vector (A, A, A n2) in the reference system (A,, x*, y*, z*). It then follows, for this same reference system and taking into account (12h that: b~v~+c~w~+Z~p*+m*~~+n*r*=O.

(14)

Experience shows that the point P, i.e. the theoretical locus of the intersection of the lines S, and a,, can be used advantageously in this reduction. In this case, the parameterization in Table 3 allows the following relation to be written, in the reference system R*, of origin P: VA’.a2+(PA2

A a,)*~=b:v:+cfw:+L*p* +A&*q*,+Iv*r*=o

(8)

and shows that p”, q”, r” satisfy, in the general case, the two constraint equations (7). However, it does not allow the precise identification of the characteristics of the costo-vertebral articulation and an examination for eventual local geometric properties is required.

(12)

(1.5)

where: L*=M*=N*=O since PA2 A n2 = 0. As a result, the constraint tion (14) is reduced to the equality: b:vf+c;w:=O.

equa(16)

In addition, the numerical calculation of equation (11) shows that S, and S, are co-planar in theory. It follows that: c$=O 1 sbfv:=O

ov:=O.

Under these conditions, the equivalent system T* at the point P, associated with the compound joint _5?tu 5?s, can be written: YB = (p*, 4*, r*, 0, 0, w?).

where:

(17)

Note that the preceding expression gives the descrip tion of an annular joint (Table 1) (or a ring joint) as: center Taking into account the notation in Table 3, (9) can be rewritten in scalar form in R,: W,,

vector N=

&)=

~‘(X~-X~~+b’(Y~~)+c”{Z~--Z;) (aoz+ b0’ + c01)1/2

P = 1/2(OoP, + O,P,) u1 A n2

IIn, A ~~11

that will be denoted by (P, N) (Fig. 5). *

01)

A numerical calculation performed using the experimental data in Table 5 shows the convergence of the lines 6, and 6,: [d(6,, 6,) = (1.24 x 10m3 m)]. This result allows the kinematical analysis of the combination Y, u .%‘sto be locally reconsidered. Kinematical aspects. The reducing elements, at the point A,, of the equivalent system VA1 associated with 5?, can be written in the reference frame

General kinematicalequivalentsystem The preceding calculations show that the combination {(S)(p)(p)} can be reduced to the combination {(S)(a)} in Fig. 6. The ring joint: ~(P,N)=~l(At,n1)u~2(A2tn2) is such that: VP= PN.

Modelling of a human costal slice It can thus be written in R,: v~=(av~)xo+(b~~~)yv+(c~~~)~*

(19)

219

As a consequence, the equivalent system V’ associated with the compound costo-vertebral articulation can be expressed at the point 0 as:

The center 0 of the spherical joint is conserved. It follows that: VP=V”+Po

A n=n

A OP

(20)

and in R,:

Y” = (0, 0, r, 0, 0, 0).

(24)

This corresponds to a revolute joint having an axis Z erectly localized in the refireme system R. tied to the vertebra.

VP=(qOx,o- r~y~)xo+(rvx~-p~r~)yo f(p”Y;-qox;)zo

where: xi, y$ z$ are the coordinates of OP in R,. Taking into account (19), it can be shown that pot q”, r” are governed by the following system of equations: q%$-rOy$-aOVP=O

20x~-p0z$-4Pv~=0.

i

Q9

pOy~-qOx$-cOvp=O

i

This system is significantly reduced if the compound joint is observed in the reference system: R=(O, x, y. z)

GEOMETRICDI?SCRII’TION

(21)

The preceding kinemati~l study has allowed the precise identification of the costo-vertebral articulation. Moreover, it will be interesting to extend this study, in the context of a complete reconstitution of the thorax, by a synthetic geometrical description. With this objective in mind, the description of the costo-vertebral joint will be made using homogeneous coordinates in the vector space R4. These coordinates allow the following quantities to be expressed as a simple matrix product: The position of the point 0 E A/R,. * The orientation of the coordinate system R with respect to Ro.

such that

l

N(a, 0, c) E x02

Let:.

PEA(O, z=~).

CaP3(Ro,’cx; y; 2; 11’ CW(,) = Ex,Ypql IIT.

and:

This reference system, represented in Fig. 7, allows the constraint equations to be written in the form:

The respective homogeneous coordinates tors 0,P in R, and OP in R become:

of the vec-

qzz,=aP -pz,=o

.

[~~=[~ila~~~

(25)

cvp=o Thus:

VP=@ qzp= -pz,=o

*

p=q=o.

where: [XE Y,O2,” l]* are the homogeneous coordinates of the point 0 in R,. (23) The problem thus consists of identifying the matrix [R], (3 x 3), using a minimal number of parameters pertaining to the orientation of R with respect to Ro. The paramete~zation in Fig. 7 allows [R] to be written in the form: cos er [R]=

sin@,

[

0

-sin 8, cos e2 cos 8, cos

8,

~0~ e2

e2

sin

sin tll sin O2 -c0se,sin&

.

1 (26)

Identifying the coefficients of this matrix with respect to the coordinates of z(2.*, PO, v”) in R,:

10

sin 6, =h

cost?, = -$ sin5,=A Fig. 7. Reduced parameterization of the compound costovertebral joint.

cos e2 = vo

0

220

P. MINOTTIand C. LEXCELLENT

where: Substituting

these relations into equation (26) yields:

--PO

AOVO

--

A

A0

[RI=

10

A /LOVO

--

x

A

0

A

I.

po VO

(27)

The experimental numerical data given in Table 6 allow the calculation of the homogeneous transformation matrix [T]: r

0.698

-0.688 1 6.98 1 -0.671 - 0.275 0

-0.197 -0.192 0.961 0

This matrix contains all the information relative to the orientation of the line A(O,z) in Ro, and gives the detailed geometric description of the costo-vertebral joint. COSTO-CHONDRAL AND CHONDRO-STERNAL JOINTS

The mechanical characteristics of these two joints seem to be better known than those which concern the costo-vertebral joint, and an extensive bibliography

. .,‘..‘.. ;, ‘,

? ,i

.*

:

‘_. ‘,..

..I

already exists on this subject. A detailed study of this work leads to the conclusion that the costo-chondral articulation is a synarthrosis (Taddei, 1982), which does not allow the cartilage to move with respect to the rib. Moreover, the chondro-sternal articulations are often described, depending on the thoracic slice, as amphiarthroses (immobile articulation) (Luschka, 1958), as synchondroses (rigid joints between the two bones connected by a cartilage), or even as diarthroses (mobile articulations, compound or simple, of the spherical type, which allow extended motions) (Morris, 1953). In the thoracic slice studied here, the chondro-sternal articulation is in principle a diarthrose of the spherical type. It should be noted, however, that the experimental conditions under which the thoracic sample was removed did not allow

Table 6 Reference

000

;‘.

OP

z=-

OP WPII

(10e3m)

R,

6.98 39.4 32.05

R

,‘..

;::.;:

0,P

system

‘., I_

.‘.,‘,_ *;I. . ::. .. ‘,

..;. ..

Fig. 8. Numerical visualization of the 7th rib.

- 6.25 26.5 26.77

- 13.25 - 12.9 - 5.28 0 0 19.21

-0.688 -0.671 -0.275

Modelling of a human costal slice

for a verification of this hypothesis, given the advanced age of the subject (around 80 yr old). RIB GEOMETRY

The complete description of the human costal slice requires the identification of the total rib surface area. The acquisitions of the measurement robot PAG (programmable automatic gauge) allowed an extremely detailed sampling of the costal surface to be obtained (800 points). The exploitation of these experimental measurements allow, for the time being, the numerical visualization of the 7th rib under study (Fig. 8). In the near future, this visualization will lead to the elaboration of a three-dimensional finite element mesh for the calculation of a costal slice model. A critical analysis of the mesh technique used should allow the definition of an optimal topographical sampling for each rib, with the objective of reducing the number of experimental acquisitions necessary for the modelization of the different thoracic slices. CONCLUSION

The topographical samples performed on the costal surface, and especially the modelization of the costovertebral articulation, were made using precise experimental acquisitions, and they contribute to a significant improvement in the geometric knowledge of the human costal slice. The obtained results permit a mathematical modelization to be envisaged which is based on reliable and concrete data. The utilization of a finite element calculation will soon allow the various boundary conditions to be simulated (frontal and lateral shocks, pressures, etc.) and also the performance of stress analysis, which is useful in the understanding of rib fractures. This work constitutes only the first step in the modelization of the thoracic skeleton. It represents an apparently new approach, long and tedious, which requires the acquisition and processing of a large quantity of data, but which nevertheless merits to be studied more deeply.

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