Image Anal Stereol 2005;24:35-40 Original Research Paper
ON THE VOLUME FROM PLANAR SECTIONS THROUGH A CURVE
Ximo Gual-Arnau
Department of Mathematics, Campus Riu Sec, s/n, University Jaume I, 12071-Castell´o, Spain e-mail: gual@mat.uji.es (Accepted October 19, 2004)
ABSTRACT
We derive a formula to obtain the volume of a compact domain from planar sections through a curve. From this formula we propose a stereological estimator for the volume which generalizes some known unbiased estimators which use a systematic sampling scheme. Moreover we formulate a Cavalieri’s principle for compact domains is spaces of constant curvature ?.
Keywords: Cavalieri, curve, planar sections, space form, unbiased estimator, volume.
INTRODUCTION
A problem of biomedical interest is how to estimate the volume of an object (bladder, prostate, ...) from planar sections. The best known method to obtain an unbiased estimation of this volume is provided by the Cavalieri’s principle, which is based on parallel equidistant plane sections; that is, orthogonal plane sections through a line. The Cavalieri’s method has been used for scanning techniques such as computed tomography (Pache et al., 1993) or magnetic resonance imaging (Roberts et al., 1994). However, it is difficult to obtain parallel sections of a given object (organ) by ultrasound scan when such obstacles as bones or air are present between the scanner and the organ. In this case, in Watanabe (1982) the volume of an object in R3 is calculated from planar sections through a curve of centroids. From this result, a numerical method has been developed to approximate the volume of an object, whose accuracy has been proved in Treece et al. (1999). However, an unbiased estimator of the volume is not available from this method and only from scanning planes which emanate from a common axis, an unbiased estimator for the volume has been derived, based on the ancient theorem of Pappus of Alexandria (Cruz-Orive and Roberts, 1993).
Here we calculate the volume of a compact domain in Rn from plane sections through a curve c(t). When c(t) is a line and the plane sections are orthogonal to the curve we obtain the Cavalieri method; when c(t) is a curve of centroids of the plane sections we obtain the formula in Watanabe (1982) and when c(t) is a circumference and the plane sections are normal to c(t), these plane sections are rotating planes that contain a vertical axis and we have the result in Cruz-Orive and Roberts (1993). Moreover, from this formula we obtain an unbiased estimator for the volume when systematic sampling through the curve is considered.
Stereological estimators of volume, using a systematic sampling scheme, are usually much more precise than those based on independent sampling. Therefore, systematic sampling is widely used in stereology as sampling scheme. Here we will consider systematic sampling through a curve to obtain unbiased estimators for the volume.
Since our principal aim is to present the mathematical foundations to derive formulas to calculate the volume from plane sections through a curve, in section “Volume of domains in Rn” we derive the general formula for domains in Rn and we obtain some important consequences of this formula. However, in section “Stereological applications” we will concentrate on the applications in volume and area estimation in R3 and R2, respectively, and we add two examples of bodies with sectional planes orthogonal to different curves. In section “Discussion” we generalize some results for compact domains in a space form M?n of dimension n and sectional curvature ?, and we formulate a Cavalieri’s principle in M?n.
VOLUME OF DOMAINS IN R n
Let D be a compact domain in the n-dimensional euclidean space Rn. Let c : I = [0,L] —> Rn be a C°° curve parametrized by its arc-length t (c{t) may be inside D or not). For every t G I, let Pt denote a (n - 1)-dimensional plane (hyperplane) in Rn through c(t) (not necessarily orthogonal to c(t)). Let {E1(t),E2(t),...,En(t)} be a smooth orthonormal frame along c{t) such that, for each t G I, {E2(t),E3(t),...,En(t)}isabasis of Tc{t)Pt = Pt.
From now on we will suppose that there exist subsets Dt cPtn D such that
D = \jDt    and    DtinDt2=0/    for   t 1^t2.  (1) teI
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Gual-Arnau X: On the volume from planar sections through a curve
Under the above hypotheses, each point xt G Dt will be given as
xt = c(t) + Lni(t)Ei(t)
(2)
i=2
Let p(t) = Yni-2Pi(t)Ei(t) = rni-Mt),Ei(t))Ei(t) andN(t) =/i(0/lM(OI; then,
xt = c(t) + rtN(t),                      (3)
where rt =dist{c{t),xt).
Theorem 1. The volume of D is given by
L Voln(D)=   0 Voln-1iDtcitlE1Wdt
-II rt{N{t),—E1{t))atdt,   (4) 0    D                 dt
where ot is the (n -1)-dimensional volume element of Pt.
Proof. Let co be the volume element in Rn and we consider on D the coordinates given by (t,/i2(t),...,/xn(0);then,
Voln(D)=     a>=       /   co(dt,E2(t),...,En(t))otdt.
D             0     D
(5) From the properties of the cross vector product we get
L
Voln(D)=           (dt,E2(t)A---AEn(t))otdt.    (6)
0     D
But
dt
Tt
dt
(c(t)+rt N(t))
c'(t) + rtN'(t),    (7)
so
(dt,E2(t)A---AEn(t)) = (dt,E1(t))
= {c'{t),E1{t)) + rt{N'{t),E1{t))
= (c'(t),E1(t))-rt(N(t),dtE1(t))
(8)
Now, substituting Eq. 8 in Eq. 6 we obtain the result.
In order to obtain some important consequences of the above theorem we will recall the concept of moment and center of masses of a domain.
MOMENTS AND CENTER OF MASSES (ORCENTROID)OFDt
Let r be a in - 2)-dimensional plane (that is, an hyperplane in Pt) through c(t), with unit normal vector field |. T separates Pt - T into two components. Let A be the component t, points to. Let e be the real function defined on Pt by
e(xt)

1       if xt G A -1    ifxtLA
(9)
The moment of Dt respect to T (Mr(Dt)) is given by the integral
MT{Dt) = f L{xt)l{xt)at ,              (10)
Dt
where l is the distance from xt to r.
From elementary trigonometric properties we have
that
Mr(Dt)= I rt($,N(t))ot
Dt
(11)
Hence, c(t) is the centroid of Dt if for every unit vector t, G Tc{t)Pt = Pt , one has thatMr(Dt) = 0.
Now we come back to Theorem 1. Suppose that
d dt
E1(t) = ? ai(t)Ei(t)
(12)
i=1

and let ri denote the hyperplane orthogonal to Ei(t) in Pt ,(i = 2,3,...,n).
Corollary 1. The volume of D is given by
Voln(D)=   0   Voln-1(Dt)(c'(t),E1(t))dt
n   /   ai(t)Mri(Dt)dt.   (13) i=1 0
Proof. Immediate from (12) and Theorem 1.
Now    we    will    consider    some    important consequences of the above corollary.
c(t) is a curve ofcentroids. (See Figs. 1, 2)
Suppose that, for each t G [0,L], c{t) is the center of masses ofDt; then, from Eq. 13,
L Voln(D)=   0 Voln -1(Dt)(c'(0,E1 t>dt
(14)
The above formula for n = 3 is a generalization of the main result in (Watanabe, 1982); however the method used here to obtain it is completely different.
_
d
d
_
_
xt
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Image Anal Stereol 2005;24:35-40
Pt is orthogonal to c. (See Figs. 2, 5)
Suppose that for each t G [0,L], Pt is an orthogonal hyperplane to c(t). Now, we shall consider that the curve c(t) has a Frenet frame {fi(t) = c'(t),f2(t),. ¦ ¦ ,fn(t)}, which is positively oriented and satisfies the Frenet equations:
f[(t) = k(t)f2(t)
fi{t) = -k{t)f\{t) + k2{t)fi{t)
.                                                       (15)
fn-l{t) = -kn-2{t)fn-2{t)+kn-\{t)fn{t) f'n{t) = -kn-i{t)fn-i{t) ,
where kjt) is called the ith curvature of c(t).
Then, from Corollary 1, substituting Ex it) by f (t) we obtain
L
Voln(D) = /   Voln-1(Dt)dt-0
L
 k(t)Mr2(Dt)dt,   (16)
where MT2{Dt) is the moment of Dt respect to the hyperplane in Pt orthogonal to f2(t).
Corollary 2. When c(t) is a straight line in Rn, (ki(t) = 0), or when Mr2(Dt) = 0, we obtain the Cavalieri’s formula
L
Voln(D) =   0 Voln-1(Dt)dt.             (17)
From Eq. 9 and Eq. 10, if D j =DtnA and Dt - = {xt €Dt/xt<L A}, we have that MT2 (Dt) = 0 means
l(xt )?t =       l(xt )?t ,              (18)
Dt+                       D-t
that is, the distance from the centroid of Dt- to ?2 and the distance from the centroid of Dt+ to ?2 coincide.
STEREOLOGICAL APPLICATIONS
In this section we will concentrate on the stereological applications of volume estimation in R3 and area estimation in R2.
Let D be a compact domain in R3 and c(t) a curve which satisfy the conditions imposed in Eq. 1 and such that Pt is the orthogonal plane to c(t). We define
f(t) =Area(Dt)-k(t)M2(Dt ),          (19)
where k(t) is the curvature of c(t) and M2(Dt) is the moment of Dt with respect to the line in Pt given by the binormal vector of the Frenet frame in c(t).
Then, Vol(D) can be expressed as an integral Vol(D) = 0f(t)dt and may be estimated from systematic sampling on [0,L]; that is, let T = L /m and to a point placed uniformly at random in [0, T], then
m-\
V(t0) = TLf(t0 + iT)                (20)
i=0
is an unbiased estimator of Vol(D).
Since c is parametrized by its arc -length it is locally injective. If we suppose that c is globally injective (c has no self-intersections), then c is an isometry and therefore the distance between c(t0 + iT) and c(tQ+(i-1)T) is T; that is, the sets DiT are orthogonal to the curve and placed at equidistant intervals with spacing T along the curve. Moreover, the point a(t0) is placed uniformly at random in [a(0),a(T)].
It is important to note that the torsion of c does not appear in Eq. 20. Properties of this kind of estimators (variance, etc.) have been widely considered in the literature (see, e.g., Gual-Arnau and Cruz-Orive (1998)).
Note that when c(t) is a curve of centroids or when c(t) is a line (k(t) = 0) the measure function f(t) is given by f(t)=Area(Dt).
For a compact domain D in R2, f(t) has the form f(t) = LengthiDt) - k(t)Mc{t) (Dt) ,        (21)
with
Mc(t){Dt)= I L{xt)rtr\t,               (22)
dt
where i}t is the line element of Pt and e(xt) is given by Eq. 9.
Now we will prove that the method developed in Eq. 3.3 of Cruz-Orive and Roberts (1993), to obtain the volume of a domain in R3 from scanning planes which emanate from a common axis Oz, is a particular case of our method when the curve c(t) is a circumference of unit radius placed in a plane orthogonal to Oz.
Using the notation in Cruz-Orive and Roberts (1993) and supposing that Dt lies entirely in the half space A, the moment MT2 (Dt) can be written as:
M2(Dt) = -li(t)Area(Dt) ,             (23)
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Gual-Arnau X: On the volume from planar sections through a curve
where l1+(t) is the distance from ?2 to the centroid of Dt . Then, from Eq. 21 and Eq. 23, we have
2%
2%
2n                             2n
Vol(D)=       Area{Dt)dt+       lUt)Area{Dt)dt 0                                   0
= \    Area(Dt)(1 + l 1t(t))dt = /     lt(t)Area(Dt)dt, o                                         0
(24)
where lL(t) is the distance from Oz to the centroid of
Dt, which is the Eq. 3.3 in Cruz-Orive and Roberts
(1993).
To finish this section we will present two examples with schematic figures of different domains, curves and planes sections.
In the first example the curve c{t) is a curve of centroids which is inside D.
Fig. 1. Domain D.
Fig. 3. Domain with some plane sections which give Dt .
In the second example the curve c(t) is outside D and the scheme is similar to that of Cavalieri.
Fig. 4. Domain D.
Fig. 2. Curve of centroids and portions of planes Pt     Fig. 5. Curve c(t) and planes Pt vertical to the x - y
orthogonal to the curve.
plane.
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Image Anal Stereol 2005;24:35-40
Fig. 6. Planes Pt from another viewpoint.
Fig. 7. Domain with some plane sections and the curve c(t).
DISCUSSION
Stereological estimators of volume using a systematic scheme, based on equidistant cuts through a line, are usually much more precise than those based on independent sampling; because any systematic sample represents the whole material better than most samples obtained by independent sampling. Our estimator Eq. 20 preserves the systematic sampling scheme and offers the possibility to choose an alternative curve to a line whose orthogonal sample planes represent better the whole material.
On the other hand, our approach may be adapted to several clinical areas; for example, in the vessel study from IVUS (Intravascular Ultrasound) images, an artery can be modelled as a tube of non-constant
section (Fig. 3 is a particular case of these tubes), where the catheter trajectory is the curve c.
When the measurement function f(t) given in Eq. 19 is not known, its estimation will depend on the information provided in each practical case.
VOLUME OF DOMAINS IN A SPACE FORM
Now we will extend Eq. 16 for compact domains D in a simply connected space form M\ of dimension n and sectional curvature X, with X / 0 (for X = 0 we have the results in Section 2). Let c : I = [0,L] —> Ml be a C°° curve parametrized by its arc-length t and, for every t G [0,L], let Pt denote a complete totally geodesic hypersurface of Ml through c(t) and orthogonal to the curve c. We suppose that there exists a Frenet frame along c with the same properties as in Eq. 15 but f((t) means, now, the covariant derivative
of fi(t) along c(t), (jtfi(t)).
Under the same assumptions as in Eq. 1, each point xt G D will be given by
xt = expc(0 rtN{t) = expc(t) /x (t) ,         (25)
where
n
ß(t) = Y,ßi(t)Fi(t).                  (26)
i=2
Supposing now that DcU, where U is the image by exp of an open set of the normal bundle of c on which exp is a diffeomorphism, we may consider on U the coordinates:
0(expc(ortN(t)) = (t,/x2(t),...,/x„(t)),       (27)
then
Voln (D)=       /    (dt, %rf2 (0 A • • • A Trf„ (0 ) Oi dt ,
o    d,
(28) where  rr  is  the  parallel  transport from c{t)  to
YN{t){rt) (7N(t) is the geodesic with YN{t)(0) = c(t) and Ym(0)=N(t)).
Now,
^ = dt (expc(t) rN(0) = Yi (rt) ,          (29)
where Yi is the Jacobi field along yN{t) satisfying:
Yi (0) = fi (t),    Y/(0) = JN(t),        (30)
39
Gual-Arnau X: On the volume from planar sections through a curve
which is given by Gray and Miquel (2000)
? Y1(rt) = c?(rt)?r f1(t)+s?(rt)?r    N(t)
dt
(31)
where, for every 1gR,sa:R^R will denote the solution of the equation s" + Xs = 0 with the initial conditions s(0) = 0 and s'(0) = 1; and cx = s'x; i. e.
si(s)
sin(s??)/??,     ?> 0
s,                        ?=0        (32)
sinh(s??)/??,   ?< 0 .
(Note that for ? = 0, Y1(rt) is given by Eq. 7). Finally, from Eq. 28 and Eq. 31 we have that
Vo ln(D)
L( I cx{r)o^]dt 0      dt
+
0  D ^(rt      N t'f 1W
0     Dt                  dt
= I   ( I cx{r)a\dt- f MTJDt)k1(t)dt,   (33) 0        Dt                                0
where
Mr,
(Dt) = / sx{rt)(f2{t),N{t))ot dt
(34)
and Eq. 33 generalizes Eq. 16 for space forms M?n.
In the particular case where the compact domain D is obtained by a motion of a domain D0 through the curve c(t); Eq. 33 has been obtained in Gray and Miquel (2000).
ABOUT THE CAVALIERI PRINCIPLE IN M?n
The Cavalieri principle, already familiar to the ancient Greeks, states that the volumes of two solids in Rn are equal if the areas of the corresponding sections drawn perpendicular to a straight line are equal. This principle is not valid when totally geodesic hypersurfaces perpendicular to a given geodesic are considered; however, from Eq. 33 it is possible to formulate this principle for compact domains in M?n as follows:
Volumes of two compact domains in M?n are equal if the integrals
c?(r)?t Dt
(35)
are equal through a geodesic c(t) in M?n where r denotes the geodesic distance from x to c(t).
(The integral Eq. 35 is equal to Voln-1(Dt) for ? = 0 and it is called the ‘modified volume’ in Choe and Gulliver (1992).)
ACKNOWLEDGEMENTS
The research was supported by the grant BSA2001-0803-C02-02. The author wishes to thank to the referees for helpful suggestions.
REFERENCES
Cruz-Orive LM, Roberts N (1993). Unbiased volume estimation with coaxial sections: an application to the human bladder. J Microsc 170:25-33.
Choe J, Gulliver R (1992). Isoperimetric inequalities on minimal submanifolds of space forms. Manuscripta Math 77:169-89.
Gray A, Miquel V (2000). Pappus type theorems on the volume in space forms. Ann Glob Anal Geom 18:241-54.
Gual-Arnau X, Cruz-Orive LM (1998). Variance prediction under systematic sampling with geometric probes. Adv Appl Probab (SGSA) 30:889-903.
Pache J C, Roberts N, Vock P, Zimmermann A, Cruz-Orive LM (1993). Vertical LM sectioning and parallel CT scanning designs for stereology: application to human lung. J Microsc 170:9-24.
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Watanabe Y (1982). A method for volume estimation by using vector areas and centroids of serial cross sections. IEEE Trans Biomed Eng 29:202-5.
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