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The growth of the organs of human embryo is changing significantly over a short period of time in the mother body. The shape of the human organs is organic and has many folds that are difficult to model or animate with conventional techniques. Convolution surface and function representation are a good choice in modelling such organs as human embryo stomach and brain. Two approaches are proposed for animating the organ growth: First, uses a simple line segment skeleton demonstrated on a stomach model and the other method uses a tubular skeleton calculated automatically from a 2D object outline. The growth speed varies with the position within the organ and thus the model is divided into multiple geometric primitives that are later glued by a blending operation. Animation of both the embryo stomach and brain organs is shown.
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Heterogeneous Modeling of Biological Organs
and Organ Growth
Roman ˇ
Durikoviˇc12, Silvester Czanner3,J´ulius Parulek2,5, and Miloˇsˇ
Sr´amek2,4
1University of Saint Cyril and Metod, Trnava, Slovakia,
2Faculty of Mathematics, Physics and Informatics, Comenius University, Slovakia
roman.durikovic@fmph.uniba.sk
3Warwick Manufacturing Group, University of Warwick, UK
S.Czanner@warwick.ac.uk
4Austrian Academy of Sciences, Austria
5Institute of Molecular Physiology and Genetics, Slovak Academy of Sciences,
Slovakia
Abstract. The growth of the organs of human embryo is changing sig-
nificantly over a short period of time in the mother body. The shape
of the human organs is organic and has many folds that are difficult
to model or animate with conventional techniques. Convolution surface
and function representation are a good choice in modelling such organs
as human embryo stomach and brain. Two approaches are proposed for
animating the organ growth: First, uses a simple line segment skeleton
demonstrated on a stomach model and the other method uses a tubular
skeleton calculated automatically from a 2D object outline. The growth
speed varies with the position within the organ and thus the model is
divided into multiple geometric primitives that are later glued by a blend-
ing operation. Animation of both the embryo stomach and brain organs
is shown.
1 Introduction and Previous Works
The purpose of this manuscript is to model the outer shape and the shape
metamorphosis during the growth of some human embryo organs, particularly
brain and digestive system. Popular methods like 3D shape reconstruction from
Computer Tomography (CT) sections or ultrasound data can not be used for this
type of modelling because the resolution of the devices used in those methods
are much higher comparing to the size of human embryo. Four weeks old embryo
is approximately 3 mm tall while the CT resolution is 1 mm giving us only three
sections for a reconstruction process. Usually, the microscopic cross-sections are
used to reconstruct the polygonal representation of an embryo, which is exact but
complicated process. In case of such destructive approach often a mouse embryo
is used instead of the human embryo [1]. To control the shape metamorphosis
between two mesh objects become a problem when they have different topology
and geometry. To create the realistically looking human organ models and to
generate the animations demonstrating the growth process requires a proposed
methodology.
2
Growing human organs can be described as dynamical systems with a dynam-
ical structure [2]. In such systems not only the values of variables characterizing
system components, but also the number of components and the connections
between them, may change over time. There is a need to construct a mathe-
matical description of a system. The model can then be used for simulation or
optimization. All models are predictive in the meaning that simulation output
predict what could occur in the real world where the system is operating. Numer-
ous interdisciplinary research initiatives are generating excellent research results
with regards to modelling, simulation and visualization of human anatomy and
physiology. These research initiatives focus on different (biological) levels [3];
molecular and cellular levels, tissue [4] and organ levels [5], [6], [7], and system
and human (organism) levels.
The simulation of human organs growing can be seen as an imitation of the
reality for studying the effect of changing parameters in a model as a means
of preparing a decision or predicting experiment results. Since the human body
is mainly made up of a variety of organs, the medical consequence of organ
modelling is very important, ranging from heart surgery to minimally-invasive
surgery.
In the area of modelling and simulation of human organs many research
works have been carried out. One class of reconstruction methodologies uses
implicit functions. They allow extracting an iso-surface either by a procedural
method, or skeletal implicit surfaces (surfaces generated by a field function and a
skeleton). Amrani introduced a method using the skeleton-based implicit surface
for implicit reconstruction [7].
Another construction method is presented by Leymarie. His approach is
based on propagation along the scaffold from initial sources of flow as a means to
efficiently construct it. The detection of these sources can be shown to be reduced
to considering pairs of input points, which then constitutes the computational
bottleneck of this method [8].
A semi-automatic reconstruction method that can be used on noisy scattered
points of a medical organ is presented by Tsingos [5]. The method is based on
implicit iso-surfaces generated by skeletons that provide a smooth and compact
representation of the surface. The user can guide the reconstruction by initializ-
ing some skeletons and their reconstruction windows, thus taking benefits of his
initial knowledge of the data.
The method developed by Attali et al. [9] computes the Voronoi graph of the
point set to build the skeleton of the object and reconstructs its surface. The
surface thus reconstructed has only fixed topological type.
Even though the convolution surfaces provide nice blending between several
parts of organs, the control of the blend shape is very limited. The functional
representation [10] is a tool that generalize the set theoretic operations and
generates full range of shapes from simple object union to smooth blend. The
animation of such surfaces follow the changes smoothly, even if the topology
changes. Because of this advantage the functional representation become a pop-
3
ular tool where the shapes to be modelled are from the natural world. We explain
here our modelling experience that can be useful for others.
2 Shape Representation for Growth Animation
The polygonal models can not capture the development of such complex process
as the growth of the digestive system. So far, we have created the skeletons of
different physiological parts, we need to blend them together to get the smooth
shapes. Even though, the convolution surfaces provide nice blending between
several parts of organs, the control of the blend shape is very limited. The func-
tional representation is a tool that generalizes the set theoretical operations and
generates full range of shapes from simple object union to smooth blending. The
animation of such surfaces follow the changes smoothly, even if the topology
changes. Because of this advantage the functional representation is an excel-
lent tool when the shapes to be modelled are from the natural world. We discuss
herein the shape modeling based on skeleton calculated from dynamic simulation
and L-system growth.
An implicit surface is defined by an isosurface of some potential field F:
R3→Rat threshold level T:S={p∈R
3:F(p)T=0}. The function F(p)
is also called an implicit function. A convolution surface is implicitly defined by
a potential function Fobtained via convolution operator between a kernel and
all the points of a skeleton. The convolution surface thus obtained is a smoothed
skeleton. The skeleton is a collection of geometric primitives such as point, line
segment, arc and plane that outline the structure of an object being modelled.
Convolution surface build from complex skeletons can be evaluated individually
by adding the local potentials for each primitive, because convolution operator
is linear.3Let us have Nskeleton primitives the above statements can be written
as the following modelling equation in an implicit form:
N
i=1
Fi(x1,x
2,x
3)T=0,(1)
where Fiis the source potential of i-th skeleton primitive and Tis the iso-
potential threshold value.
3 Function Representation
Let us consider closed subsets of n-dimensional Euclidian space Enwith the
definition:
f(x1,x
2, ..., xn)0,
where fis a real continuous function defined on En. The above inequality is
called a function representation (F-rep) of a geometric object and function f
is called the defining function. In three-dimensional case the boundary of such
a geometric object is called implicit surface. The major requirement on the
4
function is to have at least C0continuity. The set of points Xi(x1,x
2, ..., xn)
En,i=0, ..., N associated with Eq. 3 can be classified as follows:
f(Xi)>0ifXiis inside the object,
f(Xi)=0ifXiis on the boundary of the object,
f(Xi)<0ifXiis outside the object.
Let us consider from now on the defining function given by the convolution
operator between a kernel and all the points of a skeleton, i.e. function Fas
defined in the last equation of the previous section.
3.1 Set-theoretic Operations
The binary operations on geometric objects represented by functions can be also
defined in the form of function representation by
F(f1(X),f
2(X)) 0,(2)
where Fis a continuous real function of two variables .10 Such operations are
closed on the set of function representations. After set theoretic operation be-
tween two subjects defined by functions f1and f2the resulting object has the
defining function as follows:
For object union
f3=f1|f21
1+a(f1+f2+f2
1+f2
22af1f2),
for object intersection
f3=f1&f21
1+a(f1+f2f2
1+f2
22af1f2),
for object subtraction
f3=f1\f2f1&(f2),
where |,&,\are notations of so-called R-functions and parameter a=a(f1,f
2)
is the arbitrary continuous function satisfying the conditions
1<a(f1,f
2)1
a(f1,f
2)=a(f2,f
1)=a(f1,f
2)=a(f1,f2).
Please, note that even thought the resulting defining function for set above the-
oretic operations is continuous, the resulting object is not continuous in general.
5
3.2 Blending Union Operation
Intuitively the blending union operation between two initial objects from the
set of function representations is a gluing operation. It allows us to control the
gluing type in the wide range of shapes from pure set union to convolution like
summation of terms. Mathematically the blending union operation is defined by
F(f1,f
2)=f1+f2+f2
1+f2
2+a0
1+(f1
a1)2+(f2
a2)2,
where f1and f2are functions representing objects that are blended. The absolute
value a0defines the total displacement of the bending surface from two initial
surfaces. The values a0>0anda1>0 are proportional to the distance between
blending surface and the original surface defined by f1and f2, respectively. The
effect of this operation compared to other possible object connections is demon-
strated on two object primitives whose skeleton consists of two line segments
one vertical and the other one diagonal, see Figure 1 top-left. Simple plus opera-
tion between convolution functions deforms the thickness of vertical convolution
cylinders as shown in top-right image. Considering four line segments as a single
skeleton of geometric primitive results in the shape shown in top-center image.
The sequence of shapes shown on bottom of Figure 1 are the blending union
operations between two parallel geometric primitives. The geometric primitives
and their skeletons do not change but the blending parameters used to blend
them are different for each image. In orderer from left side the used parameters
are ai=0.01, ai=0.07, ai=0.3, ai=0.5, and ai=0.7, respectively. We can
conclude that in the case when the shape and size of geometric primitives must
be preserved the blending union operation with different parameters a0,a1,and
a2is a good choice. On the other hand when the blending shape is main concern
the convolution plus operation should be used. When both the shape of geomet-
ric primitives and that of blending are important the small values of blending
union parameters is a choice. The F-rep blending union operation has similar
advantages as simple convolution union with respect to minimizing unwanted
bulges.
4 Shape of Organic Models
In previous sections have been discussed the theory of F-rep and convolution
surfaces. As next, we will show a method to model the organic shapes by F-rep,
where each of the geometric primitives is defined by
N
i=1
Fi(x1,x
2,x
3)T=0,
where Fiare the source potentials of skeleton primitives i.e. points, lines or
triangles and Tis a threshold value. Therefore, what we need to design next are
the skeletons for different organs.
6
Fig. 1. Blending union operation. top: standard and bottom: Blending union operation.
4.1 Human Brain Model
First step in the model creation process is to obtain the size measurements of
brain and stomach stages from atlas of embryology. Embryological atlas contains
hand-drawing pictures and photographs of human embryo organs ordered by age.
For the purpose of this study the models from 28 - 56 days old brain were used.
The brain pictures has been scanned, stored in binary form and measured by
ruler. The model, at this stage of precessing, was divided into physiological parts
to suite the animation purposes. The outlines of physiological parts were drawn
over the pictures and photographs, see Figure 2.
4.2 Brain: Central Skeleton
The result of the measurements is a 2D planar contour, call the central skeleton,
nearly outlining the outer contour of the shape. Interior of central skeleton is
triangulated such that it crates a triangular strip. One can observe different
growth speed for different pars of embryo brain. It is therefore natural to divide
the central skeleton into those parts. Additional parts could be necessary to
model the folds and control the unwanted blending problem near the folding
areas. Figure 3, shows namely the part I corresponding to the part of brain
7
Fig. 2. The conversion of drawing human embryo brain to central skeleton.
called rhombencephalon, part II will develop to mesencephalon and part III is a
prosencephalon. The next step is to calculate the central line that will be used
as a base to define the thickness of the model along the line forming the tubular
object. Central line passes through the center of central skeleton, connecting the
mid points of vertical edges of a triangular strip.
part I
part II
part III
Fig. 3. Dividing the central skeleton to 3 parts. The line in the middle of the central
skeleton is called central line.
4.3 Brain: Skeleton
By adding the thickness to 2D central skeleton the 3D skeleton of the model is
obtained. Multiple number of copies of central skeleton are slightly scaled and
shifted to left and right sides of central skeleton. By this way the cross sections
are produced which are then connected to form the tubular skeleton, see Figure 4:
Each of side skeletons is scaled to fit the ellipses whose center is on the
central line. Radius aof the ellipse is a distance to the central line from the
border of the central skeleton. Radius bfollows the equation, b=αa, where
αis a ratio parameter.
8
As next step, for a given θthe side skeletons are translated by distance
t=ccos θ, where cis known from parametric equation of ellipse shown in
Figure 5.
Finally, side skeletons are connected with a central skeleton or with other
side skeletons by a triangular mesh.
After erasing all interior triangular patches we obtain multiple tubular shapes
forming together the entire skeleton of the brain.
central
skeleton
side
skeletons
central line
Fig. 4. Adding the thickness by scaling and shifting the central skeleton.
4.4 Model of the Human Digestive System
To approximate the shape of an organ while considering the speed and direction
of cell growth at the same time, we group the entire set of cells into a number
of cylindrical bunches (clusters). Thus, the skeleton of the organ is defined by a
chain of linear segments passing through the cluster centers, see Fig. 6. Organ
growth can then be modeled by the growth of the line skeleton, and variations
in shape thickness during the growth process can be captured by variations in
cylinder size. When a cylinder changed in size, it was understood that the organ
cells grew in the directions emanating from the cluster center. Similarly, when
the skeleton segment underwent changes in length, it was understood that the
cells included in two adjoined clusters grew in directions parallel to this segment.
Taking into account the development, the organs were divided into physiolog-
ical parts having different speed and direction of growth to suite the animation
purposes. The physiological parts of the intestine system are shown in Fig. 7
and marked I, II and III for stomach, marked IV for small intestine, marked VII
for large intestine, marked V for appendix, and marked VI for vitteline duct.
While refering to Langman’s embryology [11] we collected data that are shown in
9
central
skeleton
side
skeleton
b
ac
t
θ
y
z
y
z x
Fig. 5. A 3D skeleton for 36 days old human embryo brain.
Fig. 6. Skeleton of the organ and the clusters.
Table 1. For each available embryo age (developmental stage) of large intestine
its mean thickness and skeleton length are listed. Statistical data for a human
embryo stomach have already been summarized by ˇ
Durikoviˇc et al.[2].
The organs, at this preprocessing stage, were divided into physiological parts
having different speeds and directions of growth to suite the animation purposes.
4.5 Digestive System: Skeleton
The topology of the digestive system is expressed by a tree structure and the
development of the tree-like structure can be easily modeled with an algebraic
L-system [12, 13]. An L-system formalism was proposed by Lindenmayer [14],
and the method has been used as a general framework for plant modeling. The
L-systems are extended to by introducing continuous global time control over
10
I.
II.
III.
IV.
V.
VI.
VII.
Fig. 7. Physiological parts represented with line skeleton.
Table 1 . Shape measurements of large intestine, physiological part no. IV.
Embryo age (day) Length (mm) Thickness (mm)
28 2.76 0.30 herniation
49 15.58 0.45
58 19.49 0.52
70 21.77 0.61 reduction
83 24.04 0.82
113 28.62 1.00 fixation
the productions, stochastic rules for the capture of small variations, and explicit
functions of time used to describe continuous aspects of model behavior, in
addition to differential equations.
In some cases it is convenient to describe continuous behavior of the model
using explicit functions of time rather then differential equations. For example,
global shape transformations varying over time require a large and complicated
system of differential equations, while only few explicit functions of time are
sufficient for the description of these transformations.
4.6 Cell Model
Let’s move to a micro structure of muscle cell structures on the organelle level.
We present a modeling concept based on the theory of implicit surfaces that
allows for creation of a realistic infrastructure of the micro-world of muscle cells.
From the viewpoint of geometry, the structure of living cells is given by the three-
dimensional organization of their numerous intracellular organelles of various
sizes, shapes and locations.
11
1
2
3
4
5
6
7
Fig. 8. An example demonstrating eight consecutive sarcomeres of a muscle cell (left).
For better clarity, the sarcolemma is hidden and, also, the bottom part of the my-
ofibrillar system is clipped of by a transversal plane (middle). The complex system of
underlying skeletons is made visible by clipping with a longitudinal plane (right). The
myofibrillar system (1) is defined by means of c-graphs (2). The remaining organelles
include mitochondria (3), sarcoplasmic reticulum (4), t-tubules (5) and sarcolemma
(6); given in the basic repetitive unit, sarcomere (7).
4.7 Cell: Central Skeleton
The initial step involves creation of the central skeleton of the cell, which is
represented by a system of parallel cross-sectional graphs (c-graphs) distributed
along the longitudinal axis. We define the c-graph as a continuous planar graph
which divides the plane in a finite number of closed non-intersecting polygons.
Then we exploit the two-dimensional c-graphs to create the myofibrillar system
by means of the F-rep representation of polygons and interpolation. For better
clarity, this concept is demonstrated in Figure 8.
In the following subsections we propose approaches for creation of the most
complex structures of muscle cells, reticulum and mitochondria.
4.8 Cell: Skeleton
The basic modeling object at this step is a set of seed points distributed in
a system of several cross-sectional planes as shown in Figure 9a. Let S=
{s1,s
2,...,s
n}stand for the set of seeds in one crossection. Each seed pro-
duces an implicit circle fiwith an appropriate radius. The whole contribution
of Sis represented by CSG union:
f(S)=
sS
fs.(3)
12
(a) (b)
Fig. 9. (a) Seeds (here represented by the red spheres) are distributed in sets of parallel
planes. (b) A classical interpolation technique results in non-interconnected segments.
Similarly, we define the set Rof seeds in a neighboring crossection. To create
a smooth junction between shapes f(S)andf(R) we apply an interpolation
method.
Assume that both shapes (sets of implicit circles) contain a set of control
points, P={p1,...,p
n}for the function f(S)andQ={q1,...,q
n}for the
function f(R). Moreover, vectors of correspondence are specified between these
points, CP={q1p1,...,q
npn}and CQ={p1q1,...,p
nqn}. The set CPis
attached to the set P, and the set CQis attached to the set Q. Now, we create two
weighting displacement functions φpand φq, which represent transformations of
the given shapes in the directions defined by the vectors of CPand CQ.The
weighting displacement functions are defined by
φp(x)=x+h1(t)d1(x)
φq(x)=x+h2(t)d2(x),(4)
where h1(t), h2(t) represent weighting proportions within the interval <0,1>,
and d1(x), d2(x) represent interpolation of control points given by vectors of
CPand CQ. To interpolate the displacement d1,d2we adopt volume splines—
the so-called thin-plate function [15, 16]. The weighting factors h1and h2,i.e.
functions that specify the size of control point displacements, are defined as
h1(t)=(1ta)b
h2(t)=1h1(t),(5)
where the parameters a,bmodify the slope and curvature of the transition
(Fig. 10a).
To create the required smooth transformation without gaps, the linear inter-
polation is modified by the displacement functions, Eqs. 4:
Flt =(1t)f(S)(φp(x)) + tf(R)(φq(x)) + aw3(t),(6)
where aw3(t) is the additional blending term used to fine-tune interconnection
of shapes by adding material primarily in the central part of the interpolation
region. The parameter astands for the amount of blending and the weighting
13
0.2 0.4 0.6 0.8 1
-0.2
0.2
0.4
0.6
0.8
1h (t)
h (t)
1
2
(a)
0.2 0.4 0.6 0.8 1
-0.2
0.2
0.4
0.6
0.8
1
w (t)
3
(b)
Fig. 10. (a) Weighting functions h1and h2. (b) The weighting function w3has the
maximum in the middle.
()a()b
Fig. 11. Modeling of sarcoplasmic reticulum. (a) The final warping interpolation pro-
vides gap free interconnections. (b) The smooth junction between terminal cisterns and
tubes is obtained by the blended union.
function w3(t) is defined as
w3(t)=(1(2t1)c)d,(7)
where the parameters c,dmodify the slope and the curvature (Fig. 10b).
A result of this approach with three sets of seeds defined in three parallel
planes is depicted in Figure 11a.
The second step in the building process is formation of terminal cisterns.
These cylindrical shaped objects form a smooth junction to systems of longitu-
dinal tubules. Terminal cisterns are created as blended union of implicit cylin-
ders. Their underlying line segments are obtained by connecting the seeds in the
bottom most and the top most plane of the system of crossectional planes, see
Figure 11b.
4.9 Cell: Mitochondria
In order to capture the varying elliptical shape of mitochondria, we use implicit
sweep objects. The basic components of sweep objects are a 2D sweep template
and a 3D sweep trajectory. Here, the 2D template is a 2D implicit ellipse with
variable dimensions. Figure 12 demonstrates such a mitochondrion defined by a
trajectory specified by means of spline control points.
14
r
r
=0.8
=1.8
=-40a
r
r
=0.5
=0.4
r =0.5
=30a
r
r
=0.8
=1.1
r =0.8
=30a
r
r
=0.5
=0.6
=10a
x
x
x
x
y
y
y
z
z
y
Fig. 12. The curve, represented as a quadratic B-spline, is created from the control
points, where each has assigned corresponding radii and rotation angles (left). Note
end control points have specified also zradius for 3D ellipsoid. The resultant sweep
object is depicted on the right.
5 Organ Growth
Continuous processes such as the elongation of skeleton segments, and growth of
cell clusters, over time can easily be described by the growth functions. Growth
functions can be then included into algebraic L-systems as explicit functions
or differential equations. Growth is often slow initially, accelerating near the
maximum stage, slowing again and eventually terminating. A popular example
of the growth function [17] is the logistic function which is a solution to the
following differential equation
∂r
∂t =p1r
rmax rgrmax,p (r).(8)
Logistic function monotonically increases from initial value r0to rmax with
growth rates of zero at start and end of time interval [T0,T]. It is an Sshape
function with a steep controlled by a parameter p.
The details of the L-system tables have been described by Durikovic [13]. He
has described the skeleton elongation, the global bending of the skeleton parts,
dynamics of skeleton structures, and growth functions.
6 Shape from Skeleton
The measured organ models discussed in previous section were divided into phys-
iological parts, at preprocessing stage, having different speed and direction of
growth to suit the animation purposes. A single physiological part has the shape
defined by the skeleton based F-rep. The skeleton of the physiological part can
be animated directly by a key-frame animation or we can use a sophisticated
methods to simulate the skeleton growth based on L-system or the dynamic
L-system.
15
6.1 Brain Shape
A smooth convolution surface defined over the triangular mesh of tubular skele-
ton creates the model of embryo brain. In order to create brain model with con-
volution surfaces, we use HyperFun1,9as modelling library and POV-Ray8as
rendering software. HyperFun command hfConvTriangle generates convolution
surface over the triangles which suites our problem. Let us discuss all parame-
ter settings for one particular example, the stage3 human embryo brain shown
in Figure 13. The convolution kernel width is set to s=0.5 and iso-potential
threshold value is T=0.6. The ration parameters of brain thickness have been
set to α=1.0 at parts I and II and to α=1.2 at part III. Nice blending during
the animation can be guarantied by blend-union operation between three parts
of this model using the HyperFun command hfBlendUni. The blending parame-
ters a1=a2=a3=0.2 are used for both gluing parts I, II and parts II and III,
respectively.
Fig. 13. Stage3 human embryo brain. Left: 3D tubular skeleton, right: entire brain
model, defined by function representation.
6.2 Shape of the Digestive System
We represent the smooth shape of the digestive system in a compact way by
piecewise linear skeleton and locally defined convolution cylinders along each
linear segment of a skeleton. Thus, the resulting smooth tubular surface is rep-
resented by a real function as the blend union operation between many convolu-
tion cylinders. The shape of a convolution surface can be varied in several ways:
by varying the skeleton, by varying the thickness of convolution cylinders with
parameter sfrom Eq. 9, and by the iso-potential threshold value T:
fi(X)=Vi
1
(1 + s2r2(v))2dv T. (9)
16
For example, the small and large intestines monotonically increase their thickness
which can be modeled with the monotonically decreasing parameter sas seen
for the six developmental stages of intestine in Table 2.
Table 2 . Convolution parameters for thickness of small and large intestine.
Intestine
Embryo age (day) Small Large
sTs T
28 0.45 0.2 0.69 0.25
49 0.42 0.2 0.67 0.25
58 0.40 0.2 0.63 0.25
70 0.35 0.2 0.60 0.25
83 0.32 0.2 0.57 0.25
113 0.29 0.2 0.54 0.25
Thickness. As was already mentioned, the increasing thickness of convolu-
tion cylinders distributed along the skeleton segments is given by monotonically
decreasing the width parameter sin time as shown in Table 2. We will trans-
form the solution of Eq. 10, ˆs, that monotonically increases from 0.01 to 0.16
over the time interval [28,120] into a monotonically decreasing function sby
Eq. 11, where smax =0.7:
ˆs(t)
∂t =g0.16,0.003s),ˆs(28) = 0.01 (10)
s(t)=smax ˆs(t).(11)
Function s(t) is the growth function controlling the thickness of large intestine
with a good approximation of data from Table 1. The graph of the growth
function over the time is shown on right of Fig. 14.
7 Results
Few frames from animation of Organ growth show the embryo stomach and brain
described by embryo age and the real size scale bar, see Figures 15, 16.
Shown in Fig. 17 are several frames from a generated animation simulat-
ing digestive growth based on the proposed L-system using the above growth
functions. The environment forces and self collision were handled by the spring
representation of results obtained from L-system. The shape of the digestive
system shown in this figure undergoes global bending transformation and de-
formations resulting from gravity, animator intervention (looping process), and
collision. Some of the intermediate shapes in Fig. 17 have disjoined elements due
to aliasing in the implicit polygonizer that has difficulties to find a mesh for long
thin structures.
17
0,00
5,00
10,00
15,00
20,00
25,00
30,00
35,00
0 20 40 60 80 100 120
t [days]
length [mm]
0,5
0,52
0,54
0,56
0,58
0,6
0,62
0,64
0,66
0,68
0,7
0 20 40 60 80 100 120
t [days]
width s [mm]
Fig. 14. Graphs of growth functions. Left) Total length of large intestine in time.
Right) Change of width parameter sin time, see Eq. 9.
8 Conclusions
We have presented a method for simulation of the growth of human embryo
digestive system. The method uses the shape calculated based on F-rep using iso-
surfaces generated by skeleton segments, which provides a smooth and compact
representation of the surface usable for complex animations. We proposed a
method in which the organ growth and global bends are separate processes. The
differential growth functions are introduced for an algebraic L-system which
efficiently control the elongation of skeleton segments.
We succeeded to model structure of living cells, virtual human embryo or-
gans, namely brain, stomach and digestive system using convolution surfaces and
functional representation. The growth animation of a stomach was generated for
all 9 months of development while the brain growth animation was generated
for first 4 months of embryo development. The advantage of skeleton based ap-
proach is that it avoids the the topology artifacts that can occur when using the
nonlinear interpolation between two defining functions of F-rep models. Variable
speed of growth and shape thickness is successfully modelled by convolution plus
or blending union between model parts.
We have proposed the skeletons consisting of triangular patches which gives
us the opportunity to define the flat shapes like pillow, refer to the brain model.
Acknowledgements
Authors wishes to thank Mineo Yasuda from Medical School of Hiroshima Uni-
versity for sharing the knowledge as an embryologist. The authors are grateful to
Laboratory of Cell Morphology at the Institute of Molecular Physiology and Ge-
netics, Slovak Academy of Sciences. The images were rendered by the POVRay
ray-tracing program programmed by POVRay Team. This research was sup-
ported by a Marie Curie International Reintegration Grant within the 6th Eu-
ropean Community Framework Programme EU-FP6-MC-040681-APCOCOS.
18
Fig. 15. A single frame from the human embryo stomach animation.
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Fig. 16. A single frame from the human embryo brain animation.
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20
Fig. 17. Development of a human embryo digestive system with proposed method
taking into account the skeleton dynamics and growth functions in algebraic L-system.
Function representation is used to define a smooth shape. Indicated stages from left to
right represent 28, 34, 40, 49, 52, 58, 64, 70, 76, 79, 83, 94, 101, 107, 110, and 113 days
of animation sequence.
... For example, design and manufacturing of implant devices and scaffolds [8] need a model to tailor various conflicting properties like high hardness and high toughness, biocompatibility, and other bio-factors; drug delivery devices and wound covers [9] need the release of multiple drug molecules that can be controlled by defining various composition profiles of the drug molecules locally within different sub-domains; numerical analysis and simulation of fault-lines and rocks in the area of geo-science [6,7], simulation of growth of the organs for plants and animals [5] require the model to capture various properties across the layers and around different non-manifold entities. The main challenge in modelling these is that they are associated with random material distribution as shown in figure 2 where the material variation in material composition due to points and boundary edges are shown using different colours. ...
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