Download Rulebased Assignment of Myocardial Sheet Orientation

1
Rule Based Assignment of Myocardial Sheet
Orientation
Rolf F. Schulte, Frank B. Sachse, Christian D. Werner, Olaf Dössel
email: [email protected]
Abstract— Ventricles consist of an ordered laminar
arrangement of myocytes. This has influence on the
electrophysiological and elastomechanical properties of
the heart and cannot be neglected in realistic models.
Main objective was to assign this sheet orientation into
any given dataset of mammalian hearts. Sheet orientation is generally radial to ventricular surfaces inside
the mid-myocardium and becomes roughly tangential
to it near the epi- and endocardium.
The first step is to calculate wall normals, using the
gradient of the Gaussian function as mask. These normals are verified through various criteria, such as the
length and the direction of the normals and the distance towards boundaries of the area. Few relative
sheet angles are assigned on these wall normals equidistantly, which are used as fulcra for a latter interpolation. Due to different patterns, the left and right ventricular free walls and the interventricular septum are
treated separately. Finally, all relative sheet angles are
transformed into pairs of absolute angles.
The tool was applied on the Visible Female dataset,
which is part of the Visible Human project from the
National Library of Medicine (USA). Together with the
fibre orientation, a complete anatomical model of myocardial sheet orientation was created.
Fig. 1. Isolated mammalian myocyte; intercalated disks are
located mainly at the end, but also laterally (from [3]).
Keywords— myocardial sheet orientation; anatomical
model
I. Introduction
Although the gross morphology of mammalian
hearts is well investigated, its microscopical structure
is still unknown in great parts. Recently, a laminar arrangement of groups of myocytes was discovered [1].
One axis of these planes runs approximately radially
from endocardium towards epicardium in transmural
direction. The other axis coincides with the local fibre
orientation
This laminar arrangement influences the electrophysiological and elastomechanical properties of the
heart and cannot be neglected in realistic physiological models. Main objective of this study was to assign this sheet orientation into any given dataset of
mammalian hearts with the methods of digital image processing. Due to numerous differences in the
morphology between canine and human hearts, simple mapping is not feasible. Thus, a possible way is to
assign myocardial sheet orientation with a set of rules.
The rules used in this study are derived from [1], [2].
II. Microscopic Anatomy of Mammalian
Heart
Hearts are mainly build of myocytes (figure 1), cardiac muscle cells, which have a length of 50 to 120
µm and a diameter of 5 to 25 µm [6]. Myocytes are
interconnected with each other via intercalated disks,
Fig. 2. Visible female heart [4], including fibre orientation
(from [5]).
which provide mechanical and electrical coupling [7].
Intercalated disks are located mainly at the end of the
cells, but also laterally.
The principal axes of adjacent cells are nearly parallel, allowing to determine a mean direction, which
is generally referred to as muscle fibre orientation [2].
Fibres run approximately parallel to the outer cardiac
surface.
The heart wall can be described as a threedimensional continuum made up essentially of cardiac
muscles cells, with smooth change of direction across
the wall [8]. There are no “bundles of fibres” running
through the myocardium.
Collagen Fibres
The arrangement of cardiac myocytes is mirrored
by a complex extracellular connective tissue matrix.
An extensive network of collagen fibres is present inside the myocardium. According to Caulfield [9] these
fibres can be divided into three different types:
Fig. 3. Collagen network of the rabbit left ventricle interconnecting the myocyte (M) and the capillary (C) (×6,500; from
[9]).
Fig. 5. Schematic of cardiac microstructure. (a) A transmural
blocks cut from the ventricular wall shows the fibre orientation inside a myocardial sheet. (b) The myocytes are shown
lying three to four cells thick within a sheet. Endomysial
collagen is seen linking adjacent cells within a sheet, while
perimysial collagen links adjacent sheets (from [2]).
Fig. 4. Two orthogonal surfaces of mid-wall specimen (from
[2]).
Myocyte to myocyte collagen struts, which provide
equal stretch of contiguous cells, and prevent slippage
of adjacent cells
• Myocyte to capillary collagen struts
• Meshwork of collagen fibrils surrounding groups of
myocytes; few loose collagen connections inbetween
permit slippage and rearrangement
The latter kind of collagen fibres groups three or more
myocytes into an ovoid configuration. Its main axis is
parallel to the long axis of the cells, which is equivalent
to the fibre axis. In the mid-myocardium the second
axis is nearly perpendicular to the endo- and epicardial
surface.
•
Muscle Sheet Arrangement
LeGrice et al. [2], [1] discovered, that myocytes are
ordered laminarly with relatively extensive cleavage
planes inbetween muscle layers. Branching between
adjacent layers is relatively sparse with muscle bridges
one to two cells thick. Each of these layers consists
of tightly packed groups of cardiac myocytes. This
general arrangement is repeated throughout the ven-
tricles. There are typically four myocytes across the
thickness of a layer.
One axis of the sheets is built by the myocardial fibre orientation. The other axis is measured relative to
the local outer-wall normal in planes built by the principal axis of the heart and a vector radial to it. Near
the equator the orientation is similar for both ventricular free walls, changing from around −90◦ at the endocardium to 30−60◦ at the epicardium. Inside the interventricular septum the orientation is changing from
90◦ in the left ventricular sub-endocardium to around
−90◦ in the right ventricular sub-endocardium. In
the mid-myocardium the orientation is approximately
parallel to the normal.
Two areas which do not follow this standard pattern
are around the bases of the left ventricular papillary
muscles, where there is a complex interweaving, and
around the base of the left ventricular free wall, where
sheets angle up into the basal skeleton [2]. In the
papillary muscles, trabeculae, and the atria a laminar
structure is not visible.
Differences between Canine and Human Hearts
All gross morphological features are similar in both
mammalian hearts. The only significant differences
concern their shape. Furthermore, Caulfield used various species for his investigations. The main conclusion
is, that the collagen network is qualitatively similar in
the rodent and human hearts, but quantitatively dissimilar [9].
Although the sheet orientation is mapped only for
canine hearts up to now [2], [1], it can be assumed,
that the derived rules are valid for all mammalian
species, including human hearts. Reasons are the morphological similarities in their macroscopic and micro-
3 parts:
Left ventricular free wall
Right ventricular free wall
Interventricular septum
Detection of fulcra
Detect starting point
Calculate normal
endocardium
Assign angles
OP
mid-myocardium
N
epicardium
γ
P
Correction
(only LV)
Interpolation
Transform angles
Fig. 6. Principal functioning of the developed tool.
base
following three areas are distinguished: left and right
ventricular free wall, and the interventricular septum.
The principal functioning can be observed in figure 6.
left atrium
P
N
interventricular
septum
left ventricluar
free wall
OP
Fig. 8. Equidistant assignment of sheet angles (γ) relative to
the outer wall normal ~
n.
CG
right ventricle
apex
Fig. 7. Sketch of left ventricle describing the basic functioning.
The normal ~
n starts in P and exits in OP. The direction
points towards the centre of gravity (CG). All starting points
too close to the left atrium (base) and right ventricle (apex
and septum) are excluded.
scopic structure, common roots in evolution, and the
fact, that all mammalian hearts are highly optimized
and efficient in their functioning. Thus they are likely
to use the same mechanisms.
III. Material and Methods
Requisite is a volume based anatomical model of
the heart with the following classification: left and
right ventricle, left and right atrium, as well as fat and
other tissue. The interventricular septum is usually
classified as approximately two third belonging to the
left ventricle and one third to the right ventricle.
Two axes are required for a complete model of ventricular sheet orientation. One axis coincides with the
local myocardial fibre orientation. Models describing
this orientation already exist [5] and were not subject of this study. The other axis is generally radial
to the ventricular surfaces, but becomes roughly tangential to the epi- and endocardium. Two exceptions
are the area near the base of the left ventricular free
wall, where a correction is applied, and the bases of
the left ventricular papillary muscles, which were excluded during the segmentation.
The developed tool assigns few ventricular sheet angles on wall normals, which are used as fulcra for a succeeding interpolation. Due to different patterns, the
Left Ventricle
The first task is getting proper wall normals. Starting points are left ventricular voxels with differently
classified neighbours. To these points, normals are
calculated using the gradient of a Gaussian function.
Now the left ventricular free wall is traversed until an
exiting point to another material class is found.
Several criteria apply for verifying these normals:
• Distance towards right ventricle and left atrium:
Removes erroneous normals near boundaries towards
other cardiac areas and inside the interventricular septum.
• Length of normal: Excludes abnormal sizes, e.g.
caused by fat particles inside the myocardium.
• Direction towards centre of gravity of the left ventricle: Due to the fact that the left ventricular free wall
has roughly the shape of parts of an ellipsoid, all normals point approximately into the same direction, the
centre of gravity. For this criterion the vector starting point towards centre of gravity of left ventricle is
multiplied with the normal and checked. This also
excludes normals starting endocardial (figure 7).
On all remaining normals sheet angles are assigned
equidistantly throughout the wall from the starting
point P towards the exiting point OP.
Due to different patterns near the basis of the left
ventricular free wall, a correction for these angles is
applied. The left atrium is used as an approximation
for the area of the basis to avoid additional segmentation. All angles are increased towards a maximum
value of 90◦ , depending on their distance to the left
atrium, with the following equation:
γ 0 = γ + (90◦ − γ) · scale
0 ≤ scale ≤ 1
max − distance 6
scale =
max − min
where distance stands for the distance of the present
voxel towards the left atrium, max for the maximum
distance and, min for the closest voxel.
RV
N
LV
first step
P
γ
x
N
OP
α
α
b
second step
OP *
P*
N*
P
Fig. 9. Normals inside the interventricular septum. In a first
step the normal ~
n traverses the left ventricle (LV). In the
exiting point (OP= P∗ ), the normal is turned by 180◦ (~
n∗ =
−~
n) and passes through both, left and right ventricle.
teria are applied:
• Distance towards fat: Excludes normals near the
base or apex of the heart, resembled through fat.
• Length of normal
• Direction of normal: The reference is the vector
from the centre of gravity of the right ventricle towards
the centre of gravity of the left ventricle (figure 10).
left atrium
base
right atrium
LR
Fig. 11. Creation of ~b with the angle γ, the wall normal ~
n, and
the principal axis p
~
CGL
Normal Calculation
CGR
N
LV
RV
apex
Fig. 10. Sketch of interventricular septum. Starting points too
close to fat (base and apex) are excluded. The direction
of the normal ~
n has to match the direction of the vector
LR from the the centre of gravity of the left ventricle (CGL)
towards the centre of gravity of the right ventricle (CGR).
In a last step these relative angles are transformed
into the absolute angles ϕ and ϑ.
The assignment of sheet orientation within the right
ventricular free wall is similar to this, but no correction needs to be applied.
Interventricular Septum
Inside the interventricular septum, a different strategy is used. Because there is no separate classification for this region, the border between right and left
ventricle is used for normal calculation. This border
is relatively smooth. A starting point (P) is a voxel
classified as left ventricle with a contiguous right ventricular voxel.
First the area of the left ventricle is traversed from
the starting point towards an exiting point to another
material class. Now the normal is turned around and
the former exiting point is set as new starting point.
Then the area of both, left and right ventricle is traversed towards an exiting point to the right ventricular
cavity (figure 9).
For exterminating invalid normals the following cri-
For performance reasons the calculation of normals
on cardiac surfaces is divided into two steps. First, a
normal mask is generated. In this case the gradient of
a Gaussian function is used, which is defined as follows
[10]:
f (x)
=
1
√
σ 2π
x2
e− 2σ2
where σ is the standard deviation and equated with
the width of the mask. By using a Gaussian function,
the voxels closer to the centre are weighted stronger
and thus errors caused by a rough surface are eliminated.
The required gradient for the mask is:

 

∂f
− x2
∂x
~x
 ∂f   σy 
− σ2
grad f =  ∂y  =
f =− 2 f
σ
∂f
− σz2
∂z
In a second step this mask is used for a normal calculation around each eligible point. This point represents the centre of the mask. All values of the mask
of neighbouring voxels with different classification are
grouped in a vector, and thus result in the normal.
Transformation of Relative into Absolute Angles
Sheet angles are assigned in voxels of the normal
relative to this normal. A transformation into the two
absolute angles ϕ and ϑ is required. This is done in
two steps. First, an orientation vector ~b is created as
follows:
~b
= ~n + p~
sin γ |~n|
sin α |~
p|
where γ is the relative sheet angle and α the angle
between the orientation vector ~b and the first principal
Fig. 12. Sheet orientation in a slice of the visible female heart
[4]. The left upper part shows the left ventricular free wall,
the middle is the interventricular septum, and on the right
side is the right ventricular free wall
axis p~. For small angles α, the orientation ~b is equated
with the principal axis p~ (figure 11).
Then the orientation vector is transformed into the
two angles ϕ and ϑ with the following equation:




x
sin ϑ · cos ϕ
 y  = r  sin ϑ · sin ϕ 
z
cos ϑ
y
⇒ ϕ = arctan
x
r
x2 + y 2
ϑ = arctan
z
with 0◦ ≤ ϑ ≤ 180◦ and 0◦ ≤ ϕ ≤ 360◦ . The vector
needs to be normalized first, which means r = 1.
Interpolation
Finally, all not yet assigned points are interpolated
[11].
IV. Results
With the developed tool it is possible to assign
myocardial sheet orientation to anatomical models of
mammalian hearts. It was applied on the visible female dataset [4]. The various criteria, like the length
and the direction of normals and the exclusion of normals close to critical areas, allow a stable and robust
search for all valid wall normals. With enough fulcra an interpolation of angles without discontinuities
is assured.
Figure 12 and 13 show the results of the assignment
of sheet orientation.
V. Discussion
A rule based assignment of myocardial sheet orientation is always as good as the data and rules it is
based on. Although the microstructural properties of
canine and human hearts are similar, an investigation
for human hearts is recommended as well. With a
phase sensitive MRT it might be possible to examine
Fig. 13. Sheet orientation inside the left ventricular free wall
living human hearts without negative side effects, like
an impact on the health or a greater aberration of the
results, due to rough investigation methods. The obtained data can be used to verify and improve the developed tool. The parameter settings can be adopted.
The specimen used by LeGrice et al. [2], [1] altered
during the dehydration process, making the sheet
orientation more evident. An investigation method,
which is not based on dehydration, could further improve gained results and minimize errors.
Main conclusion is, that a rule based assignment of
myocardial sheet orientation is a feasible way. The errors are depending on the quality of the derived rules.
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