Optimization of stimulation parameters as well as shape and positioning
of electrodes are important questions in Functional Electrical Stimulation
(FES) of paraplegic patients. For that
reason a MATLAB tool, named FES-FIELD, modeling the three dimensional electric
field in human body has been developed to calculate the electric field in a
region of interest. The simulation tool provides a Graphic User Interface
(GUI).
In case of denervation of the lower extremities an important target
muscle is the m. quadriceps femoris. The electrical potential distribution
along its fibers is representative for its functional activation. For this
special application the human thigh stimulated by skin electrodes has been
modeled.
The simulation process has been done in five steps:
1) Reading the geometric information of the thigh from 50 CT-slices (256
by 256 pixels).
2) Segmentation in tissue types
by pixel value and definition of each
conductivity.
3) Selection of electrode geometry and positioning.
4) Calculating the electric field
iteratively by solving the system of
approx. 1.5mio. linear equations.
5) Visualization of the solution by equipotential lines in either cross-
or length-sections of the thigh.
The simulation requires 5-6 hours time for approx. 6000 iterations
computed with a standard PC (800MHz CPU, 512MB RAM). The graphical information
and the electrical solution can be exported to binary files for further
investigations like calculation of the activating function of representative
fibers in several muscle regions.
STATE OF THE ART
Most of the knowledge on Functional Electrical Stimulation (FES) is the
product of more than 30 years of experimental use /1/ /2/, i.e. it is empirical and subjective. For
quantitatively observation a 2D-model of the distribution of the extracellular
electric field in a length section of the human thigh has been established in
1999 /3/. It focuses on denervation of the lower extremities.
The major target muscle for
However, in a 3D-simulation the stimulation current would be passed on
to the hamstrings through the muscles enclosing the femur. Voltage distribution
would change considerably compared to the 2D-model, i.e. a stronger electric
field would be observed in the region of the hamstrings. Hence, functional
tetanic contraction of the quadriceps influenced by the co-contractions of the
hamstrings would be investigated much better.
MATERIAL AND METHODS
Geometric model
The geometric information is available as a number of either frozen cuts
or CT cross-sections each stored in a - 256 by 256 pixel - graphic file format
of *.jpg or *.tif, respectively. The MATLAB application FES-FIELD provides an
input filter for both formats, which first converts the files into gray scaled
8bit images (Fig.
1), then finds the outermost contour (skin) and
last colors the contour-pixels white and all outlying pixels black (air). In
this example, pixel size is 1 by 1mm (Dx,
Dy) and the distance between each cross-section is 10mm (Dz). This determines a 3D-grid equidistant in each direction (x, y,
z).
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relative conductivities
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Fig. 1: 8bit gray scaled cross-section (256 x 256pixel @ 1x1mm) in the upper third of the left human
thigh (left) corresponding to the framed cross-section and further
cross-sections taken lengthwise every 10mm from the hip down to the knee
(right). |
Tab. 1: Conductivities of several types of material. |
Depending on their gray values, the pixels inside the contour (skin)
represent basic types of tissue like fat, muscle, bone and connecting tissue.
The entire 8bit gray scale can be divided into several groups (i.e. segments),
where each group stands for a type of tissue. For each group a value for the
conductivity g (shown in Tab.
1) has to be defined. For example the segmentation
of muscle tissue is shown in Fig.
1.
Electrode positioning
To apply electrical stimulation (i.e. the electric field) via surface
electrodes, shape and position of electrodes have to be selected. The geometric
information of the electrodes must be added to the 2D-superficial surface
(skin) and the junction points for anode and cathode of the stimulator have to
be defined. This is shown in Fig.
2.
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Fig. 2: Skin of human thigh sliced at the bottom side representing a
2D-superficies surface with two rectangular electrodes on top (black).
Electrodes should be placed within the gray shaded area bounded by the
dash-dotted lines, because adding electrodes outside of this area increases
the risk of them loosing their shape when the 2D configuration is attached
onto the 3D thigh. The two small squares, placed onto the electrodes
represents the junctions to the anode and cathode of the stimulator. |
The electric field
After segmentation of tissue has been performed and electrodes and
voltage sources have been added into the geometric information of the thigh,
all data required for calculating the stationary electric field, i.e. a
3D-matrix of the thigh including surface electrodes, and the correlation
between pixel values and conductivities, is available. Describing the voltage
distribution in a medium with variable conductivity leads to an elliptic
boundary value problem denoting a current density
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with boundary
conditions |
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Equ. 1 |
The coefficient g is a function,
which gives the conductivity depending on the location x, and the function V determines the voltage at x. Dirichlet (D) boundary conditions define
voltage sources and Neumann (N) boundary conditions describe the behavior of
the field at the crossover from skin to air and at the hip and knee, where the
3D-matrix-representation ends. n denotes the outward pointing normal vector, i.e.
the vector orthogonal to the boundary curve. A large system of linear equations (e.g. approx 1.5 million linear
equations for 50 cross sections) can be obtained via discretization of Equ. 1.
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(a) |
(b) |
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Fig. 3: a) Interdependent voxels in a 3D-grid, equidistant in each direction
(Dx, Dy, Dz), neighboring a central voxel-i,j,k and b)
corresponding voltages V of each voxel-center connected to the neighboring
voxel by the conductance G depending on direction (L…left, R…right, D…Down,
U…up, F…front, B…back). |
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Physical interpretation of Equ.
1 (i.e. Ohm’s and Kirchhoff’s law) leads to the system
of linear equations
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Equ. 2 |
by summing the currents coming from neighboring voxels (Fig. 3). Current flows over the conductance G
depending on direction (L…left, R…right, D…Down, U…up, F…front, B…back) because
of the voltage difference. This system can be solved iteratively using the
MATLAB’s numerical method of conjugate gradients (CG).
RESULTS
In Fig.
4 an example shows the stationary electric field
calculated by FES-FIELD in one of the cross-sections and in one length-section.
The length section was extracted from all cross-sections using the data at the
corresponding vertical line.
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Fig. 4:Cross-section from Fig. 1 (left) and length-section with attached
electrodes corresponding to the vertical line in the cross-section (right)
both displaying the stationary electric field using equipotential lines. |
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DISCUSSION
In this research the tool FES-FIELD was
implemented to simulate the stationary electric field in the human thigh while
applying functional electrical stimulation to paraplegic patients. This tool could be used for other purposes as
well. It could be applied directly to the lower leg or the upper extremities
and with minor modification also to other regions or organs of the body. One of
the initial motivations for developing FES-FIELD was being able to determine
voltage distribution along muscle fibers. Using the 3D-matrix giving a voltage
value for each pixel, the voltage distribution along any fiber or path can be
determined. This process could be automated in a future version.
REFERENCES
/1/ Kern H., Funktionelle Elektrostimulation paraplegischer Patienten,
Österr. Z. Phys. Med.,
1995, 1, Supplementum
/2/ Kralj A. and Bajd T., Functional
Electrical Stimulation: Standing and Walking after Spinal Cord Injury,
1989, CRC Press, Inc.
/3/ Reichel M.,
Mayr W., Rattay F., Computer Simulation of Field Distribution and Excitation of
Denervated Muscle Fibers caused by Surface Electrodes, Artificial Organs, 1999, 23(5): 453-456
ACKNOWLEDGEMENTS
Many thanks to Prof. W. Auzinger from the Dep. of Applied and Numerical Mathematics at the
Technical University of Vienna for support in numerical mathematics and to Dr.
D. Fleischmann from the Dep. of Diagnostic Radiology at the University of
Vienna for providing the CT-dataset.