May 5, 1988 - of Single Muscle Fiber Action Potentials. BERT A. ALBERS, WIM L. C. RUTTEN, WILLEMIEN WALLINGA-DE JONGE,. AND HERMAN B. K.
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Frequency Domain Modeling of Volume Conduction of Single Muscle Fiber Action Potentials BERT A. ALBERS, WIM L. C. RUTTEN, WILLEMIEN WALLINGA-DE JONGE, AND HERMAN B. K . BOOM
Abstract-An inhomogeneous frequency dependent model of volume conduction in skeletal muscle tissue is used to calculate a transfer function between injected membrane current and extracellular action potential in the frequency domain. This model accounts for tissue structure and microscopic electrical parameters (intra- and extracellular conductivities and membrane impedance) and represents the volume conductor by an extensive electrical network. Results obtained with the model are compared with those of a conventional homogeneous volume conductor model. The comparison shows a significant influence of tissue structure and microscopic electrical parameters close to the source. As a result, in this new model the transfer function close to the source is less sensitive to the frequency content and conduction velocity of the membrane current, compared with results of the homogeneous volume conductor approach.
INTRODUCTION KELETAL muscle tissue is an inhomogeneous volume conductor with two microscopically intertwined, electrically different media (the intra- and extracellular medium). One way to describe volume conduction in such a tissue is to approximate it as a homogeneous volume conductor with conductivities parallel and normal to the fiber direction. Since these conductivities are usually obtained by measuring the electrical potential difference between points at a large distance from the current source, they can be considered as macroscopic parameters. Such a homogeneous approach has been used by a number of authors to simulate the extracellular action potential [3], [ 8 ] -
S
[ill.
Recently, it was found that the homogeneous approach is insufficient at shorter distances from a current source. Gielen et al. [6], using the four electrode method, showed that the electrical potential close to a point-shaped current electrode cannot be described adequately with homogeneous conductivities. They concluded that volume conduction at a small geometrical scale can be more correctly described if both the tissue structure and the microscopic electrical parameters (intra- and extracellular conductivities and membrane impedance) are taken into account. Gielen et al. [7] introduced a model describing the inManuscript received July 8, 1986; revised October 19, 1987. This work was supported by The Netherlands Organization for the Advancement of Pure Research (ZWO). The authors are with the Department of Electrical Engineering, Biomedical Engineering Division, University of Technology, 7500 AE Enschede, The Netherlands. IEEE Log Number 88 19591.
fluence of tissue structure and microscopic electrical parameters on volume conduction. Application of this model is restricted to a one-dimensional current injection with large plate electrodes. Albers et al. [2] introduced a discrete model, based on the same model structure as used by Gielen et al. [7]. In this model, the muscle tissue is represented by an electrical network which enables the calculation of a potential distribution due to a point-shaped current source in an inhomogeneous volume conductor. Simulation results obtained with this model show a strong influence of tissue structure and microscopic electrical parameters on volume conduction at a small geometrical scale [l], [ 2 ] . This paper compares this network model with the homogeneous approach in the frequency domain. CALCULATION OF THE TRANSFER FUNCTION Consider an infinite medium around an excited muscle fiber with infinite length. A central longitudinal portion of this medium is divided into a number of cross sections of length L , normal to the fiber direction. In a discretization of this part of the volume conductor, each point is indicated by two index numbers, viz. c , the index of the cross section and n, the index of a point in the cross section. A current source distribution is situated in cross section c = 0, injecting a total current Io * sin ( u t ) with w = 27rf, the angular frequency of the injected current. The spatial discretization of this source represents a part of the excited fiber membrane with length L. The Fourier transform of the resulting potential distribution is represented by a frequency dependent matrix 9( U ) , with elements 9c,n( U ) . Furthermore, an impedance Zc,n( U ) is defined by
The response V, (a)in point n of cross section c = 0 to a current source moving with constant velocity U and amplitude Zo along the excited fiber is given by
with
0018-9294/88/0500-0328$01.OO O 1988 IEEE
7,.=
c
*
L U'
-
(3)
~
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The total number of cross sections taken into account is 2 * Ne + 1. The transfer function H,, ( U ) is defined by
Y
This expression can be used to calculate H,, ( U ) , irrespective of the way @ ( U ) is calculated and of the model description of the volume conductor. SIMULATION MODELS Two different volume conductor models will be used in order to obtain transfer functions H,, ( U ) . In the network model (introduced by Albers [2]), the volume conductor around the source is modeled by a simplified structure of skeletal muscle tissue, consisting' of identical parallel muscle fibers with hexagonal cross section (see Fig. 1). The fiber membranes are represented by parallel circuits of membrane resistances and membrane capacitances. Parameters of the model are given in Table I. One excited fiber is surrounded by a number of passive fibers, this fibrous structure in turn being surrounded by an infinite homogeneous medium. The model is divided into a number of cross sections with length L and translated into an electrical network (see Fig. 1). A detailed description of this model structure is given by Albers et al. [2]. In cross section c = 0, a current Zo * sin ( U T ) is equally divided over the six extracellular nodes of the active fiber (Fig. 1). This source is situated at the center of the network, resulting in a high degree of symmetry. Inside the fibrous structure, the potential distribution is solved numerically from a set of linear equations, representing the Kirchhoff laws for the network, as has been described previously [2]. In the homogeneous medium surrounding this structure, the potential is calculated using an analytical expression introduced below in the description of the second model. In the second model (the homogeneous model), skeletal muscle tissue is considered as a homogeneous volume conductor with the same current source distribution as in the network model. The potential distribution is calculated using an analytical expression for the potential in an infinite, homogeneous volume conductor. For the six current source points of Fig. l , the potential at field point c , n is given by
Fig. 1. The model structure and the way in which it is transformed into an electrical network. Only seven fibers are shown in this example. Parameters of the model are listed in Table I . The model structure is given at cross section c = 0. Current is injected only in this cross section and is equally divided over the extracellular nodes of the central fiber, indicated by arrows. TABLE 1 PARAMETER VALUES U S E D IN THE SIMULATIONS
set A
set B
a.
intracellular conductivity
0.55
0.80
(%)-I
a
extracellular conductivity
2.40
2.00
P1-l
Cd
membrane capacitance per unit
0.01
0.01
Fm-2
1 .oo
,-1
Symbol
Parameter
of surface area
Gd
Unit
membrane conductance per unit 1.00
of surface area A
fiber radius
d
thichess of the extracellular layer
1.35
1.35
p
intracellular volume fraction
0.9
0.9
a
macroscopic conductivity
0.69
0.92
a
macroscopic conductivity normal to the fibers
0.13
0.11
K'
anisotropy ratio
5.48
8.74
25
parallel to the fibers
25
-2
w vm
(%)-I
where ri,zi:
K2: a,, ur:
radial and axial distance between the source point i and the field point, anisotropy ratio, K 2 = a,/a,., conductivity, parallel and normal to the fiber direction, respectively.
In the next section, both models will be applied to calculate transfer functions using ( 5 ) . The results of both models can be compared if the microscopic parameter values used in the network model can be transformed to the homogeneous parameters a, and a,.. For the model structure used in this paper, this transformation is given by [7] az = ( 1 - p ) . a, + p
*
a;
(7)
At f = 15 kHz, the second and third terms on the right
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hand side of (8) contribute only 2 percent to 1 urI and introduce a phase angle in ur of 11'. Therefore, these terms are neglected, which results in a frequency independent homogeneous parameter set. This is in accordance with the homogeneous models, used by many other authors. RESULTS Moduli of transfer functions, calculated as described in the previous section are presented in Fig. 2. The total length over which contributions from individual cross section have been summed [see relation (5)] is 33.5 mm. The network representation in the network model represents 217 muscle fibers with a length of 13.5 mm situated at the center of this total length. In order to study the way in which both models are sensitive to variations of U, and oi, two different parameter sets, listed in Table I, are used. The microscopic electrical parameter values are within the range, indicated by Cole and Curtis [4]. Realistic values of U, and U , are obtained using the microscopic parameters of Table I [ 5 ] , [13]. The homogeneous model parameters were taken to be frequency independent. Therefore, the frequency dependence of the homogeneous transfer functions is caused only by the propagation of the current source. The results of Fig. 2 lead to three major observations. First, the network model results in higher spectral amplitudes than the homogeneous model. Second, the difference between both models becomes larger with increasing frequency. High-frequency contributions are attenuated more in the homogeneous model than in the network model. Third, Fig. 2 shows how the results of the two models are affected by microscopic parameter variations in opposite ways. The homogeneous model results in equal or lower spectral amplitudes of H, ( U ) if parameters set B is used instead of set A. In contrast, the network model shows a small increase in the spectral amplitude if the values of parameter set A are replaced by those of set B. It is obvious that the propagation of the current source introduces a frequency dependence in the transfer func(U),even if the medium is frequency independent tion H,, (e.g., the homogeneous model). Therefore, the influence of a conduction velocity variation on the results of both models was studied. To this end, transfer functions are calculated with a conduction velocity U of 5 m/s using the parameter values of set B . The results of this calculation are shown in Fig. 3. Both models show higher spectral amplitudes of H , ( w ) for increased conduction velocity. Comparison of these results with those for parameter set B in Fig. 2 ( U = 3 m/s) shows that the difference between both models decreases with higher conduction velocities. In order to illustrate some practical consequences, both models have been used to calculate the extracellular action potential of a single excited muscle fiber, the so called single fiber action potential (SFAP). These SFAP's are calculated by inverse Fourier transformation of V, ( U ) , calculated with (4).Now Io ( w ) is taken to be the Fourier
0. H dB -121
-24'
-36'
(b) Fig. 2. Transfer functions calculated for the network model (a) and the homogeneous model (b). r , = 55 pm and r, = 147 pm from the center of the excited fiber. Parameters used in these calculations are listed in Table I (sets A and B ) . The conduction velocity of the intracellular action potential is 3 m/s. All transfer functions are normalized to the maximum of the transfer function, calculated with the network model, using parameter set B for r,.
I
-361
5
--2. lo
I
kHz
Fig. 3. Transfer functions calculated for r , = 55 pm and r z = 147 pm with parameter set B . The conduction velocity is 5 m/s. Results are normalized as described in the caption of Fig. 2 .
transform of the membrane current i, ( t ) , injected over a length L. This current is calculated from the intracellular action potential Vi( t) by
i,(t)
n
~
2
a2~ v,~( t )i
=2 -
U
at2
'
(9)
For the intracellular action potential, a generalization of a mathematical expression given by Rosenfalck [12] is used: V i ( t ) = At3 exp (-Bt) - C
(10)
withA = 1.66 * 10" V * sP3,B = 1.20 * lo4 s - ' , C = 0.09V. The membrane current calculated from this intracellular
ALBERS er al.: VOLUME CONDUCTION O F MUSCLE FIBER POTENTIALS
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with the network model will be sensitive to the extracellular conductivity rather strongly, whereas the homogemV neous transfer functions will be more sensitive to the intracellular conductivity. An increase of the intracellular conductivity will result in larger source amplitudes [relation (9)]. Therefore, in -1' both models, the SFAP amplitude would increase if parameter set B is used instead of set A . In the homogeneous -2. model, which is very sensitive to U, but less sensitive to ' 0 a, the increase in amplitude is partially compensated by the volume conductor effect of the increase in U,. On the other hand, in the network model, which is highly sensitive to ue, the decreased U, in set B causes an additional increase in the SFAP amplitude. Nandedkar and Stilberg [9] concluded from comparisons of simulated and recorded SFAP's that the SFAP amplitude predicted by the homogeneous model is too low. These authors mentioned the significant variation in -1 I 'id, the SFAP amplitude due to a change in the macroscopic 1 -21 parameter U,, but they did not discuss the limitations of 1 -~ homogeneous models in general. The results presented 0 1 2 t m s 3 in this paper show that these differences can be caused at (b) Fig 4. Simulated SFAP's for r , = 55 pm using the transfer functions pre- least partially by the deficiency of a homogeneous volume sented in Fig. 2 and the membrane current calculated from the intracel- conductor model in the description of SFAP's close to the lular action potential given by (10) (a) Network model. (b) Homogeexcited fiber. neous model. Gydikov and Trayanova [8] also used a homogeneous volume conduction model to study the influence of the action potential has - 3 dB points at 800 Hz and 4.4 kHz. origination and termination of the intracellular action poResulting SFAP's are presented in Fig. 4. tential on SFAP's, calculated close to the excited fiber. When using parameter set B instead of set A , the mem- The present results, however, show that along with an brane current, calculated with (9) has the same spectral accurate description of the source, the description of volcomposition but the amplitude in the time domain is about ume conduction close to the source has also to be more 45 percent higher, due to the increase of ai. The SFAP precise than is possible with the homogeneous approach. amplitude increases by 35 and 63 percent in the homo- In particular, it has to account for the influence of the geneous model and the network model, respectively. structure of the volume conductor. Thus, in the homogeneous model, the SFAP amplitude The results in Figs. 2 and 3 show that the difference increases less than i, ( t )does. In contrast, in the network between the transfer functions calculated with both models model the opposite is true. These observations directly increases for higher frequencies. Furthermore, this differreflect the fact that the results of both models are influ- ence appears to be smaller for a higher conduction velocenced in different ways by variations of microscopic prop- ity. erties. These different sensitivities to frequency and conduction velocity are due to the difference between the way D~SCUSSION both models describe volume conduction. As illustrated by Albers et al. [2], current injected by The pourier transform of the SFAP described by (2) as an active muscle fiber in the extracellular medium and well as the transfer function H,,(w) are vectorial sumconducted parallel to the fiber direction is redistributed mations of a number of individual contributions, with a over the intra- and extracellular media. This redistribution phase shift of 07, between the successive contributions. of parallel current is the main cause of the differences be- This implies that the resulting amplitude of H,, ( U ) between the homogeneous model and the network model. comes smaller for increasing phase shift, i.e., for higher In the network model, the current close to the source is frequencies and lower conduction velocities [see relation almost solely conducted by the extracellular medium since (3)]. This can be observed in results of both models. the high membrane impedance prevents a quick redistriHowever, an important difference between the models bution of current over both media. In contrast, in the is the fact that in the network model, the extracellular curhomogeneous model, where current is quickly redistri- rent density close to the current source is higher than in buted over both media, the major part of the current is the homogeneous model. This results in a faster decline conducted through the intracellular medium since the in- of the amplitudes of successive contributions to H,, ( w ) in tracellular volume is much larger than the extracellular the network model. As a consequence, the relative influvolume. Consequently, the transfer function calculated ence of contributions, originating from cross sections,
,
SFAP2'
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close to the source is larger in the network model than in the homogeneous model. In other words, in the network model, H,, ( w ) is mainly determined by contributions with a small phase shift, whereas in the homogeneous model, contributions with different phase shifts have a more uniform influence on H , ( w ) . Hence, the transfer functions calculated with the homogeneous model are more sensitive to frequency and conduction velocity than the results of the network model. Consequently, the difference between transfer functions, calculated with both models, becomes larger for increasing frequency and decreasing conduction velocity. Thus, the sensitivity of the SFAP to source properties, such as spectral composition and conduction velocity is influenced by the way in which structure and microscopic electrical properties of the volume conductor are taken into account. REFERENCES [ I ] B A . Albers, W L C Rutten, and W Wallinga-de Jonge, “Influence of inhomogeneou\ structure and frequency dependent membrane properties on volume conduction i n skeletal muscle tissue,” in Proc XIV I C M B E , VI1 ICMP, 1985, pp 930-931. 121 B A Albers, W L. C. Rutten, W. Wallinga-de Jonge, and H B K Boom, “A model study on the influence of structure and membrane capacitdnce on volume conduction in skeletal muscle tissue,” IEEE Trans Biomed Eng , vol BME-33, pp 681-689, 1986 [3] S Andreassen and A Rosenfalck, “Relationship of intracellular and extracellular action potentials of skeletal muscle fibre,” in Critical Revrew 111 Bio-Engineering Boca Raton CRC Press. vol 6. 1980, pp. 267-306. K. S . Cole and H. J. Curtis, Bio-electricity: Electric Physiology, Medicul Physics (0.Glasser, Ed.). Chicago, IL: Year Book, 1950, vol. 2, pp. 82-89. B. R. Epstein and K. R. Foster, “Anisotropy in the dielectric properties of skeletal muscle,” Mrd. Biol. Eng. Comput., vol. 21, pp. 51-55, 1983. F. L. H. Gielen, W. Wallinga-de Jonge, and K. L. Boon, “Electrical conductivity of skeletal muscle tissue: Experimental results from different muscles in vivo,” Med. Biol. Eng. Comput., vol. 22, pp. 569577, 1984. 171 F. L. H. Gielen, H . E. P. Cruts, B. A . Albers, K. L. Boon, W. WaIlinga.de Jonge, and H . B. K. Boom, “Model of electrical conductivitv of skeletal muscle based on tissue structure.” Med. B i d . Eng. Comput., vol. 24, pp. 34-40, 1986. 181 A. A. Gydikov and N. A . Trayanova, “Extracellular potentials of single active muscle fibres: Effects of finite fibre length,” B i d . Cybern., vol. 53, pp. 363-372, 1986. [9] S. D. Nandedkar and E. Stilberg, “Simulation of single muscle fibre action potentials,” Med. B i d . Eng. Comput., vol. 21, pp. 158-165, 1983. I IO] P. Plonsey, “The active fiber in a volume conductor,” IEEE Trans. Biomed. Eng., vol. BME-21, pp. 371-381, 1974. 11 I] R. Plonsey, “Action potential sources and their volume conductor fields,” Proc. IEEE, vol. 65, pp. 601-611, 1977. [ 121 P. Rosenfdlck. Intra- and Extracellular Potential Fields of Active Nerve and Muscle Fibres. A Physicomathematical Analysis of Different Models. Copenhagen, Denmark: Akademisk Forlag, 1969. 1131 E. Zheng, S . Shao, and J. C. Webster, “Impedance of skeletal muscle from I Hz to 1 MHz,” IEEE Trans. Biomed. E n g . , vol. BME3 I , pp. 477-481. 1984.
Bert A. Albers was born in Hengelo, The Netherlands, in 1958. He received the M.Sc. and the Ph.D. degrees from Twente University of Technology, Enschede, The Netherlands, in 1983. His Ph.D. study involved electrophysiology and volume conduction in skeletal muscle tissue. He joined the Biomedical Engineering Division, Department of Electrical Engineering, Twente University of Technology, in 1983. Recently, he joined Cordis Europe N.V. where he is involved with clinical research of angiographic Droduct s.
Wim L. C. Rutten was born in The Hague, The Netherlands, in 1950. He received the M Sc. degree in experimental physics in 1974 from the University of Leiden, Leiden, The Netherlands, and the Ph D. degree in physics in 1979 In 1971 he joined the Solid-state Magnetism Research Group at the Kamerlingh Onnes Laboratory, Leiden, The Netherlands Since 1979 he has been engaged mainly in experimental and clinical audiology and i n the physics of hearing at the ENT Department of the University Hospital in Leiden. Since 1985 he ha\ been a Senior Member of the Biomedical Engineering Group Department of Electrical Engineering, Twente University of Technology, Enschede, The Netherlands His main current research interests are volume conduction in muscle, development of multimicroelectrodes, and surface EMG
Willemien Wallinga-De Jonge was born in Veendam, The Netherlands, in 1945. She received the M.Sc. degree in biology from the University of Groningen, Groningen, The Netherlands, in 1968 and the Ph.D. degree in biomedical engineering from Twente University of Technology, Enschede, The Netherlands, in 1980. She is a Senior Staff Member of the Biomedical Engineering Group, Department of Electrical Engineering, Twente University of Technology. Her research activities there involve the electrical activity of muscle in general, and the intracellular and extracellular single fiber activity in particular.
Herman B. K. Boom was trained as a medical physicist at the University of Utrecht, The Netherland\, where he received the Ph D. degree in 1971. He joined the Department of Medical Physics and Medical Physiology where he was engaged in research in the field of cardiac mechanics and taught physiology and biophysics. Since 1976 he has occupied the Chair of Medical Electronics i n the Department of Electrical Engineering, Twente University of Technology, Enschede, The Netherlands. His research interests are cardiovascular system dynamics, bioelectricity, and rehabilitation technology.
