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Improved multi-scale modeling of uterine electrical activity
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www.sciencedirect.com IRBM 34 (2013) 38–42
Digital technologies for healthcare
Improved multi-scale modeling of uterine electrical activity J. Laforet b,∗ , C. Rabotti a , M. Mischi a , C. Marque b a
b
Department of Electrical Engineering, Eindhoven University of Technology, Eindhoven, Netherlands Université de technologie de Compiègne, CNRS UMR 6600, biomécanique et bioingénierie, Compiègne, France Received 9 November 2012; received in revised form 11 December 2012; accepted 12 December 2012 Available online 29 January 2013
Abstract Objective. – We aim to simulate the uterine electrical activity at the myometrium level and at the skin level where it can be recorded non-invasively. Material and methods. – We use 2D models both at the myometrium scale and for the volume conductor. The multi-scale model has been implemented in Python, as a complete package including the needed tools. To speedup the slowest step, we integrated parallel execution. Results. – We obtain realistic simulations of EHG signals as recorded by an electrode array placed on the woman abdomen. Conclusion. – These simulations are still generic, the next problem to address will be the identification of the model’s parameters to obtain patient-specific simulations. © 2013 Elsevier Masson SAS. All rights reserved.
1. Introduction The work present here is part of the BioModUE PTL european project, which aims to detect threats of preterm labor by the analysis of the uterine electrical activity. The contractile element of the uterus is the myometrium, which is composed of smooth muscle cells. The sequence of contraction and relaxation of the myometrium results from the electrical activity associated to the generation and propagation of cellular action potential bursts. The uterine electrical activity can be realistically modeled by representing the principal ionic dynamics at the cell level, the propagation of electric activity at the tissue level and to the skin surface. An important part of the modeling work about the electrophysiology of the uterine muscle has been done at the cell level. These models are based on the Hodgekin–Huxley approach: they detail the effect on the ionic channels in the cell membrane to simulate the transmembrane potential and the intracellular ionic concentrations [1,2]. Another aspect treated is the propagation of the electrical signals through the different layers of abdominal tissues until their recording by surface electrodes [3].
On-going work in the field is now focusing on multi-scale approaches, trying bridge the gap between ionic currents at the cell level and surface recordings [4,5]. The same scale hierarchy is observed for the mechanical models, with most of the work being done at the cellular scale. At this scale, mechanical models are also based on ionic currents, but focus on calcium concentration instead of the transmembrane potential. Using Hai–Murphy models of actin/myosin interaction, they derive the contractile stress from the calcium concentration [6–8]. Another approach is to model directly the intra uterine pressure evolution during contractions, at the expense of loosing the physiological link, as done in [9,10] for educational purpose. We present here the improvements to the multi-scale model and its implementation as a stand-alone simulation software, packaged following the Python standards. A simplified model based on the physiology was already presented [5]. We were able to simulate 0-dimension to 2-dimension grids of muscle cells. We developed a parallel integration based on shared memory by using python standard libraries to improve the computational effectiveness of the simulations. We also improved the spatial filtering due to the tissues between the uterus and the electrode array by using a 2D volume conductor. 2. Methods
∗
Corresponding author. CNRS UMR 7338, rue Personne-de-Roberval, 60200 Compiègne, France. E-mail address: [email protected] (J. Laforet). 1959-0318/$ – see front matter © 2013 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.irbm.2012.12.004
In this section, we will focus on describing the improvements made to our multi-scale modeling of uterine electrical activity.
J. Laforet et al. / IRBM 34 (2013) 38–42
39
2.1. 2D tissue model
Table 1 Main parameters of the presented simulations.
To model the generation of the electrical activity and its propagation within the myometrium we adopted a reaction–diffusion formalism. The reaction term corresponds to the model of cell excitability and the diffusion term to the communication between the cells. We dispose of two models of the myometrial cell electrical activity:
Parameter
Value
Number of cells Cell size Intercellular resistance Abdominal muscle thickness Fat tissue thickness Skin thickness Electrode diameter Electrode distance (centers) EHG SnR
140(+4) × 140(+4) 50 m 2000 9.36 mm 11.32 mm 2 mm 8 mm 17.5 mm 5 db
• Red6: from [1], based on Hodgkin–Huxley model. It takes into the main ionic currents through the cell membrane; • Red3: is a simplified version of Red6, enabling up to 60% faster simulations with a negligible loss in precision. The Red3 model, being much faster than the Red6 model, will be used as default model in this work. However, the Red6 will be kept for studies on small number of cells when requiring a higher accuracy. Both models are expressed as a system of ODEs, the digit in the name indicates the number of equations. For instance, Red3 can be written as: 1 dVm (Istim − ICa − IK − IKCa − Ileak ) , = dt Cm dnK Hk∞ − nK d Ca2+ = , dt TnK dt = fc −αICa − KCa Ca2+
Finally, we added a two-layer padding on the borders to avoid side effects of the spatial filtering. These ‘ghost cells’ have a coefficient of diffusion 104 times lower than the other cells to efficiently attenuate the signals at the borders.
(1)
with Vm the transmembrane potential, nK the potassium activation variable, and [Ca2+ ] the intracellular calcium concentration. At the tissue scale, the communication between the myometrial cells through gap-junctions is modeled by the spatial diffusion of the electrical potential over the cells. The cells are arranged into a Cartesian grid, which can be 0D (a single cell), 1D (a cable-like line of cells), 2D (a flat surface), or 2.5D (a flat surface with non-null thickness). The grid is modeled by a n-dimensional state array where each element represent a cell which is electrically coupled with its direct neighbors (2, 4 or 6 depending on the dimensions). For 2D and 2.5D tissue model, in order to simulate a simple model of cylindrical geometry, the border in one of the dimensions of the model may be considered as the neighbor of the opposite border. However due to limitations of the numerical code used, full 3D shapes can’t be used at this point. The dimension of a cell, the membrane resistance, and different physiologic features can be modified for each cell individually along each spatial dimension. This permits adding anisotropy to the model and also taking into account local variations of the parameters. It is also possible to represent non-muscular cells (i.e. connective tissue). These cells would be affected by the spatial diffusion as the other cells but the model defined in Eq. (1) would not be applied to generate their own response.
Fig. 1. Simulated electrical activity at the myometrium level, amplitude is colorcoded. a: stimulation corresponding to the simulated EHG signals presented after; b: a different stimulation pattern leads to spiral formation.
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J. Laforet et al. / IRBM 34 (2013) 38–42
Fig. 2. Simulated monopolar EHG signals.
2.2. 2D Volume conductor model To simulate the effect of the tissues interposed between the myometrium and the recording surface, we adopted the model proposed in [3], which allows modeling the surface EHG in the spatial frequency domain as the product between an electrical source, at the myometrium, and an analytical expression representing the effect of the volume conductor. To improve the volume conductor model described in [3] and in [5] was extended to two dimensions. The electrical source is the transmembrane potential Vm , previously computed at the tissue level using the 2D Red3 models. For the volume conductor, we extended to two dimensions the description in [3] of which we discuss here only the relevant aspects. The volume conductor is considered as made of parallel interfaces separating the different abdominal tissues, namely, the myometrium, where the source is placed at a depth z = z0 , the abdominal muscle, fat, and skin. The volume conductor effect
depends on the tissue thicknesses, their conductivities, and the source depth, z0 . All these tissues are assumed to be isotropic with the exception of the abdominal muscle. For the tissue conductivities, the values reported in the literature are used for simulating a signal propagating along the direction parallel to the vertical line of the abdomen [3,11,12]. Finally, we assume the source to be close to the myometrium-abdominal muscle interface, i.e., z0 →0. In order to visualize a simulated EHG similarly to a recorded one, we simulated the surface EHG in the spatial domain for subsequent time instants and then reconstructed the time samples from the spatial ones. Indicating by CV the conduction velocity, a sampling frequency ft = 20 Hz was chosen, in time, in order to verify the relation ft < fz CV, to avoid artifacts due to aliasing in the reconstruction of the signal in the time domain. The surface signal in Figs. 2 and 3 was simulated using tissue thickness values of 2 mm, 9.36 mm, and 11.32 mm, for the skin, the fat layer, and the abdominal muscle, respectively.
J. Laforet et al. / IRBM 34 (2013) 38–42
41
Fig. 3. Simulated bipolar EHG signals.
2.3. Electrodes grid model We included a simple model of electrode grid to generate EHG signals similar to the ones recorded experimentally. It is flexible and has been designed to be able to reproduce the different type of arrays we may use in human or animal experiments. To simulate the case of animal experiments where the grid is placed directly on the myometrium, the grid model includes basic spacial filtering to take into account the attenuation due to the distance between the cells and the electrode. When applied after the volume conductor model this is unnecessary as it’s already taken into account, then the grid model only does the summation of the signals. Finally, white Gaussian noise is added to the signals to match the typical SNR of recorded signals (5 to 10 dB). 2.4. Implementation We implemented all components using Python (version 2.7) in an object oriented approach. The code will be published under an open-source license and its required dependencies are only common Python modules beside the standard library (Numpy and Scipy). It can be executed under Mac-OS X, Windows, and GNU/Linux systems. We implemented two methods to compute the tissue models. The first method is a serial integration, which
computes the model by using only one process. It is based the Euler finite difference method. For each time sample, a discrete version of the cell model described in Eq. (1) is applied for every muscle cell of the tissue model. We also implemented an adaptive temporal step size to improve the efficiency of this simple scheme in terms of computation time. After each time step, the step size is refined according to the previously computed derivative value. To improve the computation time further, parallel computing can be used. We divide the tissue area into several parts and a different process computes each part of the tissue model and for each time step, the processes share their results. For this study, we developed an integration model based on shared memory. We used the multiprocessing module of Python standard library for the parallel implementation. It allows several processes to use a shared memory. To deal with Numpy arrays (more convenient than the standard type of array) using shared memory, we used the module shmarray, written by David Baddeley (under BSD license). Using the parallel implementation, we achieve a linear speedup (tested up to 24). In some case speedup values for both models are slightly supralinear. This can be explained by memory hierarchy effects, and mostly the use of memory cache that reduces the access time to the data. However, for smaller number of cells, the overhead will have a greater effect and reduce the speedup.
42
J. Laforet et al. / IRBM 34 (2013) 38–42
3. Results
Acknowledgements
Parameters used for the simulations presented in this section are summarized in the Table 1. Fig. 1 presents simulation results at the myometrium surface level. The Fig. 1(a) is a surface plot of the electrical potential over the myometrium used to generate the simulated EHG of Figs. 2 and 3. In the example of Fig. 1(b), we used two peacemaker areas on the opposite sides of the surface to elicit particular behaviors such as wave cancellation and spiral propagation. The simulated activity is a very simple example: an action potential wave generated near the border of the surface and propagating perpendicularly to it at a velocity of about 6 cm/s. Figs. 2 and 3 show the simulated EHG signals, respectively mono and bipolar. They correspond to the default experimental setup of the BioModUE PTL project: 16 electrodes, placed in a 7 cm × 7 cm grid on the woman’s abdomen.
This work was funded by ANR and ZonMw, partners of the ERASysBio+ initiative supported under the EU ERA-NET Plus scheme in FP7.
4. Discussion – Conclusion In this paper, we present a 2D multi-scale model of the uterine electrical activity during pregnancy. Relative to previous work, with the improvements described in the paper we reach a simulation accuracy sufficient to address comparisons with typical experimental data recorded by electrodes placed on the abdomen of a pregnant woman using the same electrode array we model in the present paper [13]. The simulated EHG is in fact consistent with experimental data in terms of SNR, frequency content and overall shape. The presented models can support understanding of the electrophysiological processes and the physical implications underlying a generic EHG signal and represent, therefore, a good reference for the validation of dedicated signal analysis tools. These simulations are based on parameters derived from the literature and the models are therefore generic and should be considered preliminary. Nevertheless, these models represent a solid basis for the development of patient-specific models and dedicated simulations aiming at more accurately describing specific physiological states or pharmacological treatments and their effects on the EHG recorded on the abdomen. Therefore, in the future we will focus on a sensitivity analysis of the complete model, with the aim to single-out the model parameters that have the most relevant effects on the properties of the EHG signal as it is recorded non-invasively on the abdomen.
References [1] Rihana S, Terrien J, Germain G, Marque C. Mathematical modeling of electrical activity of uterine muscle cells. Med Biol Engineer Comput 2009;47(6):665–75, http://dx.doi.org/10.1007/s11517-009-0433-434. [2] Tong WC, Choi CY, Karche S, Holden AV, Zhang H, Taggart MJ. A computational model of the ionic currents, Ca2+ dynamics and action potentials underlying contraction of isolated uterine smooth muscle. PLoS ONE 2011;6(4):e18685, http://dx.doi.org/10.1371/journal.pone.0018685. [3] Rabotti C, Mischi M, Beulen L, Oei S, Bergmans J. Modeling and identification of the electrohysterographic volume conductor by highdensity electrodes. IEEE Trans Biomed Eng 2010;57:519–27. [4] Aslanidi O, Atia J, Benson A, van den Berg H, Blanks A, Choi C, et al. Towards a computational reconstruction of the electrodynamics of premature and full term human labour. Progr Biophys Mol Biol 2011;107(1):183–92, http://dx.doi.org/10.1016/j. pbiomolbio.2011.07.2011.004. [5] Laforet J, Rabotti C, Terrien J, Mischi M, Marque C. Toward a multiscale model of the uterine electrical activity. IEEE on Bio-Medical Engineering 2011;58(12):3487–90, Trans http://dx.doi.org/10.1109/TBME. 2011.2167970. PMID: 21968708. [6] Maggio C, Jennings S, Robichaux J, Stapor P, Hyman J. A modified hai – murphy model of uterine smooth muscle contraction. Bull Math Biol 2012;74:143–58, http://dx.doi.org/10.1007/s11538-011-9681. [7] Bursztyn L, Eytan O, Jaffa AJ, Elad D. Modeling myometrial smooth muscle contraction. Ann N Y Acad Sci 2007;1101:110–38, http://dx.doi.org/10.1196/annals.1389.025. [8] Bursztyn L, Eytan O, Jaffa AJ, Elad D. Mathematical model of excitationcontraction in a uterine smooth muscle cell. Am J Physiol Cell Physiol 2007;292(5):C1816–29, http://dx.doi.org/10.1152/ajpcell.00478.2006. [9] Bastos LF, Lobo MF, van Meurs WL, de Campos DA. An intrauterine pressure generator for educational simulation of labour and delivery. Med Eng Phys 2010;32(7):740–5, http://dx.doi.org/10.1016/j.medengphy.2010.04.015. [10] Bastos LF, van Meurs W, de Campos DA. A model for educational simulation of the evolution of uterine contractions during labor. Comput Methods Progr Biomed 2012;107(2):242–7, http://dx.doi.org/10.1016/j.cmpb.2011.09.016. [11] Gabriel S, Lau R, Gabriel C. The dielectric properties of biological tissues: II. mesurements in the frequency range 10 Hz to 20 GHz. Phys Med Biol 1996;41:2251–69. [12] Mesin L, Merletti R. Distribution of electrical stimulation current in a planar multilayer anisotropic tissue. IEEE Trans Biomed Eng 2008;2:660–70. [13] Hassan M, Terrien J, Karlsson B, Marque C. Interactions between uterine emg at different sites investigated using wavelet analysis: Comparison of pregnancy and labor contractions. EURASIP J Adv Signal Process 2010;2010(1):918012, http://dx.doi.org/10.1155/2010/918012.
www.sciencedirect.com IRBM 34 (2013) 38–42
Digital technologies for healthcare
Improved multi-scale modeling of uterine electrical activity J. Laforet b,∗ , C. Rabotti a , M. Mischi a , C. Marque b a
b
Department of Electrical Engineering, Eindhoven University of Technology, Eindhoven, Netherlands Université de technologie de Compiègne, CNRS UMR 6600, biomécanique et bioingénierie, Compiègne, France Received 9 November 2012; received in revised form 11 December 2012; accepted 12 December 2012 Available online 29 January 2013
Abstract Objective. – We aim to simulate the uterine electrical activity at the myometrium level and at the skin level where it can be recorded non-invasively. Material and methods. – We use 2D models both at the myometrium scale and for the volume conductor. The multi-scale model has been implemented in Python, as a complete package including the needed tools. To speedup the slowest step, we integrated parallel execution. Results. – We obtain realistic simulations of EHG signals as recorded by an electrode array placed on the woman abdomen. Conclusion. – These simulations are still generic, the next problem to address will be the identification of the model’s parameters to obtain patient-specific simulations. © 2013 Elsevier Masson SAS. All rights reserved.
1. Introduction The work present here is part of the BioModUE PTL european project, which aims to detect threats of preterm labor by the analysis of the uterine electrical activity. The contractile element of the uterus is the myometrium, which is composed of smooth muscle cells. The sequence of contraction and relaxation of the myometrium results from the electrical activity associated to the generation and propagation of cellular action potential bursts. The uterine electrical activity can be realistically modeled by representing the principal ionic dynamics at the cell level, the propagation of electric activity at the tissue level and to the skin surface. An important part of the modeling work about the electrophysiology of the uterine muscle has been done at the cell level. These models are based on the Hodgekin–Huxley approach: they detail the effect on the ionic channels in the cell membrane to simulate the transmembrane potential and the intracellular ionic concentrations [1,2]. Another aspect treated is the propagation of the electrical signals through the different layers of abdominal tissues until their recording by surface electrodes [3].
On-going work in the field is now focusing on multi-scale approaches, trying bridge the gap between ionic currents at the cell level and surface recordings [4,5]. The same scale hierarchy is observed for the mechanical models, with most of the work being done at the cellular scale. At this scale, mechanical models are also based on ionic currents, but focus on calcium concentration instead of the transmembrane potential. Using Hai–Murphy models of actin/myosin interaction, they derive the contractile stress from the calcium concentration [6–8]. Another approach is to model directly the intra uterine pressure evolution during contractions, at the expense of loosing the physiological link, as done in [9,10] for educational purpose. We present here the improvements to the multi-scale model and its implementation as a stand-alone simulation software, packaged following the Python standards. A simplified model based on the physiology was already presented [5]. We were able to simulate 0-dimension to 2-dimension grids of muscle cells. We developed a parallel integration based on shared memory by using python standard libraries to improve the computational effectiveness of the simulations. We also improved the spatial filtering due to the tissues between the uterus and the electrode array by using a 2D volume conductor. 2. Methods
∗
Corresponding author. CNRS UMR 7338, rue Personne-de-Roberval, 60200 Compiègne, France. E-mail address: [email protected] (J. Laforet). 1959-0318/$ – see front matter © 2013 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.irbm.2012.12.004
In this section, we will focus on describing the improvements made to our multi-scale modeling of uterine electrical activity.
J. Laforet et al. / IRBM 34 (2013) 38–42
39
2.1. 2D tissue model
Table 1 Main parameters of the presented simulations.
To model the generation of the electrical activity and its propagation within the myometrium we adopted a reaction–diffusion formalism. The reaction term corresponds to the model of cell excitability and the diffusion term to the communication between the cells. We dispose of two models of the myometrial cell electrical activity:
Parameter
Value
Number of cells Cell size Intercellular resistance Abdominal muscle thickness Fat tissue thickness Skin thickness Electrode diameter Electrode distance (centers) EHG SnR
140(+4) × 140(+4) 50 m 2000 9.36 mm 11.32 mm 2 mm 8 mm 17.5 mm 5 db
• Red6: from [1], based on Hodgkin–Huxley model. It takes into the main ionic currents through the cell membrane; • Red3: is a simplified version of Red6, enabling up to 60% faster simulations with a negligible loss in precision. The Red3 model, being much faster than the Red6 model, will be used as default model in this work. However, the Red6 will be kept for studies on small number of cells when requiring a higher accuracy. Both models are expressed as a system of ODEs, the digit in the name indicates the number of equations. For instance, Red3 can be written as: 1 dVm (Istim − ICa − IK − IKCa − Ileak ) , = dt Cm dnK Hk∞ − nK d Ca2+ = , dt TnK dt = fc −αICa − KCa Ca2+
Finally, we added a two-layer padding on the borders to avoid side effects of the spatial filtering. These ‘ghost cells’ have a coefficient of diffusion 104 times lower than the other cells to efficiently attenuate the signals at the borders.
(1)
with Vm the transmembrane potential, nK the potassium activation variable, and [Ca2+ ] the intracellular calcium concentration. At the tissue scale, the communication between the myometrial cells through gap-junctions is modeled by the spatial diffusion of the electrical potential over the cells. The cells are arranged into a Cartesian grid, which can be 0D (a single cell), 1D (a cable-like line of cells), 2D (a flat surface), or 2.5D (a flat surface with non-null thickness). The grid is modeled by a n-dimensional state array where each element represent a cell which is electrically coupled with its direct neighbors (2, 4 or 6 depending on the dimensions). For 2D and 2.5D tissue model, in order to simulate a simple model of cylindrical geometry, the border in one of the dimensions of the model may be considered as the neighbor of the opposite border. However due to limitations of the numerical code used, full 3D shapes can’t be used at this point. The dimension of a cell, the membrane resistance, and different physiologic features can be modified for each cell individually along each spatial dimension. This permits adding anisotropy to the model and also taking into account local variations of the parameters. It is also possible to represent non-muscular cells (i.e. connective tissue). These cells would be affected by the spatial diffusion as the other cells but the model defined in Eq. (1) would not be applied to generate their own response.
Fig. 1. Simulated electrical activity at the myometrium level, amplitude is colorcoded. a: stimulation corresponding to the simulated EHG signals presented after; b: a different stimulation pattern leads to spiral formation.
40
J. Laforet et al. / IRBM 34 (2013) 38–42
Fig. 2. Simulated monopolar EHG signals.
2.2. 2D Volume conductor model To simulate the effect of the tissues interposed between the myometrium and the recording surface, we adopted the model proposed in [3], which allows modeling the surface EHG in the spatial frequency domain as the product between an electrical source, at the myometrium, and an analytical expression representing the effect of the volume conductor. To improve the volume conductor model described in [3] and in [5] was extended to two dimensions. The electrical source is the transmembrane potential Vm , previously computed at the tissue level using the 2D Red3 models. For the volume conductor, we extended to two dimensions the description in [3] of which we discuss here only the relevant aspects. The volume conductor is considered as made of parallel interfaces separating the different abdominal tissues, namely, the myometrium, where the source is placed at a depth z = z0 , the abdominal muscle, fat, and skin. The volume conductor effect
depends on the tissue thicknesses, their conductivities, and the source depth, z0 . All these tissues are assumed to be isotropic with the exception of the abdominal muscle. For the tissue conductivities, the values reported in the literature are used for simulating a signal propagating along the direction parallel to the vertical line of the abdomen [3,11,12]. Finally, we assume the source to be close to the myometrium-abdominal muscle interface, i.e., z0 →0. In order to visualize a simulated EHG similarly to a recorded one, we simulated the surface EHG in the spatial domain for subsequent time instants and then reconstructed the time samples from the spatial ones. Indicating by CV the conduction velocity, a sampling frequency ft = 20 Hz was chosen, in time, in order to verify the relation ft < fz CV, to avoid artifacts due to aliasing in the reconstruction of the signal in the time domain. The surface signal in Figs. 2 and 3 was simulated using tissue thickness values of 2 mm, 9.36 mm, and 11.32 mm, for the skin, the fat layer, and the abdominal muscle, respectively.
J. Laforet et al. / IRBM 34 (2013) 38–42
41
Fig. 3. Simulated bipolar EHG signals.
2.3. Electrodes grid model We included a simple model of electrode grid to generate EHG signals similar to the ones recorded experimentally. It is flexible and has been designed to be able to reproduce the different type of arrays we may use in human or animal experiments. To simulate the case of animal experiments where the grid is placed directly on the myometrium, the grid model includes basic spacial filtering to take into account the attenuation due to the distance between the cells and the electrode. When applied after the volume conductor model this is unnecessary as it’s already taken into account, then the grid model only does the summation of the signals. Finally, white Gaussian noise is added to the signals to match the typical SNR of recorded signals (5 to 10 dB). 2.4. Implementation We implemented all components using Python (version 2.7) in an object oriented approach. The code will be published under an open-source license and its required dependencies are only common Python modules beside the standard library (Numpy and Scipy). It can be executed under Mac-OS X, Windows, and GNU/Linux systems. We implemented two methods to compute the tissue models. The first method is a serial integration, which
computes the model by using only one process. It is based the Euler finite difference method. For each time sample, a discrete version of the cell model described in Eq. (1) is applied for every muscle cell of the tissue model. We also implemented an adaptive temporal step size to improve the efficiency of this simple scheme in terms of computation time. After each time step, the step size is refined according to the previously computed derivative value. To improve the computation time further, parallel computing can be used. We divide the tissue area into several parts and a different process computes each part of the tissue model and for each time step, the processes share their results. For this study, we developed an integration model based on shared memory. We used the multiprocessing module of Python standard library for the parallel implementation. It allows several processes to use a shared memory. To deal with Numpy arrays (more convenient than the standard type of array) using shared memory, we used the module shmarray, written by David Baddeley (under BSD license). Using the parallel implementation, we achieve a linear speedup (tested up to 24). In some case speedup values for both models are slightly supralinear. This can be explained by memory hierarchy effects, and mostly the use of memory cache that reduces the access time to the data. However, for smaller number of cells, the overhead will have a greater effect and reduce the speedup.
42
J. Laforet et al. / IRBM 34 (2013) 38–42
3. Results
Acknowledgements
Parameters used for the simulations presented in this section are summarized in the Table 1. Fig. 1 presents simulation results at the myometrium surface level. The Fig. 1(a) is a surface plot of the electrical potential over the myometrium used to generate the simulated EHG of Figs. 2 and 3. In the example of Fig. 1(b), we used two peacemaker areas on the opposite sides of the surface to elicit particular behaviors such as wave cancellation and spiral propagation. The simulated activity is a very simple example: an action potential wave generated near the border of the surface and propagating perpendicularly to it at a velocity of about 6 cm/s. Figs. 2 and 3 show the simulated EHG signals, respectively mono and bipolar. They correspond to the default experimental setup of the BioModUE PTL project: 16 electrodes, placed in a 7 cm × 7 cm grid on the woman’s abdomen.
This work was funded by ANR and ZonMw, partners of the ERASysBio+ initiative supported under the EU ERA-NET Plus scheme in FP7.
4. Discussion – Conclusion In this paper, we present a 2D multi-scale model of the uterine electrical activity during pregnancy. Relative to previous work, with the improvements described in the paper we reach a simulation accuracy sufficient to address comparisons with typical experimental data recorded by electrodes placed on the abdomen of a pregnant woman using the same electrode array we model in the present paper [13]. The simulated EHG is in fact consistent with experimental data in terms of SNR, frequency content and overall shape. The presented models can support understanding of the electrophysiological processes and the physical implications underlying a generic EHG signal and represent, therefore, a good reference for the validation of dedicated signal analysis tools. These simulations are based on parameters derived from the literature and the models are therefore generic and should be considered preliminary. Nevertheless, these models represent a solid basis for the development of patient-specific models and dedicated simulations aiming at more accurately describing specific physiological states or pharmacological treatments and their effects on the EHG recorded on the abdomen. Therefore, in the future we will focus on a sensitivity analysis of the complete model, with the aim to single-out the model parameters that have the most relevant effects on the properties of the EHG signal as it is recorded non-invasively on the abdomen.
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