Simulation of biatrial conduction via different pathways during sinus rhythm with a detailed human atrial model

Simulation of biatrial conduction via different pathways during sinus rhythm with a detailed human atrial model 1 first-author 1 1 2 3 1 1 corresp ‡Corresponding Author ‡Corresponding Author †E-mail:xialing@zju.edu.cn †E-mail:xialing@zju.edu.cn xialing@zju.edu.cn College of Biomedical Engineering and Instrument Science, Zhejiang University, Hangzhou 310027, China College of Biomedical Engineering and Instrument Science, Zhejiang University, Hangzhou 310027, China Institute of Clinical Anatomy, Southern Medical University, Guangzhou 510515, China Institute of Clinical Anatomy, Southern Medical University, Guangzhou 510515, China Biological Physics, University of Manchester, Manchester M13 9PL, UK Biological Physics, University of Manchester, Manchester M13 9PL, UK In order to better understand biatrial conduction, investigate various conduction pathways, and compare the differences between isotropic and anisotropic conductions in human atria, we present a simulation study of biatrial conduction with known/assumed conduction pathways using a recently developed human atrial model. In addition to known pathways: (1) Bachmann’s bundle (BB), (2) limbus of fossa ovalis (LFO), and (3) coronary sinus (CS), we also hypothesize that there exist two fast conduction bundles that connect the crista terminalis (CT), LFO, and CS. Our simulation demonstrates that use of these fast conduction bundles results in a conduction pattern consistent with experimental data. The comparison of isotropic and anisotropoic conductions in the BB case showed that the atrial working muscles had small effect on conduction time and conduction speed, although the conductivities assigned in anisotropic conduction were two to four times higher than the isotropic conduction. In conclusion, we suggest that the hypothesized intercaval bundles play a significant role in the biatrial conduction and that myofiber orientation has larger effects on the conduction system than the atrial working muscles. This study presents readers with new insights into human atrial conduction. Cardiac electrophysiology Conduction Human atrial model Modeling Introduction 1 Biatrial conduction during sinus rhythm (SR) has been widely studied (Anderson et al., 1979 1979 ; Boineau et al., 1988 1988 ; Bromberg et al., 1995 1995 ; Anderson et al., 2000 2000 ; Burashnikov et al., 2004 2004 ; Anderson and Cook, 2007 2007 ), and the reported conduction pathways can be summarized as Bachmann’s bundle (BB), limbus of fossa ovalis (LFO), and coronary sinus (CS) (de Ponti et al., 2002 2002 ; Markides et al., 2003 2003 ; Lemery et al., 2004 2004 ; Jurkko et al., 2010 2010 ). The experimental data showed that the BB is the most prominent pathway, but in pathologic conditions the ratio of BB conduction decreases significantly. Jurkko et al. ( 2010 2010 ) reported the quantitative measurements of the proportion of conduction pathways in the control and atrial fibrillation (AF) patients, and showed that the ratio of BB conduction decreased from 67% to 54% in AF patients. The quantitative biatrial conduction via BB has been investigated by many groups (de Ponti et al., 2002 2002 ; Markides et al., 2003 2003 ; Lemery et al., 2004 2004 ; 2007 2007 ; Jurkko et al., 2009 2009 ; Tapanainen et al., 2009 2009 ), and the results are not significantly different between the groups. However, the quantitative biatrial conduction with other pathways like CS, LFO, or their combination has been rarely reported. Recently, magnetocardiography (MCG) has been used to measure the time of biatrial conduction via different pathways, and some novel and interesting results have been presented (Jurkko et al., 2009 2009 ; 2010 2010 ; Tapanainen et al., 2009 2009 ). Tapanainen et al. ( 2009 2009 ) reported that when conduction was via BB (either through a single route or as part of combined routes), the total left atrial activation was significantly shorter, and when LFO was involved in the biatrial conduction, the left atrial activation started earlier but the duration of left atrial activation lasted longer. Furthermore, they showed that when CS was the only biatrial conduction pathway, the total atrial activation time was less than that via BB. However, such a phenomenon has not been reproduced by other researchers. In addition to vast experimental data, many computer models have been used to study biatrial conduction (Harrild and Henriquez, 2000 2000 ; Zozor et al., 2003 2003 ; Seemann et al., 2006 2006 ); however, the previous simulation study always involved only one or two conduction pathways. Harrild and Henriquez ( 2000 2000 ) applied a detailed human atrial anatomy model to simulate atrial excitation conduction, in which the BB and LFO were considered. Seemann et al. ( 2006 2006 ) used a visible female dataset (Sachse et al., 2000 2000 ) to simulate biatrial conduction, and only the BB was taken into account. Since the experimental data showed that more than one pathway existed in biatrial conduction, and that the conduction patterns may vary when different pathways were involved in biatrial conduction, the model, including only one or two pathways, seems insufficient to study the effect of varied pathways. New experimental results have shown that when CS or LFO was involved in biatrial conduction, the earliest activation happened in left atrium (LA) and the total left atrial activation time had large differences. To better understand these results and to investigate the effect of different conduction pathways, we present a detailed simulation study of the various conduction pathways in biatrial conduction. In addition to the known pathways: (1) BB, (2) LFO, and (3) CS, we also assumed that there existed two fast conduction bundles that connected the crista terminalis (CT), LFO, and CS, which were reported by the previous anatomic evidence of human heart (Wang et al., 1995 1995 ; Ho et al., 2002a 2002a ). The simulation is carried out using a detailed anatomic biatrial model incorporated with realistic measured fiber orientation. The interesting phenomenon reported by Jurkko ( 2009 2009 ) is well explained based on the simulation results. Furthermore, the differences in conduction patterns between isotropic and anisotropic conductions were also compared. Materials and methods 1 Cardiome-CN human atrial anatomic model 2 The heart specimen in this study was collected from a healthy adult male in Zhujiang Hospital, Southern Medical University, China, with approval from the ethics committee at the Southern Medical University and following the Chinese law of heart research. The heart specimen was scanned using spiral computerized tomography (Philips/Brilliance 64, Netherlands). The size is 512 pixels×512 pixels, and the spatial resolution is 0.3574 mm×0.3574 mm×0.33 mm. The software ScanIP (Simpleware Inc., Bradninch Hall, UK) was used to segment and reconstruct the heart from the scanned images. Limited by the images’ resolution, the fast conduction bundles in the atria cannot be directly obtained from the images, therefore, we used Mimics (Materialize Inc., Leuven, Belgium) to construct the conduction system referred to in the previous publications about human atrial anatomic data (Anderson et al., 1979 1979 ; 2000 2000 ; James, 2002 2002 ; Sanchez-Quintana et al., 2002 2002 ; Lemery et al., 2003 2003 ; Hucker et al., 2008 2008 ; Saremi et al., 2008 2008 ; Ho and Sanchez-Quintana, 2009 2009 ). The detailed method used to obtain and construct the heart model can be found in Deng et al. ( 2012 2012 ). The final obtained anatomical structure of the Cardiome-CN human atrial heart is displayed in Fig. 1a 1a . The conduction bundles are illustrated in Fig. 1b 1b , and the relative location of the bundles in atria is shown in Fig. 1c 1c . (a) Posterior view of the atria; (b) Conduction bundles in the atria; (c) Transparent display of the conduction bundles and atrial muscles; (d) Fiber orientation within the atrial anatomic model in the anterior view; (e) Fiber orientation within the atrial anatomic model in the posterior view; (f) The top image represents the inclination angle of one cross section of the atria in (a), the bottom image is the transverse angle of the same cross section. The 3D orientation of the atria in each figure is given in the corner, and the space scale is also listed; the following figures use the same space scale. PV: pulmonary veins; SVC: superior vena cava; IVC: inferior vena cava; LAM: left atrial muscles; RAM: right atrial muscles; LPM: left pectinate muscles; RPM: right pectinate muscles; BB: Bachmann’s bundle; SAN: sinoatrial node; CS: coronary sinus; FO: fossa ovalis; CT: crista terminalis In this model the sinoatrial node (SAN) includes the center and periphery parts and locates at the superior poster lateral wall of the right atrium (RA); its size is about 10 mm×4 mm×1 mm which is based on experimental data (Anderson et al., 2009 2009 ; Fedorov et al., 2010 2010 ). There is little quantitative data about CT and pectinate muscles (PM), so in this paper, we used qualitative description of the position and anatomic structure to construct them (Sanchez-Quintana et al., 2002 2002 ; Ho and Sanchez-Quintana, 2009 2009 ). In our model, consistent with experimental data, the length of the BB is 14.7 mm, the maximum anteroposterior diameter is 4.5 mm, and the maximum superoinferior diameter is 3.7 mm (Lemery et al., 2003 2003 ; Saremi et al., 2008 2008 ). The diameter of the fossa ovalis (FO) is 16 mm, and the shortest distance to the BB is 16 mm, which is a little longer than the experimental data (Lemery et al., 2003 2003 ). The starting position and opening diameter (7.8 mm) of the CS are based on the original images. However, because the site of left atrial insertion cannot be distinguished in the images, we set the distance from CS ostium to left atrium as 10 mm, based on the anatomic data (Chauvin et al., 2000 2000 ; Matsuyama et al., 2010 2010 ). It is worth mentioning that in this model two intercaval bundles are included: the first connects the CT and the antero-superior rim of FO, and the other connects the CT and CS. In our simulation, the excitation from SAN can conduct to CT, but not atrial muscles, since Fedorov et al. ( 2010 2010 ) showed that the SAN was functionally insulated from the surrounding atrial myocytes except for a limited number of exit sites. BB can conduct the excitation from RA to LA and it couples with atria only at the two ends, as shown by Harrild and Henriquez ( 2000 2000 ). CT and PM can spread to atrial muscles, and vice versa, but the anisotropy is set as 9:1 (Henriquez et al., 1996 1996 ; Harrild and Henriquez, 2000 2000 ; Seemann et al., 2006 2006 ). The properties of the two intercavel bundles are set as for those of CT and PM. Fiber orientation of the atria 2 Due to the complexity of atrial myoarchitecture, there are few studies that quantitatively describe atria fiber orientation and only a small number of qualitative anatomical studies are available (Wang et al., 1995 1995 ; Ho et al., 1999 1999 ; 2002a 2002a ; 2002b 2002b ; Anderson and Cook, 2007 2007 ; Ho and Sanchez-Quintana, 2009 2009 ). In order to obtain the fiber orientation, some atrial muscles were peeled off along the fiber orientation after being scanned by a computerized tomography scanner, and then the heart was scanned by the 3D laser scanner (RealScan USB Scanner model 200, 3D Digital Corporation, USA). Scan spatial resolution is 0.01 mm and the total point of the epicardium is about 0.3 million. Geomagic software (Geomagic Inc., Davis Drive, USA) was used to trace the fiber orientation. The original data for myocardial fiber orientation was from the 3D laser scanned pictures of the human heart. Because the two layers of data were not obtained at the same coordinate, registration was carried out using the method we previously described (Deng et al., 2009 2009 ). After registration, fiber orientations of endocardia and epicardia were set at the same coordinate allowing the data to be used to construct the whole atrial fiber orientation. Figs. 1d and 1e 1d and 1e show all the measured points along with fiber orientation data but they only contain the surface fiber orientation of the atria, and the remaining atrial muscles do not have the fiber orientation, so based on these measured atrial fiber data, the interpolation process was applied to all atrial points as follows. First the fiber data was projected onto the anatomic model, and then the points closest to those with fiber data were identified and assigned corresponding fiber data. For the fiber orientation of fast conduction bundles, we used Mimics software to depict vectors along the principal axis of the conduction system (data not shown), similar to the method used by Seemann et al. ( 2006 2006 ). Once the fiber orientation of conduction bundles was obtained, fiber orientation of whole atria was completed. In simulation, we used two angles to represent a vector as demonstrated by previous researchers (Rijcken et al., 1999 1999 ; Rohmer et al., 2007 2007 ). Fig. 1f 1f shows the inclination angle and the transverse angle of one cross section of the atria; the color represents the degree of the angle. Atrial cell models 2 The atria include many kinds of cells, which add to the variability of atrial cellular models. The SAN should be set up first, though it is difficult to measure normal human SAN data. Recently, by investigating the expression of ion channels in the human SAN (Chandler et al., 2009 2009 ), a human SAN action potential (AP) model was developed based on the human atrial cell model (Courtemanche et al., 1998 1998 ). Since most of the ionic parameters in the AP model come from experimental data, the electrical properties of SAN are likely to be realistically reflected, thus it was used in our simulation (Fig. 2 2 ). In our simulation, the frequency of the autorhythm of central SAN cell is 1.19 Hz, the maximum diastolic potential of the central cell is −50.82 mV, and the overshoot is 12.29 mV. As was demonstrated previously, the peripheral cell does not have autorhythmicity (Chandler et al., 2009 2009 ); its resting potential is −78.37 mV, and the overshoot is 5.77 mV. (a) AP of central SAN; (b) AP of peripheral SAN, atrial musle (AM), crista terminalis (CT), and pectinate muscles (PM) Experimental data has shown that the electrophysiological properties of atrial tissue from different anatomical regions of dogs were different (Feng et al., 1998 1998 ; Burashnikov et al., 2004 2004 ). The parameters used in this study to simulate the electrical propagation of atria were based on those used in previous work (Courtemanche et al., 1998 1998 ; Feng et al., 1998 1998 ; Seemann et al., 2006 2006 ). Although Duytschaever et al. ( 2002 2002 ) reported that in goat the absolute refractory period at middle part of BB was much longer than that at RA and LA, and Feng et al. ( 1998 1998 ) descried that the AP duration in CT was longer than that in atrial muscles, the APs of BB and CT were both longer than that in the atrial muscles, and they are both the fast conduction bundles whose conduction speed is much higher than that of the atrial muscles, a cell model of BB has not yet been developed. Therefore, in this study, the cell model of CT was used to replace the BB. Since the intercaval bundles are made up of atrial cells, the AP of atria (Courtemanche et al., 1998 1998 ) was applied to them in our simulation. Numerical analysis of excitation conduction model 2 The mono-domain equation is used to simulate the excitation conduction, which is expressed as follows (Whiteley, 2006 2006 ): $$\frac{{\partial {V_{\text{m}}}}}{{\partial t}} = \nabla (D\cdot\nabla {V_{\text{m}}}) - ({I_{{\text{ion}}}} + {I_{{\text{applied}}}})/{C_{\text{m}}},{\text{ }}D = {G_{\text{i}}}/({S_{\text{v}}}\cdot{C_{\text{m}}}),$$ where S v is the surface volume ratio of cells (μm −1 ), with the value of 4 μm −1 . Since these cell models were based on the model of Courtemanche et al. ( 1998 1998 ), all cell types in this study have the same S v values. C m is the specific capacitance (pF), G i is the bulk intracellular conductivity (mS/cm), V m is the transmembrane potential (mV), I applied is the transmembrane stimulating current density and I ion is the sum of all transmembrane ionic currents (pA/pF). In this study, we used the finite difference method to calculate Eq. ( 1 1 ), because of its simplicity and suitability for the parallel computation. Simulation protocol 2 In the following simulations, the anisotropic conduction situation was investigated first. Simulation via BB has been extensively investigated by many groups (Harrild and Henriquez, 2000 2000 ; Seemann et al., 2006 2006 ). Furthermore, the developed model in this study can be validated by comparing the corresponding simulation results. Conduction via CS, LFO, and the combination of BB, CS, and LFO were investigated next. In these conduction pathways (excluding BB), we also compared the two situations: with or without the two intercaval bundles connecting CT, LFO, and CS. The obtained simulation conduction patterns were explained in detail in each case. Finally, the isotropic conduction via BB was also investigated as an example of checking the difference between the anisotropic and isotropic conductions. Experimental data showed that the anisotropy in the atria was not obvious (Hansson et al., 1998 1998 ) and that the longitudinal and transversal ratio varied from 1.0 to 1.6 when cycle length was 160 ms (Gray et al., 1996 1996 ). In order to study how the anisotropy affects the electrical conduction, the longitudinal and transverse ratio was set as 1.3, a midrange value between 1.0 and 1.6. With these settings, the conduction in the atria for both isotropic and anisotropic conductions was in accordance with the clinical data (Lemery et al., 2004 2004 ; 2007 2007 ; Jurkko, 2009 2009 ). The details about these simulations were presented in the following sections. Results 1 Normal electrical conduction via BB 2 The simulation results for the conduction pattern via BB without fast bundles are displayed in Fig. 3 3 , and the associated parameter settings are listed in the first row of Table 1 1 . From the excitation sequence as shown in Fig. 3 3 , it can be seen that the earliest activation area in RA was located in the center of the SAN, followed by the periphery (after about 5 ms), then the wave spread arterially to the right atrial appendage (RAA) and upward to the roof of the superior vena cava (SVC). Meanwhile, the wave quickly spread downwards to the atrioventricular region along CT and leftwards to LA along BB (arrows 1 and 2 in Figs. 3a and 3b 3a and 3b ). The excitation also conducted to the PM (arrow 2 in Fig. 3c 3c ) and it can be seen that the conduction speed was slower from CT to PM than from CT itself and the wavefront was almost elliptic. At 30 ms, the wave spread to the junction of LA and RA in the posterior wall and to the upper part of RAA; most parts of the SVC were excited. At the same time, the excitation conducted to LA via BB (the solid circle in Fig. 3a 3a ), then the wave spread in all directions. At 51 ms, the muscular rim of FO in RA was excited along with half of the RA. The activation descended the lateral wall, posterior wall, and septum in a uniform manner. At 81 ms, RA was excited completely. The last activation site was in the front of CS (the asterisk in Fig. 3c 3c ). (a–c) are the excitation conduction sequences. The red lines are the isochrones, the black solid circle is the left insertion of BB and is the earliest activation position in LA. The arrows represent the wave spread directions, and they are distinguished by different numbers. The black asterisk in (b) is the last activation site of LA, the black asterisk in (c) is the last activation site of RA. The following figures used the same representations (Note: for interpretation of the references to color in this figure legend, the reader is referred to the web version of this article) BB: Bachmann’s bundle; PM: pectinate muscles; FO: fossa ovalis; LFO: limbus of the fossa ovalis; CS: coronary sinus; CT: crista terminalis; RAM: right atrial muscles; LAM: left atrial muscles; IB1: intercaval bundle connets CT and LFO; IB2: intercaval bundle connets CT and CS; IB: the combination of IB1 and IB2; Iso: isotropic conduction 2 Conduction pathway 10 Conductivity (mS/cm)/average conduction speed (cm/s) BB CT PM LFO FO CS LAM RAM IB1 IB2 Solitary BB without IB 107.2/113 125.1/115 125.1/123 0/0 0/0 0/0 19.7/76 25/85 0/0 0/0 Solitary LFO with IB 0/0 125.1/115 125.1/123 3.5/30 7.15/45 0/0 14.3/62 25/85 120/132 0/0 Solitary LFO without IB 0/0 125.1/115 125.1/123 3.5/30 7.15/45 0/0 25/85 25/85 0/0 0/0 Solitary CS with IB 0/0 125.1/115 125.1/123 0/0 0/0 10.7/50 26.8/87 25/85 0/0 120/132 Solitary CS without IB 0/0 125.1/115 125.1/123 0/0 0/0 10.7/50 19.7/76 25/85 0/0 0/0 BB+LFO with IB 107.2/113 125.1/115 125.1/123 3.5/30 7.15/45 0/0 10.72/54 25/85 120/132 0/0 BB+LFO without IB 107.2/113 125.1/115 125.1/123 3.5/30 7.15/45 0/0 14.3/62 25/85 0/0 0/0 BB+CS with IB 107.2/113 125.1/115 125.1/123 0/0 0/0 10.7/50 8.94/50 25/85 0/0 120/132 BB+CS without IB 107.2/113 125.1/115 125.1/123 0/0 0/0 10.7/50 12.51/58 25/85 0/0 0/0 LFO+CS with IB 0/0 125.1/115 125.1/123 3.5/30 7.15/45 10.7/50 14.3/62 25/85 120/132 120/132 LFO+CS without IB 0/0 125.1/115 125.1/123 3.5/30 7.15/45 10.7/50 25/85 25/85 0/0 0/0 BB+LFO+CS with IB 107.2/113 125.1/115 125.1/123 3.5/30 7.15/45 10.7/50 8.22/48 25/85 120/132 120/132 BB+LFO+CS without IB 107.2/113 125.1/115 125.1/123 3.5/30 7.15/45 10.7/50 14.3/62 25/85 0/0 0/0 Solitary BB without IB iso 53.61/110 35.74/115 35.74/123 0/0 0/0 0/0 19.7/76 25/85 0/0 0/0 For LA, after it was activated via BB, the wave in the roof of LA conducted in all directions. At 81 ms, the anterior parts of LA and the left atrial appendage (LAA) were excited completely and there were three different wavefronts conducting towards the posterior wall: the first one came from the roof of LA (arrow 6 in Figs. 3a–3c 3a–3c ); the second one from the LAA (arrow 5 in Figs. 3a–3c 3a–3c ); and the third one from the septum of LA (arrow 7 in Figs. 3a–3c 3a–3c ). After approximately 14 ms, the three wavefronts merged together and conducted to the area near mitral annular (arrows 5 and 7 in Figs. 3b and 3c 3b and 3c ). At 109 ms, the excitation wave ended in the middle of the bottom LA. The time of critical positions and the excitation time in atria were summarized in the first row of Table 2 2 . BB: Bachmann’s bundle; FO: fossa ovalis; LFO: limbus of the fossa ovalis; CS: coronary sinus; IB1: intercaval bundle connets CT and LFO; IB2: intercaval bundle connets CT and CS. IB: the combination of IB1 and IB2; Iso: isotropic conduction; LA: left atrium; RA: right atrium 2 Conduction pathway 6 Earliest activation time (ms) 3 Total activation time (ms) BB left BB right FO left FO right CS left CS right LA RA Whole atria Solitary BB with IB 30 18 79 81 109 Solitary LFO with IB 30 26 90 79 120 Solitary LFO without IB 55 51 71 81 126 Solitary CS with IB 41 34 68 81 109 Solitary CS without IB 82 75 80 81 162 BB+LFO with IB 30 18 30 26 89 79 119 BB+LFO without IB 30 18 55 51 87 81 117 BB+CS with IB 30 18 41 34 88 81 118 BB+CS without IB 30 18 82 75 88 81 118 LFO+CS with IB 30 26 41 34 85 81 115 LFO+CS without IB 55 51 82 75 71 81 126 BB+LFO+CS with IB 30 18 30 26 41 34 86 79 116 BB+LFO+CS without IB 30 18 55 51 82 75 87 81 117 Solitary BB without IB iso 32 17 77 82 109 Normal electrical conduction via LFO 2 The simulation results for this condition with fast bundles are displayed in Figs. 4a–4c 4a–4c and the associated parameters are listed in the second row of Table 1 1 . It can be seen that before the wave conducted to FO, the conduction pattern in RA was similar to the BB case, but after that, the conduction patterns in the right atrial septum changed a lot. (a–c) are the excitation conduction sequences via LFO with fast bundles, (d–f) without fast bundles At 30 ms, the muscular rim of FO in RA was excited, then the wavefront conducted upwards to meet the wavefront from SVC, forwards to activate the anterior part of RA (arrow 4 in Figs. 4a and 4c 4a and 4c ), and downwards to the isthmus (arrow 5 in Fig. 4b 4b ). At 43 ms, the FO was almost excited and the wavefronts from LFO and SVC began to merge. Later, the CS began to excite and the wavefront spread to the isthmus. At 78 ms the entire RA was excited and the last activation site was the vicinity of atrioventricular ring (AVR). For RA, the anterior wall was activated by the wavefronts from SAN and FO, but the lateral wall and most of the posterior wall were activated by the wavefront from SAN. When the wavefront conducted to LA via LFO, it began to spread in all directions (arrows 6, 7, and 8 in Fig. 4c 4c ): anterior wall, posterior wall, and the upper part of septum, then the wavefronts began to activate the anterior and posterior walls. At 90 ms, half of the LAA lateral wall and the left atrial posterior wall had been excited. The excitation sequence showed that the LAA was activated by the wavefront from the anterior wall and that the roof of LA and lateral wall were activated by the wavefronts from the anterior wall and posterior wall (arrows 8 and 9 in Fig. 4b 4b ). At 120 ms, the last activation site-left superior pulmonary vein (LSPV) was excited. The time of critical positions and the excitation time in atria were summarized in the second row of Table 2 2 . Figs. 4d–4f 4d–4f show the simulation results without the fast bundles. Compared with the FO conduction with fast conduction bundle, the right atrial conduction pattern was similar to the conduction via BB in that they both conducted the excitation from SAN downwards to the atrioventricular node (AVN) (Figs. 4d and 4e 4d and 4e ), but when the fast conduction bundle was included in the FO conduction, the wavefront from the FO spread upward to collide with the wavefront that originated from SAN, and downward to the cavotricuspid isthmus to meet the wavefront from CT, the last activation site located at the bottom of posterior and lateral walls of RA. Although the total atrial activation time (120 ms vs. 126 ms) and activation pattern in LA were similar between the two conduction types, the left atrial activation time and the conductivites differed greatly. Without fast bundle, the whole left atrial activation time was 71 ms, which was far slower than the conduction with fast bundle (90 ms). The activation time is shown in the third row in Table 2 2 . Normal electrical conduction via CS 2 For this condition, the simulation results with fast bundles are displayed in Figs. 5a–5c 5a–5c , and the associated parameter settings are listed in the fourth row of Table 1 1 . At 41 ms, the excitation conducted from SAN to LA where the distal CS and LA joined via the CS, and then the wavefront spread to the roof of LA radically (arrows 4, 5, and 6 in Figs. 5a–5c 5a–5c ). The RA activation pattern was similar to that of the BB case. (a–c) are the excitation conduction sequences via CS with fast bundles, (d–f) without fast bundles At 55 ms, the wavefront conducted to the floor of FO and then it spread to the anterior wall of LA and upward to the roof of LA (arrows 5 and 6 in Figs. 5a and 5b 5a and 5b ). At 90 ms, the wavefronts from the anterior and lateral walls met at the floor of LAA then they conducted upward to the lateral wall of LAA and to the roof of LA (arrows 4, 7, and 8 in Figs. 5a and 5c 5a and 5c ). At 104 ms, the wavefronts from the roof and floor of LAA began to merge at the lateral wall and the wavefronts from the anterior and posterior walls of LA began to merge at the roof of LA. The last activated area was the tip of LAA and right superior pulmonary vein (RSPV) which occurred at 109 ms. The activation time is shown in the fourth row of Table 2 2 . The simulation results without the intercaval bundle (IB2 in Table 1 1 ) are displayed in Figs. 5d–5f 5d–5f , and the associated conductivities are listed in the fifth row of Table 1 1 . Compared with the CS conduction with fast bundle, the RA conduction pattern was slightly different, while the LA conduction pattern was significantly different (Figs. 5d–5f 5d–5f ). In our simulation, the conduction time via CS without fast bundle was 162 ms, which was much longer than that shown by Jurkko et al. ( 2009 2009 ; 2010 2010 ). The associated conduction speed was approximately 70 cm/s, and the left atrial conduction time was 80 ms (the earliest activation time of LA was 82 ms), which was close to the left atrial conduction time (~80 ms) shown by other studies (Roithinger et al., 1999 1999 ; de Ponti et al., 2002 2002 ; Lemery et al., 2007 2007 ). If the conductivity of LA was increased to 28.6 mS/cm, the associated conduction speed was approximately 90 cm/s, but the activation time of whole atria was still more than 140 ms. The activation time is shown in the fifth row of Table 2 2 . Normal electrical conduction via BB and LFO 2 The conductivity pattern for the simulation with the intercaval bundle that connects the origins of CT and LFO is listed in the sixth row of Table 1 1 and the results are displayed in Figs. 6a–6c 6a–6c . In this case, before the merger of the two wavefronts from BB and LFO, the conduction pattern in LA was the combination of BB and LFO. At 57 ms, the wavefront from BB began to merge with the wavefront from LFO at the anterior middle wall (arrows 8 and 11 in Fig. 6a 6a ), and then the wavefront conducted to the lateral wall of LA and medial wall of LAA (arrow 11 in Fig. 6a 6a ). At 92 ms, the wavefront from the left atrial posterior wall began to merge with the wavefront from the roof of LA (arrows 6 and 10 in Fig. 6b 6b ). Meanwhile, the wavefront from the roof of LA merged with the wavefront from the anterior wall at the conjunction of LAA and LA (arrows 11 and 12 in Fig. 6c 6c ). (a–c) are excitation conduction sequences via BB and LFO with fast bundles, (d–f) without fast bundles At 106 ms, the wavefronts from the posterior wall and the roof of LA ended at the vicinity of the left inferior pulmonary vein (LIPV), and the wavefronts from the posterior and lateral walls merged and ended at the lateral wall close to the AVR at 119 ms (arrows 10, 11, and 12 in Figs. 6b and 6c 6b and 6c ). The activation time is shown in the sixth row of Table 2 2 . The results of the simulation when the intercaval bundle (IB1 in Table 1 1 ) was not introduced are displayed in Figs. 6d–6f 6d–6f , and the associated conductivities are listed in the seventh row of Table 1 1 . Since both the BB and LFO can conduct the excitation, the wave spread in the LA was the combination of the conductions via BB and FO. Compared with the conduction with intercaval bundle, the merging time of wavefronts from BB and FO was longer (57 ms vs. 65 ms) and the merging site (close to the left atrial septum) was lower. When intercaval bundle was introduced, the anterior wall was activated by both the wavefronts from BB and FO and most of the posterior wall was activated by the wavefront from BB, but if the intercaval bundle was not introduced, the most of anterior wall was activated by the wavefront from BB, and the posterior wall was activated by both wavefronts from BB and FO. At 94 ms, the wavefronts from the left atrial septum and lateral wall had merged with the wavefront from the upper part of LA. After 6 ms, the three wavefronts merged together at the middle part of the posterior wall of LA (arrows 3, 5, and 9 in Figs. 6e and 6f 6e and 6f ). At 117 ms, the wavefront conducted to the posterolateral wall under the left pulmonary vein and ended it the atrial depolarization. The activation time is shown in the seventh row of Table 2 2 . Normal electrical conduction via BB and CS 2 The conductivity pattern with the intercaval bundle that connects the origins of CT and CS is listed in the eighth row of Table 1 1 and the results are displayed in Figs. 7a–7c 7a–7c . In this case, before the merging of the two wavefronts from BB and CS, the conduction pattern in LA was a combination of BB and CS. At 86 ms, the wavefront from BB began to merge with the wavefront from CS at the anterior wall close to the septum (arrows 7 and 11 in Fig. 7c 7c ), and then the wavefront conducted to the lateral wall of LA. At 92 ms, the wavefront from BB merged with the wavefront from the lateral wall of LAA at the conjunction of LAA and LA (arrows 7 and 8 in Fig. 7c 7c ) and then it conducted upward to meet the wavefront from the roof of LA at the middle part of the posterior wall at 109 ms (arrows 5 and 11 in Fig. 7b 7b ). Meanwhile, the wavefront from the lower part of the posterior wall merged with the wavefront from LAA at the lower part of the lateral wall of LA (arrows 8 and 12 in Figs. 7a–7c 7a–7c ). The last activation site was the right inferior pulmonary vein at 118 ms. The activation time is shown in the eighth row of Table 2 2 . (a–c) are excitation conduction sequences via BB and CS with fast bundles, (d–f) without fast bundles The simulation results without the intercaval bundle (IB2 in Table 1 1 ) are displayed in Figs. 7d–7f 7d–7f , and the associated conductivities are listed in the ninth of Table 1 1 . Compared with the conduction with intercaval bundle, the merging time of wavefronts from BB and CS was much longer (86 ms vs. 97 ms) and the merging site was much lower (at the floor of FO). When intercaval bundle was introduced, the lateral and posterior walls were activated by both wavefronts from BB and CS, but if not introduced, the lateral wall was activated by the wavefront from BB and only about one third of the posterior wall was activated by the wavefront from CS. The last activation sites were different too; one located at the posterior wall close to the atrioventricular ring and the other located at the lateral and the posterior walls close to LIPV. The activation time is shown in the ninth row of Table 2 2 . Normal electrical conduction via LFO and CS 2 Pattern conductivities when the two intercaval bundles that connect the origins of CT, CS and LFO were introduced are listed in the 10th row of Table 1 1 and the results are displayed in Fig. 8 8 . In this case, before the two wavefronts from LFO and CS merged, the conduction pattern in LA was a combination of LFO and CS. (a–c) are the excitation conduction sequences via LFO and CS with fast bundles At 54 ms, the wavefronts from the LFO and CS began to merge at the floor of FO then the wavefront conducted upward to the roof of LA and leftwards to the posterior wall (arrows 11 and 12 in Fig. 8c 8c ). At 91 ms, the two wavefronts from the posterior and anterior walls began to merge. Meanwhile, the wavefronts from the roof and floor of LAA had merged at the lateral wall of LAA (arrows 8 to 11 in Figs. 8a–8c 8a–8c ). At 102 ms, the lateral wall of LA had been activated and meanwhile, the two wavefronts from the root of atria and posterior wall had already merged and conducted towards the left pulmonary vein (arrows 6 and 12 in Figs. 8a and 8b 8a and 8b ). After 14 ms, the entire atria had been activated and the last activation site was the LSPV. The activation time is shown in the 10th row of Table 2 2 . Pattern conductivities when the two intercaval bundles (IB in Table 1 1 ) were not introduced are listed in the 11th row of Table 1 1 . The conduction pattern via FO and CS was same as that via unique FO since the wavefront via CS conducted to the junction of distal CS and LA at the same time as the wavefront via FO. The activation time is shown in the 11th row of Table 2 2 . Normal electrical conduction via BB, LFO, and CS 2 Pattern conductivities when the two intercaval bundles which connect the origin of CT, CS, and LFO were introduced are listed in the 12th row of Table 1 1 and the results are displayed in Fig. 9 9 . In this case, before the three wavefronts from BB, LFO, and CS merged, the conduction pattern in LA was a combination of BB, LFO, and CS. (a–c) are the excitation conduction sequences via BB, LFO, and CS with fast bundles At 56 ms, the two wavefronts from BB and LFO began to merge at the anterior wall then the wavefront conducted leftward to the lateral wall of LA and LAA (arrows 7 to 11 in Figs. 9a and 9c 9a and 9c ). Meanwhile, the wavefront from LFO began to merge with the wavefront from CS at the floor of FO and then conducted to the posterior wall (arrows 12 and 16 in Fig. 9c 9c ). At 95 ms, the wavefront from the roof of LA began to merge with the wavefront from the posterior wall (arrows 6 and 12 in Fig. 9b 9b ) and meanwhile, the wavefront from the roof of LA had merged with the wavefront from the anterior wall at the conjunction of LAA and LA (arrows 9 and 11 in Fig. 9c 9c ). At 107 ms, the three wavefronts from the roof of atria, the posterior wall, and the lateral wall had merged together, and ended at the vicinity of LIPV at 112 ms (arrows 9, 12, and 13 in Fig. 9b 9b ). The last activation site was the lateral wall close to the AVR and the activation time of the whole heart was 116 ms (arrows 9, 11, and 13 in Fig. 9c 9c ). The activation time is shown in the 12th row of Table 2 2 . The conductivity pattern when the two intercaval bundles (IB in Table 1 1 ) were not included is listed in the 13th row of Table 1 1 and the activation time is shown in the 13th row of Table 2 2 . The conduction pattern via BB, FO, and CS was the same as that via BB and FO. Isotropic conduction via BB 2 For the isotropic conduction, we only simulated the conduction via BB, since there is no major difference between the conductions via BB and other pathways. The associated conductivities are listed in the 14th row of Table 1 1 , and the results are shown in Fig. 10 10 . The activation time is shown in the 14th row of Table 2 2 . (a–c) are the excitation conduction sequences For the conduction pattern, the isotropic and anisotropic conductions had only small differences such as the left atrial activation time and the total conduction time of LA and RA. The activation time of FO in isotropic conduction was 55 ms, which is longer than that of the anisotropic conduction (51 ms). The biggest difference is that the conductivities of the conduction system in atria were much smaller for the isotropic conduction than for the anisotropic conduction. Discussion 1 In this study, we presented simulation of the biatrial conduction via various pathways using a detailed anatomic atrial model with realistically measured myofiber orientation data. The pathways investigated in this study include BB, CS, LFO, and the two intercaval bundles, which connect CT, LFO, and CS. For the first time, the two assumed intercaval bundles were studied in detail and the anisotropic conductions with these pathways were compared. The atrial conduction time and the activation sequences in our simulation were generally in agreement with clinical data (Boineau et al., 1988 1988 ; de Ponti et al., 2002 2002 ; Lemery et al., 2004 2004 ; 2007 2007 ; Jurkko et al., 2009 2009 ; 2010 2010 ; Tapanainen et al., 2009 2009 ). Tables 3 3 and 4 4 list the experimental data from different studies. BB: Bachmann’s bundle; FO: fossa ovalis; CS: coronary sinus; LA: left atrium; RA: right atrium Value is combined earliest activation time 2 2 Reference 6 Earliest activation time (ms) 3 Total activation time (ms) BB left BB right FO left FO right CS left CS right LA RA Whole atria 2 Lemery et al., 2004 2004 41±18 19±12 43±17 24±9 80±11 81±18 120±24 2 de Ponti et al., 2002 2002 30±6 39±5 42±5 81±10 92±10 105±9 2 Markides et al., 2003 2003 26±2 65±4 134±3 2 Lemery et al., 2007 2007 31±13 52±15 45±14 95±34 84±17 93±17 116±18 11 Jurkko, 2009 2009 4  Study III BB 41±14 81±16 121±15 CS 34±9 80±9 114±13 FO 29±11 93±7 124±11 Combined 6 35±10 * * 84±12 118±11 4  Study IV BB 34±8 81±10 116±10 CS 40±11 62±14 102±3 FO 32±4 91±11 123±11 Combined 6 32±10 * * 88±14 120±13 BB: Bachmann’s bundle; PM: pectinate muscles; CT: crista terminalis; RAM: right atrial muscles; LAM: left atrial muscles Tissue name Conduction speed (cm/s) Reference BB 92–167 Dolber and Spach, 1989 1989 CT 70–130 Boineau et al., 1980 1980 ; Boineau, 1985 1985 PM 117–154 Hayashi et al., 1982 1982 LAM 68–103 Hansson et al., 1998 1998 70.2±9.9 Kojodjojo et al., 2006 2006 RAM 68–103 Hansson et al., 1998 1998 77.0±10.8 Kojodjojo et al., 2006 2006 In our study, the spatial resolution of atrial anatomic model is approximately 0.35 mm, which is finer than other models. For example, the spatial resolution of the model developed by Lorange and Gulrajani ( 1993 1993 ) is 1 mm and that of Lu et al. ( 1993 1993 ) is 1.5 mm. A detailed human atrial anatomy model was constructed by Harrild and Henriquez ( 2000 2000 ) with a mean interelement distance is 0.55 mm; however, the drawback of that model is that it was constructed from reports, not real human atria. In a human atrial monolayer model constructed using MRI (Blanc et al., 2001 2001 ), the mean edge length of the epicardial meshes in high quality is 0.434 mm. The spatial resolution of the Visible Female datasets is 0.33 mm (Sachse et al., 2000 2000 ) which is finer than that of our model, but is limited in that it cannot provide the fiber information that is important in anisotropic simulation. Although the fiber orientation in our model is monolayer and its spatial resolution is not high, it does present improvements in atrial simulation. Until recently, a set of quantitatively measured data of whole atria has been lacking, limiting the ability to conduct realistic anisotropic conduction simulation in normal conditions. In addition, our model provides realistic fiber orientation, which improves on past models. Assignment of conductivities 2 In our simulation, our conductivities were consistently larger than that published by Harrild and Henriquez ( 2000 2000 ) for two reasons. First, the anisotropy ratio in our model was larger (9.00 vs. 7.54 for CT and PM, 1.3 vs. 1.0 for atrial muscles). Since the spatial resolution can affect the conductivities, it is expected that higher spatial resolution will quicken the conduction speed (ten Tusscher et al., 2004 2004 ). In our model, the spatial resolution was 0.3574 mm, so larger conductivities should be assigned. Second, for the source sink effect, in the Harrild and Henriquez ( 2000 2000 )’s model, PM only connects to atrial muscles at their midpoints and ends but in our model the PM and atrial muscles connect with each other and the PM must excite much more atrial cells. Because this connection slows the conduction speed, larger conductivities were assigned to compensate. It is known that for LA the conduction time through different pathways varies greatly (Jurkko, 2009 2009 ), so in order to attain realistic conduction time and speed, the conductivities were also varied. For the SAN modeling, from the original paper (Chandler et al., 2009 2009 ) we can see that the paranodal area (peripheral cells in our simulation) demonstrated an intermediate pattern of expression of many ion channels similar to that seen in the periphery of the rabbit sinus node (Tellez et al., 2006 2006 ). The paranodal area, however, shows a unique pattern of expression of K + channels and accessory subunits. So the peripheral cell in Chandler et al. ( 2009 2009 ) is different from the periphery cell in Zhang et al. ( 2000 2000 ). To our knowledge, since the paper from Chandler et al. ( 2009 2009 ) is the only one that systematically studies the human SAN, more experimental data are required to validate their findings. Finally, in normal conduction, the autorhythmicity of peripheral cells in SAN is suppressed by the center cell of SAN, so autorhythmicity will have no impact on the normal biatrial conduction. Right atrial conduction 2 In normal state, the conduction via BB was the most dominant pathway (Lemery et al., 2004 2004 ; Jurkko et al., 2010 2010 ); because we set the conductivities in CS and LFO to 0, the BB was the only conduction pathway between RA and LA. In our simulation the SAN was located in the groove between the RAA and the lateral atrial wall (Anderson et al., 1979 1979 ; James, 2002 2002 ) and contained the central and peripheral parts. The center part showed a higher frequency of autorhythmicity which dominated the heart rate and only conducted the electrical activation to periphery cells. Recently, Fedorov et al. ( 2010 2010 ) showed that the SAN was functionally insulated from the surrounding atrial myocytes except for a limited number of exit sites. Therefore, in our simulation the excitation from the periphery cells can only conduct to the CT. With these settings the time of excitation from SAN to the epicardial was 7 ms and was similar to the measured clinical data ((11±18) ms) (Lemery et al., 2007 2007 ). When CT and PM were excited, the wave spread to the surrounding atrial cells. The conduction velocity in the right atrial wall was not homogenous: the velocity of the posterolateral wall with PM was 70 to 100 cm/s and the average velocity was close to 95 cm/s. These results can be explained by the fact that, in our model, the PM and atrial muscles connected with each other and the PM displayed higher conduction velocity. This phenomenon has also been reported in experimental data (Wu et al., 1998 1998 ) and another atrial model although the PM and atrial wall was set to contact at certain points (Harrild and Henriquez, 2000 2000 ). The conduction speed of the smooth septum wall and anterior wall was far slower than that of the posterolateral wall with PMs, ranging from 70 to 90 cm/s with an average value of 85 cm/s, which falls in the range of previously reported results of −68 to 103 cm/s (Hansson et al., 1998 1998 ). Left atrial conduction via BB 2 When the atrial electrical conduction was through BB, the earliest excited site was located in the anterior left atrial wall, but the earliest activation time varied. de Ponti et al. ( 2002 2002 ) reported that the value of earliest activation time of LA was (30±6) ms, while the earliest activation time was (26±2) ms in Markides et al. ( 2003 2003 ). It has previously been reported that the RA insertion of BB was activated at (41±18) ms (Lemery et al., 2004 2004 ) and that conduction over BB resulted in earliest left atrial activation endocardially at (31±13) ms (Lemery et al., 2007 2007 ). Recently, Tapanainen et al. ( 2009 2009 ) reported that if FO was involved in the atrial conduction, the earliest activation time was (31±9) ms, otherwise, the time was (39±14) ms. Another study reported that left atrial activation started at 34 ms on average (Jurkko et al., 2009 2009 ). In our simulation, the earliest activation time in LA was 30 ms, similar to another computer simulation report of 29.7 ms (Harrild and Henriquez, 2000 2000 ). There is little experimental data about the total conduction time for the BB, partly because the SAN and BB are too close and therefore it is hard to measure the early activation time in the right insertion of BB. Nevertheless, Lemery et al. ( 2004 2004 ) reported that the quantitatively measured total BB conduction time was (23±15) ms. Our simulated result was 16 ms, which was a little shorter than the average value but within the reported range. The conduction velocity of BB in our simulation was between 95 to 150 cm/s and the average velocity was 113 cm/s, falling within the range reported by Dolber and Spach ( 1989 1989 ). In our simulation, the total activation times of LA and whole atria were 79 and 109 ms, respectively. de Ponti et al. ( 2002 2002 ) reported that the durations of left atrial propagation and whole atria were (81±10) and (105±9) ms, respectively. Our study results are similar to previous reports that determined the activation times of LA and whole atria to be (80±11) and (120±24) ms, respectively (Lemery et al., 2004 2004 ), and (84±14) and (119±14) ms, respectively (Tapanainen et al., 2009 2009 ). Left atrial conduction via LFO and CS 2 The experimental data showed that the first activation time in LA via solitary CS conduction was (34±9) ms in Study III and (40±11) ms in Study IV (Jurkko, 2009 2009 ). We believe that the reason why the activation time was much earlier than that of the other experimental data (de Ponti et al., 2002 2002 ; Lemery et al., 2004 2004 ; 2007 2007 ) is that the results of the latter groups were for the atrial conduction via solitary BB or combination of BB, CS, and LFO. When the atrial conduction is via solitary CS or LFO, the earliest activation time in LA was very short (Jurkko, 2009 2009 ). In our simulation we assumed that there existed muscle bundles between the origins of CT and CS and LFO and that this would conduct excitation from SAN to CS and LFO faster than the surrounding atrial muscles. The existence of these pathways is supported by anatomical evidence. Ho et al. ( 2002a 2002a ) described that the peripheral fibers around the fossa in the antero-superior rim extend toward the origin of the terminal crest. Wang et al. ( 1995 1995 ) showed that some muscle bundles passed from the SAN to the vicinity of the FO; the authors claimed that the excitation wave may spread from the SAN to the interatrial septum along the anterior rim of the oval fossa. In addition they also showed some muscle bundles connected the origin of CT and the vicinity of CS. Tapanainen et al. ( 2009 2009 ) suggested that the earlier activation time in LA via LFO instead of BB was possible due to the extent and location of the connecting fibers and differences in conduction velocities within the atria. In addition, computer simulations have included the intercaval bundle between CT and LFO (Harrild and Henriquez, 2000 2000 ). Sun and Khoury ( 2001 2001 ) studied the electrical conduction within the inferior atrial region and showed that sinus rhythm activation initiated in superior RA at the junction of the posterior and lateral walls propagated in anterior directions and bypassed the anterior rim of the FO toward the CS. In our simulation, the geometry of the two pathways was based on general descriptions (Wang et al., 1995 1995 ; Sun and Khoury, 2001 2001 ; Ho et al., 2002a 2002a ). While detailed description of the muscle structures in the vicinities of the interatrial septum and CS was lacking, these reports contained some qualitative description and images to support their viewpoint. Nevertheless, most of anatomists agree with these points and the debate is about whether these quicker conduction muscles are made up of working atrial myocardial fibers or special myocardial cells. In our simulation we choose the geometry of the assumed pathways based on these guide lines. To our knowledge, this is the first model containing the two pathways in biatrial conduction simulation. The model by Harrild and Henriquez ( 2000 2000 ) only contained one pathway connecting the midpoint of CT and FO. Due to the lack of accurate experimental data, an accurate geometry of the two pathways cannot be obtained from our simulation, but if these two pathways had been excluded the results would have been significantly different from the experiential data (Jurkko, 2009 2009 ). We suggest that although the geometry of the two pathways may vary they should not be excluded from the model. Factors for differing activation time of LA 2 In our model, we assumed that there existed muscle bundles between the origins of CT and CS and FO and that they were composed of normal atrial muscles but that they had a high anisotropy ratio. Although our hypothesis can explain the differences between the activation time of LA and the whole atrial activation time, we cannot exclude other factors that may influence the earliest activation time of LA. First, the sinus node is normally located in a relatively limited area in the high posterolateral RA but its location can vary within a 7.5 cm×1.5 cm area along RA and caval vein junction (Boineau et al., 1988 1988 ; Cosio et al., 2004 2004 ; Lemery et al., 2007 2007 ). If SAN locates in the lower part, the conduction time from SAN to FO or CS may be shortened. Second, both the site of pace marker and the site out of the node can vary (Bromberg et al., 1995 1995 ; Fedorov et al., 2010 2010 ). Third, in the sinus node region the shift of impulse initiation alters with increasing rate upwards and with decreasing rate downwards (Boineau et al., 1988 1988 ; Fedorov et al., 2010 2010 ). In consideration of all these factors, our opinion is that the most important factor may be the existence of some bundles in the RA that can conduct the excitation faster than the working atrial myofibers. Slow conduction via LFO 2 Our simulation results and the experimental data of Jurkko ( 2009 2009 ) and Tapanainen et al. ( 2009 2009 ) showed that if the LFO with the fast bundles was part of the conduction pathway, the duration of left atrial activation lasted longer (about 90 ms in LFO conduction vs. about 80 ms in other cases in our simulation). This may be due to the functional conduction block in LA. Markides et al. ( 2003 2003 ) reported that there existed a functional conduction block in LA, which originated in the anterior interatrial groove on the anterior left atrial wall and ended at the septal mitral annulus. When LFO is the conduction pathway, the wavefront from LFO cannot conduct to the posterior wall but can only conduct to the roof of LA and then pass through the functional conduction line to activate the remaining part of LA. For BB, the left insertion is located at the anterior wall of LA (Lemery et al., 2003 2003 ; Khaja and Flaker, 2005 2005 ) and the functional conduction line was behind the anterior interatrial band (BB) (Markides et al., 2003 2003 ); therefore, it cannot block the excitation from BB. For CS, the left insertion is located behind the functional conduction line and the line close to the septal mitral annulus was less distinct or even absent so the wavefront from CS can spread to the anterior wall and posterior wall in most cases. Differences between isotropic and anistropic conditions 2 In our simulation the conductivities for BB, CT and PM in the anisotropic conduction were much greater than those in the isotropic conduction but there was almost no difference in the speed of conduction. The reason for this may be that during isotropic conduction the cells had stronger coupling and small conductivities can produce higher conduction speed. The conduction pattern was almost the same between isotropic and anisotropic conductions since the anisotropic ratio for atrial working myocardium was small. Gray et al. ( 1996 1996 ) reported that the anisotropic ratio for sheep epicardial wall was different from 1 and was 1.6 at the cycle length of stimulation (BCL)=160 ms. In our simulation, the anisotropic ratio was set to 1.3. From our simulation we can see that in the normal conduction the atrial working myocardium can be assumed to be isotropic conduction and it was used by many groups to simulate the atrial normal conduction (Harrild and Henriquez, 2000 2000 ; Seemann et al., 2006 2006 ). Koura et al. ( 2002 2002 ) reported that the anisotropic ratio of the conduction velocity increased with increasing age and that the highest ratio was more than 1.5. When the atria have structural remodeling, the fibrosis will increase (Kamkin et al., 2003 2003 ; Everett et al., 2006 2006 ; Tanaka et al., 2007 2007 ), which leads to the increase in anisotropic ratio in the atria and can make the AF easier to maintain and sustain. Therefore, the fiber orientation did not play a significant role in the normal atrial conduction but it may greatly influence wave conduction in pathologic conditions (Jacquemet et al., 2003 2003 ; Deng and Xia, 2010 2010 ). Limitations 2 Although our anatomic model contained the detailed fast conduction bundles which are important for atrial conduction, they were not constructed based on scanned computerized tomography images due to limited spatial resolution. This may have resulted in data that are inconsistent with clinical data. The fiber orientation in our simulation only contained the epicardial wall, while the anatomic dissection showed that the myoarchitecture changes from subepicardium to subendocardium in LA. Furthermore, the fiber orientation in the anterior and septum wall in our model was based on other references so the simulated results may not be very precise. Another limitation may be that the AP model of atrial cells was based on the Courtemanche et al. ( 1998 1998 ) (CRN model) rather than a newer human atrial cell model published by Maleckar et al. ( 2009 2009 ) (Maleckar model) which was based on the Nygren et al. ( 1998 1998 ) model (NYG model). 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A numerical scheme for modeling wavefront propagation on a monolayer of arbitrary geometry 10.1631/jzus.B1100339 Project supported by the National Basic Research Program (973) of China (No. 2007CB512100), the National High-Tech R & D Program (863) of China (No. 2006AA02Z307), the National Natural Science Foundation of China (Nos. 81171421 and 61101046), and the Zhejiang Provincial Natural Science Foundation of China (No. Z1080300) 2011-11-04 2012-08-09 2012-02-28 2022-01-05 Copyright © Zhejiang University and Springer-Verlag Berlin Heidelberg 2012 2012 Journal of Zhejiang University-SCIENCE B (Biomedicine and Biotechnology) 09 13