Optimal control model of tumor treatment in the context of cancer stem cell

1.   Introduction

Myocardial ischemia (MI) causes electrical alternans ^[1,2,3,4,5], which refer to beat-to-beat alternans in action potential duration (APD) or/and Ca²⁺ transient in the cellular level ^[6,7,8,9] and T wave alternans (TWA) on the electrocardiogram (ECG) in the tissue level ^[10]. Cardiac alternans have been shown to lead fatal arrhythmia, where TWA is confirmed to have promising utility in risk prediction of sudden cardiac death (SCD) and could be considered as a strong marker of arrhythmia ^[11,12,13]. To make full use of TWA, making clear of the underlying mechanism forming it is essential in clinical. As T wave reflects transmural repolarization dispersion, TWA are thought to arise from repolarization differences in tissue heterogeneity ^[14,15], conduction velocity ^[16,17] and cellular alternans ^{[3,13,18,19,20]}. Cellular alternans and TWA are investigated by a plethora of studies from various perspectives ^{[9,21,22,23,24]}.

With respect to cellular alternans, controversy arises from the unsolved chicken-or-egg problem: which of APD alternans and Ca²⁺ alternans occur first ^[24]? Ca²⁺ alternans can be induced by applying alternans AP-clamp protocol, which is membrane voltage (V_m)-driven alternans ^[25,26]. Fluctuation of sarcoplasmic reticulum (SR) Ca²⁺ ([Ca²⁺]_sr) could trigger Ca²⁺ alternans, which is Ca²⁺-driven alternans ^[9,21]. The fact that alternans arising from either genesis could happen, along with the bidirectional coupling between Ca²⁺ transient and V_m ^[9,15,21,27] complicates the chicken-or-egg problem.

As to differentiate the two mechanisms, a method of constant-diastolic-interval-pacing is put forward ^[28]. When applying constant diastolic interval (DI), Ca²⁺-driven alternans could not be eliminated until the strength of calcium instability is reduced. Therefore, finding out the factors associate with calcium stability becomes vital in investigating Ca²⁺ alternans. Furthermore, researchers expand this constant-DI-pacing method to constant-TR-pacing method, which implements constant TR interval in one cable simulation ^[29]. Both of the two methods contribute to prevent alternans with cycle length (CL) larger than 250 ms. Significantly, calcium handlings could be affected whenever CL is more or less than 250 ms due to voltage-calcium coupling and deserves more investigation.

Cellular alternans could be divided into concordant and discordant ^[9,21,27]. When a large Ca²⁺ transient gives rise to a long APD, this alternans is called concordant. Correspondingly, discordant alternans indicates a small Ca²⁺ transients paralleled by a long APD. Due to the bidirectional coupling between Ca²⁺ dynamics and V_m, Ca²⁺-related currents, such as L-type calcium current (I_CaL) and Na⁺-Ca²⁺ exchange current (I_NaCa) make a difference in forming the two different alternans ^[9,21,27]. Although the effects of the two currents on APD alternans have been investigated, how the two different alternans affect TWA forming remains unknown.

As to TWA, we mainly care about its utility and value in clinical. A lot of studies aim to build relationship between TWA and prediction of SCD ^[30,31]. Particularly, TWA is a strong marker of arrhythmia in patients with ischemic heart disease. Prognostic value of TWA test is also confirmed after MI ^[4]. The magnitude of TWA is associated with heterogeneity of repolarization, which is the key factor to predict arrhythmia ^[11,12]. Ischemia is considered to increase the magnitude of TWA ^[11]. Meanwhile, ischemia is shown to produce beat-to-beat variations in peak-systolic and end-diastolic cytosolic Ca²⁺ ([Ca²⁺]_i) ^[6,7], alternans in amplitudes and durations of V_m ^[17]. Moreover, there could be 2:1 response of monophasic potential according to unipolar extracellular electrogram ^[17]. Various methods analyzing TWA are put forward to broaden the usage of TWA in clinical. However, there is no detailed analysis of ischemia-induced alternans forming from ionic basis to cellular level and eventually to the tissue level.

The ionic basis for cellular alternans could play an important role in explaining how TWA occur, providing valuable information in investigating risk prediction of SCD. However, the underlying mechanisms forming TWA vary from abnormal ionic currents and other electrical factors ^[12]. In heart failure, a steep fractional Ca²⁺ release determines the occurrence of alternans ^[32]. The balance between calcium uptake and release could be disrupted when heart rate is elevated ^[11]. TWA induced by myocardial ischemia is considered with calcium cycling. In all, calcium cycling and Ca²⁺-related currents involved in voltage-calcium coupling exert essential influence on alternans occurrence. Finding out key factors forming cellular alternans is essential, which contributes to explain the characteristics of TWA broaden its use in therapy.

To investigate ischemia-induced TWA mechanism, we simulate cellular alternans and TWA in one dimensional cable in ischemia. MI includes hyperkalemia, acidosis, and anoxia ^[33,34]. Hyperkalemia is shown to be related with depolarization phase ^[35] and QRS wave on ECG. Acidosis decreases the amplitudes of depolarizing current I_Na, and I_CaL ^[34]. Anoxia reduces cytosolic ATP, resulting in opening ATP sensitive potassium current (I_KATP) ^[36]. Acidosis along with anoxia impair sarcoplasmic reticulum Ca²⁺ pump (SERCA) functioning ^[37] and change properties of ryanodine receptors (RyRs) ^[38]. I_NaCa is reduced or even becomes to be in reverse mode ^[34]. In addition, diastolic [Ca²⁺]_i and net magnitudes of Ca²⁺ transients are increased during ischemia ^[6,39].

On the one hand, we aim to simulate both concordant and discordant Ca²⁺-driven alternans in ischemia and analyze factors in affecting alternans forming. On the other hand, different ischemic zones and reduced conduction velocity are set to find out factors affect TWA patterns. Clinical data show TWA could be centered on the first half of T waves ^[40,41], however, no explanations are given in forming this alternans. We aim to build the link between cellular alternans and TWA patterns, broadening TWA utility in clinical.

2.   Methods

2.1.   Impaired Ca²⁺ uptake pump

The Ca²⁺ uptake rate is decreased during ischemia ^[37]. A thermodynamic Ca²⁺ uptake pump model was used to be integrated into human ventricular cell model ^[42] (Ord model). Then the integrated model was applied to one dimensional cable to get ischemic p-ECG with CL of 700 ms. The uptake rate was multiplied by an appropriate coefficient of ${C}_{up}$, making the amount of retaken Ca²⁺ minus leaked Ca²⁺ in integrated model almost as the same with that in original Ord model.

2.2.   Two types of Ca²⁺ transients

Ischemia is confirmed to induce Ca²⁺ alternans by modulating Ca²⁺ dynamics. Here Ca²⁺ alternans patterns were mainly decided by changes in properties of RyR channels. There are three properties of Ca²⁺ release channel, randomness, refractoriness and recruitment ^[43]. Either prolonging the refractory period or increasing the frequency of spark-induced sparks (which is also called secondary sparks ^[43]) contributes to alternans. According to changes in the two properties, we modified the formula of Ca²⁺ release current, respectively. In type Ⅰ, time constants of all the RyRs (τ_rel) channels were increased by 80 ms to simulate prolonged refractory period. In type Ⅱ, a new gate of O_r was added to the formula, controlling the occurrence of secondary sparks. The value of O_r depends on [Ca²⁺]_jsr and ranges from zero to one, representing the probability of secondary sparks occurring. The time constant of this gate O_r is regulated by I_CaL.

2.3.   Ischemic settings

2.3.1.   Transmural ischemia

We also modified formula of Ca²⁺ release current in Endo and M cell model, respectively. These modifications were aimed to decrease ischemic level from Endo cells to Epi cells, simulating coronary artery occlusion. Detailed transmural ischemic parameters were shown in Table 1.

2.3.2.   Epi ischemia

During Epi ischemia, we simulated TWA induced by two patterns of Ca²⁺ alternans, respectively. Except for the difference in formula of Ca²⁺ release current, we set other ischemic parameters as the same. I_Na, I_CaL and I_NaCa were reduced to 95%, 95% and 70%, respectively; [ATP]_i, [ADP]_i, [Pi]_i and [PH]_i were set 6.8 mM, 15 mM, 5 mM, and 7.1. I_KATP and Ca²⁺ uptake current (J_up) were affected by the same ischemic conditions.

2.3.3.   Endo ischemia

To investigate the effect of ischemic zone on TWA, we simulated p-ECG during Endo ischemia, where ischemic parameters are shown in Table Ⅰ and cellular alternans are induced by adding the gate O_r.

2.4.   p-ECG

One dimensional cable was used to simulate p-ECG in control, Epi ischemia, Endo ischemia and transmural ischemia. 60 Endo cells, 45 M cells and 60 Epi cells are connected to form this fiber. As the part of atrium was missing in our cable, P-wave representing the activation of atrium was lacking in p-ECG. The length of each cell was set as 0.01cm and p-ECG was calculated at the point 3.65 cm far from the first Endo cell ^[35]. Electrical coupling between neighboring cells was expressed as the following equation, where the value of D was 0.154 mm² ms⁻¹, representing the tensor of diffusion coefficient along the fiber ^[35]. In addition, I_stim, I_ion and C_m are current stimulus, ionic currents and membrane capacity respectively.

3.   Results

3.1.   Two types of Ca²⁺ alternans

We simulated concordant alternans (Type Ⅰ) and discordant alternans (Type Ⅱ). Type Ⅰ is as shown in Figure 1, diastolic Ca²⁺ concentration alternates slightly (Figure 1a) and Ca²⁺ release current (J_rel) alternates in magnitude (Figure 1c). In type Ⅱ, alternans in diastolic Ca²⁺ concentration is evident (Figure 1b) and a large Ca²⁺ release occurs in every two beats (Figure 1d).

To evaluate the degree of Ca²⁺ alternans, Ca²⁺ alternans ratio is introduced. Ca²⁺ alternans ratio is defined as the value of 1-B/A, where B and A are net magnitudes of small and large Ca²⁺ transients respectively ^[7]. The mean values of alternans ratios in two types were calculated for ten beats in stable state and given in Table 2. The value of alternans ratio in type Ⅱ is larger than in type Ⅰ, indicating that fluctuations of Ca²⁺ transients are more evident in type Ⅱ.

Steep fractional Ca²⁺ release curve plays a key role in forming Ca²⁺ alternans. Here we recorded the relationship between the fractional released Ca²⁺ and sarcoplasmic reticulum Ca²⁺ concentration ([Ca²⁺]_sr) during 100 beats, where the initial of [Ca²⁺]_sr is 1.2 mmol/l (Figure 1e). According to Figure 1e, Ca²⁺ release is small with [Ca²⁺]_sr lower than the threshold of 1.7 mmol/l. When[Ca²⁺]_sr comes to the threshold, a small increase will result in large Ca²⁺ release, which indicates O_r gate becomes massively open. Figure 1f shows that bifurcations exist from the beginning of recording.

3.2.   Corresponding APD alternans, TWA

Two types of APD alternans are shown in Figure 2a, b. In Figure 2a, although we could not tell evident alternans in APD of type Ⅰ, partially enlarged view shows there are slight difference between APDs. In type Ⅱ, a very small Ca²⁺ transient causes I_CaL to last longer (Figure 2d). Correspondingly, plateau phases alternate evidently (Figure 2b).

During Ca²⁺-driven alternans, I_CaL and I_NaCa are key factors transforming Ca²⁺ alternans to APD alternans. Both currents also affect repolarization durations in different phases. Here, durations of repolarization and repolarizing currents (I_CaL and I_NaCa) are picked up and compared. We recorded the time duration completing 60% and 90% of repolarization (APD₆₀ and APD₉₀) in every beat and calculated the difference of them (DAPD₆₀ and DAPD₉₀) between every two consecutive beats. DAPD₆₀ (DAPD₉₀) is calculated by subtracting APD₆₀ (APD₉₀) of the second beats from that of the first beats.

Table 2 gives mean values of them in two types. Positive values indicate APDs are longer in the first beats than in the second beats. Table 2 shows that DAPD₉₀ is larger than DAPD₆₀ in type Ⅰ alternans and the absolute value of DAPD₉₀ is smaller than DAPD₆₀ in type Ⅱ. Where, DAPD₆₀ and DAPD₉₀ are mainly influenced by alternans through I_CaL and I_NaCa. A small Ca²⁺ transient results in large I_CaL (Figure 2c, d) and small I_NaCa (Figure 2e, f), having contrasting effects on the repolarization duration. Note that in the second beat of type Ⅱ alternans, although the small I_NaCa contributes to shorten repolarization duration, the prolongation of plateau phase plays a dominant role in the long APD. That indicates I_CaL lasting longer than in the first beat. Whereas in type Ⅰ, the small I_NaCa plays a dominant role in all the repolarization phases in the first beat, resulting in a shorter APD. Moreover, the durations I_CaL lasting for are almost the same during the two consecutive beats. The difference in durations of I_CaL plays a dominant role in deciding alternans patterns.

In all, when there is a large Ca²⁺ transient in type Ⅱ Ca²⁺ alternans (Figure 1b), the corresponding APD is short (Figure 2b), which is shown as discordant alternans. Whereas type Ⅰ is concordant alternans.

T wave arises from the difference of repolarization durations between transmural cells. Repolarization starts from Epi cells to M and Endo cells. When T wave reaches to its peak (Figure 4a), the difference between transmural cells repolarization durations increases to the maximum. Figure 5a shows Epi ischemia induces larger T wave magnitudes, indicating the heterogeneity of transmural repolarization is increased. To investigate how the repolarization heterogeneity is affected by these two types of alternans, we compare the APDs in control conditions and that in ischemia (Figure 3). Repolarization durations are shortened by I_KATP during ischemia. Correspondingly, the time when the magnitude of T wave reaches is earlier in all ischemic conditions than under control.

To compare the effect of ischemic zone on TWA patterns, we simulated p-ECG during Epi ischemia, and Endo ischemia, where cellular alternans was concordant alternans. In control conditions, T waves show very slight alternans (Figure 4a). We set initial values of transmural cell models, making them come to stable state in single cell simulation. The cable simulation with our initial values takes time to become stable and thus there is small fluctuations of T waves (Figure 4a). Figures 5–8 show that TWA patterns change with different ischemic zones and cellular alternans patterns.

Type Ⅰ alternans of Epi induced by ischemia could not result in obvious TWA (Figure 5a). Note that the time when J point appears is different during the two consecutive beats. In the first beat, J point appears early, which means repolarization of the cable starts early. Correspondingly, Figure 5c shows in the first beat, Epi completes depolarization phase earlier than in the second beat.

Negative T wave arises from Endo ischemia (Figure 6a). Endo ischemia reduces repolarization durations of Endo cells due to I_KATP and leads Endo to complete repolarization earlier than M and Epi cells (Figure 6b). Under control, positive T wave is mainly determined by the positive vector starting from Endo to M cells. After APDs of Endo cells are reduced, the electrical vector becomes to point to Endo cell, which is opposite from the recording point of p-ECG, forming negative T wave.

TWA in type Ⅱ is mainly centered on the first half of T waves (Figure 7a). Here, TWA could be divided into two phases, separated by the point A marked asterisk. Before point A, the second T wave is higher than the previous. After crossing the point, the second T wave becomes lower than the first. Meanwhile, APDs of Epi cells during the two beats almost intersect at the point A (Figure 7b). Comparing to the first beat, plateau phase in the second beat is longer and the whole repolarization duration is shorter. Longer plateau phase of Epi cells reduce the repolarization difference between Epi cell and Endo, M cells. In addition, the total reduced repolarization duration of Epi cell increases transmural repolarization heterogeneity.

Transmural ischemia reduces APDs of transmural cells and results in APD alternans in Endo, M and Epi cells (Figure 3). Evident TWA arises from cellular alternans among the whole cable (Figure 8). The changes of TWA magnitude and morphology are owing to altered repolarization heterogeneity. Different ischemic conditions (Table 1) shorten APD of transmural cells to different degrees. Here, T wave becomes negative, indicating APDs of Endo cells are shortened the most.

3.3.   TWA with slow electrical conduction

Myocardial ischemia affects electrical conduction along myocardial fiber ^[44]. Here we simulated the effect of reduced conduction velocity on ECG in transmural ischemia. The duration of QRS increased obviously comparing to ECG in control conditions, resulting in ST and T wave postponed (Figure 9). The pattern and degree of TWA remained unchanged with slowing conduction velocity by comparing the two T waves in Figure 9.

4.   Discussion

4.1.   Ca²⁺ release in MI and Ca²⁺ alternans

Prolonging RyRs refractory would enhance Ca²⁺ alternans ^[45]. Vyacheslav et al ^[46] find that latency of Ca²⁺ release is prolonged after large Ca²⁺ transients and the restitution property of Ca²⁺ release makes a difference in forming alternans. In addition, secondary sparks are attributed to recruitment, a property of RyRs ^[43]. Investigators build a Ca²⁺ release model and prove that either increasing secondary sparks or prolonged Ca²⁺ release activities can lead to Ca²⁺ alternans ^[43]. In our simulations, we change the two properties separately and investigate their independent roles in contributing to cellular alternans. Although changes of either property could lead to Ca²⁺ alternans independently, the patterns of alternans in Ca²⁺ release current and Ca²⁺ transients are different.

Secondary sparks are the basis for forming Ca²⁺ waves. Propagation of Ca²⁺ waves is associated with Ca²⁺ alternans ^[43,47]. During Ca²⁺ alternans, the larger Ca²⁺ transient is confirmed to arise from Ca²⁺ waves propagation according to experimental measurements ^[47]. Previous studies prove the key role of [Ca²⁺]_jsr in forming Ca²⁺ alternans and Ca²⁺ waves ^[47,48]. When [Ca²⁺]_jsr is accumulated to a threshold, Ca²⁺ alternans will appear ^[48]. In alternans of type Ⅰ, we simulate the effect of Ca²⁺ waves propagating, which is govern by O_r gate. When [Ca²⁺]_jsr reaches a "threshold", the gate becomes completely open (Figure 1e), producing large Ca²⁺ transients, which are like Ca²⁺ waves propagating. MI is shown to increase Ca²⁺ sparks frequency ^[49], however, the existence of the O_r gate has not been verified.

In type Ⅰ, duration of Ca²⁺ release is short and the magnitude of Ca²⁺ release current alternates (Figure 1c). The alternating Ca²⁺ release activities are governed by the gate O_r. According to Eq 1, the value of O_r changes with [Ca²⁺]_jsr. O_r in single cell model takes time to reach to its steady-state value and the time constant depends on I_CaL, which triggers Ca²⁺ release. Here, O_r was introduced to control magnitude of Ca²⁺ release current, mimicking alternating Ca²⁺ waves propagation.

In alternans of type Ⅱ, τ_rel is increased. As RyRs are activated by I_CaL, the increased τ_rel leads to RyRs responding to I_CaL in every two beats. J_rel alternates obviously and release activities in the second beat can almost be ignored (Figure 1d). The very small Ca²⁺ release results in small amplitude of Ca²⁺ transients in the second beats (Figure 1b). Slow falling phase and broadening systolic peak of large Ca²⁺ transients are discernible, which coincides with previous studies ^[6,50,51]. Wu et al. ^[6] attribute the slow decay to decreased J_up in ischemia. Lee et al. ^[51] favor that broadening peak arises from Ca²⁺ release. In our simulations, the velocities of Ca²⁺ uptake in Ca²⁺ alternans of the two patterns are reduced by the same ischemic conditions. However, [Ca²⁺]_i decays slower in type Ⅱ (Figure 1a, b). Our results indicated that broadening peak is mainly affected by prolonged Ca²⁺ released activities. In addition, experiments show that ischemia appears to result in contrasting changes of upstroke of [Ca²⁺]_i ^[6,51,52]. When the duration of ischemia is within 2.5 min, upstroke is found to be rapid ^[6,51]. After 4 h of ischemia, upstroke includes an initial rapid phase and a following slow phase ^[52]. Our upstroke (Figure 1b) slows down due to slow and sustained Ca²⁺ release.

4.2.   The relationship between cellular alternans and TWA patterns

Ca²⁺ alternans of different patterns in ischemia result in different APD alternans and TWA patterns. In cellular alternans of type Ⅰ, although Ca²⁺-driven APD alternans are not evident, a long APD is paralleled by a large Ca²⁺ transient, which is referred to as "in-phase" or "concordant" ^[21]. Conversely, type Ⅱ Ca²⁺-driven APD alternans are discordant. A large Ca²⁺ transient arises from a large Ca²⁺ release, leading to reduced I_CaL, increased I_NaCa, having contrast effects on APD. Where reducing I_CaL shortens plateau phase and increasing I_NaCa prolongs repolarization by extruding [Ca²⁺]_i. Previous study ^[9,21,27] analyzes that whether Ca²⁺ alternans and APD alternans are concordant or discordant depends on which one of currents between I_CaL and I_NaCa plays a dominant role. In our simulation, the concordant alternans are due to the main effect of I_NaCa and the discordant alternans arise from the dominant role of I_CaL.

Note that in cable simulation, APD in the first beat is longer than in the second beat (Figure 7b). In single cell simulation, the first APD is shorter (Figure 2b). During Ca²⁺-driven alternans forming TWA, coupling between neighboring cells also affects TWA morphology.

In control, there is slight difference in p-ECG morphology between two beats (Figure 4a). This is because the electrophysiological states of the cable are not so stable. The initial parameters of transmural cell models can make sure the cell model being in stable state during single cell simulation. Our cable simulation needs more time to come into stable with these parameters.

According to the insets in figure 5a and figure 6a, TWA can be induced by cellular alternans of type Ⅰ. In cellular alternans of type Ⅱ, TWA is obvious and its morphology is closely related with cellular alternans (Figure 7a, b). Although TWA induced by cellular alternans of type Ⅰ is not evident, the magnitudes of TWA are larger than in control. In clinical, microvolt TWA is used to predict the risk of SCD ^[1,4,11]: When the value of TWA is higher than 46 µv ^[4], the risk of SCD and cardiac morality will rise sharply. In addition, experiments show that arrhythmia will occur when ischemia-induced TWA magnitude is 100-fold larger than in control ^[1]. TWA magnitude could not be recognized easily in clinical and simulations, it is meaningful to detect and evaluate the extent of alternans.

T wave magnitude increases during Epi ischemia (Figures 5a, 7a). APD of Epi is shorten by ischemia, resulting in larger repolarization dispersion within transmural cells (Figures 5b, 7b). T wave becomes negative during Endo ischemia and transmural ischemia (Figures 6a and 8a, b). Note that the maximum repolarization dispersion occurs earliest in transmural ischemia than in regional ischemia. The polarization of T wave along with the time points when T wave amplitudes arrive at can be used to identify ischemic zone.

In Epi ischemia, TWA concentrates in the first half of T waves (Figures 5a, 7a). The duration when TWA occur is confirmed to parallel the vulnerable period ^[41]. Similar findings are reported within a few minutes of occlusion ^[40]. Meanwhile, the second half of T waves remain uniform during consecutive beats ^[40]. However, within 2 to 10 days post-MI, alternans arises during the second half of T wave ^[53]. Our simulations of Endo ischemia and transmural ischemia show alternans exist during the total T wave (Figures 6a and 8a, b). According to these studies, the location of alternans in T wave on ECG can be linked to ischemic time or ischemic zone.

4.3.   The effect of slowing conduction velocity on TWA

Conduction velocity is reduced by ischemia ^[44]. The slow electrical conduction interplays with cellular alternans to lead to spatial discordant alternans ^[16], which has ability to induce large TWA magnitude. Previous studies show the importance of T wave magnitude in ventricular arrythmia under MI ^[1,54]. Here we reduced conduction velocity and investigated the effect of slow conduction on TWA magnitude. In our simulations, slowing electrical conduction along the cable postponed ST and T wave (Figure 9), however, T waves were postponed to the same extent during the two consecutive beats. Although our results favor that reduced conduction velocity has no effect on TWA pattern and magnitude, the conclusion still needs further investigation on 3D level.

Previous experiment shows activation delay increases from about 8 ms of control conditions to 35 ms in ischemic zone, which is identical during two consecutive beats ^[44]. Prolongation of activation time has been proven to have no effect on activation sequence and TWA magnitude ^[44]. Note that, delay of activation with slow conduction velocity leads to increased repolarization dispersion between ischemic zone and normal zone. The correlation between conduction and TWA magnitude should be fully investigated by simulating various reduced conduction velocity and ischemic zone.

A plethora of studies put forward that cardiac alternans in ischemia mainly arise from Ca²⁺ alternans ^{[1,6,7,15,18]}. Hyperkaleamia and slowed conduction play dominant roles in block, which also induce repolarization alternans ^[35]. As the effects of various ischemic factors on repolarization alternans are complex, cellular alternans and TWA under ischemia should be thoroughly studied.

Epi cells are proved to have a greater effect on TWA than M cells ^[22]. In our simulation, TWA in Epi ischemia (Figure 5a) is hardly to be detected, however, Endo ischemia leads to evident TWA (Figure 6a). Epi cells are closer to electrode than Endo cells, making bigger difference in forming TWA. However, various ischemic conditions should also be considered when analyzing the TWA magnitude. Ischemia leads to various Ca²⁺ alternans ratios, where the smallest Ca²⁺ alternans ratio is close to the boundary of ischemic zone ^[7]. Thus, more sever ischemia results in larger Ca²⁺ alternans ratio and correspondingly larger APD alternans. In addition, electrical coupling should also be considered in tissue simulation, which affects the cellular alternans pattern (Figure 7b). In our cable simulations, ischemic conditions were set more severe in Endo cells than Epi cells (Table 1). Correspondingly, in single cell simulations, ischemic Endo cell shows larger alternans in APD than ischemic Epi cell (Figure 3a, c).

5.   Conclusions

Changes of RyR properties result in different Ca²⁺-driven alternans and TWA patterns. Cellular alternans along with electrical coupling affect TWA morphology. Our simulations show the location of alternans in T wave can be associated with different ischemic zone. More analysis in terms of the location of TWA would provide useful insights to more detailed electrical changes during and after MI.

Reducing conduction velocity has no effect on TWA magnitude. However, slow conduction interplays with spatially concordant alternans to form SDA, indicating conduction velocity plays a key role in assisting other electrical factors to affect TWA.

6.   Limitations

In our simulation, APD alternans induced by type Ⅰ Ca²⁺ alternans are not obvious and thus TWA are not easy to be discernible. Changing ischemic settings or our formula of O_r gate may result in various APD alternans pattern. Other ischemic factors should be added to analyze TWA pattern thoroughly in the future.

Cellular concordant alternans induced by ischemia are simulated here, however, ECG of tissue with spatially discordant alternans could also provide deeply understanding in terms of the relationship between APD alternans and TWA. We only could get more comprehensive conclusion after taking consideration of as many patterns of cellular alternans and TWA as possible.

Ischemic conditions in our simulation are set according to previous experimental data. However, our modification of J_rel is not based on experiments. We only consider if Ca²⁺ alternans ratios vary within reasonable range. In the future, more experimental data should be referenced and then furtherly modify the formula of J_rel.

More importantly, we only simulated p-ECG in one dimensional fiber. Investigation of TWA should be more valuable based on 3D-personalized cardiac model.

Acknowledgments

This work was supported by the Natural Science Foundation of China (NSFC) under grant number 61527811 and 61701435, the Key Research and Development Program of Zhejiang Province under grant number 2020C03016, the Zhejiang Provincial Natural Science Foundation of China under grant number LY17H180003, and the Medical Health Science and Technology Project of Zhejiang Provincial Health Commission under grant number 2020RC094.

Conflict of interest

The authors declare that there are no conflicts of interest.