Electromechanical Single Cell Stretcher

See code in GitLab.
Author: Gernot Plank gernot.plank@medunigraz.at and Christoph Augustin christoph.augustin@medunigraz.at

Modeling EM at the cellular level

This tutorial introduces the basic steps of setting up electromechanical (EM) simulations in an isolated myocytes. For this sake an electrophysiology (EP) model of cellular dynamics must be paired with a myofilament model for generating active stresses. This is achieved by selecting an EP model of cellular dynamics and an active stress model as a plug-in. Setting up, initializing and testing of such myocyte-myofilament combinations is performed with our single cell tool bench. There are three coupling variants implemented which are suitable depending on a particular question to be addressed:

This setup aims to replicate single cell stretch experiments as they are performed with setups.

Experimental Parameters

The following parameters are exposed to steer the experiment:

--experiment {active | passive}
                      Default is active. In the passive case any stimulation is turned off
                      and the myocyte remains quiescent. This can be used in the
                      strongly coupled case to observe the effect of step changes in length
                      upon EP states.


--EP                  pick human EP model (default is tentusscherpanfilov)


--stress              pick stress model (default is stress_niederer)

--stretch STRETCH     prescribe constant stretch over each cycle (default is 1.)

--init INIT           pick state variable initialization file (default is none)

--duration DURATION   pick duration of experiment (default is 500 ms)

--bcl BCL             pick basic cycle length (default is 500 ms)

--fs FS               provide trace file to prescribe fiber stretch (default is none)

--fsDot FSDOT         provide trace file to prescribe fiber stretch rate
                      (default is none)

--ref FOLDER          folder holding .bin files of a reference solution to be plotted in comparison

Each experiment stores the state of myocyte and myofilament in the current directory in a file to be used as initial state vector in subsequent experiments.

Activation-based active stress model

Such models are suitable for being used in clinical EM modeling study as these models - owing to their small number of parameters and their fairly direct relation to clinically measurable parameters such as peak pressure, \(\hat{P}\) or the maximum rate of rise of pressure, \(dP/dt_{\mathrm {max}}\) -are easier to fit.

As an example we use the very simple niederer stress model. The model is based on activation times and constructed with exponential functions, it also accounts for length-dependent development of active tension. Our implementation of the niederer stress model is based on the PhD work of Andrew Crozier and has also been used in publications ^{1 2} .

The shape of the active stress transient is given by

\[\begin{equation} T_{\rm a} = T_{\rm peak} \phi \tanh^2 \left( \frac{t_s}{\tau_{\rm c}} \right) \tanh^2 \left( \frac{t_\text{dur} - t_{\rm s}}{\tau_{\rm r}} \right) \text{for } 0< t_s < t_\text{dur} \nonumber \end{equation}\]

with

\[\begin{aligned} \begin{eqnarray} \phi &= \tanh (\text{ld} (\lambda - \lambda_{\rm 0})) \nonumber \\ \tau_{\rm c} &= \tau_{\rm c0} + \text{ld}_\text{up} (1 - \phi) \nonumber \\ t_{\rm s} &= t - t_\text{act} - t_\text{emd} \nonumber \end{eqnarray} \end{aligned}\]

with the following parameters:

The effect of varying the time constants governing contraction \(\tau_{\rm c}\) and relaxation, \(\tau_{\rm r}\), is shown below. Duration of the active stress transient was set to 550 ms in this case.

Experiment exp01

We start with coupling the Niederer tanh stress model with the tentusscherpanfilov human myocyte model,

./run.py --EP tentusscherpanfilov --stress stress_niederer --duration 2000 --bcl 500 --ID exp01 --visualize

Experiment exp02

As the tentusscherpanfilov is not at a limit cycle, we pace the model for 20 seconds

./run.py --EP tentusscherpanfilov --stress stress_niederer --duration 20000 --bcl 500 --ID exp02 --visualize

In exp02 the states of myocyte and myofilament model are stored at the very end of the experiment at \(t=20000 ms\) in the file exp01_tentusscherpanfilov_stress_niederer_bcl_500_ms_dur_20000_ms.sv.

Experiment exp03

Using the state of the myocyte stored in the file exp01_tentusscherpanfilov_stress_niederer_bcl_500_ms_dur_20000_ms.sv in exp02 we confirm now that your trajectories are close to a limit cycle for the given bcl and stretch we run

./run.py --EP tentusscherpanfilov --stress tanh --duration 1000 --bcl 500 --ID exp03 \
         --stretch 1.0 --init exp02_tentusscherpanfilov_tanh_bcl_500_ms_dur_20000_ms.sv \
         --visualize

Observe that there is no significant difference between the traces for AP1 and AP2.

Experiment exp04

To ensure that length-dependence of active tension is working as expected re-run the experiment at a shorter cell length by setting the fiber stretch \(\lambda=0.75\) and compare with the traces stored in exp03 in the file exp03_traces.h5.

./run.py --EP tentusscherpanfilov --stress tanh --duration 1000 --bcl 500 \
         --ID exp04 --stretch 0.75 --init exp02_tentusscherpanfilov_tanh_bcl_500_ms_dur_20000_ms.sv \
         --ref exp03 --visualize

Experiment exp05

Re-run experiment exp04, but at a stretched cell length by setting the fiber stretch \(\lambda=1.25\) and compare again with the traces stored in exp03 in the file exp03_traces.h5 at the reference cell stretch \(\lambda=1.0\).

./run.py --EP tentusscherpanfilov --stress tanh --duration 1000 --bcl 500 \
         --ID exp05 --stretch 1.25 --init exp02_tentusscherpanfilov_tanh_bcl_500_ms_dur_20000_ms.sv \
         --ref exp03 --visualize

Weakly coupled Calcium-driven active stress model

As an example of a weakly coupled Calcium-driven active stress model due to Land et al for rat and rabbit myocytes ³ and for human myocytes ⁴ . Parameters used in this tutorial were derived from the study of Tondel et al ⁵ .

Experiment exp06

As before with the Tanh stress model, we pace the model for 20 seconds at a resting length of \(\lambda=1.\) to arrive at a limit cycle.

./run.py --EP tentusscherpanfilov --stress land --duration 20000 --bcl 500 --ID exp06 --visualize

Observe that, unlike on the experiments with the Tanh stress model, tension transients vary over time as tension depends on Calcium. In exp06 the states of myocyte and myofilament model are stored at the very end of the experiment at \(t=20000~ms\) in the file exp06_tentusscherpanfilov_land_bcl_500_ms_dur_20000_ms.sv.

Experiment exp07

Using the state of the myocyte stored in the file exp06_tentusscherpanfilov_land_bcl_500_ms_dur_20000_ms.sv in exp06 we confirm now that your trajectories are close to a limit cycle for the given bcl and stretch we run

./run.py --EP tentusscherpanfilov --stress land --duration 1000 --bcl 500 --ID exp07 \
         --stretch 1.0 --init exp06_tentusscherpanfilov_land_bcl_500_ms_dur_20000_ms.sv \
         --visualize

Observe that there is no significant difference between the traces for AP1 and AP2.

Experiment exp08

To verify that the Land model in combination with the tentusscherpanfilov human ventricular myocyte models produces reasonable tensions over the physiological stretch range from \(\lambda=0.75\) up to \(\lambda=1.25\) we run two cycles at with a shortened myocyte \(\lambda=0.75\).

./run.py --EP tentusscherpanfilov --stress land --duration 1000 --bcl 500 \
         --ID exp08 --stretch 0.75 --init exp06_tentusscherpanfilov_land_bcl_500_ms_dur_20000_ms.sv \
         --ref exp07 --visualize

Experiment exp09

Re-run experiment exp08, but at a stretched cell length by setting the fiber stretch \(\lambda=1.25\) and compare again with the traces stored in exp07 in the file exp07_traces.h5 at the reference cell stretch \(\lambda=1.0\).

./run.py --EP tentusscherpanfilov --stress land --duration 1000 --bcl 500 \
         --ID exp09 --stretch 1.25 --init exp06_tentusscherpanfilov_land_bcl_500_ms_dur_20000_ms.sv \
         --ref exp07 --visualize

Experiment exp10

Force development depends on both, length \(\lambda\) and the velocity of length change \(\dot{\lambda}\). Increasing velocity \(\dot{\lambda}\) lowers the force produced by a myocyte which produces the maximum force under isometric conditions, that is \(\dot{\lambda} = 0\). To illustrate these two effects indpendently we re-run exp09 with a dynamic sawtooth loading protocol shown in fig-sawtooth-loading-ld, but we keep the velocity at zero.

./run.py --EP tentusscherpanfilov --stress land --duration 1000 --bcl 500 \
         --ID exp10 --init exp06_tentusscherpanfilov_land_bcl_500_ms_dur_20000_ms.sv \
         --fs lambda_sawtooth_200ms.dat \
         --ref exp07 --visualize

Experiment exp11

In this experiment do not suppress velocity dependence anymore. The mechanical loading protocol applied is the same as in exp10, but with \(\dot{\lambda}(t) \neq 0\). The loading protocol is illustrated in fig-sawtooth-loading-ld-vd.

./run.py --EP tentusscherpanfilov --stress land --duration 1000 --bcl 500 \
         --ID exp11 --init exp06_tentusscherpanfilov_land_bcl_500_ms_dur_20000_ms.sv \
         --fs lambda_sawtooth_200ms.dat --fsDot dlambda_sawtooth_200ms.dat \
         --ref exp10 --visualize

Strongly coupled Calcium-driven active stress model

In this set of experiments we couple the recent EP model of the human ventricular myocyte due to Grandi et al ⁶ (Grandi-Pasqualini-Bers GPB) with the active tension model due to Land et al ⁷ (Land-Niederer LN). Details on the implementation and parameterization of the coupling have been reported in detail in Augustin et al ⁸ .

Briefly, to allow for strong coupling, i.e., to account for length effects on the cytosolic calcium transient, modifications were implemented to both the GPB and the LN model. The equation governing the binding of calcium to the low affinity regulatory sites on Troponin in the myofilament model (see Eq.~(114) in the Appendix of ⁹ )

\[\begin{align} \frac{\mathrm{d}[\mathrm{TnC}_{\mathrm{L}}]}{\mathrm{d} t} = k_{\mathrm{on}_{\mathrm{TnC}_{\mathrm{L}}}}[\mathrm{Ca}_{\mathrm{i}}^{2+}] \left(\hat{B}_{\mathrm{TnC}_{\mathrm{L}}} -[\mathrm{TnC}_{\mathrm{L}}] \right) - k_{\mathrm{off}_{\mathrm{TnC}_{\mathrm{L}}}} [\mathrm{TnC}_{\mathrm{L}}] \nonumber \end{align}\]

is replaced by the analogous Eq.~(1) of the LN active stress model ¹⁰ , given as

In the combined GPB+LN model eq-Ca-Shannon is replaced by eq-LandTnC, scaled by the total buffer concentration \(\hat{B}_{\mathrm{TnC}_{\mathrm{L}}}\) assumed to be 70 \(\mu M/L\) in the GPB model. This scaling is necessary to correctly relate fractional occupancy, used by the LN myofilament model, with the concentration of Troponin bound calcium, used by the GPB model. Parameters of the GPB model were left unaltered whereas parameters of the LN model were adapted to give human myocardium tension transients when coupled to the GPB calcium transient ¹¹ .

In single cell experiments, both the unaltered GPB model and the combined GPB+LN model were paced at a cycle length of 500 ms for a duration of 60 seconds. In the combined GPB+LN model the stretch ratio was kept constant at \(\lambda=1.0\) over the entire protocol. In both models, all state variable transients were plotted over the entire pacing experiment to ensure that the system had settled close to a stable limit cycle. The effect of the altered Troponin buffering equation in the GPB+LN model relative to the GPB model is illustrated in fig-GPB-LN-modified.

The effect of strong coupling in the GPB+LN model under fixed stretch conditions in the range \(\lambda\in[1.0,1.2]\), with the same pacing protocol, is illustrated in fig-GPB-length-dep.

Experiment exp12 (pace to limit cycle)

As before, we pace the model for 20 seconds at a resting length of \(\lambda=1.\) to arrive at a limit cycle.

./run.py --EP gpb_land --duration 20000 --bcl 1000 --ID exp12  \
         --no-filter --visualize

Experiment exp13 (confirm limit cycle)

Using the state of the myocyte stored in the file exp12_gpb_land_bcl_1000_ms_dur_20000_ms.sv in exp12 we confirm now that your trajectories are close to a limit cycle for the given bcl and stretch we run

./run.py --EP augustin --duration 2000 --bcl 1000 --ID exp13 --stretch 1.0 \
         --init exp12_augustin_strongly_coupled_bcl_1000_ms_dur_20000_ms.sv \
         --visualize

Observe that there is no significant difference between the traces for AP1 and AP2.

Experiment exp14 (return to rest)

As this model is strongly coupled, mechanical loading influences EP in the myocyte. To observe MEF effects we leave the limit cycle and stop pacing for 2 seconds. That is, in this experiment we keep the myocyte quiescent, we do not deliver an electrical stimulus current, nor a mechanical stretch protocol and observe the return of trajectories to a resting state.

./run.py --experiment passive --EP augustin --duration 1000 --bcl 1000 \
         --ID exp14 --stretch 1.0 \
         --init exp12_augustin_strongly_coupled_bcl_1000_ms_dur_20000_ms.sv \
         --visualize

Experiment exp15 (stretch resting myocyte, velocity dependence off)

To observe the effect of cellular stretch upon EP we apply now mechanical loading protocols to the resting myocyte. As a reference case, we apply a step change in stretch of 20% and observe only length-dependent effects, that is, velocity-dependence is turned off. This is achieved with the loading protocol shown in fig-gpb-land-mef-ld-protocol.

./run.py --experiment passive --EP augustin --duration 1000 --bcl 1000 \
         --ID exp15 --init exp14_augustin_strongly_coupled_bcl_1000_ms_dur_1000_ms.sv \
         --fs lambda_step_tr_time_100ms.dat --visualize

Experiment exp16 (stretch resting myocyte, velocity dependence on, slower transition)

We repeat experiment exp15 now, but with velocity dependence enabled. First, we use a moderately fast step change by going from \(\lambda=1.0\) to \(\lambda=1.2\) in 100 ms and compare results with exp15 where velocity dependence was disabled. See fig-gpb-land-mef-ld-vd-slow-protocol.

./run.py --experiment passive --EP augustin --duration 1000 --bcl 1000 \
         --ID exp16 --init exp14_augustin_strongly_coupled_bcl_1000_ms_dur_1000_ms.sv \
         --fs lambda_step_tr_time_100ms.dat --fsDot dlambda_step_tr_time_100ms.dat \
         --ref exp15 --visualize

Experiment exp17 (stretch resting myocyte, velocity dependence on, faster transition)

We repeat experiment exp16 now, but with velocity dependence enabled. We use a faster step change by going from \(\lambda=1.0\) to \(\lambda=1.2\) in only 10 msecs. See fig-gpb-land-mef-ld-vd-fast-protocol.

./run.py --experiment passive --EP augustin --duration 1000 --bcl 1000 \
         --ID exp17 --init exp14_augustin_strongly_coupled_bcl_1000_ms_dur_1000_ms.sv \
         --fs lambda_step_tr_time_10ms.dat --fsDot dlambda_step_tr_time_10ms.dat \
         --ref exp16 --visualize

In this set last set of experiments we couple the recent EP model of the human ventricular myocyte due to Tomek et al ¹² (Tomek-Rodriguez-O'Hara-Rudy ToR-ORd) with the active tension model due to Land et al ¹³ (Land-Niederer LN). Details on the implementation and parameterization of the coupling have been reported in Margara et al ¹⁴ .

Experiment exp18 (pace to limit cycle)

As before, we pace the model for 20 seconds to arrive at a limit cycle.

./run.py --EP torord_landhumanstresswithpassive --duration 20000 \
         --bcl 1000 --ID exp18 --no-filter --visualize

Experiment exp19 (confirm limit cycle)

Using the state of the myocyte stored in the file exp18_torord_landhumanstresswithpassive_strongly_coupled_bcl_1000_ms_dur_20000_ms.sv in exp18 we confirm now that your trajectories are close to a limit cycle for the given bcl and stretch we run

./run.py --EP torord_landhumanstresswithpassive --duration 2000 \
         --bcl 1000 --ID exp19 \
         --init exp18_torord_landhumanstresswithpassive_strongly_coupled_bcl_1000_ms_dur_20000_ms.sv \
         --visualize

Observe that there is no significant difference between the traces for AP1 and AP2.

Experiment exp20 (return to rest)

Again, as this model is strongly coupled, mechanical loading influences EP in the myocyte. To observe MEF effects we leave the limit cycle and stop pacing for 2 seconds. That is, in this experiment we keep the myocyte quiescent, we do not deliver an electrical stimulus current, nor a mechanical stretch protocol and observe the return of trajectories to a resting state.

./run.py --experiment passive --EP torord_landhumanstresswithpassive \
         --duration 1000 --bcl 4000 --ID exp20 --visualize \
         --init exp18_torord_landhumanstresswithpassive_strongly_coupled_bcl_1000_ms_dur_20000_ms.sv

Literature