Efficient computational methods for strongly coupled

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Electronic Supplement for

Efficient computational methods for strongly coupled cardiac electromechanics Sander Land, Steven A. Niederer, Nicolas P. Smith Abstract This electronic supplement is included to aid reproducability of the models and methods. It includes details of the simplified GPB model, reparametrization of the active contraction model, more details of the 3D finite element model, an pseudocode formulation of the cell model integrator as well as a more detailed analysis of the FWE∞ method.

I. D ETAILS OF SIMPLIFYING THE GPB MODEL (S ECTION II.B) This section provides details of the equation used in simplifying the GPB model as describd in section II.B of the paper. We use the same notation as in the supplement to [1]. A. Quasi-steady state formulation of the calsequestrin buffer SR The rate of change in the calcium concentration in the sarcoplasmic reticulum ( dCa ) is given by; dt   dCaSR Vmyo dCsqnb = Jserca − JSRleak + JSRCarel − dt Vsr dt V

Where Jserca , JSRleak Vmyo , JSRCarel are calcium fluxes in and out of the sarcoplasmic reticulum and Csqnb is the concentration sr of calcium bound to calsequestrin in the sarcoplasmic reticulum. By applying a rapid buffering approximation this equation is changed to:  ! dCaSR 1 Vmyo = Jserca − JSRleak + JSRCarel dt 1 + θcsqn Vsr Bmax

Kcsqn

Where θcsqn = (Kcsqncsqn +CaSR )2 and Kcsqn = kof fcsqn /koncsqn , where koncsqn , kof fcsqn are the binding and unbinding rates of calcium to calsequestrin and Bmaxcsqn is the maximum buffer capacity. dCsqnb dCsqnb SR This equation can be derived from the assumption of a quasistatic state dt = 0 resulting in dCa = θcsqn dt . See dt Keener & Sneyd[2] for details of this derivation. This approximation removes the need for the equation for Csqnb , which can then be removed. B. Fixed sodium concentrations The intracellular sodium concentrations vary little over the course of a sinlge beat, and can be set to a constant value. For dN aj ai this change we set dN = dNdtasl = 0 and remove equations for their buffers N aBj and N aBsl . dt = dt C. Myo and TnCh calcium buffers Similarly, buffering to myosin and high-affinity binding site of troponin also has a limited effect, and equations for them dT nChm dT nC can also be set to dMdtyoc = dMdtyom = = dt hc = 0. dt D. Quasi-steady state formulation of the subsarcolemmal and junctional buffers The last change involves fast buffering approximations to some of the subsarcolemmal and junctional buffers This change is applied much like the formulation of the calsequestrin buffer. Define: Ksll θsllj Kslh θslhj d[Ca2+ ]j dt

= kof fsll /konsll BmaxSLlowj Ksll =  2 Ksll + [Ca2+ ]j = kof fslh /konslh BmaxSLhighj Kslh =  2 Kslh + [Ca2+ ]j = J[Ca2+ ]j −

dSLHj dt

−

dSLLj dt

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Where we define J[Ca2+ ]j as all the fluxes in and out of the junctional cleft unrelated to buffering. The model we use only makes the assumption of a quasistatic state for the SLL buffer, using up with:   d[Ca2+ ]j 1 dSLH J[Ca2+ ]j − dt j = dt 1 + θsllj dSLH

d[Ca2+ ]j dt

Further applying the quasistatic formulation of the SLH buffer dt j = θslhj eliminated, which is: d[Ca2+ ]j 1 = J 2+ dt 1 + θsllj + θslhj [Ca ]j

dSLLj dt

= θsllj

d[Ca2+ ]j dt

to end

gives the formulation for both buffers

However, we did not use this formulation as it results in larger changes in key variables. Equivalent changes can be applied to the subsarcolemmal space (subscripts j replaced by sl in equations). II. R EPARAMETRIZATION OF THE ACTIVE CONTRACTION MODEL FOR HUMAN CELLS (S ECTION II.C) The NHS model was parametrized using experimental data from rats at 25 ◦ C. Consistency with the GPB model requires reparametrization for simulating a human heartbeat at body temperature. Currently available experimental data is insufficient to fully reparametrize the model, but does allow us to fit the most important parameters to give sufficiently physiological whole-organ function to test our solution methods. We start by adding the tension-dependent unbinding of Ca2+ from troponin C from the NHS model to the GPB model, by introducing a tension dependence in the troponin C unbinding coefficient within the GPB model:   Ta (1) koff = krefoff · max 0.1, 1 − γTref We keep the original value of γ, and now need to estimate krefoff , taking into account the lower tension-dependent unbinding Ta 27 ≈ 56.2 at λ = 1 as in the original model, and assuming the single cell values at higher levels of tension. Assuming peak Tref Ta reflect isometric tension, we can define a simple tension curve with this peak and approximate time to peak and relaxation Tref of tension. Using this as input to our coupled model, we estimated krefoff as 5% higher than the GPB value of koff , at which the tension-dependent [Ca2+ ]Trpn curve is a good match for the original one of the GPB model. Values of peak tension during a twitch from experimental data vary around 20 − 50 kPa in single cells generating maximum tension [3], [4]. We use a reference tension of Tref = 100 kPa, which has been shown to be consistent with whole-organ contraction[5]. This corresponds to around 45 kPa peak tension in an isometric twitch at resting sarcomere length. Next, we determine the Hill coefficient, which determines the steepness of the steady state relationship between force and [Ca2+ ]i [6]. Force-pCa curves vary depending on sarcomere length and temperature, with Hill coefficients ranging from 2.0-6.5 have been reported in a variety of mammals [7], [8], [9]. There is limited data on intact cardiac muscle in humans at body temperature, but Gwathmey and Hajjar [10] reported a Hill coefficient of around 5 in intact human cardiac muscle at 30 ◦ C, which is consistent with values observed at that temperature in other species. We set the Hill coefficient to n = 5 in accordance with the best available data in intact fiber at higher temperatures. Similar to our process for determining Tref , we set [Ca2+ ]50 to the same fraction of the peak [Ca2+ ]i as in the NHS model, resulting in a value of [Ca2+ ]50 = 0.25 µM. This is quite low compared to experimental data, which includes [Ca2+ ]50 in ferrets as 0.44µM at 30 ◦ C [7] and [Ca2+ ]50 in humans as 0.58µM at 30 ◦ C [10]. However these comparisons are made in the context of the general lack of data for intact human cardiac muscle at body temperature, and this low value is also a direct consequence of peak [Ca2+ ]i in the GPB model, which is at the lower end of experimentally observed values. For coupling to other electrophysiology models this parameter should be adjusted according to peak [Ca2+ ]i . Removal of the cooperative unbinding term in the equation for actin binding (αr2 = 0) gives a tension transient that can not both reach a high maximum and relax to low levels of tension, which shows the importance of this term. Including the halfactivation value Kz for cooperative unbinding in optimization results in a very low Kz , since this adds a constant unbinding factor which makes it easier to achieve better relaxation. However, this is also highly unrealistic. Setting Kz = 0.1 is sufficient to give better relaxation overall, so we do not include in further optimization. We are left with three free parameters, the binding and unbinding rates α0 , αr1 and αr2 . To constrain our optimization we use a common measure of how reaction rates in biological systems change when temperature increases, known as the Q10 value:  10/(T2 −T1 ) R2 Q10 = (2) R1 Where R1 , R2 are the reaction rates at temperatures T1 , T2 . For biological systems, the Q10 value is generally between 1 and 3, which corresponds to a maximum 270% increase when changing the temperature from 25 ◦ C to 37 ◦ C. Using simulated annealing to fit the parameters to experimental data for time to peak tension (TPT), and time to 50% and 95% relaxation (RT50 , RT95 ), as well as low levels of resting tension, restricted to these Q10 values, we arrive at values shown in Table I, which

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show consistent Q10 values across all three parameters. These values give a peak tension of 46.3 kPa, tension at 990 ms after activation of 1.9 kPa, and relaxation times consistent with experimental data, as shown in Table II (“Reparam.”). Fig. 1 shows the resulting tension transient for an isometric twitch in a single cell. Parameter Original Reparam. Q10

Tref 56.2 kPa 100 kPa

[Ca2+ ]50 1.05 µM 0.25 µM

α0 8/s 22.3/s 2.3

αr1 2/s 5.2/s 2.2

αr2 1.75/s 4.6/s 2.2

Kz 0.15 0.10

TABLE I N EW PARAMETERS FOR THE NHS ACTIVE CONTRACTION FRAMEWORK COUPLED TO THE GPB ELECTROPHYSIOLOGY MODEL TO SIMULATE HUMAN CELLS AT BODY TEMPERATURE . N OTE THAT Q10 VALUES ONLY APPLY TO BINDING RATES .

Source Mulieri et al.[3] Pieske et al.[4] Reparametrized coupled model

TPT (ms) 157 ± 10 165 ± 7 146

RT50 (ms) 117 ± 8 116 ± 6 123

RT95 (ms) 334 ± 43 330

TABLE II T IME TO PEAK TENSION (TPT) FROM ONSET OF TENSION AND TIME TO 50% (RT50 ) AND 95% (RT95 ) RELAXATION FROM PEAK TENSION . F OR THE MODELS WE USE A RATE OF INCREASE IN TENSION GREATER THAN 0.01 kPa/ms AS THRESHOLD TO DETECT ONSET OF TENSION , AND MEASURE RT 50 , RT95 AS THE TIME WHEN TENSION DECAYS BELOW 50%, 95% OF THE DIFFERENCE BETWEEN PEAK TENSION AND MINIMUM TENSION FOR 1 H Z PACING .

Fig. 1.

Tension for an isometric twitch in the reparametrized NHS model coupled to the reduced GPB model.

III. I NTEGRATION METHOD FOR THE CELL MODEL (S ECTION III.A) Below is a pseudocode formulation of the methods used for integrating the single cell model, including the FWE∞ method and the adaptive time step. In the code v is the vector of state variables and f the function which returns dv dt .

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Algorithm 1: Integration scheme with adaptive time step for the GPB cell model. v = v0 t=0 T = 1000 ∆Vm,max = 1 min J[Ca 2+ ] = −0.001 j max J[Ca 2+ ] = −0.0015 j ∆t = 1/3 while t < T do dv dt = f (t, v) ∆tcurr = ∆t m if ∆tcurr · | dV dt | > ∆Vm,max then m ∆tcurr = ∆Vm,max / dV dt end if Vm > 35 then ∆tcurr = min(∆tcurr , 0.05) end max if J[Ca2+ ]j ∆tcurr > J[Ca 2+ ] then j max ∆tcurr = J[Ca2+ ]j /J[Ca2+ ]j end min if J[Ca2+ ]j ∆tcurr < J[Ca 2+ ] then j min ∆tcurr = J[Ca2+ ]j /J[Ca2+ ]j end v = v + ∆tcurr dv dt for gate ∈ {m, d, j} do if dgate dt · (gate − gate∞ (t + ∆tcurr /2)) > 0 then gate = gate∞ (t + ∆tcurr /2) end end t = t + ∆tcurr end IV. A NALYSIS OF THE FWE∞ METHOD (S ECTIONS III.A.1 & IV.A) In an attempt to understand the mechanism behind the roughly constant conduction velocity, we have created a simplified model of the sodium channel and action potential upstroke. gN a

=

15

EN a

=

40

m∞

=

1.0/(1.0 + exp((−57 − V )/9))2

αm

=

1.0/(1.0 + exp((−60 − V )/5))

βm

=

0.1/(1.0 + exp((V + 35)/5)) + 0.1/(1 + exp((V − 50)/200))

τm dm dt dV dt

=

αm βm

=

(m∞ − m)/τm

=

−gN a m3 (V − EN a )

We apply the following integration procedure to this model to simulate embedding in a tissue model with an operator splitting method: • Integrate upstroke model for a time interval [t, t + ∆t].
• Apply an activation current Istim (t) similar to seen in a tissue model. We use Istim (t) = 1[3,6] · sin 2πt/3 See Fig. 2 for a comparison of the methods on this simplified model. Some more complicated variations such as [11] give similar results to the FWE∞ scheme in this model, but as implicit schemes they are more computationally expensive. Variations of the Rush-Larsen scheme, such as using the opening and closing rates at the midpoint, perform similarly to the Rush-Larsen method.

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Further tests on the 1D fiber model with the Rush-Larsen method and a smaller ∆Vm,max of 0.5 mV/ms or simply a very small time step (0.001 ms) shows a CV error of 0.028 mm/ms and 0.014 mm/ms respectively at ∆tmono = 0.1 ms, compared to an error of 0.004 mm/ms for the FWE∞ method and decrease of 0.036 mm/ms for the Rush-Larsen method at larger time steps of the cell model solver. Furthermore the second-order method from Maclachlan et al.[11] gives a decrease of 0.02 mm/ms. This suggests the convergence behaviour is a combination of error caused by the cell model solver and operator splitting, with the FWE∞ method compensating for the latter error. The mechanism for an approximately constant conduction velocity in tissue is likely to be related to this.

Fig. 2. Comparison of the Rush-Larsen and FWE∞ method for our simplified upstroke model. The reference solution uses ∆t = 0.01 and the other solutions use ∆t = 0.1.

V. M ECHANICS F INITE E LEMENT M ODEL (S ECTION IV.B.2) A. Mapping of elements near the apex As elements near the apex meet in a single point, a specialized approach is needed here when using hexahedral elements. Please see Fig. 3 for details. The singularity at the apex resulting from this mapping can be solved in one of several ways: • Using specialized element types such as Hermite sector elements, as in Bradley et al. [12]. • Fixing derivatives to zero to deal with the undefined ξ directions here. • Leaving a hole in the apex as in Guccione et al. [13], which requires further unknown boundary conditions. • Using a non-polar topology (e.g. [14]). Of these readily available solutions we have chosen the second, adopting the same framework as the majority of other researchers working on simulating cardiac mechanics, and the one most compatible with automated meshing tools (see Lamata et al.). In an LV model this is sufficient to fix the undefined ξ directions mentioned in [12]. B. Fiber mapping To construct the orthogonal matrix with converts between global and fiber-based coordinates we construct the fiber direction in the reference ξ space as f ξ = (cos α, sin α, 0) and mapped to the undeformed space using f x = dx dξ f ξ . In our model, ξ1 is circumferential, ξ2 apex-to-base and ξ3 transmural. VI. S UMMARY OF METHODS This section summarizes the methods used in the paper, and key results obtained by applying them.

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Fig. 3. Schematic of the collapsed elements and embedded electrics mesh near the apex. The mechanics elements at the apex are handled as standard collapsed hexahedral elements, with pairs of nodes in ξ-space both pointing to the same node in x-space. Furthermore the electrics grid (blue in the right panel) does not extend to these elements. Instead, the nearest point at the boundary of the apex and sub-apex elements (red crosses) are used instead to drive contraction in the mechanics mesh Gauss points at the apex (green dots).

Method Cell model reduction Cell model: integration scheme Cell model: adaptive time step Monodomain: operator splitting

Section II.B III.A.1 III.A.2 III.B

Monodomain: adaptive time step Mechanics: strain prediction

III.B.2 III.C.1

Mechanics: modified Newton

III.C.2

Mechanics: JFNK

III.C.3

Key result Decreases stiffness and number of equations Allows for larger time steps while keeping stability Allows for larger time steps on average with high accuracy Combines well with the FWE∞ method and cell model integration scheme Allows for larger time step in both PDE and ODE solvers Large reduction in the number of Newton iterations and overall computation time Large reduction in the number of Jacobians and LU factorizations, and overall computation time. Potential reduction in number of Newton iterations when not using modified Newton methods R EFERENCES

[1] E. Grandi, F. S. Pasqualini, and D. M. Bers, “A novel computational model of the human ventricular action potential and ca transient,” J. Mol. Cell. Cardiol., vol. 48, no. 1, pp. 112–121, 2010. [2] J. Keener and J. Sneyd, Mathematical Physiology. New York: Springer, 1998. [3] L. A. Mulieri, G. Hasenfuss, B. Leavitt, P. D. Allen, and N. R. Alpert, “Altered myocardial force-frequency relation in human heart failure,” Circulation, vol. 85, no. 5, p. 1743, 1992. [4] B. Pieske, M. S¨utterlin, S. Schmidt-Schweda, K. Minami, M. Meyer, M. Olschewski, C. Holubarsch, H. Just, and G. Hasenfuss, “Diminished post-rest potentiation of contractile force in human dilated cardiomyopathy. functional evidence for alterations in intracellular Ca2+ handling.” J. Clin. Invest., vol. 98, no. 3, pp. 764–776, 1996. [5] S. Niederer and N. Smith, “The role of the Frank-Starling law in the transduction of cellular work to whole organ pump function: A computational modeling analysis,” PLoS Computat. Biol., vol. 5, no. 4, 2009. [6] D. Bers, Excitation-Contraction Coupling and Cardiac Contractile Force, 2nd ed. Dordrecht: Kluwer Academic Publishers, 2001. [7] D. T. Yue, E. Marban, and W. G. Wier, “Relationship between force and intracellular [Ca2+ ] in tetanized mammalian heart muscle,” J. Gen. Physiol., vol. 87, no. 2, p. 223, 1986. [8] P. J. Hunter, A. D. McCulloch, and H. T. Keurs, “Modelling the mechanical properties of cardiac muscle,” Prog. Biophys. Mol. Biol., vol. 69, no. 2-3, p. 289–331, 1998. [9] S. A. Niederer, P. J. Hunter, and N. P. Smith, “A quantitative analysis of cardiac myocyte relaxation: a simulation study,” Biophys. J., vol. 90, no. 5, pp. 1697–1722, 2006. [10] J. Gwathmey and R. Hajjar, “Effect of protein kinase c activation on sarcoplasmic reticulum function and apparent myofibrillar Ca2+ sensitivity in intact and skinned muscles from normal and diseased human myocardium.” Circ. Res., vol. 67, no. 3, pp. 744–752, 1990. [11] M. C. Maclachlan, J. Sundnes, and R. J. Spiteri, “A comparison of non-standard solvers for odes describing cellular reactions in the heart,” Comput. Methods Biomech. Biomed. Engin., vol. 10, no. 5, pp. 317–326, 2007. [12] C. Bradley, A. Pullan, and P. Hunter, “Geometric modeling of the human torso using cubic Hermite elements,” Ann. Biomed. Eng., vol. 25, no. 1, pp. 96–111, 1997. [13] J. M. Guccione, K. D. Costa, and A. D. McCulloch, “Finite element stress analysis of left ventricular mechanics in the beating dog heart,” J. Biomech., vol. 28, no. 10, pp. 1167–1177, 1995. [14] V. Gurev, T. Lee, J. Constantino, H. Arevalo, and N. Trayanova, “Models of cardiac electromechanics based on individual hearts imaging data image-based electromechanical models of the heart,” Biomech Model Mechanobiol., 2010.