An analysis of deformation-dependent electromechanical coupling in the mouse heart

Abstract

To investigate the effects of the coupling between excitation and contraction on whole-organ function, we have developed a novel biophysically based multiscale electromechanical model of the murine heart. Through comparison with a comprehensive in vivo experimental data set, we show good agreement with pressure and volume measurements at both physiological temperatures and physiological pacing frequencies. This whole-organ model was used to investigate the effects of material and haemodynamic properties introduced at the tissue level, as well as emergent function of our novel cell contraction model. Through a comprehensive sensitivity analysis at both the cellular and whole organ level, we demonstrate the sensitivity of the model's results to its parameters and the constraining effect of experimental data. These results demonstrate the fundamental importance of length- and velocity-dependent feedback to the cellular scale for whole-organ function, and we show that a strong velocity dependence of tension is essential for explaining the differences between measured single cell tension and whole-organ pressure transients.

Key points

• The amount of force generated by heart cells is strongly influenced by feedback from the deformation of cardiac tissue, both from the changes in cell length and the rate at which cells are stretched.

• We analysed the effect these cellular mechanisms have on whole heart function by making a computational model of mouse heart cells, and embedding this cellular model into a representation of the heart.

• Unlike previous murine models, this model represents the heart at both body temperature and the high heart rates seen in these animals, allowing us to directly compare results from our computational model with experimental measurements.

• Results show that effects from the rate of stretch are especially important for explaining the large differences observed between force generated by isolated cells and pressure measured experimentally.

• The model also provides an important framework for future research focused on interpreting results from genetic manipulation experiments in mice.

Introduction

Tension generated by cardiac myocytes is known to be influenced by both strain and strain rate at the single cell level. These feedback mechanisms are of key importance in analysing how tension generated at the cellular level gives rise to ventricular pressure. However, the underlying mechanisms leading to these dependencies remain debated. The length dependence of cardiac cells produces an increase in tension at high sarcomere length, resulting from both a change in maximum force attributed to filament overlap, and a change in calcium sensitivity. Changes in lattice spacing, titin-based effects, and effects on sensitivity from weakly or strongly bound crossbridges have all been proposed as potential mechanisms for this length-dependent change in tension, but as of yet no full explanation is available (Campbell, 2011). The dependence of tension on velocity involves at least two different processes, known as the B process and the C process, which work on different time scales to affect the crossbridge cycle. However, the response of tension to strain rate is complex (Kawai & Brandt, 1980), and the mechanistic effects controversial, with both an increase and decrease in the rate of strongly to weakly bound crossbridges with respect to strain rate having been proposed to explain this process (Yadid & Landesberg, 2010; Palmer, 2010).

These complex cellular and genetic mechanisms which determine the coupled electrical and mechanical behaviour of the heart are increasingly being investigated in experimental models which exploit the opportunities provided by genetic manipulation (Andersson et al. 2009; Lee et al. 2010). The mouse is the most commonly used species in such research, with hundreds of inbred or genetically engineered strains often being specifically developed to elucidate particular aspects of biochemistry and physiology across a wide range of functions in both normal and disease models (Skarnes et al. 2011). It is in this data-rich context that computational models provide a unique way of analysing this data by quantitatively characterising complex phenotypes and developing a rational framework for mechanistically linking measurements to function. However, even though electromechanical models have been used to investigate the mechanisms of cardiac contraction in a variety of animals (Campbell et al. 2008; Niederer & Smith, 2009; Kerckhofs et al. 2010; Keldermann et al. 2010), such a model of a mouse heart with the ability to interpret experimental data does not yet exist. Furthermore, both electrophysiological and contraction models are often based on data obtained across a range of species and temperatures. This heterogeneous source of data underpinning the parameter sets of many electromechanical models creates significant limitations in the move towards the application of models in both wet-lab and clinical settings (Niederer et al. 2009). An electromechanical model of the mouse thus provides an important opportunity in moving beyond these limitations, integrating the data which is increasingly becoming available in both normal and transgenic mice, and using it towards a better quantitative understanding of cardiac function, both specifically in this common experimental model, and ultimately in other species.

The lack of species and body temperature consistent models is indicative of challenges in identifying model parameters from only cell measurement data. Specifically, many contraction experiments are done at lower temperatures (Stull et al. 2002) or in skinned preparations, which are known to significantly affect results (Gao et al. 1994). Also, maximal active tension at the cellular level is typically significantly less than seen in tissue, and decreases over tens of seconds in conditions of saturated calcium concentrations (Campbell, 2011), which is not well understood. These difficulties with single cell data further confounds determining the influence of length- and velocity-dependent feedback on cellular mechanisms for whole organ function, as relative effects can be difficult to extract. This difficulty is, in part due to the fact that experimental preparations, and measurements are often focused on proving the existence of particular mechanisms rather than quantifying their exact magnitude as needed for parametrizing a computational model. There are also significant differences between single cell tension transients and ventricular pressure, as single cell tension tends to be more asymmetric, with longer relaxation, compared to the more symmetric pressure transients. This has led to the use of phenomenological tension models being used at the whole organ scale (Sermesant et al. 2006), for reproducing realistic ventricular pressure transients in the absence of detailed knowledge of the coupling mechanisms involved.

Whereas these cellular coupling mechanisms are of critical importance, they are also difficult to measure. Furthermore, while high quality whole-organ data are more readily available, in isolation it gives insufficient information about the underlying mechanisms to have sufficient predictive value. This motivates an integrative computational modelling approach in which both cellular data and whole organ measurements are combined to create a more accurate model for cardiac contraction. In this study our goal is to address this challenge through the development of an electromechanical mouse model at body temperature and the physiologically faster 6 Hz heart rate. Using this framework we compare in silico experiments with recent data and provide a quantitative analysis of deformation-dependent electromechanical coupling mechanisms. Specifically, our study focuses on quantifying the significance of these coupling mechanisms in producing the observed differences between isometric tension and in vivo pressure transients. We also look at the issue of model parametrization in detail in both single-cell and whole-organ contexts, with the goal of providing clarity about the degree of confidence in these parameters given the experimental data constraining them.

Methods

Models

We have developed a new contraction model for murine cardiomyocytes, described in the next section, which is based primarily on experimental data obtained under physiological conditions, and is able to represent the fast murine contraction kinetics. We investigate the identifiability of the cell model parameters in the section ‘Parameter sensitivity and identifiability’ by using a novel visualization which relates constraints given by experimental data to the model parameter space. This cellular model is extended to the whole-organ scale in the section ‘Whole-organ simulations’ supported by data from the literature as well as MRI and pressure–volume measurements, and passive inflation on isolated hearts.

Cellular modelling

For our contraction model we have aimed to strike a balance between modelling detail and ability to parameterize the model using available data. The model contains several novel elements, though it is primarily based on two previously published models (Niederer et al. 2006; Rice et al. 2008) both of which have been used before in a whole organ context.

The first of these is the model developed by Niederer, Hunter and Smith (the ‘NHS model’) (Niederer et al. 2006), which has a relatively simple structure and clear procedures for determining most of its parameters. However, the model is unable to capture the faster relaxation kinetics of mouse cardiac muscle at higher pacing frequencies.

The second is the model published by Rice et al. (2008), which includes a more detailed four-state Markov model of the sarcomere incorporating troponin, tropomyosin and crossbridge kinetics. The processes for determining its many parameters are less clear, as are their uniqueness with respect to the fitting methods used. However, the model reproduces a wide variety of experimental data, and is better able to capture the faster kinetics of contraction. Both models are also primarily based on experimental data obtained under non-physiological temperatures ranging between 15 and 25°C at a 1 Hz stimulation frequency in animals which have physiological heart rates in the 6–10 Hz range. Figure 1 shows an overview of the cell model and interaction with the tissue scale. This model has also been made available in the CellML repository (http://www.cellml.org)

Figure 1. Cell model overview.

Calcium binds to troponin C (TRPN) in a cooperative way. This increases XB, the fraction of actively cycling crossbridges. Length and velocity dependencies are added to calculate the force generated, which is used at the tissue scale.

Electrophysiology

At the basis of contraction lies the change in intracellular calcium, for which we used the data-driven C57BL/6 mouse model developed by Li et al. (2011) based on comprehensive experimental data from Andersson et al. (2009). The only change we have made to this model is a reparametrization of the decay rate of the ryanodine receptor modulation factor, changing it to to better match experimental data for time to peak of the calcium transient. The new calcium transient (shown in Fig. S1) has time to peak, decay profile and diastolic calcium all in the range of experimental data.

Troponin binding

Active tension is initiated by calcium binding to the second calcium binding site on troponin C (TnC). There is significant evidence that shows that this reaction is cooperative with a Hill coefficient of 1.6–2.0 in experiments on isolated bovine and human actin filaments (Tobacman & Sawyer, 1990; Gillis et al. 2000; Davis et al. 2007). This is further supported by recent data from Kreutziger et al. (2011), who show that different single amino acid variants of TnC can have a significant effect on the steepness of the force-calcium relationship in mice. Our model of troponin binding captures this cooperativity using a standard cooperative binding equation which has a Hill curve as its steady state solution:

(1)

where TRPN represents the fraction of occupied regulatory sites, k_TRPN the unbinding rate, and n_TRPN the Hill coefficient. We used n_TRPN = 2 because of the higher temperature and increase in cooperativity with temperature observed in mice (Blanchard et al. 1999), and k_TRPN = 0.1 ms⁻¹ based on Davis et al. (2007) (15°C, isolated reconstituted filaments from mixed bovine/human/rat proteins). While this k_TRPN value was determined at a lower temperature, it does not have a clear temperature dependence (Niederer et al. 2006) and the current value is high enough to reproduce experimental measurements of force at a 6 Hz stimulation frequency. The value of [Ca²⁺]_T50, the calcium concentration needed for 50% bound TnC in steady state, can be taken directly from K_d values obtained in experiments (K_d= ([Ca²⁺]_T50)n_trpn). However, such data are also typically obtained at lower temperatures, and are inconsistent between different sources. Davis et al. (2007) gives K_d≍ 5, which results in significantly reduced activation for physiological calcium transients. Gillis et al. (2000) give [Ca²⁺]_T50 > 3 μm at 37°C, with a shift of 0.13 pCa units from 21°C to 37°C in bovine preparations. Tobacman & Sawyer (1990) report [Ca²⁺]_T50≍ 1 μm at 25°C in bovine thin filaments. Applying a 0.13 × (37 − 25)/(37 − 21) ≍ 0.1 pCa shift to this gives [Ca²⁺]_T50≍ 0.8 μm, which was deemed a more plausible value considering our calcium transient.

There have also been suggestions of crossbridge binding affecting the unbinding rate. However, more recent work suggests this effect appears only under rigor conditions, and is not significant under physiological conditions (Sun et al. 2009; Farman et al. 2010). For this reason we have not included it in the model. Furthermore, the troponin buffer can be either strongly or weakly coupled to the intracellular calcium, depending on whether binding directly affects intracellular calcium. Since our model of the electrophysiology includes a quasi-steady state buffer for calcium which represents a combination of several buffers present in the cell, and the total concentration of these individual buffers (including troponin) is unknown, we have thus chosen to keep the original formulation.

Tropomyosin/crossbridge kinetics

The process of calcium binding to TnC causes a conformational change in its associated tropomyosin complex, unblocking the actin sites for myosin crossbridge cycling. The six-state model of McKillop & Geeves (1993) described these transitions using a blocked, closed and open state of tropomyosin along with weakly, strongly or unbound crossbridges. While both the open and closed states allow for weak binding of myosin crossbridges, only the open state allows for transition to the force-generating strongly bound state.

Assuming the open and closed states are in rapid equilibrium, this reduces to a four-state model as in Rice et al. (2008), with tropomyosin kinetics and a crossbridge cycle. Due to the scarcity of experimental data and the intrinsic difficulties of measuring the details of subcellular processes, we further reduced this model by collapsing the crossbridge cycle to a single state, which leaves the model with the following two states. The first state is the crossbridge state, XB, in which the crossbridge is actively cycling. A constant fraction of these is considered to be in weakly/strongly/unbound states. The second state is the non-permissive state, N= 1 − XB.

As the crossbridge cycle is not faster than tropomyosin kinetics, this reduction will only be reasonable for sufficiently slowly changing calcium, such as a twitch, and is less suitable for reproducing other features such as the response to fast step changes in calcium.

The equation for transition between these two states was also chosen to give a Hill curve at steady state. There are several functions which give such a steady state which have been used previously, such as cooperativity only in the binding rate (Niederer et al. 2006) or a division of cooperativity between the binding and unbinding rates (Rice et al. 2008). Since there is insufficient data about subcellular mechanisms to distinguish between these two possibilities, we tested both in a simulation study. Using the division of cooperativity results in a 10 ms reduction in RT₉₀ and halves tension at the end of a twitch without significantly affecting time to peak tension, resulting in an improved fit for the twitch transient compared to experimental data (Stull et al. 2002).

We chose our non-linear function to ensure TRPN₅₀ has a clear interpretation, i.e. XB(TRPN = TRPN₅₀) = 0.5 at steady state, resulting in:

(2)

We set k_np= 0.1, n_xb= 5 in accordance with time to peak and Hill coefficient measurements (Blanchard et al. 1999; Stull et al. 2002), and TRPN₅₀= 0.35 to obtain a peak XB of around 1/3 which leaves sufficient room for length-dependent activation, and is consistent with levels typically seen in force–calcium loops (Stull et al. 2002).

These parameters result in a force–pCa Hill fit with coefficient of 6.7, compared 6.68 ± 0.63 at 37°C from experimental data (Blanchard et al. 1999). A biphasic Hill fit (McDonald & Moss, 1995) results in Hill coefficients of 8.3 and 4.1 for the lower and upper part of the curve, respectively, as shown in Fig. 2. Such different Hill coefficients have been observed previously in experiments (McDonald & Moss, 1995; Dobesh et al. 2002; Rice & de Tombe, 2004), and have previously been attributed to slower rates of crossbridge cycling at higher saturation (Dobesh et al. 2002). However, we found that this effect is to some extent even present in a simple [Ca²⁺]_i/TnC model, where the Hill coefficient in their lower part of the Hill curve is approximately 50% higher compared to the upper half when fit separately. Introducing cooperative binding of calcium to TnC increases this effect to the magnitude observed in experiments, with the lower part of the Hill curve having double the Hill coefficient compared to the upper half.

Figure 2. Biphasic Hill fits to model force–pCa response.

A best linear fit of log F/(1 −F) to pCa was applied separately for the upper and lower halves. The Hill coefficient n_xb for the n_trpn = 1 case was chosen to ensure the Hill coefficient fit for the entire curve matched the n_trpn = 2 case.

The model's response for tension generation and relaxation is shown in Table 1, where it is compared to experimental data.

Table 1.

Experimental data compared to model results for isometric twitch tension

Source	TPT	RT50	RT90	Measurements
Andersson et al. (2009)	33.5 ± 0.5			Unloaded shortening of isolated myocytes at 6 Hz
Stull et al. (2002)	∼35	22 ± 3	∼45	Isometric twitch in trabeculae at 6 Hz
Stull et al. (2006)		24 ± 2		Isometric twitch in trabeculae at 6 Hz
Bluhm et al. (1999)	∼30	∼27	∼60	Isometric twitch in LV papillary at 6 Hz
Bluhm et al. (2000)	37 ± 2	35 ± 2		Isometric twitch in LV papillary at 6 Hz
Stuyvers et al. (2002)	∼40	∼30	∼50	Isometric twitch in RV trabeculae at 5 Hz
Palmer et al. (2004b)	∼41/37		∼55/38	Isometric twitch in RV trabeculae at 4/10 Hz
Model results	36	23	49

TPT is time to peak tension from onset of tension in milliseconds, except for Andersson et al. where it indicates time to peak shortening. RT₅₀, RT₉₀ refer to time to 50% and 90% relaxation in milliseconds. All data are from mice at 37°C or 37.5°C. The ‘∼’ symbol indicates an approximation obtained from twitch transient graphs or bar charts shown in sources.

Parameter sensitivity and identifiability

The two stages of cooperative binding in the model introduces pairs of parameters, some of which may be highly coupled and thus problematic to distinguish during fitting: the sensitivities [Ca²⁺]_T50, TRPN₅₀, binding rates k_TRPN, k_xb, and Hill coefficients n_TRPN, n_xb. To determine the effect of this coupling on model parametrization, and how new experimental data could help improve it, we performed a parameter sensitivity analysis. We varied the pairs of parameters to identify the regions in which RT₅₀∈[19, 25], RT₉₀∈[45, 55], TPT ∈[30, 41], max XB(t) ∈[0.3, 0.4], XB(t= 167 ms) < 0.01. The Hill coefficients were further limited by the resulting Hill fit of the steady state force–pCa curve to [6, 7.5]. The ‘viable’ region for each of these parameters can then be identified as the intersection of these individual constraints, and its boundaries show which experimental data best constrains which parameters.

Results for the sensitivities are shown in Fig. 3A). These confirm the high degree of coupling between these parameters, constrained by the peak XB and relaxation times. While the viable region spans a wide range of values of either parameter, it is also very narrow, showing that determining one parameter accurately is sufficient for constraining both of them. Figure 3B shows results for the binding rates. Assuming that both processes happen on a similar time scale, we can see that these rates should be in the range of 0.07–0.15 to obtain realistic tension development and relaxation. Without such an assumption, there is a significant range of viable parameter values. Finally, results for the Hill coefficients are shown in Fig. 3C. These parameters are restricted to a fairly narrow window determined by the force–calcium Hill coefficient fit, peak XB, and relaxation kinetics. Overall, evaluating pairs of parameters in this way results in an intuitive visualization of what parameters are well constrained, which experimental data matter most, and where new data can help further constrain the model development process.

Figure 3. Viable regions for pairs of highly coupled parameters.

These regions shown as the shaded region as the intersection of all constraints, with lines showing individual constraints. The ‘X’ symbols indicate our choice of parameters. A, the parameter sensitivity for [Ca²⁺]_T50 and TRPN₅₀. The time to peak does not limit parameters within these ranges. B, k_TRPN, k_xb. C, n_TRPN, n_xb. The time to peak curve was omitted to improve the visualization as it does not contribute to restricting the parameters.

Length dependencies

Adding length- and velocity- dependent factors to the model, the normalized force including length and velocity dependence is given by:

(3)

where g(Q) determines the velocity dependence and h(λ) the change in maximum force, and λ is the extension ratio along the fibre direction.

Consistent with recent reviews (Solaro, 2007; Hanft et al. 2008), we based the increase in maximum force on filament overlap, modelled using the approach from Rice et al. (2008), and simplified as:

(4)

(5)

where β₀= 1.65, resulting in a linear length dependence near resting length, and a twice as steep decrease in tension when the sarcomere length falls below the thick filament length (at λ= 0.87).

A shift in calcium sensitivity is an additional and important mechanism for the length-dependence of tension. As details of the subcellular mechanisms which cause this are still being debated (Campbell, 2011), we use a simple phenomenological representation, where the calcium sensitivity [Ca²⁺]_T50 is directly length dependent:

(6)

Data from Stull et al. (2002, 2006) show twitch transient force roughly doubles with a 10% increase in sarcomere length. Having established h(λ), the length-dependent activation can be estimated from this data, resulting in β₁=−1.5. Figure 4 shows the length dependencies at both steady state and for twitch transients.

Figure 4. Length dependence of tension.

A, twitch transients at various sarcomere lengths (λ= 0.8, 0.85, 0.925, 1, 1.075, 1.15, 1.2 from bottom to top, where the extension ratio ). The maximum force F_dev is compared to experimental data in inset. B, force–pCa curves for different extension ratios (λ= 0.8, 0.9, 1.0, 1.1, 1.2 from bottom to top).

There are significant differences between models with regards to the relative importance of the change in maximum force and length-dependent activation such as increased calcium sensitivity. For example, our choice of β₀ is significantly smaller than in the NHS model, where β₀= 4.9. This balance between the different factors in the length dependence of tension will be further investigated later at the whole organ level.

Velocity dependence

Another well known aspect of cardiac muscle is the velocity dependence of force (Kawai & Brandt, 1980). As the exact mechanisms involved remain debated (Yadid & Landesberg, 2010; Palmer, 2010), we used a phenomenological ‘fading memory’ model (Hunter et al. 1998) as in the NHS model, which can reproduce many of the observed effects without being tied to underlying mechanisms.

The parametrization of the velocity dependence is based on sinusoidal analysis experiments (Kawai & Brandt, 1980). Our only complete source of data of this kind of analysis performed at 37°C is from Palmer et al. (2007), whose data were digitized and fitted using the equation:

(7)

to obtain the viscous and elastic modulus. See Table 2 for results, which are consistent with the c= 81.7 s⁻¹ value reported by Palmer et al.

Table 2.

Experimental data compared to model parameters and results for strain-rate response using sinusoidal analysis

Source	α1 = 2πb	α2 = 2πc	B	C	Measurements
Blanchard et al. (1999)	2π15 = 94 s−1	2π66 = 415 s−1	∼800	∼1000	27°C mouse
Blanchard et al. (1999)	2π27 = 170 s−1	2π94 = 591 s−1	—	—	37°C mouse
Palmer et al. (2007)	2π10.7 = 67 s−1	2π83.7 = 526 s−1	566	2489	37°C, Fit to data from image
Model	150 s−1	500 s−1		−4 B
Virtual experiment	153 s−1	502 s−1	642	−3.7 B

Showing experimental results from the literature for the frequency parameters of the B and C processes converted to equivalent model parameters α₁, α₂, and their magnitudes B,C. ‘Model’ shows model parameters used, while ‘Virtual experiment’ shows results from fitting the model response.

Sometimes an additional term (the ‘D process’) is added Kawai et al. 1993), although its relevance in cardiac muscle is controversial as it is difficult to measure, and involves frequencies above those of physiological interest (Campbell et al. 2001). Therefore we have chosen to omit this process, reducing the number of model parameters.

The fading memory model introduces two equations of the form:

(8)

The fading memory model is effectively a transform into the time domain of the frequency response observed with sinusoidal analysis of muscle fibres (Kawai & Brandt, 1980). The A_i parameters are directly related to the viscous and elastic moduli, and α_i to the frequencies. 2πb, 2πc, i.e. parameters α₁, A₁ are related to the slower ‘B process’, and α₂, A₂ to the faster ‘C process’ (Kawai et al. 1993).

The effect on tension g(Q) mentioned in the previous section is given by:

(9)

where a is the slope of the Hill force–velocity equation.

Data from Palmer et al. suggest , which would increase tension at a constant shortening velocity. Therefore we took into account the partial results from Blanchard et al. (1999), and set α₁= 150 s⁻¹, α₂= 500 s⁻¹. We left a at 0.35 based on the NHS model. Finally we set A₁=−4, A₂=−4A₁ to obtain a magnitude and ratio of B and C in the range of experimental data. These values are uncertain as the viscous and elastic moduli scale with isometric tension, which was not reported by Palmer et al. and can often be lower in experiments with saturating [Ca²⁺]_i (Campbell, 2011). We thus investigated the effect of this parameter further in our whole-organ simulations below.

Running a virtual sinusoidal analysis ([Ca²⁺]_i= 1 mm, 0.25% length change) and fitting the resulting curve to eqn (7) with k= 0 fixed results in values shown in Table 2. Our resulting f_min= 20 Hz, f_max= 85 Hz also closely match results from Palmer et al. (f_min= 20 Hz, f_max= 80 Hz).

Tension scaling

Finally, tension was scaled from normalized force to actual tension for a whole organ context:

(10)

where the reference tension T_ref encapsulates the total number of crossbridges and the fraction of cycling crossbridges in the force generating state at any time. For the NHS model the equivalent parameter is 56.2 kPa, but this is set to 100 kPa in later whole organ models (Niederer & Smith, 2009). In the Rice et al. model,

Experimental data often indicate much lower values for T_ref of around 20–30 kPa (Blanchard et al. 1999; Stuyvers et al. 2002; Palmer et al. 2004a) although some twitch transient data peaks at 50 kPa (Stull et al. 2002) and other results go up to 112 kPa for maximum developed tension (Kreutziger et al. 2011). We set T_ref= 120 kPa consistent with previous models, and the higher end values for maximum tension and twitches, XB ≍ 0.35, resulting in 42 kPa in a twitch transient at resting SL.

Whole organ simulations

As murine experimental models are typically primarily focused on determining the effects of changes at the cellular scale on whole-heart function, and experimental data are more readily available at this scale, it is important to extend the models to make this link between subcellular mechanisms and whole-organ function in models. This introduces several new model components, including the material properties of cardiac tissue, electrical activation sequences, mechanical boundary conditions, valves and blood flow in the chambers. Their effect on model results also needs to be characterized within their experimental constraints, to provide or exclude alternative explanations for any changes observed in heart function.

Experimental data

To develop and validate our model at the whole organ scale, we have recently obtained a range of multimodal whole-organ data sets in mice, from the same strain as used to parametrize the electrophysiological cell model (Li et al. 2011). The Supplemental Material includes details of the materials and methods used to obtain these measurements. Briefly, the data include in vivo MRI data and pressure–volume measurements and passive inflation of ex vivo hearts. MRI data used for mesh generation and volume estimation were obtained at a 0.2 × 0.2 × 1 mm resolution during a whole heart cycle. Pressure and volume were measured in vivo using a pressure–volume catheter, along with ECG, and flow measurements to accurately determine start and end of ejection. These data were used to obtain end-diastolic pressure (EDP), aortic pressure at start of ejection, and pressure and volume transients for comparison.

Passive inflation data were obtained from echo measurements in isolated hearts at 0, 5 and 10 mmHg, and used to estimate the increase in volume at a given pressure. Using manual segmentation of the short-axis slices as well as a measurement of apex–base distance in a long-axis slice, the volume change V(p)/V(0) was estimated at A(p)/A(0) ×D(p)/D(0), where A(p) is the area of the cavity in a short-axis slice and D(p) the apex–base distance. This was done separately for two slices to give two estimates for 5 and 10 mmHg.

Simulation set-up

For our whole-organ simulations we applied the framework for electromechanics as described previously (Land et al. 2012). For mechanics we used a 112 element cubic Hermite mesh derived from the MRI data, at a time point half-way through diastole to approximately match the geometry of a stress-free ventricle. We estimated mechanical properties based on passive inflation data, as described below, and adjusted the reference state as needed to match EDP and EDV from PV and MRI data. Since this process converges quickly, the second choice of reference geometry was consistent with available data. We used a dense electrophysiology grid with a spacing of ∼0.1 mm. The smaller size of the mouse allows for this high resolution, although the ∼10⁵ nodes still incur a significant computational cost due to the fast time scales involved in the detailed cellular model. The ODE solver used is described in the Supplemental Material. Due to faster contraction, a smaller time step than the conventional 1 ms for mechanics is required. Specifically, we used a 0.1 ms time step for mechanics and monodomain equations, and half this for monodomain equations during the depolarization, to ensure convergence (Land et al. 2012). The mesh and mechanical boundary conditions are shown in Fig. 6.

Figure 6. Mesh generation process and boundary conditions.

A, the raw MRI data, numbered from base to apex, with segmentation in blue. B, the fitting of a cubic Hermite finite element mesh to this data. C, the application of boundary conditions: blue nodes at the apex are part of multiple collapsed elements and have their derivatives fixed to zero. All nodes at the base (green) are fixed along the apex to base z axis. Furthermore, one of the red nodes at the base is fixed along the x and y directions, and the other along the y axis, which prevents free rotation and translation of the mesh during the solution process without limiting the deformation. Nodes at the apex and base have increased stiffness, interpolated throughout their neighbouring elements. The region of electrical activation is indicated in gold.

The framework was further extended with a full heart cycle model. We inflated to the end diastolic pressure (EDP = 0.7kPa), and stayed in the diastolic phase until volume flow reversed. Next, we simulated isovolumetric contraction until the cavity pressure was high enough to open the aortic valve (≥p_a= 9 kPa as in experimental data). Ejection was then simulated using a three element Windkessel model, which consisted of the equation:

(11)

where R is the peripheral resistance, C is the total arterial compliance, and Z is the aortic characteristic impedance. When flow reversed once again, we switched to the isovolumetric constraint to reproduce isovolumetric relaxation. Finally, when pressure relaxed below 0.1 kPa, we switched back to diastole by solving a simple phenomenological model for diastole given by . This was implemented by solving the equation monolithically in the mechanics Newton iteration process, with coefficients m_i varying depending on the heart cycle to represent isovolumetric phases , diastolic filling, or the Windkessel model.

Electrical activation

Two important factors in the electrical activation of tissue are the activation pattern and the conductance. The latter was set using D_ff= 0.33 mm² ms⁻¹ to obtain a conduction velocity of around 75 cm ms⁻¹ (Tamaddon et al. 2000), with consistent with the anisotropy observed in mice (Baker et al. 2000).

We initiated electrical activation with a stimulus midway between apex and base near the endocardium (60 mV ms⁻¹ for 3 ms). This was compared to the simpler case of inducing full activation homogeneously, which gives a large decrease in EF (ejection fraction, 36.5%vs. 56.3%). These differences are caused by a difference in calcium transient, which is increased in cells which act as a source for activation of nearby cells. This effect can be reproduced by inducing homogeneous activation with a stimulus current of I_stim=−40sin(2t) for t∈[0,π], which increases EF back up to 53.1%.

Since the shape of the stimulus current can change simulation results significantly, and is significantly different in tissue from the stimulus protocol used in single cells (I_stim=−50 for 1 ms), we paced the cell model again for 5000 beats with the above sinusoidal current, so as to better prepare it for a tissue environment. This changes EF to 61.2% for endocardial activation, and aids in an earlier decrease of the larger calcium transient seen in tissue.

Fibres

Our choice of fibre distribution throughout the tissue is −60 deg at the epicardium to +90 deg at the endocardium, based on mouse DTMRI data from Jiang et al. (2004). To investigate the sensitivity of our model to this choice, we briefly compared it to several other choices of fibre directions. By looking at the position of the apex, we are able to clearly show some small but significant differences in deformation. When using a −70 deg to +70 deg variation throughout the wall, ejection is similar, but the deformation was more realistic in the case of using mouse DTMRI data, as the apex-to-base distance did not increase. With detailed fibre directions based on dog data (Usyk et al. 2000) the ventricle inflates slightly more compared to the ±70 deg case, and generates more force because of this, but the model produces an approximately 10% lower ejection fraction, as the apex moves less during ejection. Details of the apex movement for these three cases are shown in Fig. 7, which demonstrates the steep fibre angles are important for preventing a large increase in apex–base distance during IVC. The mouse DTMRI fibre directions produce a similarly high ejection, but also does not suffer from the unrealistically large increase in apex–base distance during isovolumetric contraction.

Figure 7. Location of the apex for three different fibre distributions.

Simulation time includes inflation, isovolumetric contraction, ejection and the beginning of isovolumetric relaxation. The legend shows a slab of tissue with transmural fibre directions.

Material and haemodynamic properties

What remains to be determined are the material properties of cardiac tissue, and the Windkessel parameters which determine the relation between blood flow and pressure during ejection.

For our material properties we used the Guccione constitutive law (Guccione et al. 1991):

(12)

where E is the Lagrangian strain tensor for a local coordinate system aligned with the fibre direction. We use C₁= 1.1, C₂= 8, C₃= 2.0, C₄= 3.7 based on work by Omens et al. (1994), with C₂ adjusted from 9.2 to 8 to better match passive inflation data. Fig. 5 shows the estimates for volume change in experimental data, compared to parameters from Omens et al. (1994) (C₂= 9.2) and adjusted parameters (C₂= 8). We only estimated C₂ since it is highly coupled to other constitutive parameters (Xi et al. 2011a), and passive inflation data are insufficient to estimate all parameters. Stiffness at the base and apex is increased (C₁= 8) as in previous work (Land et al. 2012).

Figure 5. Constitutive parameters and passive inflation data.

Figure shows the relative increase in volume as a function of cavity pressure in simulations (line) compared to experimental data (two different short-axis slices indicated by different symbols). Inset shows example segmentation of echo data, where apex–base distance is indicated in the long-axis slices and the cavity is segmented in the short-axis slices.

For the Windkessel model we used R= 400 mmHg s ml⁻¹, Z= 20 mmHg s ml⁻¹, C= 0.002 ml mmHg⁻¹. Aortic characteristic impedance and peripheral resistance are compatible with extrapolations from rat data (Westerhof & Elzinga, 1991) assuming the mass of a mouse is 20% that of a rat, as well as data from Segers et al. (2005). A simple estimate using the cardiac output and aortic blood pressure results in R= 535 mmHg s ml⁻¹, which is known to be an overestimate (Toorop et al. 1987), and therefore also consistent with our choice. Compliance was based on differences between different Windkessel model fits (Segers et al. 2008) and data from four-element fits (Segers et al. 2005) to constrain C to a moderate increase from the value used in four element Windkessel models, and then adjusted according to PV data.

Results

Parameter variation

Table 3 shows a summary of parameter variations at the whole-organ level. We have chosen a set of key metrics available from pressure–volume data for comparison. Of these, ejection time and peak systolic pressure are important for characterizing ejection, easily determined, and show little variation across experiments. The other measures, end-systolic pressure and stroke volume, are more difficult to compare directly, as end-systolic pressure is more variable in the literature, and the volume measurements differ between MRI and PV data. However, these are still included to give a more complete picture of the relative influence of different parameters.

Table 3.

Summary of parameter sensitivity results at the whole organ level

	Ejection	Peak SP	ESP	SV
PV data	43 ms	11.8 kPa	6.9 kPa	28.8 μl
Default	27.3 ms	16.0 kPa	11.3 kPa	38.5 μl
Whole-organ parameters
C2−12.5%	↑ 0.3 ms	↑ 0.3 kPa	↑ 0.1 kPa	↑ 1.9 μl
C4+370%	↓ 0.5 ms	↑ 0.3 kPa	↑ 0.1 kPa	↑ 1.1 μl
edp+0.1 kPa	↑ 0.5 ms	↑ 0.4 kPa	↑ 0.1 kPa	↑ 2.6 μl
pa+1 kPa	↑ 0.1 ms	↑ 0.5 kPa	⇑ 0.7 kPa	↓ 1.8 μl
R−50%	↑ 0.1 ms	↓ 0.1 kPa	↓ 0.4 kPa	↑ 0.5 μl
Z+50%	⇑ 3.8 ms	⇑ 1.3 kPa	↓ 0.3 kPa	↓ 2.6 μl
C−50%	=	↑ 0.5 kPa	⇑ 1.6 kPa	⇓ 4.6 μl
Cellular parameters
β0–50%	↑ 0.1 ms	↑ 0.2 kPa	=	↑ 2.4 μl
β1–50%	⇑ 2.5 ms	↓ 0.5 kPa	↓ 0.1 kPa	↓ 0.2 μl
Tref−20 kPa	↑ 1.3 ms	⇓ 1.3 kPa	↓ 0.5 kPa	⇓ 5.2 μl
A1=−29	⇑ 10.8 ms	⇓ 2.7 kPa	↓ 0.5 kPa	⇓ 4.9 μl
a+50%	↑ 0.5 ms	↓ 0.2 kPa	↓ 0.1 kPa	=

Results shown for duration of ejection, peak systolic pressure (SP), end systolic pressure (ESP) and stroke volume (SV). The magnitude of parameter variation is based on how constrained it is given available data, and generally taken in the direction to increase duration of ejection. See Tables S1 and S2 in the Supplemental Material for detailed results.

The constitutive parameters C₁, C₂ and C₃ have similar effects on overall stiffness, where a higher stiffness decreases diastolic volume and ejection as expected. Surprisingly, increasing C₄ results in a slightly higher SV, and duration of ejection can both slightly increase or decrease depending on the geometry used. Overall the model is very insensitive to large changes in this parameter. The model is also not very sensitive to large changes in R or C.

There are clearly some significant differences between simulations and PV data, in both the duration of ejection and peak systolic pressure. The only parameter which extends ejection by >1 ms for our default parameter variation is the aortic characteristic impedance Z. However, this parameter is well constrained from data, and furthermore increases peak systolic pressure when extending ejection, increasing the difference between model results and pressure data from measurements.

Coupling mechanisms

Even after careful model parametrization, significant differences remain between experimental data and simulations. The new parameters and boundary conditions introduced when embedding the cellular model within the whole organ framework have been constrained, and we have found no explanation at this scale. This motivated an investigation into those aspects of the cell model that were less constrained during the cellular modelling stage, and are influenced by the mechanical deformation. This includes the balance between the mechanisms of length dependence, as well as aspects of the velocity dependence.

Length dependence of tension

For the length dependence of tension, we included a filament overlap factor as well as a shift in calcium sensitivity. We assumed the latter factor was dominant, but had no direct data in mice to constrain their relative influence.

To test the assumptions made in our cell model, we applied two parametrizations which give a roughly identical increase in tension for 10% stretch: our parametrization β₀= 1.65, β₁=−1.5 with dominant length-dependent activation (LDA) factors, and β₀= 4.9, β₁=−0.8 closer to the NHS model's settings. The comparison in Fig. 8A shows the former parameter set maintains higher pressures for longer, as it takes some time for tension to decrease when shortening, which leads to a higher EF and faster relaxation. These differences show that our initial choice is closer to experimental data, and suggests keeping the parametrization unchanged.

Figure 8. Effect of length and velocity dependence on whole organ function.

A, a comparison for two different length-dependent tension mechanisms, with weak velocity dependence. B, a comparison of two magnitudes for velocity-dependent effects. C, the same comparison of length dependence as panel A, but using a strong velocity dependence.

Velocity dependence

Another important aspect of the cell model, in the context of ongoing debates about the mechanisms underlying the coupling between mechanical deformation and cell function, is the velocity dependence. We parametrized this model component using data from sinusoidal analysis. The moduli A₁ and A₂ involved were fitted to experimental data using the viscous and elastic modulus, but these scale with isometric tension, which can often be lower in experimental measurements.

To further investigate the effect of these parameters, in light of the differences between experimental data and our results, we again compared two different cases in detail. We compared using our default parametrization as outlined previously to using A₁=−29 as in the NHS model (keeping A₂=−4A₁). Figure 8B shows the difference is clear, and the higher velocity dependence is much closer to PV catheter pressure data. This confirms our previous suspicion that the elastic and viscous modulus may not always be suitable for parametrizing single cell models directly. The results also show a reduction in several differences between model and experiments, most significantly the duration of ejection, which was much lower. Other cellular scale parameters which can reduce this difference include a decrease in the length-dependent binding parameter β₁. However, β₀ has the opposite effect and the combination of these parameters is well constrained from cellular data, as shown in the previous section.

A higher a results in a slightly longer ejection phase, and lowers peak systolic pressure, though the effects are very small considering experimental data are centred on the 0.2–0.4 range (Niederer et al. 2006). We also briefly tested violating the constraint mentioned in the section on cellular modelling (by setting α₂= 750 s⁻¹ or α₁= 75 s⁻¹). This results in oscillating tension and strain, as well as very short ejection time, confirming that the adjustment to meet this constraint was indeed needed. Figure 8C shows that in this corrected case, our conclusions about the length dependence are still valid, with 4 μl higher ejection and better relaxation in the case of high length-dependent activation.

Tension scale

Finally, we considered the tension scale T_ref. For our initial case, decreasing reference tension to T_ref= 100 kPa decreased ejection by 5.2 μl, lowering overall pressure without significantly impacting duration of ejection. In the case of the proposed high velocity dependence, ejection similarly decreased, taking it out of the range of experimental values. Increasing T_ref to 140 kPa increased both ejection fraction and overall pressure. This shows that T_ref= 120 kPa is a more realistic choice than T_ref= 100 kPa, in line with isometric twitch force used in other models. However, it cannot rule out a value which is slightly higher, considering the variability in peak pressure found in experimental data (Joho et al. 2007). Figure 9 shows a pressure–volume loop for the cases T_ref= 120 kPa, A₁=−29 and T_ref= 140 kPa, A₁=−35, compared to experimental data. Both of these cases are well within experimental variability, with end-systolic volume between estimates from echo and MRI (∼25 μl). Figure 10 shows the T_ref= 120 kPa, A₁=−29 case in more detail. As there was a significant difference in IVC duration, we also performed a simulation at 8 Hz, closer to the MRI data, at which this difference was reduced as pressure transient minima coincided.

Figure 9. Pressure–volume loop, compared to echo data.

The volume for echo data has been shifted to the EDV of MRI data, to account for elements like papillary muscles being included in volume in both simulation and MRI data, but not in echo. End systolic volume from MRI data coincides approximately with the left axis limits (25 μl).

Figure 10. Detailed results for the T_ref = 120, A₁ = −29 case.

Left panels show velocity and pressure for simulations at both 6 Hz and 8 Hz compared to experimental data, with PV catheter pressure offset to match EDV from MRI data. Note that heart rates differ slightly between PV and MRI data. Bottom left panel shows the mean value of g(Q), which indicates the relative effect of the velocity dependence on tension. Panels on the right show simulation results (8 Hz) at key points: end of inflation, beginning of ejection, mid ejection, end of ejection and end of isovolumetric relaxation. Both the fibre arrows and the endocardial surface are coloured to represent the fibre strain λ. Transmural surfaces is coloured depending on the calcium concentration, ranging from green (diastolic) to black (peak in single cell preparations with standard pacing) to white (hyper-activated in tissue).

Discussion

To investigate electromechanical coupling, we have developed an electromechanical model, which can reproduce the fast contraction and relaxation kinetics observed in mouse hearts. This model is computationally efficient to solve and includes length and velocity dependence, making it suitable for use in multiscale modelling.

Furthermore, our model is based mainly on experiments performed at body temperature. Since such data are currently still limited, a few parameters are still based on some data obtained at lower temperatures, most notably some of the Ca²⁺–TnC binding rates and sensitivities. However, we have used several methods to address this issue through refining the model development at the cellular scale, and subsequently using multiscale modelling to enhance our understanding and constrain the model parametrization of coupling mechanisms. Firstly, the parameter identifiability shows which parameters are already well determined, and where new experimental data could help further model development. The narrow valley of viable parameters shown in Fig. 3 means that more detailed data on calcium to TnC binding at physiological temperatures would further constrain the parameter choice for both [Ca²⁺]_T50 and TRPN₅₀. Unbinding rates for TnC would also help determine k_TRPN, and perhaps allow k_xb to be parametrized from force development rates. Although such binding coefficients might also aid in determining Hill coefficients to greater precision, experimental errors are unlikely to be much smaller than the small region already determined. At the cellular scale we also found that the strongly biphasic force–pCa relation previously reported (McDonald & Moss, 1995; Dobesh et al. 2002; Rice & de Tombe, 2004) can be explained by cooperative binding of calcium to TnC.

Secondly, we have shown that running whole-organ simulations can assist in the inference and interpretation of aspects of length- and velocity-dependent feedback at the cellular scale, for which data are still insufficient or contradictory. As outlined in our introduction, there is ongoing debate in the literature about the basic mechanisms involved in the strong length dependence of tension. Even without trying to model the details of these mechanisms, it can be difficult to understand the balance between filament overlap and length-dependent activation, as maximum twitch force in single cells cannot readily be used to distinguish between them. Investigating the dynamics of ejection and isovolumetric relaxation in the ventricle strongly suggests that length-dependent activation factors are dominant, considering their relative effect on ejection and dp/dt_min. This is most likely related to the delay involved in our LDA mechanism, which ensures that the strong decrease in tension influences late ejection and isovolumetric relaxation. Filament overlap is modelled as an instantaneous effect, and causes tension to decrease at the start of ejection, decreasing stroke volume. Strong delays in the response of force to length changes have been shown before in experiments (Levy & Landesberg, 2006), but to our knowledge have not been measured in mice at body temperature, and will likely be difficult to separate from velocity-dependent effects under these conditions.

Perhaps more importantly, we were able to use emergent function in whole-organ simulations to show the importance of a strong velocity-dependent effect on tension in reproducing pressure transients seen in vivo. Introducing a stronger velocity dependence resulted in a longer duration of ejection, with a lower and extended pressure plateau during ejection, rather than a pressure transient which is similar to single-cell tension transients, as shown in Fig. 8. Results from other simulations (Table 3) show that this is a necessary change to explain pressure transients, as other parameters in the model are either constrained from experimental data or do not explain differences between model results and experimental data. Furthermore this result confirms our suspicion that isometric tension in these single cell experiments was likely to be much lower than that generated in vivo. An explicit mention of the isometric tension during sinusoidal analysis experiments would make such experimental data more useful for direct use in cell model parametrization.

The combination of length- and velocity-dependent effects is shown to be responsible for the difference between tension transients at the single cell level and pressure transients seen at the whole organ level. The flatter plateau in pressure transients is mainly a velocity-dependent effect: when the aortic valve opens there is a phase of fast shortening which slows further generation of tension. At the end of ejection, tension is briefy increased by the ‘B process’ known from sinusoidal analysis (Kawai et al. 1993), resulting in higher pressures at the end of ejection, as can be seen from the velocity dependence effects in Fig. 10. The end of this phase, combined with the faster relaxation at a shorter sarcomere length, is likely to lead to the high dp/dt_min values seen. We were also able to test the choice of T_ref in a whole-organ context, confirming that isometric twitch transients at resting SL should generate around 40–45 kPa peak tension, as used in previous whole-organ work and cell models. The detailed whole-organ simulations uncovered significant effects of the electrical propagation on the calcium transient in tissue. In the model this appears as an increase in the L-type calcium current just after the action potential upstroke, when cells are driving action potential propagation and act as a source for neighbouring tissue. This raises questions about whether such effects play a major role in vivo, as the dependence of calcium regulation on action potential propagation was not validated in Li et al. (2011). Both novel experimental measurements and simulations with different cell models would help further determine the importance of these effects. Fibre directions were also shown to be important, and can significantly affect deformation and whole-heart function, most significantly apex to base shortening during IVC and ejection.

Whereas previous studies using electromechanical models have usually been based on a few results or required large supercomputing facilities, the smaller organ size combined with efficient computational methods (Land et al. 2012) have allowed us to run hundreds of whole-organ simulations on an average quad-core desktop PC. Using such a large number of results we can look at the effect of a broader range of parameters, and obtain a better understanding of the important components in a ventricular model, thereby allowing us to identify or exclude other potential mechanisms which might also explain differences between experimental and simulation data.

In our parameter variation study, we were able to estimate overall stiffness based on passive inflation data, arriving at similar constitutive parameters as used by Omens et al. (1994). The relatively small effect of C₄ explains the large variations in this parameter between different studies, and difficulty of determining it using optimization methods (see e.g. Wang et al. 2009). We also identified pressure at the beginning of ejection as an important parameter which can significantly influence simulation results. Taking this parameter directly from experimental results prevents incorrect fitting of Windkessel models and velocity-dependent effects, as they are highly coupled with pressure at the beginning of ejection in their effects on peak systolic pressure. Overall, experimental data from passive inflation and in vivo pressure–volume loops, combined with data in the literature, can be used to determine most parameters introduced at the whole-organ level. Such parameters tend to be either well constrained by these data, or have a relatively minor effect on results.

Modelling cardiac function at a higher heart rate and physiological temperature allowed for more quantitative comparison with data from in vivo measurements compared to previous murine models at low temperatures or pacing frequencies, as differences between models and experiments cannot simply be rationalised by appealing to temperature differences. Using more physiological conditions has already shown that previous contraction models do not always handle relaxation well at these pacing frequencies, which can easily go unnoticed when running simulations at 1 Hz or comparing to experimental data obtained at lower temperatures. As models progress towards simulating increasingly physiological cases, they can reveal gaps in our current understanding of cardiac physiology. An example of this relevant to the current study are the differences seen between model and experiments during late IVR and the rate of relaxation, which are still to be fully explained.

As with any simplified model of a complex multiphysics system, there are a number of limitations, as a result of choices to reduce model complexity, as well as lack of complete knowledge and data. We have used a data-driven approach to inform model complexity, using models suitable for the amount of available data whose structure is consistent with the current physiological consensus on the relevant underlying mechanisms. Because the model is modular, further complexity has the potential to be introduced as data become available, or as necessary for specific studies. Length dependence properties were assumed to be homogeneous throughout the tissue. This is likely to be an oversimplifcation (e.g. Cazorla & Lacampagne, 2011, though there are not sufficient data available to characterise this spatial variation in more detail. This question could be investigated in future work using our framework, when more experimental data become available.

For our whole organ results, we have used a simple model for diastole, as our study focuses on earlier cardiac phases, which are more sensitive in changes to our parameters. The determination of an optimal choice of mechanical boundary conditions remains a challenge. Previous solutions have included similar increased stiffness (Nickerson et al. 2005), a fxed base plane (Keldermann et al. 2010) or epicardium (Niederer & Smith, 2009), and prescribing this boundary based on MRI data (Xi et al. 2011a). Such interaction with the rest of the heart is only one of several mechanical factors not included in these models, including papillary muscles, the pericardium and interaction with the rest of the body. We have used transversely isotropic material properties, which may be limiting the prediction of tissue deformation. Ideally, fully orthotropic material should be used, including structural sheet data, as suggested both by the tissue microstructure and shown in experiments (Holzapfel & Ogden, 2009). However, such orthotropic constitutive laws cannot be constrained even when using more detailed data than currently available in mice (Xi et al. 2011b). The three-element Windkessel model does not include an inertance element, unlike more detailed models of haemodynamics (Segers et al. 2008). This may have caused ejection to end earlier in our model, with a higher end systolic pressure. Finally, there are significant differences between experimental measurements from conscious mice, compared to anaesthetized mice (Joho et al. 2007). In this study we have focused on the latter, consistent with our recently obtained experimental measurements and most data in the literature. However, to what extent these differences involve changes in function at the cellular scale remains an important open question.

In conclusion, we have developed the first electromechanical model of the mouse heart, allowing in silico investigation of many experiments done in these animals. This framework has already shown potential for improving our understanding of cardiac function at both the single-cell and whole-organ level, by revealing the role of coupling mechanisms in explaining the striking differences between the shapes of cellular tension and ventricular pressure transients. In future work we plan to use this modelling framework to investigate genetic knockout mice, extending the work of Li et al. (2011) on electrophysiology to whole-organ electromechanics, thereby linking changes in genes to whole-heart function.

Glossary

DTMRI

diffusion-tensor magnetic resonance imaging

EDP

end-diastolic pressure

EDV

end-diastolic volume

EF

ejection fraction

IVC

isovolumetric contraction

IVR

isovolumetric relaxation

LDA

length-dependent activation

PV

pressure–volume

TnC

troponin C

Author contributions

SL, SAN, NPS: Conception and design of the mathematical models, computational simulations and analysis of simulation results. JMA, EKSE, LZ, WEL, IS, OMS: Conception and design of the MRI, PV and passive inflation experiments, experimental measurements and data processing. All authors approved the final version of the manuscript.

Supplemental Data

Figure S1

Figure S2

Figure S3

Table S1

Table S2