STACOM Challenge: Simulating Left Ventricular Mechanics in the ...

STACOM Challenge: Simulating Left Ventricular Mechanics in the Canine Heart Liya Asner(B) , Myrianthi Hadjicharalambous, Jack Lee, and David Nordsletten Department of Biomedical Engineering, King’s College London, London, UK [email protected]

Abstract. In this paper we outline our approach for creating subjectspecific whole-cycle canine left-ventricular models, as part of the 2014 STACOM Challenge. Each canine heart was modeled using the principle of stationary potential energy, with the myocardium treated as a nearly incompressible hyperelastic material. Given incomplete data on the motion and behavior of each canine heart, we decreased model complexity by employing reduced–parameter constitutive laws. Additionally, base plane motion and left ventricular volume input data were integrated into the cardiac cycle model through the inclusion of novel external energy potentials (using Lagrange multipliers), which allow for relaxed adherence to the constraints and minimize spurious energy modes stemming from model simplification and data noise. Subsequently, using the available data we employ the reduced-order unscented Kalman filter (ROUKF) approach to estimate the myocardial passive parameters and active tension. Finally, along with model predictions for each canine, we assess the spatial convergence and robustness of our model. Keywords: Cardiac mechanics · Canine heart · Parameter estimation · Data assimilation

1

Introduction

Aspects of left ventricular mechanics – including motion, constitutive relations, microarchitecture and contractile function – have been the focus of intense research. Complementing this effort, the mathematical modeling of the full cycle mechanics has become well–established [13], incorporating varying degrees of the inherent physiological complexity into passive and active tissue constitutive models [9] as well as advanced physiological boundary conditions [4]. While these models are idealised, they provide a platform for assessing patient-specific myocardial function as well as prediction through tuning a series of model parameters. The model-tuning process has raised interest in inverse problems in whole heart mechanical modelling, and currently a range of tractable methods to quantify the parameters underpinning function have been proposed [1,5,11]. This approach has been applied within the cardiac modelling community to estimate both passive and active mechanical parameters [4,11,14,15]. c Springer International Publishing Switzerland 2015  O. Camara et al. (Eds.): STACOM 2014, LNCS 8896, pp. 123–134, 2015. DOI: 10.1007/978-3-319-14678-2 13

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Various inverse estimation techniques have been proposed in the literature to date, including sequential quadratic programming, which has roots in nonlinear optimisation [1,14], the unscented transform algorithm [11] and families of data assimilation methods including Kalman filter-based sequential assimilation, and adjoint-based variational assimilation [5], although a systematic comparison of the techniques is currently lacking. Further to the estimation methodology used, the accuracy and quality of prediction is clearly limited by the selected models, their construction as well as the data available for parameterization. For a unique parameterization, the well-posedness and conditioning of the problem is critical, requiring careful selection of the appropriate forward model so available input data may be judiciously integrated. The common approach in the literature has been to limit the number of parameters estimated and use a much richer set of image data that covers the full spatial and temporal extent of the pertinent cardiac phases to constrain the parameterization process [7]. For instance, full Lagrangian displacement field extracted from tagged MRI has been the basis for estimating 4 parameters [14,15], while the surface motion derived from cine MRI has been used for estimation of 4 parameters [11], or 1 parameter in 6 different regions of myocardium [4]. In this paper we employ these developed approaches for constructing canine left ventricular models as part of the 2014 STACOM Challenge. The problem is a challenging one due to its underconstrained nature, whereby the input data consists of the surface geometry provided at three time points and a pressurevolume loop. Accordingly, to produce accurate deformation, the solution approach requires a corresponding increase in the ”regularisation” – this may be provided by further reduction of model complexity to prevent over-fitting and sensibly-posed boundary conditions to constrain the motion of the ventricle to be physiological. Our approach employs a reduced-parameter constitutive law adapted from the literature, and a generic strain-dependent active law. For each of the cases the models are personalised based on the available data. The methods used are described in Section 2, and the results are shown in Section 3. We were able to match pressure-volume loops, compute displacements and stresses through the cycle, and ascertain that numerical convergence was achieved in these variables.

2 2.1

Methods Cardiac Mechanics Model

The mechanical deformation and pressure (u, p) in the canine left ventricle were modeled using the principle of stationary potential energy which minimizes a free energy functional, Π [2]. Momentum in the heart was neglected due to the dominance of mechanical stresses through the cycle. The myocardial muscle was modeled as a hyperelastic material, as described by equation (1). The strain energy comprised passive and active terms (subscript p and a, respectively). Viscoelastic effects were also neglected due, in part, to the lack of validated material models. As the myocardial volume was observed to change up to 12%

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Fig. 1. Schematic of the forward model and given data for STACOM 2014

through the cardiac cycle (percentage volume change between end–diastole and end–systole was 3.6%, 8.9%, 12.0% and 8.3% for the four cases), the material was modeled as nearly incompressible (with bulk modulus κ) as shown in equation 1.  p  (1) W (u, p) = Wp (C) + Wa (C) + p J − 1 − 2κ Here C and J denote the right Cauchy Green strain and the Jacobian respectively [2]. The passive strain energy Wp was modeled using a reduced form of the Holzapfel-Ogden model [7,8] characterized by two scaling parameters {a1 , a2 } and two exponential parameters {b1 , b1 }, with indices 1 and 2 corresponding to the fibre and the cross-fibre terms: a1 a2 {exp[b1 (IC f − 1)2 ] − 1} + exp[b2 (ICˆ − 3)]. Wp (C) = 2b1 2b2 The active generation of internal energy was modeled via a simple time-dependent scaling: Wa (C) = α(t)(IC f − 1). Both the passive and the active strain energy functions are directly dependent on fibre strains, and therefore on the choice of the fibre distribution in the myocardium. The use of DTI-based fibres provided for the four cases consistently led to non-physiological lengthening of the ventricle in systole, even with a uniform relaxation (see Figure 6). Instead we have constructed idealised fibre fields based on the geometries, with the angles between sheet planes and fibre vectors varying linearly in the transmural direction between −80◦ and 80◦ from epi- to endocardium. As shown in Section 3, this distribution allowed us to simulate long axis shortening of the LV, which is normally observed in systole. From the definition of the strain energy given by equation (1), the energy functional Π can be written as the sum of the internal energy and external

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(boundary-based) energies. Additional Lagrange multipliers λb and λl are introduced along the base plane and lumen surfaces respectively to ensure adherence of the model to base plane motion and volume change data, as discussed in the following sections.   W (u, p) dX − Πkext (u, λk ), (2) Π(u, p, λb , λl ) := Ω0

k=b,l

where Πbext denotes the base plane energy, and Πlext the lumen wall boundary energy. The final solution for any point in time t ∈ [0, T ] in the heart cycle is determined as the saddle point of the free energy functional Π, ensuring adherence to boundary conditions whilst minimizing the internal energy, i.e. Π(u, p, λb , λl ) = inf {sup{Π(v, q, q, μ), (q, q, μ) ∈ X}, v ∈ U }

(3)

Here U is the space of admissible displacement solutions and X = W × γb U × R the space of Lagrange multipliers, with W being the space of hydrostatic pressure solutions, and γb U the trace space of U on the base plane (see Figure 1). 2.2

Base Plane and Endocardial Boundary Energies

In our model the specific external energy terms were applied on the boundaries to enable assimilation of the data provided. To ensure proper base plane motion of the model Πbext was selected as,    1 b b b ext λ · u − ud (t) − K(t)λ Πb (u, λ ) := dX, (4) 2 Γ0b where ud (t) is the displacement of the base plane and K(t) a constraint relaxation matrix. The motion of the base plane was given through the position of four points in space X = (x1 , x2 , x3 , x4 ) at three points in time (DS, ED and ES). Therefore the displacement fields ud (t) applied on the whole base through the cycle had to be interpolated. Specifically we used singular value decompositions to construct affine mappings A(DS to ED) and A(ED to ES) ∈ R3 that accommodated the given transformation of points X, i.e. A(DS to ED) XDS = XED , A(ED to ES) XED = XES , The mappings were then applied on the whole set of base points xb : xbED = A(DS to ED) xbDS , xbES = A(ED to ES) xbED = A(ED to ES) A(DS to ED) xbDS .

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At any time during the heart cycle [0, T ] (with tDS = 0) the displacement of the base points can be linearly interpolated in time using the DS, ED and ES values: t  b xED − xbDS , t ∈ [0, tED ] : ud (t) = tED t − tED  b t ∈ [tED , tES ] : ud (t) = xES − xbED + xbED − xbDS , tES − tED T −t  b xES − xbDS . t ∈ [tES , T ] : ud (t) = T − tES Strict adherence to the interpolated base condition gives rise to unphysiological stress / strain distributions. The additional term − 21 K(t)λb in the base boundary energy expression (4) allowed us to relax the constraint and lower the artificially induced boundary stresses. The matrix was chosen as K = ε(I − nb (t) ⊗ nb (t)) with nb denoting the base plane normal through time, so that (u − ud ) · nb (t) = 0 (i.e. translation of the base plane is exact), while (u − ud − ελb ) · v = 0 for any vector v in the plane (i.e. in-plane motion is relaxed). Sending ε → ∞ requires no adherence to the in-plane motion, while sending ε → 0 requires strict adherence (ε ∼ 10−7 m/Pa was used). Incorporation of the lumen volume data was achieved using the boundary energy term Πlext given as,     1 1 x · n dx + x · nb (t) dx − V (t) , (5) Πlext (u, λl ) := λl 3 Γl 3 Γ lb where V (t) is the given volume data and Γ lb is the base plane area at the top of the lumen, completing the boundary integral over the LV cavity. By divergence theorem,     1 1 1 1 x · n dx + x · nb (t) dx = x · n dx = ∇ · x dx = V (t), 3 Γl 3 Γ lb 3 Γ lv 3 Ωlv with Ωlv the corresponding LV cavity volume domain and Γ lv the corresponding LV cavity boundary. Equation (5) enforces strict adherence of the model to the given data as any deviation sends Π → ∞. Dependence on Γ lb in equation (5) can be removed by introducing the matrix I b (t) = (1/2)(I − nb (t) ⊗ nb (t)). As I b nb = 0, by divergence theorem,    I b (t) : x ⊗ n dx = I b (t) : x ⊗ n dx = ∇ · (I b (t)x) dx Γl

Γ lv

= (I b (t) : I)V (t) = V (t),

Ωlv

(6)

providing a means for measuring volume without requiring base plane area (even when base plane translation / rotation is observed). 2.3

Finite Element Solution

The solution to equation (3) was approximated using the finite element method (see [6] for discussion). Specifically Q2 −Q1 Taylor-Hood element [3] interpolation

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was used for uh − ph , as well as surface Q2 for λbh and a single constant for λl . Taking the directional derivatives of equation (3) with respect to all state variables gives the following form of the minimization problem: Duh Π[v h ]+Dph Π[qh ]+Dλbh Π[q h ]+Dλl Π[μ] = 0 ∀ (v h , qh , q h , μ) ∈ U h ×X h . The final model solution at any time point is given as the (uh , ph , λbh , λl ) ∈ U h × X h which satisfies,  ∇F W (uh , ph ) : ∇X v h + qh (Jh − 1) dX Ω0     l −λ v h · n dx − μ I b (t) : x ⊗ n dx − V (t) l Γl  Γ    − λbh · v h dX − q h · uh − ud (t) − K(t)λbh dX = 0, Γ0b

Γ0b

∀ (v h , qh , q h , μ) ∈ U h × X h

(7)

Equation (7) was solved for each load state through the cardiac cycle enabling the reconstruction of the myocardial motion. 2.4

Passive and Active Material Parameter Estimation

In order to simulate the cycle for each of the cases we required a suitable set of constitutive law parameters. These parameters were estimated using the reduced-order unscented Kalman filter (ROUKF) [4,12]. The method is particularly efficient since the number of parameters is low, and consequently few forward model runs are required in the estimation process. Assimilation was based on pressure observations through the whole cycle. We first considered the passive stage independently. This was taken to start at the points marked in the PV loop as DS, and end at or just before ED. The available pressure-volume data was normalised and coarsely fit to the experimental results presented in Klotz [10]. The parameters b1 = b2 = 5.0 were chosen so that inflation of an idealized LV of average size (tuning cross-sectional area, long axis length and wall thickness) across the group produced the degree of exponential growth observed. More comprehensive estimates of {b1 , b2 } were not made due to their coupling with outer stiffness parameters {a1 , a2 }. The nature of this coupling meant that adjusting {a1 , a2 } for any reasonable pair {b1 , b2 } allowed a good fit to the data. In the purely passive stage of the cycle (DS-ED), the absence of the Wa term in the total strain energy meant that the remaining passive parameters {a1 , a2 } could be estimated. All diastolic pressures were observed simultaneously in a ROUKF iteration, and the estimation was repeated with updated states and covariance matrices until convergence was achieved (as measured by the closeness of the determinant of the background error covariance matrix to 1). In some cases the fitting process highlighted pressure outliers at or before ED, which were treated as containing some active tension.

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In the second stage of the cycle (ED-DS) the passive parameters were fixed, and the magnitude of the active tension was estimated as a time-varying scalar α(t), t ∈ [ted , T ]. A number of models (quadratic, cubic Hermite and tanh2 ) were evaluated as possible parametrisable expressions for the active tension, but neither of these gave a reasonable fit to the pressure-volume data. We have therefore made no assumptions on the functional form of α(t), and estimated it as a piecewise constant in time. Two ROUKF iterations were run per time step to ensure filter convergence, and the initial guess for each step was updated based on the estimate at the previous time step. All of the parameters were estimated globally in space, since no localised data could be used in the estimation process. For each case the final simulation results were computed using the estimates for a1 , a2 and α(t).

3

Results

The computed deformation of meshes consisting of 4608 quadratic hexahedral elements for all cases is illustrated by Figure 2. Characteristic twist and shortening are observed in systole. 0912

0917

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1024

DS

ED

ES

Fig. 2. LV configurations for all cases at DS, ED and ES. The endocardial surface is coloured by magnitude of displacement from DS. The colour is scaled from blue to red per row, with the highest ED value being 3.7 mm and the highest ES value 8.9 mm. Mesh lines are shown on the epicardial surface to illustrate deformation directions.

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0917

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ED

ES

Fig. 3. Fibre stresses σ f for all cases at ED and ES. The surfaces are coloured by the value of the stress from blue (0 kPa) to red (4 kPa at ED and 40kPa at ES).

The computed fibre stresses at ED and ES are shown in Figure 3. Extreme values were observed on the base plane, which suggests that the condition which propagates motion of four epicardial points to the whole of the base surface could be introducing inaccurate displacements. Even though the magnitude of the stresses on the base was reduced through the use of the relaxed condition, this region still exhibited likely non-physiological stresses. 3.1

Data Matching

The pressure-volume loops obtained on meshes are shown in Figure 4. Volumes were prescribed through the model, and were matched exactly. Only minor pressure errors are present, mostly at the corners of the loops. Since surface data was not used for estimation in the current setup, the process could not fully account for local effects in deformation. However, a comparison was performed between the location of the endo- and epicardial surfaces predicted by the model, and those provided as the data. Figure 5 shows several views of the surfaces in the original image coordinates ES for the case 0912. High-error regions are located either near the points of attachment between the left and the right ventricles, or on the septal wall. This suggests that including the right ventricle or a corresponding boundary condition in the model has the potential to improve the results. Figure 6 illustrates the effect of fibre orientations on myocardial deformations at ES, as discussed above. The DTI fibres provided, as well as the commonly used −60◦ to 60◦ distribution resulted in non-physiological and non-data-compliant lengthening of the ventricle in systole.

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(a) case 0912

(b) case 0917

(c) case 1017

(d) case 1024

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Fig. 4. Pressure-volume loops: data in solid, and simulation in dash-dotted line

Fig. 5. Deformation of endo- and epicardial surfaces of the mesh for case 0912, shown in image coordinates at ES. Colour marks distances to nearest points on the data surfaces (0..4 mm from purple to red).

3.2

Verification

In order to assess the reliability of the method in modelling left ventricular mechanics we performed a number of additional tests. Mesh convergence of the solutions was studied on a series of uniform refinements of a coarse quadratic hexahedral mesh: mesh 1 was composed of 72 elements, mesh 2 of 576, mesh 3 of 4608, and mesh 4 of 36864. In the absence of a ground truth solution, simulation results on mesh 4 were taken as reference. This allowed us to compute relative errors in the main problem variables: e = v − v ref / v ref ,

v = u, p, σ f .

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(a) DTI

(b) −60◦ ..60◦

(c) −80◦ ..80◦

Fig. 6. ES deformation of the LV mesh for case 0912 produced using different fibre orientations (in colour), as compared to DS configuration (in black).

The errors on meshes 1, 2 and 3 are shown in Figure 7. In all cases the error on mesh 3 (used as the challenge submission) is less than 5%.

(a) u, L2

(b) u, H 1

(c) p, L2

(d) σ f , L2

Fig. 7. Relative error in problem variables on progressively refined meshes for case 0917. The red and blue lines correspond to errors at ED and ES respectively.

Passive parameter estimates for a given set of ROUKF inputs were obtained on meshes 1, 2 and 3. The values were consistent at every ROUKF iteration, and the relative error in the parameters after 10 assimilation steps was < 5% between meshes 1 and 2, and < 2% between meshes 2 and 3. Filter convergence was observed in all cases, both for the passive and active estimation procedures. At the same time, changes in ROUKF initial state and error covariance matrices in passive estimation resulted in a significant variation in the final estimates. Figure 8a illustrates how pressures computed using these estimates, a1 = (1.4, 0.9) kPa, a2 = (0.9, 2.6) kPa, a3 = (0.7, 3.4) kPa, compare to the data. The map of the errors in diastolic pressures P − λl (a1 , a2 ) over the passive parameter space (Figure 8b) shows that instead of a clear minimum there is a valley of parameter combinations that minimize the l2 -norm pressure functional. This indicates a coupling between the parameters, suggesting that the model requires additional data to achieve reliable passive parameter estimation. Finally, the estimated active tension curves are presented in Figure 8c. All cases exhibit fast growth post-activation, and sustained (sometimes increasing) tension through systole, with a tailing off in the deactivation stage.

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(b) P − λl 

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(c) α(t)

Fig. 8. Estimation based on pressure data in diastole and systole: (a) matching of cavity pressure data P by the Lagrange multiplier λl in diastole using different parameter combinations, (b) map of distances between the vectors of simulated and observed cavity pressures over the passive parameter space (a1 , a2 ), (c) estimates for the active tension over time

4

Discussion

The mechanics model incorporated cavity pressure-volume data to simulate deformation of the left ventricles for four canines through the whole cardiac cycle. Model personalisation was achieved through estimation of passive model parameters and time-dependent active tension scaling. A good fit for PV data was obtained, as well as reasonable physiological deformations, although total exclusion of the right ventricle from the model appears to produce higher errors in attachment areas. Simulation and estimation results were verified through a series of convergence tests. In the setup presented there is limited potential for model personalisation based on bulk measures. Incorporating distances to surfaces at ED and ES could potentially result in a better fit, but was impractical to implement in a short time span and was only used for validation. An augmented set of data, e.g. time-resolved images of cardiac motion, could be included in the workflow to compute reliable estimates, allowing the use of parameters as biomarkers. In addition, assimilation of local displacements would enable better data fitting and stress computation, as well as studying the effects of tissue heterogeneity. Acknowledgments. The authors would like to acknowledge funding from the BHF New Horizons program (NH/11/5/29058) and support from the Wellcome Trust-EPSRC Centre of Excellence in Medical Engineering (WT 088641/Z/09/Z) and the NIHR Biomedical Research Centre at Guy’s and St. Thomas’ NHS Foundation Trust and KCL. The views expressed are those of the authors and not necessarily those of the NHS, the NIHR, or the DoH. The authors would also like to thank Radomir Chabiniok for his input.

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