Models

ConstantCoefficient(value)

Evaluates to the same value in space and time everywhere.

FieldCoefficient(data, interpolation)

A constant in time data field, interpolated per element with a given interpolation.

AnalyticalCoefficient(f::Function, cs::CoordinateSystemCoefficient)

A coefficient given as the analytical function f(x,t) in the specified coordiante system.

SpectralTensorCoefficient(eigenvector_coefficient, eigenvalue_coefficient)

Represent a tensor A via spectral decomposition ∑ᵢ λᵢ vᵢ ⊗ vᵢ.

SpatiallyHomogeneousDataField(timings::Vector, data::Vector)

A data field which is constant in space and piecewise constant in time.

The value during the time interval [tᵢ,tᵢ₊₁] is dataᵢ, where t₀ is negative infinity and the last time point+1 is positive infinity.

create_microstructure_model(coordinate_system::CoordinateSystemCoefficient, ip::VectorInterpolationCollection, parameters)

Create a rotating fiber field by deducing the circumferential direction from apicobasal and transmural gradients.

Linear transmural distribution of the microstructure with three angles to describe all reachable physiological angles.

RobinBC(α, boundary_name::String)

\[\bm{P}(\bm{u}) \cdot \bm{n}_0 = - \alpha \bm{u} \quad \textbf{x} \in \partial \Omega_0,\]

NormalSpringBC(kₛ boundary_name::String)

\[\bm{P}(\bm{u}) \cdot \bm{n}_0 = - k_s \bm{u} \cdot n_0 \quad \textbf{x} \in \partial \Omega_0,\]

BendingSpringBC(kᵇ, boundary_name::String)

\[\bm{P}(\bm{u}) \cdot \bm{n}_0 = - \partial_F \frac{1}{2} k_b \left (cof(F) n_0 - n_0 \right) \quad \textbf{x} \in \partial \Omega_0,\]

ConstantPressureBC(p::Real, boundary_name::String)

\[\bm{P}(\bm{u}) \cdot \bm{n}_0 = - p n_0 \quad \textbf{x} \in \partial \Omega_0,\]

PressureFieldBC(pressure_field, boundary_name::String)

\[\bm{P}(\bm{u}) \cdot \bm{n}_0 = - k_s \bm{u} \cdot n_0 \quad \textbf{x} \in \partial \Omega_0,\]

QuasiStaticModel(displacement_sym, mechanical_model, facet_models)

A generic model for quasi-static mechanical problems.

ExtendedHillModel(passive_spring_model, active_spring_model, active_deformation_gradient_model,contraction_model, microstructure_model)

The extended (generalized) Hill model as proposed by Ogiermann et al. [9]. The original formulation dates back to Stålhand et al. [10] for smooth muscle tissues.

In this framework the model is formulated as an energy minimization problem with the following additively split energy:

\[W(\mathbf{F}, \mathbf{F}^{\rm{a}}) = W_{\rm{passive}}(\mathbf{F}) + \mathcal{N}(\bm{\alpha})W_{\rm{active}}(\mathbf{F}\mathbf{F}^{-\rm{a}})\]

Where $W_{\rm{passive}}$ is the passive material response and $W_{\rm{active}}$ the active response respectvely. $\mathcal{N}$ is the amount of formed crossbridges. We refer to the original paper [9] for more details.

GeneralizedHillModel(passive_spring_model, active_spring_model, active_deformation_gradient_model,contraction_model, microstructure_model)

The generalized Hill framework as proposed by Göktepe et al. [11].

In this framework the model is formulated as an energy minimization problem with the following additively split energy:

\[W(\mathbf{F}, \mathbf{F}^{\rm{a}}) = W_{\rm{passive}}(\mathbf{F}) + W_{\rm{active}}(\mathbf{F}\mathbf{F}^{-\rm{a}})\]

Where $W_{\rm{passive}}$ is the passive material response and $W_{\rm{active}}$ the active response respectvely.

ActiveStressModel(material_model, active_stress_model, contraction_model, microstructure_model)

The active stress model as originally proposed by Guccione et al. [2].

In this framework the model is formulated via balance of linear momentum in the first Piola Kirchhoff $\mathbf{P}$:

\[\mathbf{P}(\mathbf{F},T^{\rm{a}}) := \partial_{\mathbf{F}} W_{\rm{passive}}(\mathbf{F}) + \mathbf{P}^{\rm{a}}(\mathbf{F}, T^{\rm{a}})\]

where the passive material response can be described by an energy $W_{\rm{passive}$ and $T^{\rm{a}}$ the active tension generated by the contraction model.

PK1Model(material, coefficient_field)
PK1Model(material, internal_model, coefficient_field)

Models the stress formulated in the 1st Piola-Kirchhoff stress tensor. If the material is energy-based, then the term is formulated as follows: $\int{\Omega0} P(u,s) \cdot \delta F dV = \int{\Omega0} \partial_{F} \psi(u,s) \cdot \delta \nabla u $

PrestressedMechanicalModel(inner_model, prestress_field)

Models the stress formulated in the 1st Piola-Kirchhoff stress tensor based on a multiplicative split of the deformation gradient $F = F_{\textrm{e}} F_{0}$ where we compute $P(F_{\textrm{e}}) = P(F F^{-1}_{0})$.

Please note that it is assumed that $F^{-1}_{0}$ is the quantity computed by prestress_field.

A simple dummy energy with $\Psi = 0$.

A simple linear fiber spring model for testing purposes.

\[\Psi^{\rm{a}} = \frac{a^{\rm{f}}}{2}(I_e^{\rm{e}}-1)^2\]

https://onlinelibrary.wiley.com/doi/epdf/10.1002/cnm.2866

The well-known orthotropic material model for the passive response of cardiac tissues by Holzapfel and Ogden [12].

\[\Psi = \frac{a}{2b} e^{b(I_1-3)} + \sum_{i\in\{\rm{f},\rm{s}\}} \frac{a^i}{2b^i}(e^{b^i<I_4^i - 1>^2}-1) + \frac{a^{\rm{fs}}}{2b^{\rm{fs}}}(e^{b^{\rm{fs}}{I_8^{\rm{fs}}}^2}-1)\]

This is the Fung-type transverse isotropic material model for the passive response of cardiac tissue proposed by Lin and Yin [13].

\[\Psi = C_1(e^{C_2(I_1-3)^2 + C_3(I_1-3)(I_4-1) + C_4(I_4-1)^2}-1)\]

This is the transverse isotropic material model for the active response of cardiac tissue proposed by Lin and Yin [13].

\[\Psi=C_0 + C_1*(I_1-3)(I_4-1) + C_2(I_1-3)^2 + C_3*(I_4-1)^2 + C_3*(I_1-3) + C_5*(I_4-1)\]

This is the transverse isotropic material model for the active response of cardiac tissue proposed by Humphrey et al. [14].

\[\Psi = C_1(\sqrt{I_4}-1)^2 + C_2(\sqrt{I_4}-1)^3 + C_3(\sqrt{I_4}-1)(I_1-3) + C_3(I_1-3)^2\]

An orthotropic material model for the passive myocardial tissue response by Guccione et al. [15].

\[\Psi = B^{\rm{ff}} {E^{\rm{ff}}}^2 + B^{\rm{ss}}{E^{\rm{ss}}}^2 + B^{\rm{nn}}{E^{\rm{nn}}}^2 + B^{\rm{ns}}({E^{\rm{ns}}}^2+{E^{\rm{sn}}}^2) + B^{\rm{fs}}({E^{\rm{fs}}}^2+{E^{\rm{sf}}}^2) + B^{\rm{fn}}({E^{\rm{fn}}}^2+{E^{\rm{nf}}}^2)\]

The default parameterization is taken from from [16].

BioNeoHookean

A simple isotropic Neo-Hookean model of the form

\[\Psi = \alpha (\bar{I_1}-3)\]

SimpleActiveSpring

A simple linear fiber spring as for example found in [11].

\[\Psi^{\rm{a}} = \frac{a^{\rm{f}}}{2}(I_e^{\rm{e}}-1)^2\]

A simple helper to use a passive material model as an active material for GeneralizedHillModel, ExtendedHillModel and ActiveStressModel.

The active deformation gradient formulation by Göktepe et al. [11].

\[F^{\rm{a}} = (\lambda^{\rm{a}}-1) f_0 \otimes f_0\]

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See also [9] for a further analysis.

An incompressivle version of the active deformation gradient formulation by Göktepe et al. [11].

\[F^{\rm{a}} = \lambda^{\rm{a}} f_0 \otimes f_0 + \frac{1}{\sqrt{\lambda^{\rm{a}}}}(s_0 \otimes s_0 + n_0 \otimes n_0)\]

See also [9] for a further analysis.

The active deformation gradient formulation by Rossi et al. [17].

\[F^{\rm{a}} = \lambda^{\rm{a}} f_0 \otimes f_0 + (1+\kappa(\lambda^{\rm{a}}-1)) s_0 \otimes s_0 + \frac{1}{1+\kappa(\lambda^{\rm{a}}-1))\lambda^{\rm{a}}} n_0 \otimes n_0\]

Where $\kappa \geq 0$ is the sheelet part.

See also [9] for a further analysis.

A simple active stress component.

\[T^{\rm{a}} = T^{\rm{max}} \, [Ca_{\rm{i}}] \frac{(F \cdot f_0) \otimes f_0}{||F \cdot f_0||}\]

The active stress component described by Piersanti et al. [18] (Eq. 3).

\[T^{\rm{a}} = T^{\rm{max}} \, [Ca_{\rm{i}}] \left(p^f \frac{(F \cdot f_0) \otimes f_0}{||F \cdot f_0||} + p^{\rm{s}} \frac{(F \cdot s_0) \otimes s_0}{||F \cdot s_0||} + p^{\rm{n}} \frac{(F \cdot n_0) \otimes n_0}{||F \cdot n_0||}\right)\]

The active stress component as described by Guccione et al. [2].

\[T^{\rm{a}} = T^{\rm{max}} \, [Ca_{\rm{i}}] (F \cdot f_0) \otimes f_0\]

A simple dummy compression model with $U(I_3) = 0$.

A compression model with $U(I_3) = \beta (I_3 -1 - 2\log(\sqrt{I_3}))^a$.

An isochoric compression model where

\[U(I_3) = \beta (I_3^b + I_3^{-b} -2)^a\]

with $a,b \geq 1$.

Entry 1 from table 3 in [19].

An isochoric compression model where

\[U(I_3) = \beta (\sqrt{I_3}-1)^a\]

with $a > 1$.

Entry 2 from table 3 in [19].

An isochoric compression model where

\[U(I_3) = \beta (I_3 - 2\log(\sqrt{I_3}) + 4\log(\sqrt{I_3})^2) - 1)\]

Entry 3 from table 3 in [19].

TransientDiffusionModel(conductivity_coefficient, source_term, solution_variable_symbol)

Model formulated as $\partial_t u = \nabla \cdot \kappa(x) \nabla u + f$

SteadyDiffusionModel(conductivity_coefficient, source_term, solution_variable_symbol)

Model formulated as $\nabla \cdot \kappa(x) \nabla u = f$

Simplification of the bidomain model with the structure

χCₘ∂ₜφₘ = ∇⋅κ∇φₘ + χ(Iᵢₒₙ(φₘ,𝐬) + Iₛₜᵢₘ(t)) ∂ₜ𝐬 = g(φₘ,𝐬)

(TODO citation). Can be derived through the assumption (TODO), but also when the assumption is violated we can construct optimal κ (TODO citation+example) for the reconstruction of φₘ.

The original model formulation (TODO citation) with the structure

χCₘ∂ₜφₘ = ∇⋅κᵢ∇φᵢ + χ(Iᵢₒₙ(φₘ,𝐬,x) + Iₛₜᵢₘ,ᵢ(x,t)) χCₘ∂ₜφₘ = ∇⋅κₑ∇φₑ - χ(Iᵢₒₙ(φₘ,𝐬,x) + Iₛₜᵢₘ,ₑ(x,t)) ∂ₜ𝐬 = g(φₘ,𝐬,x) φᵢ - φₑ = φₘ

Transformed bidomain model with the structure

χCₘ∂ₜφₘ = ∇⋅κᵢ∇φₘ + ∇⋅κᵢ∇φₑ + χ(Iᵢₒₙ(φₘ,𝐬,x) + Iₛₜᵢₘ(x,t)) 0 = ∇⋅κᵢ∇φₘ + ∇⋅(κᵢ+κₑ)∇φₑ + Iₛₜᵢₘ,ₑ(t) - Iₛₜᵢₘ,ᵢ(t) ∂ₜ𝐬 = g(φₘ,𝐬,x) φᵢ = φₘ + φₑ

This formulation is a transformation of the parabolic-parabolic form (c.f. TODO citation) and has been derived by (TODO citation) first.

ReactionDiffusionSplit(model)
ReactionDiffusionSplit(model, coeff)

Annotation for the classical reaction-diffusion split of a given model. The second argument is a coefficient describing the input x for the reaction model rhs, which is usually some generalized coordinate.

A dummy protocol describing the absence of stimuli for a simulation.

Supertype for all stimulation protocols fulfilling $I_{\rm{stim,e}} = I_{\rm{stim,i}}$.

Describe the transmembrane stimulation by some analytical function on a given set of time intervals.

The classical neuron electrophysiology model independently found by FitzHugh [20] and Nagumo et al. [21]. This model is less stiff and cheaper than any cardiac electrophysiology model, which maks it a good choice for quick testing if things work at all.

The canine ventricular cardiomyocyte electrophysiology model by Pathmanathan et al. [22].

DummyLumpedCircuitModel(volume_fun)

Lock the volume at a certain value.

MTKLumpedCicuitModel

A lumped (0D) circulatory model for LV simulations as presented in Regazzoni et al. [3].

RSAFDQ2022LumpedCicuitModel

A lumped (0D) circulatory model for LV simulations as presented in Regazzoni et al. [3].

Abstract supertype for all interface coupling schemes.

Abstract supertype for all volume coupling schemes.

Helper to describe the coupling between problems.

A descriptor for a coupled model.

Enforce the constraints that chamber volume 3D (solid model) = chamber volume 0D (lumped circuit) via Lagrange multiplied, where a surface pressure integral is introduced such that ∫ ∂Ωendo Here chamber_volume_method is responsible to compute the 3D volume.

This approach has been proposed by Regazzoni et al. [3].

Chamber volume estimator as presented in [23].

Compute the chamber volume as a surface integral via the integral

• ∫ (x + d) det(F) cof(F) N ∂Ωendo

where it is assumed that the chamber is convex, zero displacement in apicobasal direction at the valvular plane occurs and the plane normal is aligned with the z axis, where the origin is at z=0.

Compute the chamber volume as a surface integral via the integral -∫ det(F) ((h ⊗ h)(x + d - b)) adj(F) N ∂Ωendo

as proposed by Regazzoni et al. [3].

Annotation for the split described by Regazzoni et al. [3].

The split model described by Regazzoni et al. [3] alone.