Parameters

This module defines the parameters used in the bone cell population models. Each model has a specific set of parameters governing its behavior, interaction dynamics, and biological processes. For the bone cell population models, the parameters are named consistently across models to facilitate comparison and understanding. In each parameter file, the parameter names are matched with the corresponding model nomenclature and units.

Base Parameters

This module contains the base parameters for the bone cell population models. The base model is not intended to be used as a standalone model but as a base class for other models. The base parameters are thus all set to None; a value is assigned to them in the derived models.

class bone_models.bone_cell_population_models.parameters.base_parameters.Base_Parameters[source]

Bases: object

This class defines the parameters of the bone model.

class bone_models.bone_cell_population_models.parameters.base_parameters.activation_coefficient[source]

Bases: object

This class defines the activation coefficients of respective receptor-ligand binding.

class bone_models.bone_cell_population_models.parameters.base_parameters.apoptosis_rate[source]

Bases: object

This class defines the apoptosis rates of the different cell types.

class bone_models.bone_cell_population_models.parameters.base_parameters.binding_constant[source]

Bases: object

This class defines the binding constants of RANK RANKL and OPG.

class bone_models.bone_cell_population_models.parameters.base_parameters.bone_volume[source]

Bases: object

This class defines the parameters relevant for bone volume of the bone model.

class bone_models.bone_cell_population_models.parameters.base_parameters.concentration[source]

Bases: object

This class defines fixed concentrations.

class bone_models.bone_cell_population_models.parameters.base_parameters.correction_factor[source]

Bases: object

This class defines the correction factors.

class bone_models.bone_cell_population_models.parameters.base_parameters.degradation_rate[source]

Bases: object

This class defines the degradation rates of the different factors.

class bone_models.bone_cell_population_models.parameters.base_parameters.differentiation_rate[source]

Bases: object

This class defines the differentiation rates of the different cell types.

class bone_models.bone_cell_population_models.parameters.base_parameters.external_dosage[source]

Bases: object

This class defines the external dosages of the different cell types. They are 0 but can be altered in a load case scenario to simulate external injections.

class bone_models.bone_cell_population_models.parameters.base_parameters.production_rate[source]

Bases: object

This class defines the intrinsic/ endogenous production rates of the different factors.

class bone_models.bone_cell_population_models.parameters.base_parameters.repression_coefficient[source]

Bases: object

This class defines the repression coefficients of respective receptor-ligand binding.

Lemaire Parameters

This module contains the parameters for the model by Lemaire et al. (2004). In each parameter class, the parameter names are matched with the corresponding model nomenclature and units.

class bone_models.bone_cell_population_models.parameters.lemaire_parameters.Lemaire_Parameters[source]

Bases: object

This class defines the parameters of the Lemaire bone cell population model.

Parameters:

differentiation_rate (differentiation_rate) – differentiation rates of the different cell types, see differentiation_rate for details
apoptosis_rate (apoptosis_rate) – apoptosis rates of the different cell types, see apoptosis_rate for details
activation_coefficient (activation_coefficient) – activation coefficients of the different cell types, see activation_coefficient for details
repression_coefficient (repression_coefficient) – repression coefficients of the different cell types, see repression_coefficient for details
correction_factor (correction_factor) – correction factors for the model, see correction_factor for details
degradation_rate (degradation_rate) – degradation rates of the different factors, see degradation_rate for details
concentration (concentration) – fixed concentrations of the different factors, see concentration for details
binding_constant (binding_constant) – binding constants of the receptor-ligand interactions, see binding_constant for details
unbinding_constant (unbinding_constant) – unbinding constants of the receptor-ligand interactions, see unbinding_constant for details
production_rate (production_rate) – intrinsic production rates of the different factors, see production_rate for details
bone_volume (bone_volume) – parameters relevant for bone volume, see bone_volume for details
differentiation_rate.OBp (float) – corrected differentiation rate of precursor osteoblasts

class bone_models.bone_cell_population_models.parameters.lemaire_parameters.activation_coefficient[source]

Bases: object

This class defines the activation coefficients of the different cell types in the Lemaire bone cell population model.

class bone_models.bone_cell_population_models.parameters.lemaire_parameters.apoptosis_rate[source]

Bases: object

This class defines the apoptosis rates of the different cell types in the Lemaire bone cell population model.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
OBa	\(k_B\)	1/day
OCa	\(D_A\)	pM/day

Parameters:

OBa (float) – apoptosis rate of active osteoblasts
OCa (float) – apoptosis rate of active osteoclasts

class bone_models.bone_cell_population_models.parameters.lemaire_parameters.binding_constant[source]

Bases: object

This class defines the binding constants of the possible receptor-ligand interactions in the Lemaire bone cell population model.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
RANKL_OPG	\(k_1\)	1/(pM·day)
RANKL_RANK	\(k_3\)	1/(pM·day)
TGFb_OC	\(C^s\)	pM
PTH_OB	\(k_5\)	1/(pM·day)

Parameters:

RANKL_OPG (float) – Association binding constant for RANKL-OPG
RANKL_RANK (float) – Association binding constant for RANKL-RANK
TGFb_OC (float) – dissociation binding coefficient of TGFb with its receptor
PTH_OB (float) – rate of PTH binding with its receptor on OB

Constructor method.

class bone_models.bone_cell_population_models.parameters.lemaire_parameters.bone_volume[source]

Bases: object

This class defines the parameters relevant for bone volume of the bone model.

Parameters:

formation_rate (float) – rate of bone formation
resorption_rate (float) – rate of bone resorption

class bone_models.bone_cell_population_models.parameters.lemaire_parameters.concentration[source]

Bases: object

This class defines the fixed concentrations of the different factors in the Lemaire bone cell population model.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
OPG_max	\(OPG_{\max}\)	pM
RANK	\(K\)	pM

Parameters:

OPG_max (float) – Maximum concentration of OPG
RANK (float) – fixed concentration of RANK

Constructor method.

class bone_models.bone_cell_population_models.parameters.lemaire_parameters.correction_factor[source]

Bases: object

This class defines the correction factors for the Lemaire bone cell population model. They are used to adjust the model parameters.

Parameters:: f0 (float) – correction factor for OBp differentiation rate and TGFb activation function

Constructor method.

class bone_models.bone_cell_population_models.parameters.lemaire_parameters.degradation_rate[source]

Bases: object

This class defines the degradation rates of the different factors in the Lemaire bone cell population model.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
PTH	\(k_P\)	1/day
OPG	\(k_O\)	1/day

Parameters:

PTH (float) – degradation rate of PTH
OPG (float) – degradation rate of OPG

class bone_models.bone_cell_population_models.parameters.lemaire_parameters.differentiation_rate[source]

Bases: object

This class defines the differentiation rates of the different cell types in the Lemaire bone cell population model.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
OBu	\(D_R\)	pM/day
OBp	\(d_B\)	pM/day
OCp	\(D_C\)	pM/day

Parameters:

OBu (float) – differentiation rate of uncommitted osteoblasts
OBp (float) – differentiation rate of precursor osteoblasts
OCp (float) – differentiation rate of precursor osteoclasts

Constructor method.

class bone_models.bone_cell_population_models.parameters.lemaire_parameters.production_rate[source]

Bases: object

This class defines the intrinsic/ endogenous production rates of the different factors in the Lemaire bone cell population model.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
intrinsic_PTH	\(S_P\)	pM/day
intrinsic_RANKL	\(r_L\)	pM/day
min_OPG_per_cell	\(K^P_O\)	pM/day
max_RANKL_per_cell	\(K^P_L\)	pM/pM

Parameters:

intrinsic_PTH (float) – Intrinsic production rate of PTH
intrinsic_RANKL (float) – Intrinsic production rate of RANKL
min_OPG_per_cell (float) – Minimal rate of OPG production per cell
max_RANKL_per_cell (float) – Production rate of RANKL per cell

Constructor method.

class bone_models.bone_cell_population_models.parameters.lemaire_parameters.repression_coefficient[source]

Bases: object

This class defines the repression coefficients of the different cell types in the Lemaire bone cell population model.

class bone_models.bone_cell_population_models.parameters.lemaire_parameters.unbinding_constant[source]

Bases: object

This class defines the unbinding constants of the possible receptor-ligand interactions in the Lemaire bone cell population model.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
RANKL_OPG	\(k_2\)	1/day
RANKL_RANK	\(k_4\)	1/day
PTH_OB	\(k_6\)	1/day

Parameters:

RANKL_OPG (float) – Unbinding constant for RANKL-OPG
RANKL_RANK (float) – Unbinding constant for RANKL-RANK
PTH_OB (float) – rate of PTH unbinding with its receptor on OB

Martonova Parameters

This module contains the parameters for the model by Martonova et al. (2023). In each parameter class, the parameter names are matched with the corresponding model nomenclature and units.

class bone_models.bone_cell_population_models.parameters.martonova_parameters.Martonova_Parameters[source]

Bases: object

This class contains all parameters of the two-state receptor model by Martonova et al.

Parameters:

kinematics (kinematics) – kinematic parameters of the model, see kinematics for details
activity (activity) – activity parameters of the model, see activity for details
basal_PTH_pulse (basal_PTH_pulse) – basal PTH pulse parameters of the model, see basal_PTH_pulse for details
injected_PTH_pulse (injected_PTH_pulse) – injected PTH pulse parameters of the model, see injected_PTH_pulse for details
pharmacokinetics (pharmacokinetics) – pharmacokinetic parameters of the model, see pharmacokinetics for details

class bone_models.bone_cell_population_models.parameters.martonova_parameters.activity[source]

Bases: object

This class contains the activity parameters of the two-state receptor model by Martonova et al.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
active_receptor	\(a_1\)	nM
active_complex	\(a_2\)	nM
inactive_complex	\(a_3\)	nM
inactive_receptor	\(a_4\)	nM

Parameters:

active_receptor (float) – activity constant of active receptor
active_complex (float) – activity constant of active complex
inactive_complex (float) – activity constant of inactive complex
inactive_receptor (float) – activity constant of inactive receptor

Constructor method.

class bone_models.bone_cell_population_models.parameters.martonova_parameters.basal_PTH_pulse[source]

Bases: object

This class contains the basal PTH pulse parameters of the two-state receptor model by Martonova et al.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
min	\(\gamma_{off}\)	nM
max	\(\gamma_{on}\)	nM
off_duration	\(\tau_{off}\)	min
on_duration	\(\tau_{on}\)	min
period	\(T\)	min

Parameters:

min (float) – concentration of non-pulsatile share of PTH pulse
max (float) – concentration of pulsatile share of PTH pulse
off_duration (float) – duration of non-pulsatile share of PTH pulse
on_duration (float) – duration of pulsatile share of PTH pulse
period (float) – duration of the PTH pulse period (pulse + non-pulse)

Constructor method.

class bone_models.bone_cell_population_models.parameters.martonova_parameters.injected_PTH_pulse[source]

Bases: object

This class contains the injected PTH pulse parameters of the two-state receptor model by Martonova et al.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
min	\(\gamma_{off}\)	nM
max	\(\gamma_{on}\)	nM
on_duration	\(\tau_{on}\)	min

Parameters:

min (float) – concentration of non-pulsatile share of PTH pulse
max (float) – concentration of pulsatile share of PTH pulse
on_duration (float) – duration of pulsatile share of PTH pulse

Constructor method.

class bone_models.bone_cell_population_models.parameters.martonova_parameters.kinematics[source]

Bases: object

This class contains the kinematic parameters of the two-state receptor model by Martonova et al.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
receptor_desensitized	\(k_1\)	1/min
receptor_resensitized	\(k_{-1}\)	1/min
complex_desensitized	\(k_2\)	1/min
complex_resensitized	\(k_{-2}\)	1/min
active_complex_binding	\(k_r\)
active_complex_unbinding	\(k_{-r}\)
inactive_complex_binding	\(k_d\)
receptor	\(K_1\)
complex	\(K_2\)
active_binding_unbinding	\(K_r\)	nM
inactive_complex_unbinding	\(k_{-d}\)	1/min
inactive_binding_unbinding	\(K_d\)	nM

Parameters:

receptor_desensitized (float) – rate of receptor desensitization
receptor_resensitized (float) – rate of receptor resensitization
complex_desensitized (float) – rate of complex desensitization
complex_resensitized (float) – rate of complex resensitization
active_complex_binding (float) – rate of active complex binding
active_complex_unbinding (float) – rate of active complex unbinding
inactive_complex_binding (float) – rate of inactive complex binding
receptor (float) – combined kinetic receptor constant
complex (float) – combined kinetic complex constant
active_binding_unbinding (float) – active binding unbinding constant
inactive_complex_unbinding (float) – rate of inactive complex unbinding
inactive_binding_unbinding (float) – inactive binding unbinding constant

Constructor method.

class bone_models.bone_cell_population_models.parameters.martonova_parameters.pharmacokinetics[source]

Bases: object

This class contains the pharmacokinetic parameters of the two-state receptor model by Martonova et al. for injected PTH.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
absorption_rate	\(k_a\)	1/h
elimination_rate	\(k_e\)	1/h
volume_of_distribution	\(V\)	L
bioavailability	\(F\)

Parameters:

absorption_rate (float) – rate of absorption of the injected PTH
elimination_rate (float) – rate of elimination of the injected PTH
volume_of_distribution (float) – volume of distribution of the injected PTH
bioavailability (float) – bioavailability of the injected PTH

Constructor method.

Modiz Parameters

This module contains the parameters for the model by Modiz et al. (2025). In each parameter class, the parameter names are matched with the corresponding model nomenclature and units.

class bone_models.bone_cell_population_models.parameters.modiz_parameters.Calibration_Parameters[source]: Bases: object

class bone_models.bone_cell_population_models.parameters.modiz_parameters.Calibration_Parameters_Only_For_Healthy_State[source]: Bases: object

class bone_models.bone_cell_population_models.parameters.modiz_parameters.Modiz_Parameters[source]: Bases: Lemaire_Parameters

Pivonka Parameters

This module contains the parameters for the model by Pivonka et al. (2008). In each parameter class, the parameter names are matched with the corresponding model nomenclature and units.

class bone_models.bone_cell_population_models.parameters.pivonka_parameters.Pivonka_Parameters[source]

Bases: object

This class defines the parameters of the Pivonka bone cell population model.

Parameters:

differentiation_rate (differentiation_rate) – differentiation rates of the different cell types
apoptosis_rate (apoptosis_rate) – apoptosis rates of the different cell types
activation_coefficient (activation_coefficient) – activation coefficients of respective receptor-ligand bindings
repression_coefficient (repression_coefficient) – repression coefficients of respective receptor-ligand binding
correction_factor (correction_factor) – correction factors, see correction_factor for details
degradation_rate (degradation_rate) – degradation rates of the different factors
concentration (concentration) – fixed concentrations
binding_constant (binding_constant) – binding constants of receptor-ligand bindings
unbinding_constant (unbinding_constant) – unbinding constants of receptor-ligand bindings
production_rate (production_rate) – intrinsic/ endogenous production rates of the different factors
bone_volume (bone_volume) – parameters relevant for bone volume

class bone_models.bone_cell_population_models.parameters.pivonka_parameters.activation_coefficient[source]

Bases: object

This class defines the activation coefficients of respective receptor-ligand bindings in the Pivonka bone cell population model.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
TGFb_OBu	\(K_{D1, \text{TGF-}\beta}\)	pM
TGFb_OCa	\(K_{D3, \text{TGF-}\beta}\)	pM
PTH_OB	\(K_{D4, \text{PTH}}, K_{D5, \text{PTH}}\)	pM
RANKL_RANK	\(K_{D8, \text{RANKL}}\)	pM

Parameters:

TGFb_OBu (float) – activation coefficient related to TGF-beta binding on OBu
TGFb_OCa (float) – activation coefficient related to TGF-beta binding on OCa
PTH_OB (float) – activation coefficient related to PTH binding to osteoblasts
RANKL_RANK (float) – activation coefficient related to RANKL binding on RANK

class bone_models.bone_cell_population_models.parameters.pivonka_parameters.apoptosis_rate[source]

Bases: object

This class defines the apoptosis rates of the different cell types in the Pivonka bone cell population model.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
OBa	\(A_{OB_a}\)	1/day
OCa	\(A_{OC_a}\)	pM/day

Parameters:

OBa (float) – apoptosis rate of active osteoblasts
OCa (float) – apoptosis rate of active osteoclasts

class bone_models.bone_cell_population_models.parameters.pivonka_parameters.binding_constant[source]

Bases: object

This class defines the binding constants of receptor-ligand bindings in the Pivonka bone cell population model.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
RANKL_OPG	\(K_{A1,\text{RANKL}}\)	1/(pM day)
RANKL_RANK	\(K_{A2,\text{RANKL}}\)	1/(pM day)
TGFb_OC	\(C^s\)	pM
PTH_OB	\(k_5\)	1/(pM day)

Parameters:

RANKL_OPG (float) – association binding constant for RANKL-OPG
RANKL_RANK (float) – association binding constant for RANKL-RANK
TGFb_OC (float) – dissociation binding coefficient of TGFb with its receptor
PTH_OB (float) – rate of PTH binding with its receptor on OB

class bone_models.bone_cell_population_models.parameters.pivonka_parameters.bone_volume[source]

Bases: object

This class defines the parameters relevant for bone volume of the bone model in the Pivonka bone cell population model.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
formation_rate	\(k_{\text{form}}\)	1/day
resorption_rate	\(k_{\text{res}}\)	1/day
stored_TGFb_content	\(\alpha\)

Parameters:

formation_rate (float) – formation rate of bone volume
resorption_rate (float) – resorption rate of bone volume
stored_TGFb_content (float) – proportionality constant expressing the TGF-β content stored in bone volume

class bone_models.bone_cell_population_models.parameters.pivonka_parameters.concentration[source]

Bases: object

This class defines fixed concentrations in the Pivonka bone cell population model.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
OPG_max	\(C^{\max}_{\text{OPG}}\)	pM
RANK	\(C^{\text{RANK}}\)	pM

Parameters:

OPG_max (float) – maximum concentration of OPG
RANK (float) – fixed concentration of RANK

class bone_models.bone_cell_population_models.parameters.pivonka_parameters.correction_factor[source]

Bases: object

This class defines the correction factors in the Pivonka bone cell population model. Remark: f0 is implemented in the model (as in Lemaire 2004), but not explicitly given in the publication.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
f0	\(f_0\)

Parameters:: f0 (float) – correction factor for OBp differentiation rate and TGFb activation function

class bone_models.bone_cell_population_models.parameters.pivonka_parameters.degradation_rate[source]

Bases: object

This class defines the degradation rates of the different factors in the Pivonka bone cell population model.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
PTH	\(\tilde{D}_{\text{PTH}}\)	1/day
OPG	\(\tilde{D}_{\text{OPG}}\)	1/day
RANKL	\(\tilde{D}_{\text{RANKL}}\)	1/day
TGFb	\(\tilde{D}_{\text{TGFb}}\)	1/day

Parameters:

PTH (float) – degradation rate of PTH
OPG (float) – degradation rate of OPG
RANKL (float) – degradation rate of RANKL
TGFb (float) – degradation rate of TGFb

class bone_models.bone_cell_population_models.parameters.pivonka_parameters.differentiation_rate[source]

Bases: object

This class defines the differentiation rates of the different cell types in the Pivonka bone cell population model.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
OBu	\(D_{OB_u}\)	pM/day
OBp	\(D_{OB_p}\)	pM/day
OCp	\(D_{OC_p}\)	pM/day

Parameters:

OBu (float) – differentiation rate of uncommitted osteoblasts
OBp (float) – differentiation rate of precursor osteoblasts
OCp (float) – differentiation rate of precursor osteoclasts

class bone_models.bone_cell_population_models.parameters.pivonka_parameters.production_rate[source]

Bases: object

This class defines the intrinsic/ endogenous production rates of the different factors in the Pivonka bone cell population model.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
intrinsic_PTH	\(\beta_{PTH}\)	pM/day
intrinsic_RANKL	\(\beta_{RANKL}\)	pM/day
min_OPG_per_cell	\(\beta_{1,OPG}, \beta_{2,OPG}\)	pM/day
bool_OBp_produce_OPG
bool_OBa_produce_OPG
max_RANKL_per_cell	\(R_1^{RANKL}, R_2^{RANKL}\)	pM/pM
bool_OBp_produce_RANKL
bool_OBa_produce_RANKL

Parameters:

intrinsic_PTH (float) – intrinsic production rate of PTH
intrinsic_RANKL (float) – intrinsic production rate of RANKL
min_OPG_per_cell (float) – minimal rate of OPG production per cell
bool_OBp_produce_OPG (int) – boolean variable determining if OBp produce OPG
bool_OBa_produce_OPG (int) – boolean variable determining if OBa produce OPG
max_RANKL_per_cell (float) – production rate of RANKL per cell
bool_OBp_produce_RANKL (int) – boolean variable determining if OBp produce RANKL
bool_OBa_produce_RANKL (int) – boolean variable determining if OBa produce RANKL

class bone_models.bone_cell_population_models.parameters.pivonka_parameters.repression_coefficient[source]

Bases: object

This class defines the repression coefficients of respective receptor-ligand binding in the Pivonka bone cell population model.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
TGFb_OBp	\(K_{D2, \text{TGF-}\beta}\)	pM
PTH_OB	\(K_{D6, \text{PTH}}, K_{D7, \text{PTH}}\)	pM

Parameters:

TGFb_OBp (float) – repression coefficient related to TGF-beta binding on OBp
PTH_OB (float) – repression coefficient for OPG production related to PTH binding on osteoblasts

class bone_models.bone_cell_population_models.parameters.pivonka_parameters.unbinding_constant[source]

Bases: object

This class defines the unbinding constants of receptor-ligand bindings in the Pivonka bone cell population model.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
RANKL_OPG	\(k_2\)	1/day
RANKL_RANK	\(k_4\)	1/pM
PTH_OB	\(k_6\)	1/day

Parameters:

RANKL_OPG (float) – unbinding constant for RANKL-OPG
RANKL_RANK (float) – unbinding constant for RANKL-RANK
PTH_OB (float) – unbinding constant for PTH binding with its receptor on OB

Scheiner Parameters

This module contains the parameters for the model by Scheiner et al. (2013). In each parameter class, the parameter names are matched with the corresponding model nomenclature and units.

class bone_models.bone_cell_population_models.parameters.scheiner_parameters.Scheiner_Parameters[source]

Bases: object

This class defines the parameters of the bone model.

Parameters:

differentiation_rate (differentiation_rate) – differentiation rates of the different cell types
apoptosis_rate (apoptosis_rate) – apoptosis rates of the different cell types
activation_coefficient (activation_coefficient) – activation coefficients of respective receptor-ligand binding
repression_coefficient (repression_coefficient) – repression coefficients of respective receptor-ligand binding
degradation_rate (degradation_rate) – degradation rates of the different factors
concentration (concentration) – fixed concentrations
binding_constant (binding_constant) – binding constants of RANK RANKL and OPG
production_rate (production_rate) – intrinsic/ endogenous production rates of the different factors
mechanics (mechanics) – parameters relevant for mechanics of the bone model
bone_volume (bone_volume) – parameters relevant for bone volume of the bone model

class bone_models.bone_cell_population_models.parameters.scheiner_parameters.activation_coefficient[source]

Bases: object

This class defines the activation coefficients of respective receptor-ligand binding.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
TGFb_OBu	\(K^{TGFb}_{act,OBu}\)	pM
TGFb_OCa	\(K^{TGFb}_{act,OCa}\)	pM
PTH_OB	\(K^{PTH}_{act,OB}\)	pM
RANKL_RANK	\(K_{d, [RANKL-RANK]}\)	pM

Parameters:

TGFb_OBu (float) – activation coefficient related to TGF-beta binding on OBu
TGFb_OCa (float) – activation coefficient related to TGF-beta binding on OCa
PTH_OB (float) – activation coefficient related to PTH binding to osteoblasts
RANKL_RANK (float) – activation coefficient related to RANKL binding on RANK

class bone_models.bone_cell_population_models.parameters.scheiner_parameters.apoptosis_rate[source]

Bases: object

This class defines the apoptosis rates of the different cell types.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
OBa	\(A_{OB_a}\)	1/day
OCa	\(A_{OC_a}\)	1/day

Parameters:

OBa (float) – apoptosis rate of active osteoblasts
OCa (float) – apoptosis rate of active osteoclasts

class bone_models.bone_cell_population_models.parameters.scheiner_parameters.binding_constant[source]

Bases: object

This class defines the binding constants of RANK RANKL and OPG.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
RANKL_OPG	\(K_{a, [RANKL-OPG]}\)	1/pM
RANKL_RANK	\(K_{a, [RANKL-RANK]}\)	1/pM

Parameters:

RANKL_OPG (float) – association binding constant for RANKL-OPG
RANKL_RANK (float) – association binding constant for RANKL-RANK

class bone_models.bone_cell_population_models.parameters.scheiner_parameters.bone_volume[source]

Bases: object

This class defines the parameters relevant for bone volume of the bone model.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
formation_rate	\(k_{form}\)	1/day
resorption_rate	\(k_{res}\)	1/day
stored_TGFb_content	\(\alpha\)
vascular_pore_fraction	\(f_{vas}\)	%
bone_fraction	\(f_{bm}\)	%

Parameters:

formation_rate (float) – formation rate of bone volume
resorption_rate (float) – resorption rate of bone volume
stored_TGFb_content (float) – proportionality constant expressing the TGF-β content stored in bone volume
vascular_pore_fraction (float) – fraction of vascular pores in bone volume in percentage
bone_fraction (float) – fraction of bone matrix in bone volume in percentage

class bone_models.bone_cell_population_models.parameters.scheiner_parameters.concentration[source]

Bases: object

This class defines fixed concentrations.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
OPG_max	\(C^{max}_{OPG}\)	pM
RANK		pM
OCp		pM

class bone_models.bone_cell_population_models.parameters.scheiner_parameters.degradation_rate[source]

Bases: object

This class defines the degradation rates of the different factors.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
PTH	\(\check{D}_{PTH}\)	1/day
OPG	\(\check{D}_{OPG}\)	1/day
RANKL	\(\check{D}_{RANKL}\)	1/day
TGFb	\(\check{D}_{TGF-\beta}\)	1/day

Parameters:

PTH (float) – degradation rate of PTH
OPG (float) – degradation rate of OPG
RANKL (float) – degradation rate of RANKL
TGFb (float) – degradation rate of TGF-beta

class bone_models.bone_cell_population_models.parameters.scheiner_parameters.differentiation_rate[source]

Bases: object

This class defines the differentiation rates of the different cell types.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
OBu	\(D_{OB_u}^{Pivonka}\)	1/day
OBp	\(D_{OB_p}\)	1/day
OCp	\(D_{OC_p}\)	1/day

Parameters:

OBu (float) – differentiation rate of uncommitted osteoblasts
OBp (float) – differentiation rate of precursor osteoblasts
OCp (float) – differentiation rate of precursor osteoclasts

class bone_models.bone_cell_population_models.parameters.scheiner_parameters.mechanics[source]

Bases: object

This class defines the parameters relevant for mechanics of the bone model.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
strain_effect_on_OBp	\(\check{\Pi}_{act, OB_p}^{mech}\)
strain_energy_density	\(\check{\psi}_{bm}\)
update_OBp_proliferation_rate
OBu_differentiation_rate	\(a_{P_{OB_p}}\)
RANKL_production	\(P_{RANKL}\)
bulk_modulus_water	\(k_{H_2O}\)	GPa
shear_modulus_water	\(\mu_{H_2O}\)	GPa
volumetric_part_of_unit_tensor	\(\mathbb{J}\)
unit_tensor_as_matrix	\(\mathbb{I}\)
deviatoric_part_of_unit_tensor	\(\mathbb{K}\)
stiffness_tensor_vascular_pores	\(\mathbb{c}_{vas}\)
stiffness_tensor_bone_matrix	\(\mathbb{c}_{bm}\)
step_size_for_Hill_tensor_integration
hill_tensor_cylindrical_inclusion	\(\mathbb{P}_{r}^{bm}\)
stress_tensor_normal_loading	\(\mathbb{\Sigma}_{cort}\)

Parameters:

strain_effect_on_OBp_steady_state (float) – strain effect on OBp steady state
strain_energy_density_steady_state (float) – strain energy density steady state
update_OBp_proliferation_rate (bool) – update OBp proliferation rate
fraction_of_OBu_differentiation_rate (float) – fraction of OBu differentiation rate
RANKL_production (float) – RANKL production
bulk_modulus_water (float) – bulk modulus of water
shear_modulus_water (float) – shear modulus of water
volumetric_part_of_unit_tensor (numpy.ndarray) – volumetric part of unit tensor
unit_tensor_as_matrix (numpy.ndarray) – unit tensor as matrix
deviatoric_part_of_unit_tensor (numpy.ndarray) – deviatoric part of unit tensor
stiffness_tensor_vascular_pores (numpy.ndarray) – stiffness tensor of vascular pores
stiffness_tensor_bone_matrix (numpy.ndarray) – stiffness tensor of bone matrix
step_size_for_Hill_tensor_integration (float) – step size for Hill tensor integration
hill_tensor_cylindrical_inclusion (numpy.ndarray) – Hill tensor of cylindrical inclusion
stress_tensor_normal_loading (numpy.ndarray) – stress tensor of normal/ habitual loading

class bone_models.bone_cell_population_models.parameters.scheiner_parameters.production_rate[source]

Bases: object

This class defines the intrinsic/ endogenous production rates of the different factors.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
intrinsic_PTH	\(\beta_{PTH}\)	pM/day
intrinsic_RANKL	\(\beta_{RANKL}\)	pM/day
min_OPG_per_cell	\(p_{OB}^{OPG}\)	pM
bool_OBp_produce_OPG
bool_OBa_produce_OPG
max_RANKL_per_cell	\(N_{RANKL}^{OB}\)	pM
max_RANK_per_cell	\(C_{RANK}\)	pM
bool_OBp_produce_RANKL
bool_OBa_produce_RANKL

Parameters:

intrinsic_PTH (float) – intrinsic production rate of PTH
intrinsic_RANKL (float) – intrinsic production rate of RANKL
min_OPG_per_cell (float) – minimal rate of OPG production per cell
bool_OBp_produce_OPG (int) – boolean variable determining which cells produce OPG
bool_OBa_produce_OPG (int) – boolean variable determining which cells produce OPG
max_RANKL_per_cell (float) – production rate of RANKL per cell
max_RANK_per_cell (float) – production rate of RANK per cell
bool_OBp_produce_RANKL (int) – boolean variable determining which cells produce RANKL
bool_OBa_produce_RANKL (int) – boolean variable determining which cells produce RANKL

class bone_models.bone_cell_population_models.parameters.scheiner_parameters.proliferation_rate[source]

Bases: object

This class defines the proliferation rates. The prolifaration rate of OBp is depends the mechanics effect and is thus computed in the model (Eq. (16) in the paper).

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
OBp	\(P_{OB_p}\)	1/day

Parameters:: OBp (float) – proliferation rate of precursor osteoblasts

class bone_models.bone_cell_population_models.parameters.scheiner_parameters.repression_coefficient[source]

Bases: object

This class defines the repression coefficients of respective receptor-ligand binding.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
TGFb_OBp	\(K^{TGFb}_{rep,OBp}\)	pM
PTH_OB	\(K^{PTH}_{rep,OB}\)	pM

Parameters:

TGFb_OBp (float) – repression coefficient related to TGF-beta binding on OBp
PTH_OB (float) – repression coefficient for OPG production related to PTH binding on osteoblasts

Martinez-Reina Parameters

This module contains the parameters for the model by Martinez-Reina et al. (2019). In each parameter class, the parameter names are matched with the corresponding model nomenclature and units.

class bone_models.bone_cell_population_models.parameters.martinez_reina_parameters.Martinez_Reina_Parameters[source]

Bases: object

This class defines the parameters of the bone model.

Parameters:

differentiation_rate (differentiation_rate) – differentiation rates of the different cell types
apoptosis_rate (apoptosis_rate) – apoptosis rates of the different cell types
activation_coefficient (activation_coefficient) – activation coefficients of respective receptor-ligand binding
repression_coefficient (repression_coefficient) – repression coefficients of respective receptor-ligand binding
degradation_rate (degradation_rate) – degradation rates of the different factors
concentration (concentration) – fixed concentrations
binding_constant (binding_constant) – binding constants of RANK RANKL and OPG
production_rate (production_rate) – intrinsic/ endogenous production rates of the different factors
mechanics (mechanics) – parameters relevant for mechanics of the bone model
bone_volume (bone_volume) – parameters relevant for bone volume of the bone model

class bone_models.bone_cell_population_models.parameters.martinez_reina_parameters.PMO[source]

Bases: object

This class defines the parameters of the postmenopausal osteoporosis model.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
increase_in_RANKL	\(P_{RANKL}^{PMO,ini}\)	pM
reduction_factor	\(\xi\)
characteristic_time	\(\tau_{PMO}^{RANKL}\)	days

Parameters:

increase_in_RANKL (float) – initial increase in RANKL concentration
reduction_factor (float) – reduction factor
characteristic_time (float) – characteristic time (900 days in the paper, but doesn’t agree with the results -> corrected here)

class bone_models.bone_cell_population_models.parameters.martinez_reina_parameters.activation_coefficient[source]

Bases: object

This class defines the activation coefficients of respective receptor-ligand binding.

The following table provides a mapping between the model parameters and their original names from the publication by Scheiner et al., 2013:

Parameter Name	Symbol	Units
TGFb_OBu	\(K^{TGFb}_{act,OBu}\)	pM
TGFb_OCa	\(K^{TGFb}_{act,OCa}\)	pM
PTH_OB	\(K^{PTH}_{act,OB}\)	pM
RANKL_RANK	\(K_{d, [RANKL-RANK]}\)	pM

Parameters:

TGFb_OBu (float) – activation coefficient related to TGF-beta binding on OBu
TGFb_OCa (float) – activation coefficient related to TGF-beta binding on OCa
PTH_OB (float) – activation coefficient related to PTH binding to osteoblasts
RANKL_RANK (float) – activation coefficient related to RANKL binding on RANK

class bone_models.bone_cell_population_models.parameters.martinez_reina_parameters.apoptosis_rate[source]

Bases: object

This class defines the apoptosis rates of the different cell types.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
OBa	\(A_{OB_a}\)	1/day
OCa	\(A_{OC_a}\)	1/day

Parameters:

OBa (float) – apoptosis rate of active osteoblasts
OCa (float) – apoptosis rate of active osteoclasts

class bone_models.bone_cell_population_models.parameters.martinez_reina_parameters.binding_constant[source]

Bases: object

This class defines the binding constants of RANK RANKL and OPG.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
RANKL_OPG	\(K_{a, [RANKL-OPG]}\)	1/pM
RANKL_RANK	\(K_{a, [RANKL-RANK]}\)	1/pM
RANKL_denosumab	\(K_{a, [RANKL-den]}\)	1/pM

Parameters:

RANKL_OPG (float) – association binding constant for RANKL-OPG
RANKL_RANK (float) – association binding constant for RANKL-RANK
RANKL_denosumab (float) – association binding constant for RANKL-denosumab

class bone_models.bone_cell_population_models.parameters.martinez_reina_parameters.bone_volume[source]

Bases: object

This class defines the parameters relevant for bone volume of the bone model.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
formation_rate	\(k_{form}\)	1/day
resorption_rate	\(k_{res}\)	1/day
stored_TGFb_content	\(\alpha\)
vascular_pore_fraction	\(f_{vas}\)	%
bone_fraction	\(f_{bm}\)	%

Parameters:

formation_rate (float) – formation rate of bone volume
resorption_rate (float) – resorption rate of bone volume
stored_TGFb_content (float) – proportionality constant expressing the TGF-β content stored in bone volume
vascular_pore_fraction (float) – fraction of vascular pores in bone volume in percentage
bone_fraction (float) – fraction of bone matrix in bone volume in percentage

class bone_models.bone_cell_population_models.parameters.martinez_reina_parameters.concentration[source]

Bases: object

This class defines fixed concentrations.

The following table provides a mapping between the model parameters and their original names from the publication by Scheiner et al., 2013:

Parameter Name	Symbol	Units
OPG_max	\(C^{max}_{OPG}\)	pM
RANK		pM
OCp		pM

Parameters:

OPG_max (float) – maximum concentration of OPG
RANK (float) – fixed concentration of RANK
OCp (float) – fixed concentration of preosteoclasts

class bone_models.bone_cell_population_models.parameters.martinez_reina_parameters.degradation_rate[source]

Bases: object

This class defines the degradation rates of the different factors.

The following table provides a mapping between the model parameters and their original names from the publication by Scheiner et al., 2013:

Parameter Name	Symbol	Units
PTH	\(\check{D}_{PTH}\)	1/day
OPG	\(\check{D}_{OPG}\)	1/day
RANKL	\(\check{D}_{RANKL}\)	1/day
TGFb	\(\check{D}_{TGF-\beta}\)	1/day

Parameters:

PTH (float) – degradation rate of PTH
OPG (float) – degradation rate of OPG
RANKL (float) – degradation rate of RANKL
TGFb (float) – degradation rate of TGF-beta

class bone_models.bone_cell_population_models.parameters.martinez_reina_parameters.denosumab[source]

Bases: object

This class defines the parameters of the denosumab model.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
accessibility_factor	\(\chi\)
molar_mass	\(M_{den}\)	g/mol
absorption_rate	\(k_a\)	1/day
volume_central_compartment	\(V_c\)	ml/kg
bioavailability	\(F\)
maximum_volume	\(V_{max}\)	ng/(kg day)
michaelis_menten_constant	\(K_m\)	ng/ml
elimination_rate	\(k_{el}\)	1/day
reference_body_weight	\(-\)	kg

Parameters:

accessibility_factor (float) – denosumab site-specific accessibility factor
molar_mass (float) – molar mass of denosumab, given in kDa and converted to g/mol
absorption_rate (float) – absorption rate of denosumab
volume_central_compartment (float) – volume of the central compartment, multiplied by the reference body weight
bioavailability (float) – bioavailability of denosumab
maximum_volume (float) – maximum volume of denosumab, multiplied by the reference body weight
michaelis_menten_constant (float) – Michaelis-Menten constant of denosumab
elimination_rate (float) – elimination rate of denosumab
reference_body_weight (float) – reference body weight

class bone_models.bone_cell_population_models.parameters.martinez_reina_parameters.differentiation_rate[source]

Bases: object

This class defines the differentiation rates of the different cell types.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
OBu	\(D_{OB_u}\)	1/day
OBp	\(D_{OB_p}\)	1/day
OCp	\(D_{OC_p}\)	1/day

Parameters:

OBu (float) – differentiation rate of uncommitted osteoblasts
OBp (float) – differentiation rate of precursor osteoblasts
OCp (float) – differentiation rate of precursor osteoclasts

class bone_models.bone_cell_population_models.parameters.martinez_reina_parameters.mechanics[source]

Bases: object

This class defines the parameters relevant for mechanics of the bone model.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
strain_effect_on_OBp_steady_state	\(\breve{\Pi}_{act, OB_p}^{mech}\)
strain_energy_density_steady_state	\(\breve{\psi}_{bm}\)
update_OBp_proliferation_rate
fraction_of_OBu_differentiation_rate	\(a_{P_{OB_p}}\)
RANKL_production	\(P_{RANKL}\)	pM/day
stress_tensor_normal_loading	\(\Sigma_{cort}\)	GPa
poissons_ratio	\(\nu\)

Parameters:

strain_effect_on_OBp_steady_state (float) – strain effect on OBp steady state
strain_energy_density_steady_state (float) – strain energy density steady state
update_OBp_proliferation_rate (bool) – boolean variable determining if the OBp proliferation rate is updated
fraction_of_OBu_differentiation_rate (float) – fraction of OBu differentiation rate
RANKL_production (float) – production rate of RANKL
stress_tensor_normal_loading (numpy.ndarray) – stress tensor under normal loading
poissons_ratio (float) – Poisson’s ratio

class bone_models.bone_cell_population_models.parameters.martinez_reina_parameters.mineralisation[source]

Bases: object

This class defines the parameters relevant for mineralisation of the bone model.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
density_organic	\(\rho_o\)	g/cm^3
density_mineral	\(\rho_m\)	g/cm^3
volume_fraction_organic	\(\nu_o\)
lag_time	\(t_{mlt}\)	days
primary_phase_duration	\(t_{prim}\)	days
primary_mineral_content	\(v_{m, prim}\)
maximum_mineral_content	\(v_{m, max}\)
length_of_queue	\(-\)
rate	\(\kappa\)

Parameters:

density_organic (float) – density of organic material
density_mineral (float) – density of mineral material
volume_fraction_organic (float) – volume fraction of organic material
lag_time (float) – lag time
primary_phase_duration (float) – primary phase duration
primary_mineral_content (float) – primary mineral content
maximum_mineral_content (float) – maximum mineral content
length_of_queue (int) – length of ageing queue
rate (float) – mineralisation rate

class bone_models.bone_cell_population_models.parameters.martinez_reina_parameters.production_rate[source]

Bases: object

This class defines the intrinsic/ endogenous production rates of the different factors.

The following table provides a mapping between the model parameters and their original names from the publication by Scheiner et al., 2013:

Parameter Name	Symbol	Units
intrinsic_PTH	\(\beta_{PTH}\)	pM/day
intrinsic_RANKL	\(\beta_{RANKL}\)	pM/day
min_OPG_per_cell	\(p_{OB}^{OPG}\)	pM
bool_OBp_produce_OPG
bool_OBa_produce_OPG
max_RANKL_per_cell	\(N_{RANKL}^{OB}\)	pM
max_RANK_per_cell	\(C_{RANK}\)	pM
bool_OBp_produce_RANKL
bool_OBa_produce_RANKL

Parameters:

intrinsic_PTH (float) – intrinsic production rate of PTH
intrinsic_RANKL (float) – intrinsic production rate of RANKL
min_OPG_per_cell (float) – minimal rate of OPG production per cell
bool_OBp_produce_OPG (int) – boolean variable determining which cells produce OPG
bool_OBa_produce_OPG (int) – boolean variable determining which cells produce OPG
max_RANKL_per_cell (float) – production rate of RANKL per cell
max_RANK_per_cell (float) – production rate of RANK per cell
bool_OBp_produce_RANKL (int) – boolean variable determining which cells produce RANKL
bool_OBa_produce_RANKL (int) – boolean variable determining which cells produce RANKL

class bone_models.bone_cell_population_models.parameters.martinez_reina_parameters.proliferation_rate[source]

Bases: object

This class defines the proliferation rates. The prolifaration rate of OBp is depends the mechanics effect and is thus computed in the model (Eq. (16) in the paper by Scheiner et al., 2013).

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
OBp	\(P_{OB_p}\)	1/day

Parameters:: OBp (float) – proliferation rate of precursor osteoblasts

class bone_models.bone_cell_population_models.parameters.martinez_reina_parameters.repression_coefficient[source]

Bases: object

This class defines the repression coefficients of respective receptor-ligand binding.

The following table provides a mapping between the model parameters and their original names from the publication by Scheiner et al., 2013:

Parameter Name	Symbol	Units
TGFb_OBp	\(K^{TGFb}_{rep,OBp}\)	pM
PTH_OB	\(K^{PTH}_{rep,OB}\)	pM

Parameters:

TGFb_OBp (float) – repression coefficient related to TGF-beta binding on OBp
PTH_OB (float) – repression coefficient for OPG production related to PTH binding on osteoblasts

Lerebours Parameters

This module contains the parameters for the model by Lerebours et al. (2016). In each parameter class, the parameter names are matched with the corresponding model nomenclature and units.

class bone_models.bone_cell_population_models.parameters.lerebours_parameters.Lerebours_Parameters[source]

Bases: object

This class defines the parameters of the bone model.

Parameters:

differentiation_rate (differentiation_rate) – differentiation rates of the different cell types
apoptosis_rate (apoptosis_rate) – apoptosis rates of the different cell types
activation_coefficient (activation_coefficient) – activation coefficients of respective receptor-ligand binding
repression_coefficient (repression_coefficient) – repression coefficients of respective receptor-ligand binding
degradation_rate (degradation_rate) – degradation rates of the different factors
concentration (concentration) – fixed concentrations
binding_constant (binding_constant) – binding constants of RANK RANKL and OPG
production_rate (production_rate) – intrinsic/ endogenous production rates of the different factors
mechanics (mechanics) – parameters relevant for mechanics of the bone model
proliferation_rate (proliferation_rate) – proliferation rates of the different cell types
bone_volume (bone_volume) – parameters relevant for bone volume of the bone model
calibration (calibration) – parameters relevant for calibration of the bone model

class bone_models.bone_cell_population_models.parameters.lerebours_parameters.activation_coefficient[source]

Bases: object

This class defines the activation coefficients of respective receptor-ligand binding.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
TGFb_OBu	\(k^{TGFb}_{OB_u}\)	pM
TGFb_OCa	\(k^{TGFb}_{OC_a}\)	pM
PTH_OB	\(k^{PTH}_{OB}\)	pM
RANKL_RANK	\(k_{RANK}^{RANKL}\)	pM
MCSF_OCu	\(k_{OC_u}^{MCSF}\)	pM

Parameters:

TGFb_OBu (float) – parameter for TGF-beta binding on OBp
TGFb_OCa (float) – parameter for TGF-beta binding on OCa
PTH_OB (float) – parameter for PTH binding to osteoblasts (activation)
RANKL_RANK (float) – parameter for RANKL binding on RANK
MCSF_OCu (float) – parameter for MCSF binding on OCu

class bone_models.bone_cell_population_models.parameters.lerebours_parameters.apoptosis_rate[source]

Bases: object

This class defines the apoptosis rates of the different cell types.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
OBa	\(A_{OB_a}\)	1/day
OCa	\(A_{OC_a}\)	1/day

Parameters:

OBa (float) – apoptosis rate of active osteoblasts
OCa (float) – apoptosis rate of active osteoclasts

class bone_models.bone_cell_population_models.parameters.lerebours_parameters.binding_constant[source]

Bases: object

This class defines the binding constants of RANK RANKL and OPG.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
RANKL_OPG	\(k^{RANKL}_{OPG}\)	1/pM
RANKL_RANK	\(k^{RANKL}_{RANK}\)	1/pM

Parameters:

RANKL_OPG (float) – binding constant for RANKL-OPG
RANKL_RANK (float) – binding constant for RANKL-RANK

class bone_models.bone_cell_population_models.parameters.lerebours_parameters.bone_volume[source]

Bases: object

This class defines the parameters relevant for bone volume of the bone model.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
formation_rate	\(k_{form}\)	1/day
resorption_rate	\(k_{res}\)	1/day
stored_TGFb_content	\(n_{TGF\beta}^{bone}\)	pM

Parameters:

formation_rate (float) – formation rate of bone volume
resorption_rate (float) – resorption rate of bone volume
stored_TGFb_content (float) – proportionality constant expressing the TGF-β content stored in bone volume

class bone_models.bone_cell_population_models.parameters.lerebours_parameters.calibration[source]

Bases: object

This class defines the parameters relevant for calibration of the bone model. The following table provides a mapping between the model parameters and the original names from the publication:

Parameter Name	Symbol	Units
turnover	\(\tau_{turnover}\)
steady_state_turnover	\(\delta\)
OCa	\(\beta\)
OBa	\(\gamma\)

Parameters:

turnover (float) – calibration for turnover to specific surface
steady_state_turnover (float) – calibration for steady state turnover rate of the bone model
OCa (float) – calibration coefficient for OCa of the bone model
OBa (float) – calibration coefficient for OBa of the bone model

class bone_models.bone_cell_population_models.parameters.lerebours_parameters.concentration[source]

Bases: object

This class defines fixed concentrations.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
OPG_max	\({OPG}_{sat}\)	pM
MCSF		pM
RANK		pM

Parameters:

OPG_max (float) – maximum concentration of OPG
MCSF (float) – concentration of MCSF
RANK (float) – concentration of RANK

class bone_models.bone_cell_population_models.parameters.lerebours_parameters.degradation_rate[source]

Bases: object

This class defines the degradation rates of the different factors.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
PTH	\({D}_{PTH}\)	1/day
OPG	\({D}_{OPG}\)	1/day
RANKL	\({D}_{RANKL}\)	1/day
TGFb	\({D}_{TGF-\beta}\)	1/day

Parameters:

PTH (float) – degradation rate of PTH
OPG (float) – degradation rate of OPG
RANKL (float) – degradation rate of RANKL
TGFb (float) – degradation rate of TGF-beta

class bone_models.bone_cell_population_models.parameters.lerebours_parameters.differentiation_rate[source]

Bases: object

This class defines the differentiation rates of the different cell types.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
OBu	\(D_{OB_u}^{Pivonka}\)	1/day
OBp	\(D_{OB_p}\)	1/day
OCp	\(D_{OC_p}\)	1/day
OCu	\(D_{OC_u}\)	1/day

Parameters:

OBu (float) – differentiation rate of uncommitted osteoblasts
OBp (float) – differentiation rate of precursor osteoblasts
OCp (float) – differentiation rate of precursor osteoclasts
OCu (float) – differentiation rate of uncommitted osteoclasts

class bone_models.bone_cell_population_models.parameters.lerebours_parameters.mechanics[source]

Bases: object

This class defines the parameters relevant for mechanics of the bone model.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
strain_effect_on_OBp_steady_state	\(\mu(r,t)\)
strain_effect_on_OBp	\(\mu(r,t)\)
strain_energy_density	\(\check{\psi}(r,t))\)
update_OBp_proliferation_rate
RANKL_production	\(\beta_{RANKL}^{mech}\)
unit_tensor_as_matrix	\(\mathbb{I}\)
deviatoric_part_of_unit_tensor	\(\mathbb{K}\)
stiffness_tensor_vascular_pores	\(\mathbb{c}^{micro}_{vas}\)	GPa
stiffness_tensor_bone_matrix	\(\mathbb{c}^{micro}_{bm}\)	GPa
step_size_for_Hill_tensor_integration
hill_tensor_cylindrical_inclusion	\(\mathbb{P}_{r}^{bm}\)
stress_tensor_normal_loading	\(\sigma^{tissue}\)
biomech_transduction_strength	\(\lambda\)
biomech_transduction_strength_RANKL	\(\kappa\)	pM/day
correction_factor	\(K\)	GPa

class bone_models.bone_cell_population_models.parameters.lerebours_parameters.production_rate[source]

Bases: object

This class defines the intrinsic/ endogenous production rates of the different factors.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
intrinsic_PTH	\(P_{PTH}\)	pM/day
intrinsic_RANKL	\(\beta^{RANKL}_{OB_p}\)	pM/day
min_OPG_per_cell	\(\beta^{OPG}_{OB_a}\)	pM
bool_OBp_produce_OPG	code-specific - set to 0
bool_OBa_produce_OPG	code-specific - set to 1
max_RANKL_per_cell	\(N_{RANKL}^{OB}\)	pM
max_RANK_per_cell	\(N^{RANK}_{OC_p}\)	pM
bool_OBp_produce_RANKL	code-specific - set to 1
bool_OBa_produce_RANKL	code-specific - set to 0

Parameters:

intrinsic_PTH (float) – systemic concentration of PTH
intrinsic_RANKL (float) – intrinsic production rate of RANKL
min_OPG_per_cell (float) – minimal rate of OPG production per cell
bool_OBp_produce_OPG (int) – boolean variable determining which cells produce OPG
bool_OBa_produce_OPG (int) – boolean variable determining which cells produce OPG
max_RANKL_per_cell (float) – production rate of RANKL per cell
max_RANK_per_cell (float) – production rate of RANK per cell
bool_OBp_produce_RANKL (int) – boolean variable determining which cells produce RANKL
bool_OBa_produce_RANKL (int) – boolean variable determining which cells produce RANKL

class bone_models.bone_cell_population_models.parameters.lerebours_parameters.proliferation_rate[source]

Bases: object

This class defines the proliferation rates. The proliferation rate of OBp depends the mechanics effect and is thus computed in the model (Eq. (22) in the paper).

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
OBp	\(P_{OB_p}\)	1/day

Parameters:: OBp (float) – proliferation rate of precursor osteoblasts

class bone_models.bone_cell_population_models.parameters.lerebours_parameters.repression_coefficient[source]

Bases: object

This class defines the repression coefficients of respective receptor-ligand binding.

The following table provides a mapping between the model parameters and their original names from the publication:

Parameter Name	Symbol	Units
TGFb_OBp	\(k^{TGFb}_{OBp}\)	pM
PTH_OB	\(k^{PTH}_{OB}\)	pM

Parameters:

TGFb_OBp (float) – parameter for TGF-beta binding on OBp
PTH_OB (float) – parameter for PTH binding on osteoblasts (repressor)