PD models

class BIS_model(hill_model: str = 'Bouillon', hill_param: list | None = None, random: bool | None = False, truncated: float | None = None, ts: float = 1, **kwargs)

Bases: object

Model to link Propofol effect site concentration to BIS.

The equation is:

\[BIS = E0 - Emax * \frac{U^\gamma}{1+U^\gamma}\]

If only the effect of propofol is considered the equation represents a sigmoid function, where:

\[U = \frac{C_{p,es}}{C_{p,50}}\]

If the interaction with remifentanil is considered the equation represents a Surface Response model, where:

\[U = \frac{U_p + U_r}{1 - \beta \theta + \beta \theta^2}\]

for Minto-type surface model and:

\[U = U_p + U_r + \beta U_p U_r\]

for Greco-type surface model, with:

\[U_p = \frac{C_{p,es}}{C_{p,50}}\]

\[U_r = \frac{C_{r,es}}{C_{r,50}}\]

\[\theta = \frac{U_p}{U_r+U_p}\]

Parameters:

hill_modelstr, optional

‘Vanluchene’[Vanluchene2004], do not consider the synergistic effect of remifentanil. ‘Eleveld’[Eleveld2018], do not consider the synergistic effect of remifentanil. ‘Bouillon’[Bouillon2004], considers the synergistic effect of remifentanil (Minto-type surface model). ‘Fuentes’[Fuentes2018], considers the synergistic effect of remifentanil (Greco-type surface model). ‘Yumuk’[Yumuk2024], considers the synergistic effect of remifentanil (Greco-type). Ignored if hill_param is specified. Default is ‘Bouillon’.

hill_paramlist, optional

Parameters of the model list [c50p_BIS, c50r_BIS, gamma_BIS, beta_BIS, E0_BIS, Emax_BIS, Delay_BIS]:

• c50p_BIS: Concentration at half effect for propofol effect on BIS (µg/mL).

• c50r_BIS: Concentration at half effect for remifentanil effect on BIS (ng/mL). If it is equal to zero the interaction with remifentanil is not considered.

• gamma_BIS: Slope coefficient for the BIS model.

• beta_BIS: Interaction coefficient for the BIS model (beta_BIS = 0 signifies an additive interaction, beta_BIS > 0 indicates synergy).

• E0_BIS: Initial BIS.

• Emax_BIS: Max effect of the drugs on BIS.

• Delay_BIS: Delay affecting the BIS (s)

The default is None. If Delay_BIS is not provided it is assumed equal to 0.

randombool, optional

Add uncertainties in the parameters. Ignored if hill_param is specified. The default is False.

tsfloat

Sampling time, in s.

truncatedfloat, optional

If not None it correspond to the number of standard deviation after which the distribution are truncated for generating uncertain parameters. The default is None.

Attributes:

c50pfloat

Concentration at half effect for propofol effect on BIS (µg/mL).

c50rfloat

Concentration at half effect for remifentanil effect on BIS (ng/mL). If it is equal to zero the interaction with remifentanil is not considered.

gammafloat

Slope coefficient for the BIS model.

betafloat

Interaction coefficient for the BIS model (beta_BIS = 0 signifies an additive interaction, beta_BIS > 0 indicates synergy).

E0float

Initial BIS.

Emaxfloat

Max effect of the drugs on BIS.

bis_delayfloat

Delay time on the output of the BIS model (s)

hill_paramlist

Parameters of the model list [c50p_BIS, c50r_BIS, gamma_BIS, beta_BIS, E0_BIS, Emax_BIS, Delay_BIS]

c50p_initfloat

Initial value of c50p, used for blood loss modelling.

hill_modelstr

‘Vanluchene’[Vanluchene2004], do not consider the synergistic effect of remifentanil. ‘Eleveld’[Eleveld2018], do not consider the synergistic effect of remifentanil. ‘Bouillon’[Bouillon2004], considers the synergistic effect of remifentanil (Minto-type). ‘Fuentes’[Fuentes2018], considers the synergistic effect of remifentanil (Greco-type). ‘Yumuk’[Yumuk2024], considers the synergistic effect of remifentanil (Greco-type).

tsfloat

Sampling time, in s.

References

[Bouillon2004] (1,2)

T. W. Bouillon et al., “Pharmacodynamic Interaction between Propofol and Remifentanil Regarding Hypnosis, Tolerance of Laryngoscopy, Bispectral Index, and Electroencephalographic Approximate Entropy,” Anesthesiology, vol. 100, no. 6, pp. 1353–1372, Jun. 2004, doi: 10.1097/00000542-200406000-00006.

[Vanluchene2004] (1,2)

A. L. G. Vanluchene et al., “Spectral entropy as an electroencephalographic measure of anesthetic drug effect: a comparison with bispectral index and processed midlatency auditory evoked response,” Anesthesiology, vol. 101, no. 1, pp. 34–42, Jul. 2004, doi: 10.1097/00000542-200407000-00008.

[Eleveld2018] (1,2)

D. J. Eleveld, P. Colin, A. R. Absalom, and M. M. R. F. Struys, “Pharmacokinetic–pharmacodynamic model for propofol for broad application in anaesthesia and sedation” British Journal of Anaesthesia, vol. 120, no. 5, pp. 942–959, mai 2018, doi:10.1016/j.bja.2018.01.018.

[Fuentes2018] (1,2)

R. Fuentes et al. “Propofol pharmacokinetic and pharmacodynamic profile and its electroencephalographic interaction with remifentanil in children.” Pediatric Anesthesia 28.12 (2018): 1078-1086. doi: 10.1111/pan.13486

[Yumuk2024] (1,2)

E. Yumuk et al. “Data-driven identification and comparison of full multivariable models for propofol–remifentanil induced general anesthesia.” Journal of Process Control 139 (2024): 103243. doi: 10.1016/j.jprocont.2024.103243

compute_bis(c_es_propo, c_es_remi=None)

Compute BIS function from propofol (and optionally remifentanil) effect site concentration.

If the BIS model chosen considers only the effect of propofol the effect site concentration of remifentanil is ignored. Inputs can be either nd.array or float, the format of the output will be the same as the input

Parameters:

c_es_propofloat

Propofol effect site concentration µg/mL.

c_es_remifloat, optional

Remifentanil effect site concentration ng/mL. The default is 0.

Returns:

BISfloat

Bis value.

full_sim(c_es_propo: ndarray, c_es_remi: ndarray | None = None) → ndarray

Simulate BIS model with a given input.

Parameters:

c_es_proponp.ndarray

List of propofol effect site concentrations (µg/ml).

c_es_reminp.ndarray, optional

List of remifentanil effect site concentrations (ng/ml).

Returns:

np.ndarray

List of the output BIS values during the simulation.

inverse_hill(BIS: float, c_es_remi: float | None = 0) → float

Compute Propofol effect site concentration from BIS (and optionally Remifentanil effect site concentration if the BIS model chosen takes into acount interaction) .

Parameters:

BISfloat

BIS value.

cerfloat, optional

Effect site Remifentanil concentration (ng/mL). The default is 0.

Returns:

cepfloat

Effect site Propofol concentration (µg/mL).

one_step(c_es_propo: float, c_es_remi: float | None = 0) → float

Compute one step time of the BIS model

Parameters:

c_es_propofloat

Propofol effect site concentration (µg/mL).

c_es_remifloat, optional

Remifentanil effect site concentration (ng/mL). The default is 0.

Returns:

BISfloat

Bis value.

plot_surface()

Plot the 3D-Hill surface of the BIS related to Propofol and Remifentanil effect site concentration or the 2-D Hill curve of the BIS related to Propofol effect site concentration according to the BIS model chosen

update_param_blood_loss(v_ratio: float)

Update PK coefficient to mimic a blood loss.

Update the c50p parameters thanks to the blood volume ratio. The values are estimated from [Johnson2003].

Parameters:

v_lossfloat

blood volume as a fraction of init volume, 1 mean no loss, 0 mean 100% loss.

Returns:

None.

References

[Johnson2003]

K. B. Johnson et al., “The Influence of Hemorrhagic Shock on Propofol: A Pharmacokinetic and Pharmacodynamic Analysis,” Anesthesiology, vol. 99, no. 2, pp. 409–420, Aug. 2003, doi: 10.1097/00000542-200308000-00023.

class Hemo_meca_PD_model(age: float, ts: float, model: str = 'Su', nore_model: str = 'Beloeil', random: bool = False, truncated: bool = False, hr_base: float = None, sv_base: float = None, map_base: float = None)

Bases: object

This class implements the mechanically based model of Haemodynamics proposed in [Su2023].

See section Hemodynamics for detail about the model implemented here.

Parameters:

agefloat

Age of the patient in years.

tsfloat

Sampling time in seconds.

modelstr, optional

Model to use, ‘Su’ (see [Su2023]) and ‘VitalDB’ (see Disturbance Modelling and Identification) are available. The default is ‘Su’.

nore_modelstr, optional

Model to use for norepinephrine, ‘Beloeil’ and ‘Oualha’ are available. The default is ‘Beloeil’.

randombool, optional

Add uncertainties in the parameters. The default is False.

hr_basefloat, optional

Baseline heart rate (bpm). The default is None, which will use the value from the Su model.

sv_basefloat, optional

Baseline stroke volume (mL). The default is None, which will use the value from the Su model.

map_basefloat, optional

Baseline mean arterial pressure (mmHg). The default is None, which will use the value from the Su model.

truncatedbool, optional

Use truncated normal distribution (between [-3, +3] std) for the random parameters. The default is False.

References

[Su2023] (1,2)

H. Su, J. V. Koomen, D. J. Eleveld, M. M. R. F. Struys, and P. J. Colin, “Pharmacodynamic mechanism-based interaction model for the haemodynamic effects of remifentanil and propofol in healthy volunteers,” British Journal of Anaesthesia, vol. 131, no. 2, pp. 222–233, Aug. 2023. doi:10.1016/j.bja.2023.04.043.

[Beloeil2005]

H. Beloeil, J.-X. Mazoit, D. Benhamou, and J. Duranteau, “Norepinephrine kinetics and dynamics in septic shock and trauma patients,” BJA: British Journal of Anaesthesia, vol. 95, no. 6, pp. 782–788, Dec. 2005. doi:10.1093/bja/aei259.

[Oualha2014]

M. Oualha et al., “Population pharmacokinetics and haemodynamic effects of norepinephrine in hypotensive critically ill children,” British Journal of Clinical Pharmacology, vol. 78, no. 4, pp. 886–897, 2014. doi:10.1111/bcp.12412.

continuous_dynamic(x: ndarray, u: ndarray) → ndarray

Define the continuous dynamic of the haemodynamic system.

For implementation details see supplementary material nb 6 of the paper of Su and co-authors.

Parameters:

xnp.ndarray

state array composed of tpr, sv, hr, ltde_sv, ltde_hr.

u: np.ndarray

u = [cp_propo, cp_remi, map_wanted, sv_wanted, tpr_stim, sv_stim, hr_stim], plasma concentration of propofol (µg/ml) and remifentanil (ng/ml), map wanted (for norepinephrine simulation), sv_wanted (for blood loss simulation) and stimuli value for stimuli application.

Returns:

d x / dtnp.ndarray

temporal derivative of the state array.

continuous_dynamic_sys(t, x, u)

Same as continuous_dynamic but with time as first arguments (for scipy simulation).

full_sim(cp_propo: ndarray, cp_remi: ndarray, cp_nore: ndarray, disturbances: ndarray = None, x0: ndarray | None = None) → ndarray

Simulate hemodynamic model with a given input.

Parameters:

c_proponp.ndarray

list of plasma concentration of propofol (µg/ml).

cp_reminp.ndarray

list of plasma concentration of remifentanil (ng/ml).

cp_norenp.ndarray

list of plasma concentration of norepinephrine (ng/ml).

disturbancenp.ndarray

N*3 array of distrubance signal for TPR, SV and HR.

x0np.ndarray, optional

Initial state. The default is None.

Returns:

np.ndarray

List of the output value during the simulation.

initialized_at_given_concentration(cp_propo_eq: float = 0, cp_remi_eq: float = 0, cp_nore_eq: float = 0) → None

Initialize the haemodynamic model at a given concentration.

Parameters:

c_propofloat

plasma concentration of propofol at equilibrium (µg/ml).

cp_remifloat

plasma concentration of remifentanil at equilibrium (ng/ml).

cp_norefloat

plasma concentration of norepinephrine at equilibrium (ng/ml).

Returns:

None

The haemodynamic model is initialized at the given concentration.

nore_map_effect(cp_nore: float)

Compute Norepinephrine effect on MAP.

Parameters:

c_norefloat

Concentration of Norepinephrine (ng/mL)

one_step(cp_propo: float = 0, cp_remi: float = 0, cp_nore: float = 0, v_ratio: float = 1, disturbances: list = None) → ndarray

Compute one step time of the hemodynamic system.

It use Runge Kutta 4 to compute the non-linear integration.

Parameters:

c_propofloat

current plasma concentration of propofol (µg/ml), default is 0.

cp_remifloat

current plasma concentration of remifentanil (ng/ml), default is 0.

cp_norefloat

current plasma concentration of norepinephrine (ng/ml), default is 0.

v_ratiofloat

blood volume as a fraction of init volume, 1 mean no loss, 0 mean 100% loss, default is 1.

disturbanceslist

disturbance on TPR, SV, and HR (in this order). The default is [0]*3.

output_function(x: ndarray, dist: ndarray = array([0., 0., 0.])) → ndarray

Compute final signals value from state and disturbance value.

Parameters:

xnp.ndarray

state of the system

Returns:

np.ndarray

Total peripheral resistance (mmHg min/ mL), Stroke volume (ml), heart rate (beat / min), mean arterial pressure (mmHg), cardiac output (L/min)

state_at_equilibrium(cp_propo_eq: float = 0, cp_remi_eq: float = 0, cp_nore_eq: float = 0, disturbances: list = None, x0: ndarray = None) → ndarray

Solve the problem f(x,u)=0 for the continuous dynamique with a given u.

Parameters:

c_propofloat

plasma concentration of propofol at equilibrium (µg/ml).

cp_remifloat

plasma concentration of remifentanil at equilibrium (ng/ml).

cp_norefloat

plasma concentration of norepinephrine at equilibrium (ng/ml).

disturbance: list

disturbance on TPR, SV, and HR (in this order). The default is [0]*3.

x0np.ndarray, optional

Initial state. The default is None.

Returns:

np.ndarray

List of the output value at equilibrium.

class LOC_model(hill_model: str = 'Kern', hill_param: list | None = None, random: bool | None = False, ts: float = 1, truncated: float | None = None)

Bases: object

Propofol + Remifentanil -> LOC (Loss of Consciousness) model (Greco-type interaction).

The equation is:

\[LOC = \frac{U^\gamma}{1+U^\gamma}\]

with the Greco-type surface response model:

\[U = U_p + U_r + \beta U_p U_r\]

where

\[U_p = \frac{C_{p,es}}{C_{p,50}}\]

\[U_r = \frac{C_{r,es}}{C_{r,50}}\]

Parameters:

hill_modelstr, optional

‘Kern’[Kern2004], considers the synergistic effect of remifentanil (Greco-type surface model). ‘Mertens’[Mertens2003], considers the synergistic effect of remifentanil (Greco-type surface model). ‘Johnson’[Johnson2008], considers the synergistic effect of remifentanil (Greco-type surface model). Ignored if hill_param is specified. Default is ‘Kern’.

hill_paramlist, optional

Parameters of the model list [c50p_LOC, c50r_LOC, gamma_LOC, beta_LOC]: - c50p_LOC: Concentration at half effect for propofol effect on LOC (µg/mL). - c50r_LOC: Concentration at half effect for remifentanil effect on LOC (ng/mL). - gamma_LOC: Slope coefficient for the LOC model. - beta_LOC: Interaction coefficient for the LOC model (beta_LOC = 0 signifies an additive interaction, beta_LOC > 0 indicates synergy).

randombool, optional

Add uncertainties in the parameters. Ignored if hill_param is specified. The default is False.

tsfloat

Sampling time, in s.

truncatedfloat, optional

If not None it correspond to the number of standard deviation after which the distribution are truncated for generating uncertain parameters. The default is None.

Attributes:

c50pfloat

Concentration at half effect for propofol effect on LOC (µg/mL).

c50rfloat

Concentration at half effect for remifentanil effect on LOC (ng/mL).

gammafloat

Slope coefficient for the LOC model.

betafloat

Interaction coefficient for the LOC model (beta_LOC = 0 signifies an additive interaction, beta_LOC > 0 indicates synergy).

hill_paramlist

Parameters of the model list [c50p_LOC, c50r_LOC, gamma_LOC, beta_LOC]

hill_modelstr

‘Kern’ [Kern2004], considers the synergistic effect of remifentanil (Greco-type). ‘Mertens’ [Mertens2003], considers the synergistic effect of remifentanil (Greco-type). ‘Johnson’ [Johnson2008], considers the synergistic effect of remifentanil (Greco-type).

tsfloat

Sampling time, in s.

References

[Kern2004] (1,2)

S. E. Kern et al. “A response surface analysis of propofol-remifentanil pharmacodynamic interaction in volunteers.” Anesthesiology 100.6 (2004): 1373-1381. doi : 10.1097/00000542-200406000-00007

[Mertens2003] (1,2)

M. J. Mertens et al. “Propofol reduces perioperative remifentanil requirements in a synergistic manner: response surface modeling of perioperative remifentanil–propofol interactions.” Anesthesiology 99.2 (2003): 347-359. doi : 10.1097/00000542-200308000-00016

[Johnson2008] (1,2)

K. B. Johnson et al. “Validation of remifentanil propofol response surfaces for sedation, surrogates of surgical stimulus, and laryngoscopy in patients undergoing surgery.” Anesthesia and analgesia 106.2 (2008): 471. doi : 10.1213/ane.0b013e3181606c62

compute_loc(c_es_propo, c_es_remi)

Compute LOC function (0-1) from propofol and remifentanil effect site concentration.

LOC = 0 means fully awake, LOC = 1 deep LOC

Parameters:

c_es_propofloat

Propofol effect site concentration µg/mL.

c_es_remifloat, optional

Remifentanil effect site concentration ng/mL.

Returns:

LOCfloat

LOC value.

plot_surface()

Plot the 3D-Hill surface of the LOC related to Propofol and Remifentanil effect site concentration

class TOF_model(hill_model: str = 'Weatherley', hill_param: dict | None = None)

Bases: object

Model to link Atracurium effect site concentration to train-of-four (TOF).

The equation is:

\[TOF = \frac{100*C_{50}^\gamma}{C_{50}^\gamma + C_e^\gamma}\]

Parameters:

hill_modelstr, optional

‘Weatherley’ [Weatherley1983] Ignored if hill_param is specified. Default is ‘Weatherley’.

hill_paramdict, optional

Parameters of the model:

• ‘c50’: Half effect concentration (µg/mL).

• ‘gamma’: Stepness of the Hill curve.

If it is not provided default values are used.

Attributes:

c50pfloat

Concentration at half effect for atracurium effect on TOF (µg/mL).

gammafloat

slope coefficient for the Hill curve.

hill_modelstr

‘Weatherley’ [Weatherley1983]

References

[Weatherley1983] (1,2)

B. Weatherley et al., “Pharmacokinetics, Pharmacodynamics and Dose-Response Relationship of Atracurium Administered i.v.” British Journal of Anesthesia, vol. 55, Suppl. 1, pp. 39S-45S, Jan. 1983.

compute_tof(Ce)

Compute TOF from atracurium effect site concentration.

Parameters:

Cefloat

Atracurium effect site concentration (µg/mL).

Returns:

TOFfloat

TOF value.

plot_surface()

Plot the 2D-Hill curve of the train-of-four (TOF) related to Atracurium effect site concentration

class TOL_model(model: str | None = 'Bouillon', model_param: list | None = None, random: bool | None = False, truncated: float | None = None)

Bases: object

Hierarchical model to link drug effect site concentration to Tolerance of Laringoscopy.

The equation are:

\[postopioid = preopioid * \left(1 - \frac{C_{r,es}^{\gamma_r}}{C_{r,es}^{\gamma_r} + (C_{r,50} preopioid)^{\gamma_r}}\right)\]

\[TOL = \frac{C_{p,es}^{\gamma_p}}{C_{p,es}^{\gamma_p} + (C_{p,50} postopioid)^{\gamma_p}}\]

Parameters:

modelstr, optional

Only ‘Bouillon’ is available. Ignored if model_param is specified. The default is ‘Bouillon’.

model_paramlist, optional

Model parameters, model_param = [c50p, c50p, gammaP, gammaR, Preopioid intensity]. The default is None.

randombool, optional

Add uncertainties in the parameters. Ignored if model_param is specified. The default is False.

truncatedfloat, optional

If not None it correspond to the number of standard deviation after which the distribution are truncated for generating uncertain parameters. The default is None.

Attributes:

c50pfloat

Concentration at half effect for propofol effect on BIS (µg/mL).

c50rfloat

Concentration at half effect for remifentanil effect on BIS (ng/mL).

gamma_pfloat

Slope of the Hill function for propofol effect on TOL.

gamma_rfloat

Slope of the Hill function for remifentanil effect on TOL.

pre_intensityfloat

Preopioid intensity.

compute_tol(c_es_propo: float, c_es_remi: float) → float

Return TOL from Propofol and Remifentanil effect site concentration.

Compute the output of the Hirarchical model to predict TOL from Propofol and Remifentanil effect site concentration. TOL = 1 mean very relaxed and will tolerate laryngoscopy while TOL = 0 mean fully awake and will not tolerate it.

Parameters:

cepfloat

Propofol effect site concentration µg/mL.

cerfloat

Remifentanil effect site concentration ng/mL

Returns:

TOLfloat

TOL value.

plot_surface()

Plot the 3D-Hill surface of the BIS related to Propofol and Remifentanil effect site concentration.

fsig(x, c50, gam)