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Evaluation and Calibration of In Silico Models of Thrombin Generation Using Experimental Data from Healthy and Haemophilic Subjects

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Abstract

The coagulation cascade comprises numerous chemical reactions between many proteins, that finally lead to the formation of a clot to stop bleeding. Many numerical models have attempted to translate understanding of this cascade into mathematical equations that simulate the chain reactions. However, their predictions have not been validated against clinical data stemming from patients. In this paper, we propose an extensive validation of five available models, by comparing in healthy and haemophilic subjects, thrombin generation measured in vitro to thrombin generation predicted by the models in silico. In order to render the models more predictive, we calibrated the models to have an acceptable agreement between the experimental and estimated data. Optimization processes based on genetic algorithms were developed to search for those calibrated kinetic parameters. Our results show that the thrombin generation kinetics are so complex that they cannot be predicted by a unique set of kinetic parameters for all patients: the calibration of only three parameters in a subject-specific way allows reaching good model estimations for different experimental conditions realized on the same patient.

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Abbreviations

CAT:

Calibrated automated thrombogram

CTI:

Corn trypsin inhibitor

CV:

Coefficient of variation

ETP:

Endogenous thrombin potential

GA:

Genetic algorithm

HA:

Haemophilia A

HB:

Haemophilia B

LT:

Lag time

M:

\(\hbox {mol.\,L}^{-1}\)

ODE:

Ordinary differential equation

PPP:

Platelet-poor plasma

ST:

Start tail

TG:

Thrombin generation

ttP:

Time to peak

II:

Prothrombin

IIa:

Thrombin

V:

Factor V

Va:

Activated factor V

VII:

Factor VII

VIIa:

Activated factor VII

VIII:

Factor VIII

VIIIa:

Activated factor VIII (anti-haemophilic A factor)

IX:

Factor IX

IXa:

Activated factor IX (anti-haemophilic B factor)

X:

Factor X

Xa:

Activated factor X

XI:

Factor XI

XIa:

Activated factor XI

XII:

Factor XII

XIIa:

Activated factor XII

AT:

Antithrombin

Fbg:

Fibrinogen

Fbn:

Fibrin

K:

Kallikrein

PC:

Protein C

PK:

Prekallikrein

PS:

Protein S

TF:

Tissue factor

TFPI:

Tissue factor pathway inhibitor

TM:

Thrombomodulin

\(\mathbf{C }\) :

Vector of the concentrations of the chemical species in the plasma

e :

Relative error between model predictions and experimental data

i :

Subject index

k :

Kinetic parameter

\({\tilde{k}}\) :

Normalized kinetic parameter

\(\mathbf{k }\) :

Vector of kinetic parameters

\(\tilde{\mathbf{k }}\) :

Vector of normalized kinetic parameters

\(L_1\) :

Cost function used to search for optimized sets of kinetic parameters

\(L^{exp}_{i,l}\) :

Variance of the experimental TG curve of subject i in the l-th experimental condition

M :

Number of discrete timepoints at which TG was measured

n :

Number of parameters characterizing the TG curves

N :

Number of subjects in a subgroup of the study population

\(N_\mathrm{p}\) :

Number of kinetic parameters in a model

\(N_\mathrm{reactions}\) :

Number of reactions in a model

\(N_\mathrm{s}\) :

Number of subjects included in the study

\(N_\mathrm{var}\) :

Number of variables in a model

\(p_0\) :

Weight parameter

r :

Function of the change in concentration of a chemical species

\(\mathbf{R }\) :

Vector-valued function of the changes in concentrations of all the chemical species introduced in a model

\(R^2\) :

Coefficient of determination

\(t_m\) :

Discrete time instants over which the TG assays run

\(\epsilon \) :

Set of experimental conditions

\(\sigma \) :

Standard deviation of the normalized kinetic parameter \({\tilde{k}}\) over all sets of kinetic parameters in \({\varOmega }\)

\({\varOmega }\) :

Set of vectors of kinetic parameters providing a good accuracy

[a]:

Concentration of protein a

|a|:

Cardinality of set a

\({\bar{a}}\) :

Average of the variable a

\({\tilde{a}}\) :

Normalized value of a

\(_i\) :

... related to subject i

\(_j\) :

... related to the parameter \(p_j\) of the TG curve

\(_l\) :

... related to the lth experimental condition

\(_m\) :

... related to the timepoint m of the TG curve

\(^0\) :

... at time \({t}=0\)

\(^\mathrm{calib}\) :

... used for the calibration process

\(^\mathrm{exp}\) :

... determined experimentally

\(^\mathrm{model}\) :

... as introduced in the original model

\(^{p}\) :

... related to the parameter p of the TG curve

\(^\mathrm{pop}\) :

... uniform over a subgroup of subjects

\(^\mathrm{pred}\) :

... predicted by the model

\(^\mathrm{valid}\) :

... used for the validation process

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Acknowledgements

The authors are grateful to Pfizer for its financial support for the experimental procedures and to Diagnostica Stago for its material support.

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Authors and Affiliations

  1. Mines Saint-Etienne, Univ Lyon, Univ Jean Monnet, INSERM, U 1059 Sainbiose, Centre CIS, F - 42023, Saint-Etienne, France

    Pierre Chelle, Claire Morin & Michel Cournil

  2. INSERM, U1059, SAINBIOSE, F-42000, Saint Etienne, Univ Lyon, Univ Jean Monnet, INSERM, 42023, Saint-Étienne, France

    Aurélie Montmartin, Michèle Piot & Brigitte Tardy-Poncet

Authors
  1. Pierre Chelle
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  2. Claire Morin
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  3. Aurélie Montmartin
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  4. Michèle Piot
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  5. Michel Cournil
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  6. Brigitte Tardy-Poncet
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Corresponding author

Correspondence to Claire Morin.

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Chelle, P., Morin, C., Montmartin, A. et al. Evaluation and Calibration of In Silico Models of Thrombin Generation Using Experimental Data from Healthy and Haemophilic Subjects. Bull Math Biol 80, 1989–2025 (2018). https://doi.org/10.1007/s11538-018-0440-4

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  • DOI: https://doi.org/10.1007/s11538-018-0440-4

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