Khajanchi2019 - Stability Analysis of a Mathematical Model forGlioma-Immune Interaction under OptimalTherapy

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Model Identifier
BIOMD0000000891
Short description 
Stability Analysis of a Mathematical Model for Glioma-Immune Interaction under Optimal Therapy
Subhas Khajanchi

Abstract

We investigate a mathematical model using a system of coupled ordinary differential equations, which describes the interplay of malignant glioma cells, macrophages, glioma specific CD8+T cells and the immunotherapeutic drug Adoptive Cellular Immunotherapy (ACI). To better understand under what circumstances the glioma cells can be eliminated, we employ the theory of optimal control. We investigate the dynamics of the system by observing biologically feasible equilibrium points and their stability analysis before administration of the external therapy ACI. We solve an optimal control problem with an objective functional which minimizes the glioma cell burden as well as the side effects of the treatment. We characterize our optimal control in terms of the solutions to the optimality system, in which the state system coupled with the adjoint system. Our model simulation demonstrates that the strength of treatment u1(t) plays an important role to eliminate the glioma cells. Finally, we derive an optimal treatment strategy and then solve it numerically.

Keywords: malignant gliomas; stability analysis; optimal control; adoptive cellular immunotherapy
Format
SBML (L2V4)
Related Publication
  • Stability Analysis of a Mathematical Model for Glioma-Immune Interaction under Optimal Therapy
  • Subhas Khajanchi
  • International Journal of Nonlinear Sciences and Numerical Simulation , 3/ 2019 , Volume 20 , Issue 3-4 , DOI: 10.1515/ijnsns-2017-0206
  • Department of Mathematics, Presidency University
  • We investigate a mathematical model using a system of coupled ordinary differential equations, which describes the interplay of malignant glioma cells, macrophages, glioma specific CD8+T cells and the immunotherapeutic drug Adoptive Cellular Immunotherapy (ACI). To better understand under what circumstances the glioma cells can be eliminated, we employ the theory of optimal control. We investigate the dynamics of the system by observing biologically feasible equilibrium points and their stability analysis before administration of the external therapy ACI. We solve an optimal control problem with an objective functional which minimizes the glioma cell burden as well as the side effects of the treatment. We characterize our optimal control in terms of the solutions to the optimality system, in which the state system coupled with the adjoint system. Our model simulation demonstrates that the strength of treatment u1(t) plays an important role to eliminate the glioma cells. Finally, we derive an optimal treatment strategy and then solve it numerically.
Contributors
Submitter of the first revision: Mohammad Umer Sharif Shohan
Submitter of this revision: Mohammad Umer Sharif Shohan
Modellers: Mohammad Umer Sharif Shohan

Metadata information

Curation status
Curated
Modelling approach(es)
ordinary differential equation model
Tags
Immuno-oncology

Connected external resources

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Model files
Khajanchi2019.xml SBML L2V4 Representation of Khajanchi2019 - Stability Analysis of a Mathematical Model forGlioma-Immune Interaction under OptimalTherapy 32.92 KB Preview | Download

Additional files
Khajanchi2019.cps COPASI version 4.24 (Build 197) Khajanchi2019 - Stability Analysis of a Mathematical Model forGlioma-Immune Interaction under OptimalTherapy 58.68 KB Preview | Download
Khajanchi2019.sedml SEDML L1V2 Khajanchi2019 - Stability Analysis of a Mathematical Model forGlioma-Immune Interaction under OptimalTherapy 2.65 KB Preview | Download

  • Model originally submitted by : Mohammad Umer Sharif Shohan
  • Submitted: Dec 13, 2019 11:18:56 AM
  • Last Modified: Jan 2, 2020 11:03:52 AM
Revisions
  • Version: 4 public model Download this version
    • Submitted on: Jan 2, 2020 11:03:52 AM
    • Submitted by: Mohammad Umer Sharif Shohan
    • With comment: SBML edited to add isDerivedFrom
  • Version: 3 public model Download this version
    • Submitted on: Dec 13, 2019 11:18:56 AM
    • Submitted by: Mohammad Umer Sharif Shohan
    • With comment: Automatically added model identifier BIOMD0000000891

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Legends
: Variable used inside SBML models


Species
Species Initial Concentration/Amount
u

glioma cell 		0.1 mmol
w

T-lymphocyte 		0.2 mmol
v

macrophage 		0.55 mmol
Reactions
Reactions Rate Parameters
=> u compartment*r_1*u*(1-u) r_1 = 0.4822
w => ; u compartment*(mu_1*w+alpha_4*u*w/(u+k_4)) k_4 = 0.378918; alpha_4 = 0.01694; mu_1 = 0.0074
v => ; u compartment*alpha_3*u*v/(u+k_2) alpha_3 = 0.0194; k_2 = 0.030584
=> v compartment*r_2*v*(1-v) r_2 = 0.3307
u => ; v, w compartment*(alpha_1*v+alpha_2*w)/(u+k_1)*u k_1 = 0.90305; alpha_1 = 0.069943; alpha_2 = 2.74492
=> w; u compartment*gamma_1*u*w/(k_3+u) gamma_1 = 0.1245; k_3 = 2.8743
Curator's comment:
(added: 13 Dec 2019, 11:18:47, updated: 13 Dec 2019, 11:18:47)
The model has been encoded in COPASI 4.24 (Build 197) and the figure 4 from the publication has been reproduced using COPASI