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A Numerical Handling of the Boundary Conditions Imposed by the Skull on an Inhomogeneous Diffusion-Reaction Model of Glioblastoma Invasion Into the Brain: Clinical Validation Aspects

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ABSTRACT

A novel explicit triscale reaction-diffusion numerical model of glioblastoma multiforme tumor growth is presented. The model incorporates the handling of Neumann boundary conditions imposed by the cranium and takes into account both the inhomogeneous nature of human brain and the complexity of the skull geometry. The finite-difference time-domain method is adopted. To demonstrate the workflow of a possible clinical validation procedure, a clinical case/scenario is addressed. A good agreement of the in silico calculated value of the doubling time (ie, the time for tumor volume to double) with the value of the same quantity based on tomographic imaging data has been observed. A theoretical exploration suggests that a rough but still quite informative value of the doubling time may be calculated based on a homogeneous brain model. The model could serve as the main component of a continuous mathematics-based glioblastoma oncosimulator aiming at supporting the clinician in the optimal patient-individualized design of treatment using the patient’s multiscale data and experimenting in silico (ie, on the computer).

No MeSH data available.


Related in: MedlinePlus

Error variation for different values of Δt at 180 simulated days after the start of the simulation by comparing the results from the proposed model with those obtained through the use of exponential growth (golden standard). The initial condition of Figure 9 is assumed. The space step size has been taken equal to 0.1 cm.
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f15-10.1177_1176935116684824: Error variation for different values of Δt at 180 simulated days after the start of the simulation by comparing the results from the proposed model with those obtained through the use of exponential growth (golden standard). The initial condition of Figure 9 is assumed. The space step size has been taken equal to 0.1 cm.

Mentions: The computational error of the numerical method is obtained using equation (31) where c is the concentration calculated using exponential growth and capprox is the numerically calculated concentration. Figure 15 shows the error variation for different values of dt. For all Δt values, the error is essentially the same. It should be noted that the complexity of the brain structures bounded by the skull is very high. This unavoidably leads to spatial discretization errors. The latter can explain an error of approximately 5% after the relatively large time interval of 180 simulated days.


A Numerical Handling of the Boundary Conditions Imposed by the Skull on an Inhomogeneous Diffusion-Reaction Model of Glioblastoma Invasion Into the Brain: Clinical Validation Aspects
Error variation for different values of Δt at 180 simulated days after the start of the simulation by comparing the results from the proposed model with those obtained through the use of exponential growth (golden standard). The initial condition of Figure 9 is assumed. The space step size has been taken equal to 0.1 cm.
© Copyright Policy - open-access
Related In: Results  -  Collection

License 1 - License 2
Show All Figures
getmorefigures.php?uid=PMC5392020&req=5

f15-10.1177_1176935116684824: Error variation for different values of Δt at 180 simulated days after the start of the simulation by comparing the results from the proposed model with those obtained through the use of exponential growth (golden standard). The initial condition of Figure 9 is assumed. The space step size has been taken equal to 0.1 cm.
Mentions: The computational error of the numerical method is obtained using equation (31) where c is the concentration calculated using exponential growth and capprox is the numerically calculated concentration. Figure 15 shows the error variation for different values of dt. For all Δt values, the error is essentially the same. It should be noted that the complexity of the brain structures bounded by the skull is very high. This unavoidably leads to spatial discretization errors. The latter can explain an error of approximately 5% after the relatively large time interval of 180 simulated days.

View Article: PubMed Central - PubMed

ABSTRACT

A novel explicit triscale reaction-diffusion numerical model of glioblastoma multiforme tumor growth is presented. The model incorporates the handling of Neumann boundary conditions imposed by the cranium and takes into account both the inhomogeneous nature of human brain and the complexity of the skull geometry. The finite-difference time-domain method is adopted. To demonstrate the workflow of a possible clinical validation procedure, a clinical case/scenario is addressed. A good agreement of the in silico calculated value of the doubling time (ie, the time for tumor volume to double) with the value of the same quantity based on tomographic imaging data has been observed. A theoretical exploration suggests that a rough but still quite informative value of the doubling time may be calculated based on a homogeneous brain model. The model could serve as the main component of a continuous mathematics-based glioblastoma oncosimulator aiming at supporting the clinician in the optimal patient-individualized design of treatment using the patient’s multiscale data and experimenting in silico (ie, on the computer).

No MeSH data available.


Related in: MedlinePlus