Three-Dimensional Simulation of Neutron Flux Distribution in Boron Neutron Capture Therapy (BNCT)

TUMELERO, F.; WEYMAR, G. J.; PETERSEN, C. Z.; CAURIO JR, J. L. M.

doi:10.5540/tcam.2026.027.e01903

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ORIGINAL ARTICLE • Trends Comput. Appl. Math. 27 • 2026 • https://doi.org/10.5540/tcam.2026.027.e01903 linkcopy

Three-Dimensional Simulation of Neutron Flux Distribution in Boron Neutron Capture Therapy (BNCT)

Authorship SCIMAGO INSTITUTIONS RANKINGS

Glioblastoma Multiforme (GBM) is one of the most aggressive and difficult-to-treat forms of malignant brain tumor, presenting a high incidence and resistance to conventional treatment methods. Boron Neutron Capture Therapy (BNCT) stands out as an innovative and promising approach for treating complex tumors such as GBM, as it enables the selective destruction of tumor cells with minimal impact on healthy tissues. In this study, the multigroup neutron diffusion equation is solved in a three-dimensional domain using four energy groups. The neutron source is represented as a boundary condition, and after diagonalizing the system of equations, the method of Separation of Variables can be applied to obtain an closed-form eigenfunction expansion. The validation of the proposed approach was carried out through numerical simulations in a water phantom, whose results indicated that thermal and epithermal neutron fluxes are predominant.

Keywords:
BNCT; neutron flux; neutron diffusion equation; closed-form eigenfunction expansion; method of separation of variables

Authorship

person F. TUMELERO *

schoolUniversidade Federal do Rio Grande, Instituto de Matemática, Estatística e Física, RS, Brazil - E-mail: fernandatumelero@furg.brUniversidade Federal do Rio GrandeBrazilRS, BrazilUniversidade Federal do Rio Grande, Instituto de Matemática, Estatística e Física, RS, Brazil - E-mail: fernandatumelero@furg.br

0000-0001-8905-7860

person G. J. WEYMAR

schoolUniversidade Federal do Rio Grande, Instituto de Matemática, Estatística e Física, RS, Brazil - E-mail: guilhermejahnecke@gmail.comUniversidade Federal do Rio GrandeBrazilRS, BrazilUniversidade Federal do Rio Grande, Instituto de Matemática, Estatística e Física, RS, Brazil - E-mail: guilhermejahnecke@gmail.com

0000-0001-8216-9122

person C. Z. PETERSEN

schoolUniversidade Federal de Pelotas, Instituto de Física e Matemática, RS, Brazil - E-mail: claudio.petersen@ufpel.edu.brUniversidade Federal de PelotasBrazilRS, BrazilUniversidade Federal de Pelotas, Instituto de Física e Matemática, RS, Brazil - E-mail: claudio.petersen@ufpel.edu.br

0000-0002-4720-6888

person J. L. M. CAURIO JR

schoolUniversidade Federal de Pelotas, Instituto de Física e Matemática, RS, Brazil - E-mail: juniorcaurio@gmail.comUniversidade Federal de PelotasBrazilRS, BrazilUniversidade Federal de Pelotas, Instituto de Física e Matemática, RS, Brazil - E-mail: juniorcaurio@gmail.com

0009-0008-1878-7354

person*Corresponding author: Fernanda Tumelero - E-mail:
fernandatumelero@furg.br

Associate editor:

Mustapha Rachidi

SCIMAGO INSTITUTIONS RANKINGS

Universidade Federal do Rio Grande, Instituto de Matemática, Estatística e Física, RS, Brazil - E-mail: fernandatumelero@furg.brUniversidade Federal do Rio GrandeBrazilRS, BrazilUniversidade Federal do Rio Grande, Instituto de Matemática, Estatística e Física, RS, Brazil - E-mail: fernandatumelero@furg.br

Universidade Federal do Rio Grande, Instituto de Matemática, Estatística e Física, RS, Brazil - E-mail: guilhermejahnecke@gmail.comUniversidade Federal do Rio GrandeBrazilRS, BrazilUniversidade Federal do Rio Grande, Instituto de Matemática, Estatística e Física, RS, Brazil - E-mail: guilhermejahnecke@gmail.com

Universidade Federal de Pelotas, Instituto de Física e Matemática, RS, Brazil - E-mail: claudio.petersen@ufpel.edu.brUniversidade Federal de PelotasBrazilRS, BrazilUniversidade Federal de Pelotas, Instituto de Física e Matemática, RS, Brazil - E-mail: claudio.petersen@ufpel.edu.br

Universidade Federal de Pelotas, Instituto de Física e Matemática, RS, Brazil - E-mail: juniorcaurio@gmail.comUniversidade Federal de PelotasBrazilRS, BrazilUniversidade Federal de Pelotas, Instituto de Física e Matemática, RS, Brazil - E-mail: juniorcaurio@gmail.com

Figures | Tables | Formulas

Figures (7)
Tables (1)
Formulas (34)

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Figure 1: Boron capture reaction.

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Figure 2: Cross-sectional view of the water phantom.

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Figure 3: Flowchart of the computational procedure adopted to solve the three-dimensional multi-group neutron diffusion equation.

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Figure 4: Flux distribution as a function of depth x in the phantom, in the middle of the y and z grid.

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Figure 5: Heat maps of the variables ϕ ₁, ϕ ₂, ϕ ₃, and ϕ ₄ in the plane at the midpoint of the mesh in the z-direction.

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Figure 6: Contribution of the series terms (l, m) to the solution ϕ _g for each energy group, evaluated at the point (x, y, z) = (0.2, 1.5, 1.5).

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Figure 7: Convergence of the truncated modal expansion for each energy group, evaluated through the relative change in the neutron flux between successive truncation levels (L = M).

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Table 1: Parameters used in the numerical simulation. Source: Adapted from ¹¹^{), (}¹².

(2.1)

(2.2)

(2.3)

(2.4)

imageFigure 1: Boron capture reaction. open_in_new

Source: the authors.

imageFigure 2: Cross-sectional view of the water phantom. open_in_new

Source: the authors.

imageFigure 3: Flowchart of the computational procedure adopted to solve the three-dimensional multi-group neutron diffusion equation. open_in_new

Source: the authors.

imageFigure 4: Flux distribution as a function of depth x in the phantom, in the middle of the y and z grid. open_in_new

Source: the authors.

imageFigure 5: Heat maps of the variables ϕ ₁, ϕ ₂, ϕ ₃, and ϕ ₄ in the plane at the midpoint of the mesh in the z-direction. open_in_new

Source: the authors.

imageFigure 6: Contribution of the series terms (l, m) to the solution ϕ _g for each energy group, evaluated at the point (x, y, z) = (0.2, 1.5, 1.5). open_in_new

imageFigure 7: Convergence of the truncated modal expansion for each energy group, evaluated through the relative change in the neutron flux between successive truncation levels (L = M). open_in_new

table_chartTable 1: Parameters used in the numerical simulation. Source: Adapted from ¹¹^{), (}¹².

g	D_g [cm]	Σ_R,g [1/cm]	Σ_s,g′−g [1/cm]	Σ_s,g′−g [1/cm]
g	D_g [cm]	Σ_R,g [1/cm]	(g-1→g)	(g-2→g)
1	2.0 × 10⁻¹	1.572 × 10⁻¹	-	-
2	2.5 × 10⁻¹	1.1 × 10⁻¹	6 × 10⁻²	-
3	2.2 × 10⁻¹	9.7 × 10⁻²	6.1 × 10⁻²	-
4	2.1 × 10⁻²	2.2 × 10⁻³	0	5.228 × 10⁻²

- D_{g} \nabla^{2} Φ_{g} + Σ_{R, g} Φ_{g} = \sum_{g' - 1}^{g - 1} Σ_{s, g' \to g} Φ_{g'} + S_{g},

(2.1)

- D_{g} \nabla^{2} Φ_{g} + Σ_{R, g} Φ_{g} = \sum_{g' - 1}^{g - 1} Σ_{s, g' \to g} Φ_{g'} .

(2.2)

- D \nabla^{2} Φ + Σ Φ = \bar{0},

D = [\begin{matrix} D_{1} & 0 & 0 & 0 \\ 0 & D_{2} & 0 & 0 \\ 0 & 0 & D_{3} & 0 \\ 0 & 0 & 0 & D_{4} \end{matrix}], Σ = [\begin{matrix} Σ_{R, 1} & 0 & 0 & 0 \\ - Σ_{s, 1 \to 2} & Σ_{R, 2} & 0 & 0 \\ - Σ_{s, 1 \to 3} & - Σ_{s, 2 \to 3} & Σ_{R, 3} & 0 \\ - Σ_{s, 1 \to 4} & - Σ_{s, 2 \to 4} & - Σ_{s, 3 \to 4} & Σ_{R, 4} \end{matrix}],

Λ = [\begin{matrix} λ_{1} & 0 & 0 & 0 \\ 0 & λ_{2} & 0 & 0 \\ 0 & 0 & λ_{3} & 0 \\ 0 & 0 & 0 & λ_{4} \end{matrix}] = [\begin{matrix} Σ_{R, 1} & 0 & 0 & 0 \\ 0 & Σ_{R, 2} & 0 & 0 \\ 0 & 0 & Σ_{R, 3} & 0 \\ 0 & 0 & 0 & Σ R, 4 \end{matrix}] .

- D \nabla^{2} Ψ + Λ Ψ = \bar{0},

- D_{1} \nabla^{2} Ψ_{1} + λ_{1} Ψ_{1} = 0, - D_{2} \nabla^{2} Ψ_{2} + λ_{2} Ψ_{2} = 0, - D_{3} \nabla^{2} Ψ_{3} + λ_{3} Ψ_{3} = 0, - D_{4} \nabla^{2} Ψ_{4} + λ_{4} Ψ_{4} = 0,

Ψ_{g} (x, y, z) = X (x) Y (y) Z (z) .

- D_{g} \nabla^{2} [X (x) Y (y) Z (z)] + λ_{i} X (x) Y (y) Z (z) = 0 .

Y (y) Z (z) \frac{d^{2} X (x)}{d x^{2}} + X (x) Z (z) \frac{d^{2} Y (y)}{d y^{2}} + X (x) Y (y) \frac{d^{2} Z (z)}{d z^{2}} - \frac{λ_{i}}{D_{g}} X (x) Y (y) Z (z) = 0 .

\frac{1}{X (x)} \frac{d^{2} X (x)}{d x^{2}} + \frac{1}{Y (y)} \frac{d^{2} Y (y)}{d y^{2}} + \frac{1}{Z (z)} \frac{d Z (z)}{d z^{2}} = \frac{λ_{i}}{D_{g}} .

\frac{1}{Z (z)} \frac{d^{2} Z (z)}{d z^{2}} = σ_{1} .

\frac{1}{X (x)} \frac{d^{2} X (x)}{d x^{2}} + \frac{1}{Y (y)} \frac{d^{2} Y (y)}{d y^{2}} + σ_{1} = γ_{g} .

\frac{1}{X (x)} \frac{d^{2} X (x)}{d x^{2}} + (σ_{1} - γ_{g}) = - \frac{1}{Y (y)} \frac{d^{2} Y (y)}{d y^{2}} = σ_{2} .

\frac{d^{2} X (x)}{d x^{2}} + (σ_{1} - γ_{g} - σ_{2}) X (x) = 0, \frac{d^{2} Y (y)}{d y^{2}} + σ_{2} Y (y) = 0, \frac{d^{2} Z (z)}{d z^{2}} + σ_{1} Z (z) = 0 .

Ψ_{g} = \sum_{m = 1}^{\infty} \sum_{l = 1}^{\infty} A_{l, m} [- e^{2 ξ_{3, l, m^{L_{x}}}} e^{ξ_{3, l, m^{x}}} + e^{- ξ_{3, l, m^{x}}}] \sin (ξ_{2, m} y) \sin (ξ_{1, l} z),

P^{- 1} f = \sum_{m = 1}^{\infty} \sum_{l = 1}^{\infty} A_{l, m} [- e^{2 ξ_{3, l, m} L_{x}} + 1] \sin (ξ_{2, m} y) \sin (ξ_{1, l z}) .

A_{l, m} = \frac{\int_{0}^{L_{y}} \int_{0}^{L_{z}} P^{- 1} f \sin (ξ_{2, m} y) \sin (ξ_{1, l} z) d z d y}{[- e^{- 2 ξ_{3, l, m} L_{x}} + 1] \int_{0}^{L_{y}} \sin^{2} (ξ_{2, m} y) d y \int_{0}^{L_{z}} \sin^{2} (ξ_{1, l} z) d z} .

- D_{g} \nabla^{2} Ψ_{g} + λ_{i} Ψ_{g} = 0,

Ψ_{g}^{(L, M)} (x, y, z) = \sum_{m = 1}^{M} \sum_{l = 1}^{L} A_{l, m}^{(g)} [- e^{- 2 ξ_{3, l, m}^{(g)} L_{x}} e^{ξ_{3, l, m}^{(g)} x} + e^{- ξ_{3, l, m}^{(g)} x}] \sin (ξ_{2, m} y) \sin (ξ_{1, l} z),

\frac{\partial^{2}}{\partial x^{2}} [- e^{- 2 ξ_{3, l, m}^{(g)} L_{x}} e^{ξ_{3, l, m}^{(g)} x} + e^{ξ_{3, l, m}^{(g)} x}] = {(ξ_{3, l m}^{(g)})}^{2} [- e^{- 2 ξ_{3, l, m}^{(g)} L_{x}} e^{ξ_{3, l, m}^{(g)} x} + e^{ξ_{3, l, m}^{(g)} x}] .

\frac{\partial^{2}}{\partial y^{2}} \sin (ξ_{2, m} y) = - ξ_{2, m}^{2} \sin (ξ_{2, m} y), \frac{\partial^{2}}{\partial z^{2}} \sin (ξ_{1, l} z) = - ξ_{1, l}^{2} \sin (ξ_{1, l} z) .

{(ξ_{3, l, m}^{(g)})}^{2} - ξ_{2, m}^{2} - ξ_{1, l}^{2} .

{(ξ_{3, l, m}^{(g)})}^{2} - ξ_{2, m}^{2} - ξ_{1, l}^{2} = γ_{g} .

\nabla^{2} Ψ_{g}^{(L, M)} = γ_{g} Ψ_{g}^{(L, M)} .

R_{Ψ, g}^{(L, M)} (x, y, z) = - D_{g} \nabla^{2} Ψ_{g}^{(L, M)} (x, y, z) + λ_{i} Ψ_{g}^{(L, M)} (x, y, z),

(2.3)

R_{Ψ, g}^{(L, M)} (x, y, z) = - (D_{g} γ_{g} + λ_{i}) Ψ_{g}^{(L, M)} (x, y, z) .

R_{Ψ, g}^{(L, M)} (x, y, z) = 0

{\tilde{ε}}_{R_{g}}^{(L, M)} = \frac{{||R_{Ψ, g}^{(L, M)}||}_{L^{2} (Ω)}}{{||Ψ_{g}^{(L, M)}||}_{L^{2} (Ω)}}, g = 1, . . ., G .

{\tilde{ε}}_{R}^{(L, M)} = \frac{{(\sum_{g = 1}^{G} {||R_{Ψ, g}^{(L, M)}||}_{L^{2} (Ω)}^{2})}^{1 / 2}}{{(\sum_{g = 1}^{G} {||Ψ_{g}^{(L, M)}||}_{L^{2} (Ω)}^{2})}^{1 / 2}} .

(2.4)

ε_{Φ}^{(L, M)} = \frac{{||Φ^{(L, M)} - Φ^{(L - 1, M - 1)}||}_{L^{2} (Ω)}}{{||Φ^{(L, M)}||}_{L^{2} (Ω)}},

{||Φ^{(L, M)}||}_{L^{2} (Ω)} = {(\sum_{g = 1}^{G} \int_{Ω} {|Φ_{g}^{(L, M)} (x, y, z)|}^{2} d Ω)}^{1 / 2} .

ε_{Φ_{g}}^{(L, M)} = \frac{{||Φ_{g}^{(L, M)} - Φ_{g}^{(L - 1, M - 1)}||}_{L^{2} (Ω)}}{{||Φ_{g}^{(L, M)}||}_{L^{2} (Ω)}}, g = 1, . . ., G .

ε_{Φ_{g}}^{(L, M)} < t o l_{Φ} and {\tilde{ε}}_{R_{g}}^{(L, M)} < t o l_{R},

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