TY  - JOUR
AU  - Farias, Matheus dos Santos
PY  - 2026
DA  - 2026/10/16
TI  - A Brain Uptake Constant for Mathematical Modeling of Cerebral Radiopharmaceutical Uptake: A Synthetic Computational Study
JO  - ICCK Journal of Applied Mathematics
T2  - ICCK Journal of Applied Mathematics
JF  - ICCK Journal of Applied Mathematics
VL  - 2
IS  - 4
SP  - 293
EP  - 301
DO  - 10.62762/JAM.2026.965290
UR  - https://www.icck.org/article/abs/JAM.2026.965290
KW  - Brain Uptake Constant
KW  - radiopharmaceutical kinetics
KW  - mathematical modeling
KW  - synthetic simulation
KW  - PET
KW  - SPECT
AB  - Compartmental and tracer-kinetic models are widely used to describe radiopharmaceutical uptake in brain PET and SPECT, but in simple analytical models the fraction of transferred activity that reaches the brain is seldom isolated as a parameter of its own. This study defines a dimensionless Brain Uptake Constant, $\psi$, as the uptake fraction in a two-stage first-order model of cerebral radiopharmaceutical activity and examines its behaviour in a purely synthetic setting. Activity in the systemic pool, $A(t)$, is lost at a first-order rate $k$; cerebral activity, $A_c(t)$, receives a source term $\psi k A(t)$ and is lost at an effective decay rate $\lambda$. The closed-form solution was evaluated directly for $A_0 = 100$~MBq, $k = 0.25$~min$^{-1}$, $\lambda = 0.05$~min$^{-1}$ and $\psi = 1.00$, 0.75, 0.50 and 0.25 over 0-60~min. Deterministic Gaussian fields were used to show the spatial effect of $\psi$ as surface plots and heat maps. Each curve rises, passes through a single maximum at $t_{\mathrm{peak}} = 8.047$~min and then decays. The peak time does not depend on $\psi$, whereas the peak activity (66.87~MBq for $\psi = 1$) and the 0-60~min time-integrated activity scale linearly with $\psi$. In the spatial fields, $\psi$ changes the amplitude but not the prescribed Gaussian widths. No clinical, PET or SPECT data were used, so the results say nothing about clinical validity. The model is offered as a simple parameterization that still has to be tested against measured dynamic imaging data with independently estimated kinetic parameters.
SN  - 3068-5656
PB  - Institute of Central Computation and Knowledge
LA  - English
ER  - 

@article{Farias2026A,
  author = {Matheus dos Santos Farias},
  title = {A Brain Uptake Constant for Mathematical Modeling of Cerebral Radiopharmaceutical Uptake: A Synthetic Computational Study},
  journal = {ICCK Journal of Applied Mathematics},
  year = {2026},
  volume = {2},
  number = {4},
  pages = {293-301},
  doi = {10.62762/JAM.2026.965290},
  url = {https://www.icck.org/article/abs/JAM.2026.965290},
  abstract = {Compartmental and tracer-kinetic models are widely used to describe radiopharmaceutical uptake in brain PET and SPECT, but in simple analytical models the fraction of transferred activity that reaches the brain is seldom isolated as a parameter of its own. This study defines a dimensionless Brain Uptake Constant, \$\psi\$, as the uptake fraction in a two-stage first-order model of cerebral radiopharmaceutical activity and examines its behaviour in a purely synthetic setting. Activity in the systemic pool, \$A(t)\$, is lost at a first-order rate \$k\$; cerebral activity, \$A\_c(t)\$, receives a source term \$\psi k A(t)\$ and is lost at an effective decay rate \$\lambda\$. The closed-form solution was evaluated directly for \$A\_0 = 100\$~MBq, \$k = 0.25\$~min\$^{-1}\$, \$\lambda = 0.05\$~min\$^{-1}\$ and \$\psi = 1.00\$, 0.75, 0.50 and 0.25 over 0-60~min. Deterministic Gaussian fields were used to show the spatial effect of \$\psi\$ as surface plots and heat maps. Each curve rises, passes through a single maximum at \$t\_{\mathrm{peak}} = 8.047\$~min and then decays. The peak time does not depend on \$\psi\$, whereas the peak activity (66.87~MBq for \$\psi = 1\$) and the 0-60~min time-integrated activity scale linearly with \$\psi\$. In the spatial fields, \$\psi\$ changes the amplitude but not the prescribed Gaussian widths. No clinical, PET or SPECT data were used, so the results say nothing about clinical validity. The model is offered as a simple parameterization that still has to be tested against measured dynamic imaging data with independently estimated kinetic parameters.},
  keywords = {Brain Uptake Constant, radiopharmaceutical kinetics, mathematical modeling, synthetic simulation, PET, SPECT},
  issn = {3068-5656},
  publisher = {Institute of Central Computation and Knowledge}
}