Single-cell functional analysis of parathyroid adenomas reveals distinct classes of calcium sensing behaviour in primary hyperparathyroidism.

Primary hyperparathyroidism (PHPT) is a common endocrine neoplastic disorder caused by a failure of calcium sensing secondary to tumour development in one or more of the parathyroid glands. Parathyroid adenomas are comprised of distinct cellular subpopulations of variable clonal status that exhibit differing degrees of calcium responsiveness. To gain a clearer understanding of the relationship among cellular identity, tumour composition and clinical biochemistry in PHPT, we developed a novel single cell platform for quantitative evaluation of calcium sensing behaviour in freshly resected human parathyroid tumour cells. Live-cell intracellular calcium flux was visualized through Fluo-4-AM epifluorescence, followed by in situ immunofluorescence detection of the calcium sensing receptor (CASR), a central component in the extracellular calcium signalling pathway. The reactivity of individual parathyroid tumour cells to extracellular calcium stimulus was highly variable, with discrete kinetic response patterns observed both between and among parathyroid tumour samples. CASR abundance was not an obligate determinant of calcium responsiveness. Calcium EC50 values from a series of parathyroid adenomas revealed that the tumours segregated into two distinct categories. One group manifested a mean EC50 of 2.40 mM (95% CI: 2.37-2.41), closely aligned to the established normal range. The second group was less responsive to calcium stimulus, with a mean EC50 of 3.61 mM (95% CI: 3.45-3.95). This binary distribution indicates the existence of a previously unappreciated biochemical sub-classification of PHPT tumours, possibly reflecting distinct etiological mechanisms. Recognition of quantitative differences in calcium sensing could have important implications for the clinical management of PHPT.

Solitons in Exciton-Polariton System: Reduced Modelling, Analysis, and Numerical Studies

In this dissertation, we study the behavior of exciton-polariton quasiparticles in semiconductor microcavities, under the sourceless and lossless conditions.

First, we simplify the original model by removing the photon dispersion term, thus effectively turn the PDEs system to an ODEs system,

and investigate the behavior of the resulting system, including the equilibrium points and the wave functions of the excitons and the photons.

Second, we add the dispersion term for the excitons to the original model and prove that the band of the discontinuous solitons now become dark solitons.

Third, we employ the Strang-splitting method to our sytem of PDEs and prove the first-order and second-order error bounds in the $H^1$ norm and the $L_2$ norm, respectively.

Using this numerical result, we analyze the stability of the steady state bright soliton solution. This solution revolves around the $x$-axis as time progresses

and the perturbed soliton also rotates around the $x$-axis and tracks closely in terms of amplitude but lags behind the exact one. Our numerical result shows orbital

stability but no $L_2$ stability.