# A minimal mathematical model of calcium homeostasis

#### Model Status

This CellML model runs in both OpenCell and COR but unfortunately it does not replicate the published results. The units have been checked and they are consistent. Several parameter values were missing from the model description in the original paper. Where a value for a variable is unknown it has been defined as 1.000 in this CellML model (these added values are distinct from any value which is definitely known to be 1 by including 3 decimal places).

#### Model Structure

ABSTRACT: A mathematical model of calcium homeostasis is presented in which the controlling factors are the plasma concentrations of calcium, PTH, and calcitriol, and the effector organs are the parathyroids, bone, kidney, and intestine. Other factors can be added as the need arises. The model is aimed at simulating what happens in a single individual, but its parameters and variables were adjusted to the corresponding published average values. Simulations of published observations in humans undergoing the infusion of calcium or its chelators are presented. With a single exception, these simulations provided a good fit to the data. The response of the system to extrinsic perturbations was characterized by simulating chronic infusions of calcium, PTH, and calcitriol. Finally, the steady state response to perturbations in some of its parameters (the secretory mass of the parathyroids and the affinity and/or sensitivity of the calcium, PTH, and calcitriol receptors) and to renal failure were also investigated in an attempt to analyze the pathogenesis of clinical hypo- or hypercalcemias. In its present form the model cannot be used to base clinical decisions in individual cases. However, it requires modest computational resources, and clinicians with a modest mathematical background can manipulate it. It is a useful tool for the analysis of general mechanisms of the diseases of calcium metabolism and for the design of clinical experiments aimed at characterizing these diseases. The model can also be the core of future autoadaptive extensions to be used in individual patients.

Block diagram of calcium homeostasis. Arrows represent the equations of the model; solid lines are feed forward actions; dashed lines are feedback loops; dotted lines are calcium and phosphate millimole_per_hours; shaded rounded rectangles are pools; open rounded rectangles are target organs; open rectangles are relevant quantities; crossed circle represents an instantaneous difference between phosphate intake and phosphate excretion; starred blocks are pools introduced to produce adjustable delays; triple arrow represents coupling between calcium and phosphate millimole_per_hours in and out of the bone. |

The original paper reference is cited below:

A minimal mathematical model of calcium homeostasis, Raposo JF, Sobrinho LG, Ferreira HG, 2002, *The Journal of Clinical Endocrinology and Metabolism*, 87, 4330-4340. PubMed ID: 12213894