A non-linear mathematical model for the in vivo evaluation of the RES phagocytic function

A new non-linear mathematical model was constructed in order to perform in vivo quantification of the RES phagocytic function. This method is based on the same technical facilities as used for the routine liver-spleen scintigraphy with radiocolloids [1, 2]. But kinetic modeling of dynamic Tc-99m-sulfur colloid data produced estimations of the functional RE-parameters: the clearance rate of the colloidal particles, the rate of phagocytosis, and the RES functional volume, which can not be obtained by classical approaches. This non-linear model was designed on the basis of the principal characteristics of particulate material interaction with macrophages (attachment, phagocytosis, digestion) [3, 4, 5]. The theoretically examined behavior of this in vivo mathematical model corresponds with the experimental behavior of the RES. The mathematical expression of the dynamics is the system of non-linear differential equations with constant coefficients that have no analytical solution. Fitting of the normalized heart blood time-activity curve was obtained to identify the unknown model parameters via non-linear regression. For this purpose general interactive PASCAL procedure IDPAR for a PDP-11/34 computer was used (an IBM PC version is also available). Two to three iterations were needed to estimate the set of unknown parameters for any patient study (1-1.5 min). A very good fitting was obtained between experimental and model curves in every case of different pathologies (error of the approximation is about 2-3%). Studies were performed using an in vivo bolus injection of 3.6 mg/80 kg commercially available colloid KOREN labeled with 3m-Ci 99m-Tc (analog of TCK-1). Our method was used to determine the RES functional parameters for patient groups with different levels of the RES dysfunction. Obtained results illustrate the possibilities of our technique to quantitatively estimate not only great pathology (portal cirrhosis), but also small changes of the RE-function (case of hyperlipidemia and ulcer gaster). In all patient groups marked changes of Tc-99m-sulfur colloid turnover were observed. In general, tracer clearance from the circulation was decreased, and the rate of phagocytosis and the RES volume were diminished compared with controls. The effect of a reduction of phagocytosis increases when the RES dysfunction becomes stronger. It can be shown that a non-parametric Wilcoxon-Mann-Whitney test gives a significant difference (P95%) for these patient groups. Further, we represent the possibility of using the model for monitoring changes of the RES-function parameters during and after therapy. The quantitative test of the RES function can significantly enhance the diagnosis and management of different diseases. Serial colloidal studies may document changes in the RES-function for the tumors, cirrhosis, hyperlipidemia, reticulosis, hepatitis, thrombosis, infection, AIDS, burn injury, shock and trauma patients. The technique may be useful for the different RES investigations with laboratory animals. Created computer software can be used as a tool for kinetic models, simulation, and unknown parameters identification.