مدلسازی ریاضی واکنش ایمنی در برابر گلیوم مغزی: تجزیه و تحلیل حساسیت و نفوذ تاخیر

In the paper we considered a model of immune reaction against malignant glioma. The model proposed by Kronik et al. (Cancer Immunol. Immunother., 2008) describes simplified interactions between tumour cells and five components of the immune system. We studied the effects of uncertainties of the parameters values to the system behaviour. We showed that the tumour growth rate is one of the most important parameters only in case of fast growing tumours, that is for GBM in our case. On the basis of the performed sensitivity analysis we proposed a reduced model in which the role of time delays in loops appearing in the described interactions is considered. The proposed model includes only two main components of the reaction, that is tumour cells and cytotoxic T-lymphocytes. It occurs that although the reduced system is described by several non-linear terms with three time delays, its dynamics is simple and time delays have hardly any influence on it. Both considered models confirmed that the non-linearities present in interactions between tumour cells and CTLs play a major role in the system dynamics, while other components or delays can be taken into account as supplementary elements only.

According to the American Cancer Society report, [1], it was estimated that about 22,340 new cases of brain and other nervous system cancers have been expected to be detected in the US alone during 2011. Moreover, an estimated 13,110 deaths caused by brain and other nervous system cancers should occur, [1]. One could say that it is not such a big number concerning all estimated new cases, i.e. over 1.5 million, [1]. However, the brain is the organ one needs to be very careful with while planing the treatment. Standard radiotherapy, chemotherapy and surgery treatment have major limitations due to the cancer’s genomic instability, heterogeneity, and their locations beyond the blood–brain barrier. Within brain cancers, malignant glioma (MG) is one of the most dangerous. It is a highly aggressive cancer since its five-year relative survival rates are very low comparing with other cancers. According to [2] the median survival for the high grade MG varies from 1 year for grade IV to 3–5 years for grade III. In this paper we follow the ideas of modelling of immune reaction against the MG presented in [3]. The original mathematical model, proposed by Kronik et al. in [3] and later generalised in [4] and [5], consists of six ordinary differential equations which describe the dynamics of the main components of the analysed processes, that is tumour cells (T(t))(T(t)), cytotoxic T-lymphocytes CTLs (C(t))(C(t)), two cytokines: transforming growth factor beta 1 TGF-View the MathML sourceβ(Fβ) and interferon-gamma INF-View the MathML sourceγ(Fγ), and also the molecules of major histocompatibility complex MHC of class I (MI)(MI) and II (MII)(MII). The key two players, they are tumour cells and CTLs, interact with each other through complicated pathways encompassing other considered components, compare Fig. 1. Activated cytotoxic T-lymphocyte can annihilate a tumour cell only if it recognised the latter by identifying modified surface molecule MHC class I. Before that event, activated CTL needs to reach the tumour site starting from the thymus, where it is generated and trained to recognise specific changes in MHC class I molecules. On its way the blood–brain barrier stands which separates circulating blood and the brain extracellular fluid. Unfortunately, transforming growth factor TGF-ββ, which is actively secreted by tumour cells, decreases its permeability, and hence it decreases the amount of CTLs that can reach the tumour site. TGF-ββ also has the negative influence on the expression of MHC II molecules and the efficacy of tumour cell lysis by a CTL. MHC class II molecules are necessary to trigger the immune response, as they are recognised by T helper cells which are responsible for the activation and growth of CTLs. To add to this complexity, activated CTLs secrete IFN-γγ, which induces MHC class I and II expression on the surface of tumour cells and antigen-presenting cells. Full-size image (29 K) Fig. 1. Scheme of interactions between the components of the human immune system in the brain as described in [3]. The dashed and solid lines indicate the negative and positive influence, respectively. By “treatment” we mean the injection of additional CTLs.

In this paper we considered a modification of the model of immune reaction against malignant glioma proposed in [3] that describes the interactions between tumour cells and five components of the immune system. We reproduced and extended the basic sensitivity analysis, performed in [3], where the authors checked how the variation in a single parameter value would influence the success of the tested treatment protocol. Our results suggest that the treatment in case of the MG grade IV tumour is most sensitive to the parameters rr and μCμC, while for the MG grade III the treatment is most sensitive to parameters hThT and aTaT. Interestingly, parameters that are crucial for the treatment’s outcome are different for both tumour types. Next, to extend sensitivity analysis we used the FAST procedure to ascertain the effect of parameters. It appeared that for tumour cell number, that the majority of variations for both the MGs grade III and IV tumour is caused by the uncertainty in: CTLs decay rate, maximal efficiency of CTL, accessibility of the tumour cells to CTL and maximal influence of FβFβ on tumour growth, respectively. Our analysis also indicates that the variation in the tumour growth rate rr has a huge influence on the variation in tumour size only in case of the MG grade IV tumour. Hence, we conclude that the suggested treatment is sensitive to the tumour growth rate only in the case of fast spreading diseases. Following the suggestion that came out from sensitivity analysis, using quasi-stationary approximation we reduced system (1.1) obtaining the system of two delay differential equations (3.2) for which we investigated the role of time delays in loops appearing in the described interactions. The mathematical analysis of the reduced system was presented, including the existence, uniqueness and non-negativity of solution as well as the existence of steady states depending on the value of the treatment function. Moreover, we studied the stability of steady states and the possibility of stability switches. We proved that the qualitative behaviour of the reduced model (with parameters given by (3.3)) is the same for both grades of the MGs, however the quantitative results strongly differs, compare Fig. 11 and Fig. 12. The same system behaviour for was observed for system (1.1) considered in [3], see Fig. 9, Fig. 10, Fig. 11 and Fig. 12. In [5], for the full system (1.1), it was proved that there exists a global compact attractor. Moreover, for the specific functions, and parameter values estimated in [3] and [5], it was shown that solutions tend to one of the existing steady states. The first simplification of model (1.1) considered in [4] was based on quasi-steady approximation for TGF-ββ dynamics. In this case the full system (1.1) was reduced to the system of five equations, and in the most simplified version considered by [4] it was assumed that the natural recruitment of CTLs is very small comparing to the external infusion during the treatment, such that it can be neglected. In the simplified model presented in [44] it was assumed that since the total amount of MHC class II receptors on the surface of APCs (antigen presenting cells) is usually sufficient to bind tumour antigen [3] the concentration of MHC class II receptors is constant. Although some components of the immune reaction can be assumed much faster than the tumour doubling time, one could over-simplify the model neglecting the time needed for the feedback loops action. Hence, combining time delays (describing the length of feedback loops actions) with the quasi-stationary procedure (as in [4]), the system of differential equations was obtained. However, it differs from the system considered in this paper. For the system considered in [44] the mathematical properties of the model were investigated. Moreover, in it was shown that for the biologically relevant values of parameters, the influence of delays on the model dynamics is negligible. In case of system (3.2) proposed in this paper we observe the similar behaviour. Clearly, due to the delays present in the model one could expect destabilisation of the stable (for case without delay) steady state. However, this is not the case and we observe that one of the steady states attracts the solutions. The analysis of both reduced models confirms that the non-linearities present in the interactions between tumour cells and CTLs play a major role in the system dynamics, while other components or delays can be taken into account as supplementary elements only. We conclude that delays have no significant influence on the reduced models’ dynamics, they have similar and simple dynamics, as studied in [5] and [4]. What is interesting, for all the models (the original and the reduced ones) we observe similar qualitative behaviour.