Mathematical modeling of therapeutic neural stem cell migration in mouse brain with and without brain tumors

Abstract

Neural stem cells (NSCs) offer a potential solution to treating brain tumors. This is because NSCs can circumvent the blood-brain barrier and migrate to areas of damage in the central nervous system, including tumors, stroke, and wound injuries. However, for successful clinical application of NSC treatment, a sufficient number of viable cells must reach the diseased or damaged area(s) in the brain, and evidence suggests that it may be affected by the paths the NSCs take through the brain, as well as the locations of tumors. To study the NSC migration in brain, we develop a mathematical model of therapeutic NSC migration towards brain tumor, that provides a low cost platform to investigate NSC treatment efficacy. Our model is an extension of the model developed in Rockne et al. (PLoS ONE 13, e0199967, 2018) that considers NSC migration in non-tumor bearing naive mouse brain. Here we modify the model in Rockne et al. in three ways: (i) we consider three-dimensional mouse brain geometry, (ii) we add chemotaxis to model the tumor-tropic nature of NSCs into tumor sites, and (iii) we model stochasticity of migration speed and chemosensitivity. The proposed model is used to study migration patterns of NSCs to sites of tumors for different injection strategies, in particular, intranasal and intracerebral delivery. We observe that intracerebral injection results in more NSCs arriving at the tumor site(s), but the relative fraction of NSCs depends on the location of injection relative to the target site(s). On the other hand, intranasal injection results in fewer NSCs at the tumor site, but yields a more even distribution of NSCs within and around the target tumor site(s).

Keywords: LM-NSC008; agent based modeling; glioma; intranasal drug administration; mathematical oncology; neural stem cell therapy.

Figure 1.

Summary diagram of the NSC migration model with chemotaxis.

Figure 2.

NSC distribution comparing different values of grey matter migration speed, d_g, using d_w = 5, d_g = 0.5 (top) and d_w = d_g = 5 (bottom). Note that using d_w = d_g results in scattered NSC with many of them located outside the white matter. When NSCs migrate slower in gray matter, such as d_g = 0.5, they stay mostly in the white matter. Furthermore, NSCs only arrive at the anterior commesure when the gray matter migration speed is large enough, e.g., d_g = 5. The boxplot shows the distance NSCs traveled from injection site, where we observe the NSC migration distance approximately doubles when d_g = 5 compared to d_g = 1.

Figure 3.

Distance NSCs traveled from injection site comparing different values of stochastic migration speed, β_w = 1 (left) and β_w = 4 (right). We observe an increasing number of NSC outliers both sooner and in total as β_w increases, while the median distance decreases.

Figure 4.

Cell trajectories of NSC migration without cancer (left) and with cancer (middle, right). The injection site and cancer is marked with a black star and a red circle, respectively. In the normal case, NSCs migrate along white matter tracts, however, when cancer is present, the cells robustly migrate to the cancer site when the cancer is in the frontal lobe, close to the injection site. In case the tumor further away past the anterior commissure, NSCs travel first along the white matter tract before they get close enough to the cancer site and pick up the chemotaxis signal. The minimal migration path is marked in white line.

Figure 5.

Percentage of NSCs arrived at the tumor site comparing different target locations, frontal lobe (Tumor 1) and further away pass anterior commissure (Tumor 2). As chemosensitivity λ_c increases, the amount of NSC that reaches the cancer site also increases.

Figure 6.

Comparison of intranasal (left) and intracerebral (right) administration of NSCs without the presence of cancer. The white matter tract is marked by blue dots. Selected few trajectories of NSCs (top), the location of NSCs on day 30 (middle), and the percentage of NSCs on white matter tracts (bottom) are shown. NSCs injected by intranasal administration slowly migrate to the center of the brain following the white matter tract in the lower part. On the other hand, NSCs injected in the cerebrum migrate along the white matter tract of the corpus callosum and spread out more easily across the brain. The percentage of intranasal NSCs on white matter tracts trends slowly up to over 16%. In case of intracerebral NSCs, the percentage decays from 70 to 55%, since the cells are injected near the white matter tract, but spread throughout the brain.

Figure 7.

Case 1: Location of NSCs on day 30 (top) and distance NSCs travel from injection site (bottom) starting from intranasal injection site (left) and the intracerebral injection site (right). The NSCs from the intranasal injection spread evenly throughout the simulation. On the other hand, majority of the NSC from the intracerebral injection rapidly migrates towards the cancer site while leaving some outliers near the injection site.

Figure 8.

Case 1: The migration paths of selected NSCs towards the tumor centered at the front side of right putamen (top) and the percentage of NSCs that reach the cancer site (bottom). The cells are injected either intranasally (left) and intracerebrally (right). The red line represents the NSC that traveled the shortest path from its initial position to the cancer site. The shortest path taken by the intracerebrally injected NSCs directly follows the major white matter tract, while intranasally injected NSCs have to navigate through a longer distance. NSCs from the intranasal injection site travel gradually to the cancer site, with about 14% reaching their destination on day 30. Meanwhile, NSCs from the intracerebral injection site swiftly travel to the location of the tumor and a much greater percent arrive at around 90%.

Figure 9.

Case 2: The migration paths of selected NSCs towards the tumor centered at the rear side of the right putamen (top) and the percentage of NSCs that reach the cancer site (bottom). The cells are injected either intranasally (left) and intracerebrally (right). We observe that when injected intranasally, only about 1% ever arrive on day 30 while when injected intracerebrally, this number jumps up to 5% total.

Figure 10.

Case 3: Percentage of NSC that arrive at the cancer site on the left putamen (top) and at the cancer site on the right putamen (bottom) over time (days). The starting points are the intranasal injection site (left) and the intracerebral injection site (right). We can observe that more NSC reaches both cancer sites when injected intracerebrally, which is likely because this injection site is overall closer the cancer sites and begins on white matter. However, the distribution of NSCs among the two cancer site is more uniform when intranasally injected.