Choindroitinase ABC I-Mediated Enhancement of Oncolytic Virus Spread and Anti Tumor Efficacy: A Mathematical Model

Oncolytic viruses are genetically engineered viruses that are designed to kill cancer cells while doing minimal damage to normal healthy tissue. After being injected into a tumor, they infect cancer cells, multiply inside them, and when a cancer cell is killed they move on to spread and infect other cancer cells. Chondroitinase ABC (Chase-ABC) is a bacterial enzyme that can remove a major glioma ECM component, chondroitin sulfate glycosoamino glycans from proteoglycans without any deleterious effects in vivo. It has been shown that Chase-ABC treatment is able to promote the spread of the viruses, increasing the efficacy of the viral treatment. In this paper we develop a mathematical model to investigate the effect of the Chase-ABC on the treatment of glioma by oncolytic viruses (OV). We show that the model's predictions agree with experimental results for a spherical glioma. We then use the model to test various treatment options in the heterogeneous microenvironment of the brain. The model predicts that separate injections of OV, one into the center of the tumor and another outside the tumor will result in better outcome than if the total injection is outside the tumor. In particular, the injection of the ECM-degrading enzyme (Chase-ABC) on the periphery of the main tumor core need to be administered in an optimal strategy in order to infect and eradicate the infiltrating glioma cells outside the tumor core in addition to proliferative cells in the bulk of tumor core. The model also predicts that the size of tumor satellites and distance between the primary tumor and multifocal/satellite lesions may be an important factor for the efficacy of the viral therapy with Chase treatment.

Burst size of infected cells (b): We take b = 50 virus/cell from Friedman et al. [4].
Secretion rate of ECM degrading enzyme (λ C ): This parameter value is largely unknown. We take λ C = 3.0 × 10 1 mU/(h.g), but also ran several trials ( Figure 5 in the main text) in order to gauge the effect of this parameter. Table 2 in the main text lists the reference values in the model. We define L = 3 mm following experimental setup in [5] and take the characteristic diffusion coefficient D = 1.5 × 10 −5 cm 2 /s so that T = 1.67 h. We determine the reference values for x, y, n, v, E, C as follows:
ECM density (CSPG (E * ), Tumor ECM (ρ * )): Various CSPG concentrations (0-500 µg/ml) and 5 µg/ml laminin were used for the invasive and noninvasive coculture spot assays in a study of the role of CSPG in regulation of glioma invasion and CSPG was shown to be a potent activator of microglia in vivo and to play as a key organizer of the brain tumor microenvironment [20]. High molecular weight Cat-301 CNS CSPG from brain in the density of 1.4 g/ml [21] was shown to have similar properties as aggrecan, the high molecular weight CSPG from cartilage with typical low-buoyant-densities 1.35-1.4g/ml [22]. Using isoforms of one of major CSPG components, version (Intact versicans V1, V2, and amixture of V0 and V1) isolated from calf aorta, bovine spinal cord, and the spent culture medium of the human glioma cell line U251MG, respectively, Dutt et al. investigated the role of versican V0 and V1 in the range of 0-100 µg/ml in regulation of neural crest cell migration. They found that even low levels of version V0/V1 (> 25 µg/ml) can inhibit neural crest stem cell migration [23,24]. Also, total expression of noncleaved isoforms of BEHAB/Brevican was found to be >4-fold higher in human malignant gliomas compared with normal brain tissue [25]. Isolated versican from brain tissue was estimated to be 3mg/100g wet tissue [26]. We take E * = 1.0 mg/cm 3 . Glial HA-binding protein (GHAP), a brain-specific protein mainly localized in white matter, is present in high concentrations in CNS tissues, 8.2 mg/100g in human white matter (wet tissue) [26] compared to the concentration of glial fibrillary acidic protein (GFAP) in buffered extracts of human spinal cord, 3.5 mg/100g [27]. We take ρ * =1.0 mg/cm 3 .
Concentration of Chase-ABC (C * ): High-dose infusions of Chase (2 − 1, 000 U/mL) is necessary in order to get diffusion of Chase-ABC into deep regions of the spinal cord when it is delivered intrathecally because of attendant dilution and overflow beyond the intrathecal space [17]. In a study of delivery of thermostabilized Chase for functional recovery after injury, Lee et al. [17] found that trahalose-assisted Chase in the concentration of 2 U/0.5mL was enough to digest CSPG decorin. In a study of effects of Chase ABC on the morphology of neural precursor cells (NPCs) expanded in spheres, Gu et al. [14] used a wide concentration range of Chase ABC (0.5-50 mU/mL) in addition to 20 ng/ml EGF and 20 ng/ml bFGF. In a study of effect of Chase-ABC on acute and long-lasting changes in CSPG, injected Chase ABC at a concentration of 0.25 U/µl led to significant degradation of ECM in the adult rat brain [12]. The protease-free Chase-ABC in the concentration of 50 U/ml showed digestion of CSPGs around the injection site (leading to low GAG content (1,200-2,000 µg/mg) compared to high peak values (4, 000 µg/mg) with penicillinase-treatment) and promoted axon regeneration [13]. We take C * = 50 mU/ml.
We nondimensionalize the variables and parameters in the partial differential equations (A.1)-(A.9) as follows:t The governing equations in a dimensionless form are ∂C ∂ν = 0, on ∂Ω 0 .
Velocities are defined at cell boundaries while x, y, n, E, ρ, v, C, and p are defined at the cell centers. Let a computational domain be partitioned in Cartesian geometry into a uniform mesh with mesh spacing h. The center of each cell, Ω ij , is located at (x i , y j ) = ((i−0.5)h, (j−0.5)h) for i = 1, · · · , N x and j = 1, · · · , N y . N x and N y are the numbers of cells in x and y-directions, respectively. The cell vertices are located at (x i+ 1 2 , y j+ 1 2 ) = (ih, jh). Let ∆t be a time step and k be a time step index. At the beginning of each time step, given x k , y k , n k , E k , ρ k , v k , C k , and u k , we want to find x k+1 , y k+1 , n k+1 , E k+1 , ρ k+1 , v k+1 , C k+1 , u k+1 , and p k+1 which solve the following temporal discretization of equations (2)-(10): The outline of the main procedures in one time step is: Step 1. Initialize x 0 , y 0 , n 0 , E 0 , ρ 0 , v 0 , C 0 , and u 0 .
Step 2. Solve x k+1 , y k+1 , n k+1 , E k+1 , and ρ k+1 . The resulting finite difference equations (14)-(18) are written out explicitly. They take the form where the advection term, ∇ · (x k u k ), is defined by: where u and w are horizontal and vertical velocity components, respectively. The quantities ∇ · (y k u k ), ∇ · (n k u k ), ∇ · (E k u k ), and ∇ · (ρ k u k ) are computed in a similar manner.
A pointwise Gauss-Seidel relaxation scheme is used as the smoother in the linear geometric and coupled block-implicit multigrid methods.
These complete the one time step.
Program is written in C. Calculations were performed on a Intel Core i3 CPU (3.20 GHz) with 2 GB of RAM.