We integrate in vivo lymphoma data with computational modeling to develop a basic model of Non-Hodgkin's lymphoma. Through this work we seek a deeper quantitative understanding of the dynamics of lymphoma growth in the inguinal lymph node and associated physical transport barriers to effective treatment. We obtain histology data by very fine sectioning across whole lymph node tumors, thus providing detailed three-dimensional lymphoma information. We develop a computational model that is calibrated from these cell-scale data and show that the model can independently predict the tissue-scale tumor size observed in vivo without fitting to the data. We further show that this approach can shed insight into the tumor progression within the node, particularly regarding the physical reasons why some tumors might be resistant to drug treatment – a critical consideration when attempting to quantify and predict the treatment response. We envision that the modeling and functional relationships derived in this study could contribute with further development to patient-specific predictors of lymphoma growth and drug response.
Although the number of mice used for the experimental in vivo
validation is limited, the model results are consistent with previous work. For example, a well-studied mechanism of physiological resistance is the dependence of cancer cell sensitivity to many chemotherapeutic agents on the proliferative state of the cell 
. This physical mechanism is likely important in the difference in drug-sensitivity between the tumors formed from the two cell lines and will be explored in further studies. We found that the Eμ-myc Arf-/-
cells tend to congregate at the periphery of the tumor (), even though there are vessels in the interior of the tumor. This suggests the hypothesis that the more drug-sensitive Eμ-myc Arf-/-
cells maintain better oxygenation at the expense of higher drug sensitivity by growing less compactly in the interior of the tumor – where there would be stronger competition for oxygen and cell nutrients – whereas the Eμ-myc p53-/-
lymphoma cells may enhance their survival by closer packing in the core of the tumor. Cell packing density may present a barrier to effective drug penetration 
, which we have also modeled previously 
. Closer packing could further increase the number of cells that would be quiescent due to depletion of oxygen and nutrients, as we specify in the model (Materials and Methods
) and as we have simulated in previous work 
. However, the proportion of chemoresistance inherent with Eμ-myc p53-/-
that can be attributed to resistance at the genetic level compared to what can be attributed to suboptimal drug delivery and quiescence is unclear. In follow-up work we plan to measure drug amounts near various cells in order to begin answering this question, and to perform sensitivity analyses of the IC50 of each cell line with the computational model. This would provide a (model-generated) measure of how much of an effect suboptimal delivery could be attributed to resistance as compared to genetic effects (as measured by IC50).
Lymphoma cells are known to retain cell-cell adhesion, with strength associated with the lymphoma's originating cell type (B- or T-cell) 
. Mechanisms of cell packing related to drug resistance may include weaker cell adhesion in Eμ-myc Arf-/-
than in Eμ-myc p53-/-
leading to higher cell density as well as a denser extra-cellular matrix in the latter 
. Loss of ARF has been linked to increased cancer cell migration and invasion, and hence weaker cell-cell adhesion 
, associated with the binding of ARF to the transcriptional corepressor CtBP2 and promoting CtBP2 degradation 
Perhaps surprisingly, the experimental data indicate minimal presence of hypoxia within the tumor (). This may be due to the fact that lymphoma cells may associate with other cells including stromal cells in the tumor, and the consequent cytokine stimulation (e.g., IL-7) may also trigger proliferation 
. We note that the oxygen diffusion length estimate is subject to variation, as calculated to be directly proportional to the hypoxic distances observed from the IHC; this may be improved by directly measuring the diffusing substances, e.g., oxygen. The simulated elastic tumor boundary may also introduce some variation into the size calculation. Nevertheless, even taking these variations into account, the model-calculated average ratio of apoptosis to proliferation, established from cell-scale measurements, implies that the tumor sizes fall within the range of the sizes estimated from the diameter measured with calipers in vivo
. The hypothesis we test with the model by successful comparison to the experimental data is that the growth and eventual slowdown of these tumors is the balance of proliferation and death, which we have also previously observed for ductal carcinoma in situ 
. Experimental evidence using bioluminescence imaging of living mice 
demonstrates that lymphoma cells seed the tumor in the inguinal lymph node from other sites (e.g., spleen and bone marrow) in the mouse body at earlier times during the tumor growth. The model results are robust, however, because the tumor size by Day 21 predicted by the theory is independent of the earlier times; any influx of cells only provides an initial (transient) condition.
The staining also shows that apoptosis seems highest for drug-sensitive cells at the periphery of the tumor (Sections S1 and S5) compared to the center (Section S3) (both p-values
0.04 using a Student's t-test with α
0.05), and for drug-resistant cells it is highest in the more central regions (). In accordance with biological observations 
, the model hypothesizes that increased hypoxia may lead to higher cell quiescence and hence drug resistance. In the experiments, angiogenesis is higher in the central regions, and is more pronounced for drug-resistant cells, suggesting that these cells are in a more angiogenic environment as a result of ongoing hypoxic stimulus. Higher tumor cell density around blood vessels suggests a functional relationship of cell viability as a function of nutrients, as we have implemented in the model (see Materials and Methods
). However, apoptosis may not necessarily be driven solely by hypoxia, since lymphoma cells are known to have a cellular turnover rate that is on the order of days 
. We further note that angiogenesis is not necessarily triggered only by hypoxia. Lymphoma as well as stromal cells (such as tumor associated macrophages) may produce factors promoting angiogenesis (e.g., vascular endothelial growth factor or VEGF) under otherwise normoxic conditions.
The present work calibrates a computational model of lymphoma with experimental data from drug-sensitive and drug-resistant tumors. This data was derived from detailed IHC analysis of whole tumors, and validation of the model was performed via intravital microscopy measurements. The results suggest that differences in spatial localization of cells and vasculature, as well as in the transport phenomena in the tumor microenvironment may play a nontrivial role in the tumor behavior. This suggests that the genetic differences (Eμ-myc Arf-/- and Eμ-myc p53-/-) may provide a substantial compensation mechanism for these phenomena at the tissue scale in addition to the molecular as it relates to their drug resistance. We plan to verify this hypothesis in the future by assessing model predictions for therapeutic response of drug-sensitive and drug-resistant tumors in terms of cellular parameters such as proliferation, apoptosis, and hypoxia via both IHC and intravital microscopy.