An Integrated Computational/Experimental Model of Lymphoma Growth

<div><p>Non-Hodgkin's lymphoma is a disseminated, highly malignant cancer, with resistance to drug treatment based on molecular- and tissue-scale characteristics that are intricately linked. A critical element of molecular resistance has been traced to the loss of functionality in proteins such as the tumor suppressor <i>p53</i>. We investigate the tissue-scale physiologic effects of this loss by integrating <i>in vivo</i> and immunohistological data with computational modeling to study the spatiotemporal physical dynamics of lymphoma growth. We compare between drug-sensitive <i>Eμ-myc Arf-/-</i> and drug-resistant <i>Eμ-myc p53-/-</i> lymphoma cell tumors grown in live mice. Initial values for the model parameters are obtained in part by extracting values from the cellular-scale from whole-tumor histological staining of the tumor-infiltrated inguinal lymph node <i>in vivo</i>. We compare model-predicted tumor growth with that observed from intravital microscopy and macroscopic imaging <i>in vivo</i>, finding that the model is able to accurately predict lymphoma growth. A critical physical mechanism underlying drug-resistant phenotypes may be that the <i>Eμ-myc p53-/-</i> cells seem to pack more closely within the tumor than the <i>Eμ-myc Arf-/-</i> cells, thus possibly exacerbating diffusion gradients of oxygen, leading to cell quiescence and hence resistance to cell-cycle specific drugs. Tighter cell packing could also maintain steeper gradients of drug and lead to insufficient toxicity. The transport phenomena within the lymphoma may thus contribute in nontrivial, complex ways to the difference in drug sensitivity between <i>Eμ-myc Arf-/-</i> and <i>Eμ-myc p53-/-</i> tumors, beyond what might be solely expected from loss of functionality at the molecular scale. We conclude that computational modeling tightly integrated with experimental data gives insight into the dynamics of Non-Hodgkin's lymphoma and provides a platform to generate confirmable predictions of tumor growth.</p> </div

Vasculature and angiogenesis in the lymph node tumor.

<p>Observations in living mice using intravital microscopy (A, B, C: red – functional blood vessels; shown for <i>Eμ-myc p53-/-</i> tumor) provide information to qualitatively compare the vessel formation (D, E, F: red – highest flow; white – lowest; dots indicate vessel points of origin from pre-existing vasculature (not shown)) in the computational model (calibrated from other data, see <b><a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003008#pcbi.1003008.s004" target="_blank">Text S1</a></b>). The modeling of diffusion of cell substrates (e.g., oxygen and cell nutrients) within the tumor enables prediction of the spatial distribution of lymphoma cells (inset, shown for one vessel cross-section; brown: highest concentration of cells; white: lowest concentration of cells) as their viability is modulated by access to the oxygen and nutrients diffusing from the vasculature into the surrounding tissue (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003008#pcbi.1003008.e002" target="_blank"><b>Eq. 2</b></a>).</p

Representation of the lymph node by the computational model.

<p>(A) Diagram highlighting a typical lymph node structure. (B) Simulation output from the model showing an incipient tumor (dark red) forming in the center of the node. Afferent lymphatic vessels are collectively represented as one incoming tube on the top, and the efferent vessel is at the bottom. (C) The simulated distribution of oxygen (brown color) released by the blood vasculature within the node remains uniform at this initial stage.</p

Schematic showing integrated computational/experimental modeling strategy involving both cell- and tumor-scale measurements.

<p>(<b>A</b>) Functional relationships involving cell-scale parameters such as proliferation (Ki-67), apoptosis (Caspase-3), and hypoxia (HIF-1α) are defined based on experimental observations, e.g., from immunohistochemistry the density of viable tissue as a function of vascularization is shown in the third panel (red: highest density; yellow: lowest; blue: vessels). These functional relationships as well as parameter values measured experimentally are then used as input to the model to create simulations of lymphoma growth. A sample simulated tumor cross-section showing vascularized viable tissue (highest density in red, lowest in yellow, with vessel cross-sections as small blue dots) is shown at the far right. (<b>B</b>) Lymphoma observations regarding size, morphology, and vasculature from macroscopic imaging of an inguinal lymph node in live mice provide part of the tumor-scale information to validate the model simulations. Note the pre-existing vasculature in the lymph node (in the center of each frame) from which oxygen and nutrients are supplied to the tissue. For comparison, a control group of lymph nodes in animals without tumors is also shown.</p

Scheme to obtain the cellular-scale experimental data.

<p>Lymphomas (shown as large orange sphere) were grown <i>in vivo</i> by tail vein injection of either drug-sensitive <i>Eμ-myc/Arf-/-</i> or drug-resistant <i>Eμ-myc/p53-/-</i> lymphoma cells. The inguinal lymph node tumor was excised, fixed, and sliced for histology sections (5 µm apart) every 100 µm along the tumor. A total of five sets (S1 through S5) of histology sections were obtained (for simplicity, the figure only shows three sets). The sections in each set were stained for cell viability (H&E), hypoxia (HIF-1α), proliferation (Ki-67), apoptosis (Caspase-3), and vascularization (CD-31).</p

Lymphoma tumor cell viability.

<p>Viability per area was measured along the five sets (S1 through S5) of histology sections for <i>Eμ-myc Arf-/-</i> (black) and <i>Eμ-myc p53-/-</i> (gray) tumors. All error bars represent standard deviation from at least n = 3 measurements in each section. Asterisks show level of statistical significance determined by Student's t-test with α = 0.05 (one asterisk, p<0.05; two asterisks, p<0.01).</p

Algorithm flowchart.

<p>Refer to <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003008#s2" target="_blank"><b>Materials and Methods</b></a> and <b><a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003008#pcbi.1003008.s004" target="_blank">Text S1</a></b> for equations. Using the cellular-scale data, we measured values for proliferation and apoptosis for both drug-sensitive and drug-resistant tumors and calculated corresponding values for the model mitosis and apoptosis parameters <i>λ</i>_M and <i>λ</i>_A. We solved <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003008#pcbi.1003008.e002" target="_blank">Eq. (2)</a> for the local levels of cell substrates <i>n</i> at each time step of simulation of tumor growth. The parameters were input into <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003008#pcbi.1003008.e010" target="_blank">Eq. (3)</a> to numerically calculate the source mass terms <i>S_i</i>, which were then used in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003008#pcbi.1003008.e001" target="_blank">Eq. (1)</a> to compute the volume fractions of viable <i>ρ</i>_V and <i>ρ</i>_D dead tissue. These fractions were used in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003008#pcbi.1003008.e011" target="_blank">Eq. (4)</a> to obtain the tumor tissue growth velocity.</p

Prediction of lymphoma growth based on the calibrated model parameters.

<p>Simulated mean tumor diameter (solid red line) bounded by variation in the measured oxygen diffusion distance (dashed red lines) falls within the range of values measured for the tumor growth observed <i>in vivo</i> (denoted by the triangles and squares with vertical error bars). Note that the simulated growth is the same for both <i>Eμ-myc Arf-/-</i> and <i>Eμ-myc p53-/-</i> tumors.</p

Lymphoma tumor characteristics.

<p>Histological measurements are shown <i>Eμ-myc Arf-/-</i> (black) and <i>Eμ-myc p53-/-</i> (gray) tumors along the five sets of sections (S1 through S5) of the lymphoma: (A) Endothelial cells per area; (B) hypoxic cells per area; (C) proliferating cells per area; (D) apoptotic cells per area. Sections S1 and S5 are at the tumor top and bottom, respectively, while the other sections are in the interior with S3 being in the middle. Dashes in panels (A) and (C) indicate that no data was obtained; in panel (C), no proliferation was detected for <i>Eμ-myc p53-/-</i> cells in sets S4 and S5, and none for <i>Eμ-myc Arf-/-</i> in set S5, probably due to sample defects. All error bars represent standard deviation from at least n = 3 measurements in each section; asterisk indicates statistical significance (p<0.05) determined by Student's t-test with α = 0.05. The data shows that for <i>Eμ-myc p53-/-</i> there is higher vascularization in the center, higher hypoxic density on the periphery, and higher overall apoptotic density compared to <i>Eμ-myc Arf-/-</i>.</p

Theory and Experimental Validation of a Spatio-temporal Model of Chemotherapy Transport to Enhance Tumor Cell Kill

<div><p>It has been hypothesized that continuously releasing drug molecules into the tumor over an extended period of time may significantly improve the chemotherapeutic efficacy by overcoming physical transport limitations of conventional bolus drug treatment. In this paper, we present a generalized space- and time-dependent mathematical model of drug transport and drug-cell interactions to quantitatively formulate this hypothesis. Model parameters describe: perfusion and tissue architecture (blood volume fraction and blood vessel radius); diffusion penetration distance of drug (i.e., a function of tissue compactness and drug uptake rates by tumor cells); and cell death rates (as function of history of drug uptake). We performed preliminary testing and validation of the mathematical model using <i>in vivo</i> experiments with different drug delivery methods on a breast cancer mouse model. Experimental data demonstrated a 3-fold increase in response using nano-vectored drug <i>vs</i>. free drug delivery, in excellent quantitative agreement with the model predictions. Our model results implicate that therapeutically targeting blood volume fraction, e.g., through vascular normalization, would achieve a better outcome due to enhanced drug delivery.</p><p>Author Summary</p><p>Cancer treatment efficacy can be significantly enhanced through the elution of drug from nano-carriers that can temporarily stay in the tumor vasculature. Here we present a relatively simple yet powerful mathematical model that accounts for both spatial and temporal heterogeneities of drug dosing to help explain, examine, and prove this concept. We find that the delivery of systemic chemotherapy through a certain form of nano-carriers would have enhanced tumor kill by a factor of 2 to 4 over the standard therapy that the patients actually received. We also find that targeting blood volume fraction (a parameter of the model) through vascular normalization can achieve more effective drug delivery and tumor kill. More importantly, this model only requires a limited number of parameters which can all be readily assessed from standard clinical diagnostic measurements (e.g., histopathology and CT). This addresses an important challenge in current translational research and justifies further development of the model towards clinical translation.</p></div