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Phys Biol. Author manuscript; available in PMC 2016 July 1.

Published in final edited form as:

Published online 2015 June 4. doi: 10.1088/1478-3975/12/4/046006

PMCID: PMC4486062

NIHMSID: NIHMS699087

David A. Hormuth, II,^{1,}^{3} Jared A. Weis,^{1,}^{2} Stephanie L. Barnes,^{1,}^{2} Michael I. Miga,^{1,}^{2,}^{3,}^{6} Erin C. Rericha,^{4} Vito Quaranta,^{5} and Thomas E. Yankeelov^{}^{1,}^{2,}^{3,}^{4,}^{5}

Thomas E. Yankeelov: ude.tlibrednav@voleeknay.samoht

Reaction-diffusion models have been widely used to model glioma growth. However, it has not been shown how accurate this model can predict future tumor status using model parameters (i.e., tumor cell diffusion and proliferation) estimated from quantitative *in vivo* imaging data. Towards this end, we used *in silico* studies to develop the methods needed to accurately estimate tumor specific reaction-diffusion model parameters, and then tested the accuracy with which these parameters can predict future growth. The analogous study was then performed in a murine model of glioma growth. The parameter estimation approach was tested using an *in silico* tumor “grown” for ten days as dictated by the reaction-diffusion equation. Parameters were estimated from early time points and used to predict subsequent growth. Prediction accuracy was assessed at global (total volume and Dice value) and local (concordance correlation coefficient, CCC) levels. Guided by the *in silico* study, rats (n=9) with C6 gliomas, imaged with diffusion weighted magnetic resonance imaging (DW-MRI), were used to evaluate the model’s accuracy for predicting *in vivo* tumor growth. The *in silico* study resulted in low global (tumor volume error < 8.8 %, Dice > 0.92) and local (CCC values > 0.80) level errors for predictions up to six days into the future. The *in vivo* study showed higher global (tumor volume error > 11.7%, Dice < 0.81) and higher local (CCC < 0.33) level errors over the same time period. The *in silico* study shows that model parameters can be accurately estimated and used to accurately predict future tumor growth at both the global and local scale. However, the poor predictive accuracy in the experimental study suggests the reaction-diffusion equation is an incomplete description of *in vivo* C6 glioma biology and may require further modeling of intra-tumor interactions including segmentation of (for example) proliferative and necrotic regions.

Mathematical models have been constructed to describe tumor growth and invasion over a large range of spatial scales (nm to cm) and temporal scales (ns to years). Substantial discussions have focused on translating these models to clinical care with the long term goal of providing clinicians with patient-specific predictions of future tumor growth and therapy response in order to optimally select and guide patient therapy [1–3]. Approaches for patient-specific predictions may focus on changes in a single property such as tumor volume, or changes in tumor growth as a function of several related properties (e.g., cellularity, vascularity, nutrient distribution). Models that focus on the change in a single tumor property can be parameterized readily with experimental data [4,5], but may fail to capture spatial and temporal tumor heterogeneity of, for example, cellularity, vasculature density, proliferation rates, and the level of response (or lack thereof) of cells to treatment that is observed within tumors [6,7]. Patient-specific models that capture a tumor’s spatial and temporal heterogeneity could be used to more accurately describe the delivery of treatment and subsequent response [8–11]. Unfortunately, modeling these characteristics frequently requires knowledge of parameters that can only be measured by highly invasive methods or within idealized (*in vitro*) settings [12–15]. The reliance of the existing modeling literature on parameters that are either extraordinarily difficult or impossible to measure non-invasively fundamentally limits their clinical application. Recasting these models in terms of parameters measured *via* non-invasive imaging measurements would dramatically improve the clinical relevance of patient-specific tumor growth predictions [1].

Magnetic resonance imaging (MRI) and positron emission tomography (PET) can be used to provide an array of non-invasive, quantitative, and functional measurements in 3D and at multiple time points of tumor growth. More specifically, MRI and PET can provide measurements of cellularity [16], blood volume [17,18], blood flow [17,18], hypoxia [19], oxygen saturation [20], and metabolism [21]. Additionally, the ability to make repeatable, non-invasive, spatially discretized, quantitative measurements of tumor growth supports the development, testing, and refinement of mathematical descriptions of *in vivo* tumor growth. Several groups [1,5,22–29] have incorporated imaging measurements from MRI, PET, and x-ray computed tomography into mathematical models of tumor growth. Preliminary efforts in both breast [23] and pancreatic [24] cancers, have shown that patient specific imaging data can potentially accurately predict future tumor growth. This, however, has not been demonstrated for gliomas.

One common model for glioma growth is the reaction-diffusion model, whereby the spatio-temporal change in tumor cellularity is due to proliferation and invasion (described by random diffusion) of tumor cells. The proliferation and invasion of cells are typically characterized with a proliferation rate and a diffusion coefficient, respectively. The reaction-diffusion model of glioma growth described by Swanson *et al* [29], uses proliferation and diffusion coefficients of tumor cells estimated from *T _{2}*-weighted and post-contrast

In this work we use a reaction-diffusion model of glioma growth with proliferation and diffusion values estimated from quantitative *in vivo* imaging data to predict future tumor growth and then validate (or refute) that prediction by direct comparison to future *in vivo* measurements. Using an *in silico* tumor we first developed the means to accurately estimate model parameters and assessed the accuracy of tumor growth predictions. We then performed the analogous *in vivo* study, where model parameters were estimated from serial diffusion-weighted MRI data in a murine model of glioma to predict future tumor status which could then be directly compared to experimental outcome. The *in silico* experiments show that model parameters can be accurately estimated from tumor growth datasets and then used to predict future tumor growth with low global and local errors. However, when the approach is applied to *in vivo* glioma measurements, it is shown that the reaction-diffusion model provides poor predictive ability of future tumor growth.

The reaction-diffusion equation describes the spatio-temporal rate of change in tumor cell number and distribution due to the random movement of tumor cells (diffusion; the first term on the right hand side of Eq. (1)), and proliferation (reaction; the second term on the right hand side of Eq. (1)):

$$\frac{\partial N(\overline{x},t)}{\partial t}=\nabla \u2022\left(D(\overline{x})\nabla N(\overline{x},t)\right)+k(\overline{x})N(\overline{x},t)\left(1-\frac{N(\overline{x},t)}{\theta}\right),$$

(1)

where *N* (,*t*) is the number of tumor cells at three-dimensional position and time *t*, *D*() is the tumor cell diffusion coefficient at position , *k* () is the net tumor cell proliferation at position , and *θ* is the tumor cell carrying capacity. Note that the proliferation term varies temporally as a function of cell density, *N* (,*t*), although it is assumed that the proliferation rate, *k* (), is temporally constant. As described below, quantitative DW-MRI provides estimates of *N* (,*t*). These data are obtained at multiple time points, early in the tumor’s life cycle, and used to solve an inverse problem using Eq. (1) to return estimates of *k* () for each voxel within the tumor, and two *D*() values: one for white matter (*D _{wm}*) and one for gray matter (

Figures 1 and and22 show the approach for the *in silico* experiments. An initial distribution of tumor cells, *N* (,*t*_{0}), was seeded within a rat brain domain. A spatial map of *k* was determined from DW-MRI estimates of *N* (,*t*) from a C6 glioma bearing rat using Eq. (2):

$$k(\overline{x})=-log\left(N(\overline{x},{t}_{1})/N(\overline{x},{t}_{0})\right)/({t}_{1}-{t}_{0}),$$

(2)

where *N* (,*t*_{0}) and *N* (, *t*_{1}) represent the distribution of cells at time *t _{0}* and

Figure 2 shows the modeling approach for predicting future tumor cell distributions. For each set of optimized parameters (*P _{1}*,

All experimental procedures were approved by Vanderbilt University’s Institutional Animal Care and Use Committee. Female Wistar rats (n = 9, 236–263 g) were anesthetized, given analgesics, and inoculated with C6 glioma cells (1 × 10^{5}) *via* stereotaxic injection. During each MRI procedure body temperature was maintained near 37° C by a flow of warm air directed over the animal and respiration was monitored using a pneumatic pillow. Each rat was anesthetized using 2% isoflurane in 98% oxygen for all surgical and imaging procedures. Rats were imaged beginning 10 days post-surgery (defined as day 0). Rats were imaged up to 10 days after the first imaging time point. The first three imaging measurements for all rats occurred on days 0, 2, and 4. Rats 1–3 were then imaged on days 5, 8, and 10. Rats 4–5 were imaged on days 5, 6, and 9. Rat 6 was imaged on days 5, 6, 8 and 10, while rats 7–8 were imaged only on days 5 and 6. Rat 9 was only imaged at one additional time on day 5.

MRI was performed on a 9.4 T horizontal-bore magnet (Agilent, Santa Clara, CA, USA). The animal’s head was positioned in a 38 mm diameter Litz quadrature coil (Doty Scientific, Columbia, SC, USA) and was secured by a bite bar. All MR images were sampled with a 128 × 128 × 16 matrix acquired over a 32 × 32 × 16 mm^{3} field of view. In order to facilitate the modeling, the imaging volumes obtained at time points two through the end of the experiment were registered to the first time point *via* a mutual information based rigid registration algorithm performed at the scanner [31]; this ensures that the image volumes obtained at each time point are very nearly identical (See Supplementary Figures 1–3 for example registration results). *T _{1}* map was produced using data from an inversion-recovery snapshot experiment with

DW-MRI was acquired using a pulsed fast spin echo diffusion sequence in three orthogonal diffusion encoding directions with *b*-values of 0, 300, 500, 700, 900, and 1100 s/mm^{2}, and Δ/δ = 25 ms/2 ms. The apparent diffusion coefficient (ADC) was estimated on a voxel basis using a two parameter fit of the DW-MRI data [32]. To determine *N* (,*t*), the ADC values from the DW-MRI data are then transformed to estimate cell number [33,34] using Eq. (3):

$$N(\overline{x},t)=\theta \left(\frac{{\mathit{ADC}}_{w}-\mathit{ADC}(\overline{x},t)}{{\mathit{ADC}}_{w}-{\mathit{ADC}}_{\mathit{min}}}\right),$$

(3)

where *θ* represents the tumor cell carrying capacity, *ADC _{w}* is the ADC of free water at 37° C (2.5 × 10

Tumor regions-of-interest (ROI) were manually placed at each time point using the *T _{1}* maps. ADC measurements within these ROI’s were then transformed to tumor cell number using Eq. (3).

Similar to the *in silico* experiments, three sets of model parameters (*P _{1}*,

The three different time point combinations from the *in vivo* data sets were also fit to a model substituting a spatially invariant proliferation rate (*k _{ROI}*) for the spatially variant proliferation rate (

A Levenberg-Marquardt weighted least squares nonlinear optimization, implemented with a regularization parameter described in Joachimowicz *et al* [37,38], was used to estimate model parameters (*P _{1}*,

$$\sum _{t={t}_{i}}^{{t}_{f}}\left({\left(\sum _{\overline{x}=1}^{{\overline{x}}_{i}=n}(N(\overline{x},t))\right)}^{-1}\xb7\left(\sum _{\overline{x}=1}^{\overline{x}=n}{({N}_{\mathit{est}}\phantom{\rule{0.16667em}{0ex}}(\overline{x},t)-N(\overline{x},t))}^{2}\right)\right),$$

(4)

where *t _{i}* is the initial time point,

To assess the effect of noise in ADC measurements on estimates of *P*, *in silico* parameter optimization was repeated (N = 100) for each set of parameters (*P _{1}*,

After optimization of *P*, these values were used in a FD implementation of Eq. (1), initialized with the tumor cell distribution at day 4 (*N* (,*t*_{4})), to “grow” a predicted tumor from day 4 thru 10 (600 iterations). Throughout the FD simulation, as the tumor expanded into regions where an estimate of *k* was unavailable, *k* was assigned using a local average of available non-zero *k’s* within a 3 × 3 × 3 kernel.

The accuracy of *P* estimated from the *in silico* dataset was evaluated by computing the percent error between the true, *P _{true}*, and the estimated parameter sets (

Illustrative results of the *in silico* experiments are shown Figures 3 – 5 and summarized in Table 1. Figure 3 shows the true distribution of *k* (panel (a)) used to “grow” the *in silico* tumor, the estimated values of *k* (panels (b – d)), and plots of the true voxel values of *k* against the estimated values of *k* (panels (e – g)). Panels (b) and (e) show the results from parameters estimated from days 0 and 4. Both a low level of agreement (CCC = 0.38) and a weak linear relationship (PCC = 0.54) is observed between the true *k* and the *k* estimated using the *P _{1}* data sets (

Figure 4 shows the true (panel (a)) and the predicted tumor cell distributions (top rows in panels (b) – (d)) and the percent difference between the true and predicted distributions (bottom rows in panels (b) – (d)). Panel (b) shows the predicted *N* at day 5 using *P _{1}*,

The results of the ROI and voxel level analysis are shown in Figure 5. Error in tumor volume generally increases the further out in time a prediction is made (panel (a)). Percent error in tumor volume ranged from 11.9 – 36.4% for *P _{1}* based predictions, 3.1 – 13.6% for

Table 1 shows the mean percent error between the true values and estimated values of *k*, *D _{wm}*, and

The results of the *in vivo* experiments are shown in Figures 6 – 8 and Tables 2–3. Figure 6 shows the PCC (a) and CCC (b) analysis for all nine rats, as well as *k _{P1}*,

Figure 7 shows the true (panel (a)) and the predicted tumor cell distributions (top rows in panels (b – d)) and the percent difference between the true and predicted distributions (bottom rows in panels (b – d)) for rat 1. The left column in panel (a) shows *T _{2}*-weighted images with a white box indicating the simulation domain and a black outline around the tumor. Panel (b) shows estimated

Figure 8 presents the global and local level error analysis for the *in vivo* experiments. Panels (a–c) show the results when a spatially variant *k* (i.e., *k* ()) is estimated and panels (d–f) show the results for the spatially invariant estimated *k* (i.e., *k _{ROI}*). For the spatially variant

The average diffusion estimates and the nRMS error for both the spatially variant and invariant *k* models are shown in Table 2. For the spatially variant *k* model fits, the mean *D _{wm}* ranged from 1.49 × 10

Table 3 shows the average proliferation rate for each rat from the spatially variant (*k* (), where
$\overline{k(\overline{x})}$ is the average voxel-wise *k* estimated within the tumor) and the estimated spatially invariant proliferation rate (*k _{ROI}*).
$\overline{k(\overline{x})}$ ranged from 0.51 to 4.06 day

The results of the *in silico* experiments indicate that the parameters within the reaction-diffusion equation (i.e., *D _{wm}*,

The results of the *in silico* study show, the strength of both the estimated parameters and the forward evaluation algorithm, as exhibited in the high level of overlap of tumor volumes (Dice values greater than 0.83 for *P _{1}*,

The *in vivo* experiments demonstrated greater error at both the global and local level compared to the *in silico* experiments. The increased error suggests that the reaction-diffusion equation is an incomplete description of C6 biology. The overestimation of tumor volume estimates suggests that overall tumor growth properties are changing between the estimation time points and the prediction time points. At the global level, the expansion of the tumor may be less restricted at earlier time points compared to later time points. At the local level, these changes may be the result of an increase or decrease in proliferation due to changes in the viability of the cells within a particular voxel. The very poor CCCs (less than 0.33, Figure 8) similarly suggest that the reaction-diffusion equation provides a poor description of local properties. The reaction-diffusion equation does, however, provide tumor growth predictions that co-localize (Dice values greater than 0.62) with the true tumor volumes. Different from the *in silico* study, *P _{1}* predictions had lower percent error in tumor volume (less than 34.4%) compared to

There are several limitations in this current approach. One limitation is the assumption that all tumor cells within a given voxel (spatially variant *k*) or within the tumor (spatially invariant *k*) follow the same proliferation rules. Within tumors there may be groups of actively proliferating cells as well as cells that are quiescent or necrotic [42]. In particular, necrotic tissues (which can be relatively large compared to total tumor volume) can strongly influence tumor growth [42] and patient prognosis [43,44]. Models incorporating different proliferation rules have the potential to more accurately describe *in vivo* proliferation [45,46]; however, it is challenging to initialize these models using non-invasive measurements. Although the proliferation model in this approach is limited, the spatially variant *k* lessens the potential error (relative to *k _{ROI}*) in this assumption by discretizing the tumor into individual regions that can have a proliferation rate that more closely captures local behavior. A second limitation is that proliferation rates,

In conclusion, parameters can be accurately estimated and used to predict future tumor growth with low error at the global and local levels, provided that the tumor’s growth is described by the reaction-diffusion equation. However, the *in vivo* experiments suggest that the reaction-diffusion model consisting of just tumor cell diffusion and logistic growth described by Eq. (1) provides an incomplete description of tumor growth and must be amended to provide better descriptions of *in vivo* C6 glioma growth in rats.

The authors thank Zou Yue for performing the animal surgeries. We thank the National Institutes of Health for funding through NCI R01CA138599, NCI R21CA169387, NCI U01CA174706, NCI R25CA092043, and the Vanderbilt-Ingram Cancer Center Support Grant (NIH P30CA68485). We thank the Kleberg Foundation for the generous support of our institute’s imaging program. ECR holds a Career Award from the BWF. This work was partially supported by a pilot project from the Physical Sciences in Oncology Center at the H. Lee Moffitt Cancer Center and Research Institute (PIs: Robert Gatenby, M.D., Robert Gillies, Ph.D.).

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