David A. Hormuth

Clinically relevant modeling of tumor growth and treatment response.

Noninvasive imaging technologies can help create patient-specific mathematical models to predict tumor growth. Current mathematical models of tumor growth are limited in their clinical application because they require input data that are nearly impossible to obtain with sufficient spatial resolution in patients even at a single time point—for example, extent of vascularization, immune infiltrate, ratio of tumor-to-normal cells, or extracellular matrix status. Here we propose the use of emerging, quantitative tumor imaging methods to initialize a new generation of predictive models. In the near future, these models could be able to forecast clinical outputs, such as overall response to treatment and time to progression, which will provide opportunities for guided intervention and improved patient care.

A mechanically coupled reaction-diffusion model that incorporates intra-tumoural heterogeneity to predict in vivo glioma growth

While gliomas have been extensively modelled with a reaction–diffusion (RD) type equation it is most likely an oversimplification. In this study, three mathematical models of glioma growth are developed and systematically investigated to establish a framework for accurate prediction of changes in tumour volume as well as intra-tumoural heterogeneity. Tumour cell movement was described by coupling movement to tissue stress, leading to a mechanically coupled (MC) RD model. Intra-tumour heterogeneity was described by including a voxel-specific carrying capacity (CC) to the RD model. The MC and CC models were also combined in a third model. To evaluate these models, rats (n = 14) with C6 gliomas were imaged with diffusion-weighted magnetic resonance imaging over 10 days to estimate tumour cellularity. Model parameters were estimated from the first three imaging time points and then used to predict tumour growth at the remaining time points which were then directly compared to experimental data. The results in this work demonstrate that mechanical–biological effects are a necessary component of brain tissue tumour modelling efforts. The results are suggestive that a variable tissue carrying capacity is a needed model component to capture tumour heterogeneity. Lastly, the results advocate the need for additional effort towards capturing tumour-to-tissue infiltration.

Selection and validation of predictive models of radiation effects on tumor growth based on noninvasive imaging data

The use of mathematical and computational models for reliable predictions of tumor growth and decline in living organisms is one of the foremost challenges in modern predictive science, as it must cope with uncertainties in observational data, model selection, model parameters, and model inadequacy, all for very complex physical and biological systems. In this paper, large classes of parametric models of tumor growth in vascular tissue are discussed including models for radiation therapy. Observational data is obtained from MRI of a murine model of glioma and observed over a period of about three weeks, with X-ray radiation administered 14.5 days into the experimental program. Parametric models of tumor proliferation and decline are presented based on the balance laws of continuum mixture theory, particularly mass balance, and from accepted biological hypotheses on tumor growth. Among these are new model classes that include characterizations of effects of radiation and simple models of mechanical deformation of tumors. The Occam Plausibility Algorithm (OPAL) is implemented to provide a Bayesian statistical calibration of the model classes, 39 models in all, as well as the determination of the most plausible models in these classes relative to the observational data, and to assess model inadequacy through statistical validation processes. Discussions of the numerical analysis of finite element approximations of the system of stochastic, nonlinear partial differential equations characterizing the model classes, as well as the sampling algorithms for Monte Carlo and Markov chain Monte Carlo (MCMC) methods employed in solving the forward stochastic problem, and in computing posterior distributions of parameters and model plausibilities are provided. The results of the analyses described suggest that the general framework developed can provide a useful approach for predicting tumor growth and the effects of radiation.

A Multi-Institutional Comparison of Dynamic Contrast-Enhanced Magnetic Resonance Imaging Parameter Calculations

Dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) provides quantitative metrics (e.g. Ktrans, ve) via pharmacokinetic models. We tested inter-algorithm variability in these quantitative metrics with 11 published DCE-MRI algorithms, all implementing Tofts-Kermode or extended Tofts pharmacokinetic models. Digital reference objects (DROs) with known Ktrans and ve values were used to assess performance at varying noise levels. Additionally, DCE-MRI data from 15 head and neck squamous cell carcinoma patients over 3 time-points during chemoradiotherapy were used to ascertain Ktrans and ve kinetic trends across algorithms. Algorithms performed well (less than 3% average error) when no noise was present in the DRO. With noise, 87% of Ktrans and 84% of ve algorithm-DRO combinations were generally in the correct order. Low Krippendorff’s alpha values showed that algorithms could not consistently classify patients as above or below the median for a given algorithm at each time point or for differences in values between time points. A majority of the algorithms produced a significant Spearman correlation in ve of the primary gross tumor volume with time. Algorithmic differences in Ktrans and ve values over time indicate limitations in combining/comparing data from distinct DCE-MRI model implementations. Careful cross-algorithm quality-assurance must be utilized as DCE-MRI results may not be interpretable using differing software.Dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) provides quantitative metrics (e.g. Ktrans, ve) via pharmacokinetic models. We tested inter-algorithm variability in these quantitative metrics with 11 published DCE-MRI algorithms, all implementing Tofts-Kermode or extended Tofts pharmacokinetic models. Digital reference objects (DROs) with known Ktrans and ve values were used to assess performance at varying noise levels. Additionally, DCE-MRI data from 15 head and neck squamous cell carcinoma patients over 3 time-points during chemoradiotherapy were used to ascertain Ktrans and ve kinetic trends across algorithms. Algorithms performed well (less than 3% average error) when no noise was present in the DRO. With noise, 87% of Ktrans and 84% of ve algorithm-DRO combinations were generally in the correct order. Low Krippendorff’s alpha values showed that algorithms could not consistently classify patients as above or below the median for a given algorithm at each time point or for differences in values between time points. A majority of the algorithms produced a significant Spearman correlation in ve of the primary gross tumor volume with time. Algorithmic differences in Ktrans and ve values over time indicate limitations in combining/comparing data from distinct DCE-MRI model implementations. Careful cross-algorithm quality-assurance must be utilized as DCE-MRI results may not be interpretable using differing software.

A comparison of individual and population-derived vascular input functions for quantitative DCE-MRI in rats☆

Dynamic contrast enhanced magnetic resonance imaging (DCE-MRI) can quantitatively and qualitatively assess physiological characteristics of tissue. Quantitative DCE-MRI requires an estimate of the time rate of change of the concentration of the contrast agent in the blood plasma, the vascular input function (VIF). Measuring the VIF in small animals is notoriously difficult as it requires high temporal resolution images limiting the achievable number of slices, field-of-view, spatial resolution, and signal-to-noise. Alternatively, a population-averaged VIF could be used to mitigate the acquisition demands in studies aimed to investigate, for example, tumor vascular characteristics. Thus, the overall goal of this manuscript is to determine how the kinetic parameters estimated by a population based VIF differ from those estimated by an individual VIF. Eight rats bearing gliomas were imaged before, during, and after an injection of Gd-DTPA. K(trans), ve, and vp were extracted from signal-time curves of tumor tissue using both individual and population-averaged VIFs. Extended model voxel estimates of K(trans) and ve in all animals had concordance correlation coefficients (CCC) ranging from 0.69 to 0.98 and Pearson correlation coefficients (PCC) ranging from 0.70 to 0.99. Additionally, standard model estimates resulted in CCCs ranging from 0.81 to 0.99 and PCCs ranging from 0.98 to 1.00, supporting the use of a population based VIF if an individual VIF is not available.

Incorporating drug delivery into an imaging-driven, mechanics-coupled reaction diffusion model for predicting the response of breast cancer to neoadjuvant chemotherapy: Theory and preliminary clinical results

Clinical methods for assessing tumor response to therapy are largely rudimentary, monitoring only temporal changes in tumor size. Our goal is to predict the response of breast tumors to therapy using a mathematical model that utilizes magnetic resonance imaging (MRI) data obtained non-invasively from individual patients. We extended a previously established, mechanically coupled, reaction-diffusion model for predicting tumor response initialized with patient-specific diffusion weighted MRI (DW-MRI) data by including the effects of chemotherapy drug delivery, which is estimated using dynamic contrast-enhanced (DCE-) MRI data. The extended, drug incorporated, model is initialized using patient-specific DW-MRI and DCE-MRI data. Data sets from five breast cancer patients were used-obtained before, after one cycle, and at mid-point of neoadjuvant chemotherapy. The DCE-MRI data was used to estimate spatiotemporal variations in tumor perfusion with the extended Kety-Tofts model. The physiological parameters derived from DCE-MRI were used to model changes in delivery of therapy drugs within the tumor for incorporation in the extended model. We simulated the original model and the extended model in both 2D and 3D and compare the results for this five-patient cohort. Preliminary results show reductions in the error of model predicted tumor cellularity and size compared to the experimentally-measured results for the third MRI scan when therapy was incorporated. Comparing the two models for agreement between the predicted total cellularity and the calculated total cellularity (from the DW-MRI data) reveals an increased concordance correlation coefficient from 0.81 to 0.98 for the 2D analysis and 0.85 to 0.99 for the 3D analysis (p  <  0.01 for each) when the extended model was used in place of the original model. This study demonstrates the plausibility of using DCE-MRI data as a means to estimate drug delivery on a patient-specific basis in predictive models and represents a step toward the goal of achieving individualized prediction of tumor response to therapy.

Mechanically Coupled Reaction-Diffusion Model to Predict Glioma Growth: Methodological Details

Biophysical models designed to predict the growth and response of tumors to treatment have the potential to become a valuable tool for clinicians in care of cancer patients. Specifically, individualized tumor forecasts could be used to predict response or resistance early in the course of treatment, thereby providing an opportunity for treatment selection or adaption. This chapter discusses an experimental and modeling framework in which noninvasive imaging data is used to initialize and parameterize a subject-specific model of tumor growth. This modeling approach is applied to an analysis of murine models of glioma growth.

An adjoint-based method for a linear mechanically-coupled tumor model: application to estimate the spatial variation of murine glioma growth based on diffusion weighted magnetic resonance imaging

We present an efficient numerical method to quantify the spatial variation of glioma growth based on subject-specific medical images using a mechanically-coupled tumor model. The method is illustrated in a murine model of glioma in which we consider the tumor as a growing elastic mass that continuously deforms the surrounding healthy-appearing brain tissue. As an inverse parameter identification problem, we quantify the volumetric growth of glioma and the growth component of deformation by fitting the model predicted cell density to the cell density estimated using the diffusion-weighted magnetic resonance imaging data. Numerically, we developed an adjoint-based approach to solve the optimization problem. Results on a set of experimentally measured, in vivo rat glioma data indicate good agreement between the fitted and measured tumor area and suggest a wide variation of in-plane glioma growth with the growth-induced Jacobian ranging from 1.0 to 6.0.

Publisher Correction: A Multi-Institutional Comparison of Dynamic Contrast-Enhanced Magnetic Resonance Imaging Parameter Calculations (Scientific Reports (2017) DOI: 10.1038/s41598-017-11554-w)

A correction to this article has been published and is linked from the HTML and PDF versions of this paper. The error has been fixed in the paper.A correction to this article has been published and is linked from the HTML and PDF versions of this paper. The error has been fixed in the paper.

Erratum: Dynamic contrast-enhanced magnetic resonance imaging for head and neck cancers

This corrects the article DOI: 10.1038/sdata.2018.8.