While there is a mature literature on biomathematical and biophysical modeling in cancer, many of the existing approaches are not of clinical utility, as they require input data that are extremely difficult to obtain in an intact organism, and/or require a large number of assumptions on the free parameters included in the models. Thus, there has only been very limited application of such models to solve problems of clinical import. More recently, however, there has been increased activity at the interface of quantitative, noninvasive imaging data, and tumor mathematical modeling. In addition to reporting on bulk tumor morphology and volume, emerging imaging techniques can quantitatively report on for example tumor vascularity, glucose metabolism, cell density and proliferation, and hypoxia. In this paper, we first motivate the problem of predicting therapy response by highlighting some (acknowledged) shortcomings in existing methods. We then provide introductions to a number of representative quantitative imaging methods and describe how they are currently (and potentially can be) used to initialize and constrain patient specific mathematical and biophysical models of tumor growth and treatment response, thereby increasing the clinical utility of such approaches. We conclude by highlighting some of the exciting research directions when one integrates quantitative imaging and tumor modeling.
The ability to identify—early in the course of therapy—patients that are not responding to a given therapeutic regimen is extraordinarily important. In addition to limiting patients’ exposure to the toxicities associated with unsuccessful therapies, it would allow patients the opportunity to switch to a potentially more efficacious approach. Unfortunately, existing methods of determining early response are inadequate. In particular, the current standard-of-care imaging assessment of tumor response to treatment is based on the Response Evaluation Criteria in Solid Tumors (RECIST, [
High-resolution images (typically computed tomography (CT) or magnetic resonance imaging (MRI)) are acquired at baseline before treatment as commenced. In these image sets, “target lesions” are determined, and the sum of their longest dimensions is recorded. Additional scans are then acquired during or after therapy and similarly analyzed. The change in the sum of the longest diameters from baseline to the follow-up studies are then calculated and then used to divide treatment response into one of the following four categories [ Complete response (CR)—disappearance of all target lesions. Partial response (PR)—at least a 30% decrease in the sum of the longest diameters of the target lesions, taking as the reference the baseline sum longest diameter. Progressive disease (PD)—at least a 20% increase in the sum of the longest diameters of the target lesions, taking as the reference the smallest sum longest diameter recorded since the baseline measurements or the appearance of one or more new lesions. Stable Disease (SD)—neither sufficient shrinkage to qualify for partial response nor sufficient increase to qualify for progressive disease.
While almost all clinical trials of solid tumors employ the RECIST criteria, it is well recognized that this approach needs to be significantly improved for a number of reasons. For example, the metric for positive response is based on one-dimensional changes (sum of longest diameters of target lesions) which can be grossly misleading in trying to describe a complex object that is almost certainly changing in all three spatial dimensions. Furthermore, this metric is based on anatomical and morphological changes which are (temporally) downstream manifestations of underlying physiological, cellular, or molecular changes; that is, RECIST based evaluations generally do not indicate whether a tumor is responding until the patient has received several treatment cycles. This limitation is particularly problematic in the era of targeted therapies; in fact, it is a matter for debate as to whether the RECIST criteria are even relevant in assessing the activities of non-cytotoxic anti-cancer drugs if changes in morphology may not be the most appropriate method to assess response. What is needed are methods to characterize those underlying physiological, cellular, and molecular changes as they are highly likely to offer
Independent of the recent developments in the quantitative imaging of cancer, a mature literature has developed in the biomathematical and biophysical modeling of tumor growth (for reviews of this field see, e.g., [
In this paper we aim to show how biomedical imaging data can be used to initialize and constrain patient specific mathematical models of tumor growth to predict tumor status at later time points. To accomplish this task, we first provide brief introductions of a selection of emerging, quantitative, and
We limit our discussion of quantitative imaging metrics to the (currently) most widely available MRI and PET measures. Additionally, these imaging measures are those that have already shown encouraging data in their ability to predict treatment response and are, therefore, natural candidates to integrate into biomathematical and biophysical models of tumor growth and treatment response.
DCE-MRI involves the serial acquisition of MR images of a tissue of interest (e.g., a tumor) before and after an intravenous injection of paramagnetic contrast agent [
The microscopic thermally-induced behavior of molecules moving in a random pattern is referred to as self-diffusion or Brownian motion [
Diffusion tensor imaging (DTI) is a method to noninvasively characterize the structural connectivity between brain cortical regions. DTI provides, for each voxel, a tensor matrix that describes the constraints on local Brownian motion of water molecules. Originally proposed to assess tissue properties such as diffusion anisotropy [
Fluorodeoxyglucose is a glucose analogue that accumulates in areas of increased glycolytic activity which is a near universal property of cancer. It is well known that the activity of cell surface glucose transporters GLUT-1 and GLUT-3 and the intracellular enzyme hexokinase are upregulated in malignant cancer cells. The transporters function is to transport FDG into tumor cells where it is phosphorylated by hexokinase and trapped (and hence accumulated) in malignant cells. FDG-PET has been used extensively in both the preclinical and clinical settings to study treatment response [
Fluorodeoxythymidine (FLT) was developed as a surrogate marker of cellular proliferation [
It is well known that hypoxia in malignant tumors can affect the outcome of anticancer treatments, as malignant tumors are more resistant to chemotherapy and irradiative therapy because of their lack of oxygen, which is a potent radiosensitizer [
Another common PET-based method for studying at hypoxia is based on copper(II)-diacetyl-bis(
As should be apparent from the above discussions, many of the imaging modalities are complimentary in nature, and this realization has spurred the development of multimodal assessments of cancer. For example, combining the measurements available from DCE-MRI and FMISO-PET can provide insight into the relationship between vascular status and tissue oxygenation. Another obvious paring is DW-MRI and FLT-PET to explore the (temporal and spatial) relationships between cell proliferation and cellularity. To date the overwhelming majority of multimodal studies have been correlative in nature and there have been only minimal efforts at integrating such data into an appropriate biomathematical or biophysical model. (We review such investigations below in Section
We divide our review of the literature on integrating modeling and imaging into subsections in which a single imaging technique is employed, and a section where multiple imaging techniques are employed. We note that the order of the manuscripts reviewed below was determined by their publication date.
As it is well known that glioma cells preferentially migrate along white matter fibers of the brain, it is natural to incorporate the structural
A similar approach to linking tumor growth and restricted movement was investigated by Clatz et al., who simulated the growth of glioblastoma multiforma (GBM) by incorporating the mechanical restrictions (based on classical continuum mechanics) presented by certain brain structures [
Bondiau et al. [
Also using MRI data to drive the reaction-diffusion equation in GBM patients, Wang et al. employed serial pretreatment MRI data to generate patient specific rates of net glioma cell proliferation and invasion in 32 patients [
Rockne et al. extended (
Our own group has recently contributed a simple approach to integrating imaging and modeling, whereby serial DW-MRI data obtained before and early after the initiation of therapy are used to populate the logistic growth model [
We also applied this approach to breast cancer data acquired during neoadjuvant chemotherapy. Six patients received DW-MRI before, after one cycle, and after all cycles of neoadjuvant chemotherapy. Again, the proliferation rates were estimated using the ADC data from the first two time points and then used with (
Another effort that has incorporated serially acquired DW-MRI data into a mathematical model was contributed by Ellingson et al. [
One of the first studies to include multimodality imaging in a mathematical model of tumor growth was put forth by Titz and Jeraj in 2008 [
The first study to merge modeling with clinical MRI and PET data was contributed by Szeto et al. [
Building on these efforts, this group has more recently extended the model described by (
Before leaving this section, we stress two points of interest. The first is that the overwhelming majority of the studies linking mathematical modeling and tumor growth or treatment response have occurred in the brain. While the reasons for this are debatable (though the rigid structure of the skull and the presence of many internal fiducial markers facilitates image registration and, therefore, the application of several of the models described above), it is clearly a limitation in the current literature. Moving forward, the mathematical and biophysical modeling communities will need to address the current paucity of application of their techniques at disease sites outside of the brain. The second point worth noting is that the studies described above all have, at their core, the reaction-diffusion equation. As the excellent efforts described above convincingly indicate, this approach has much merit. But integrating the myriad of data available from quantitative imaging will most likely require new modeling approaches.
There are three key areas (two theoretical and one experimental) that are primary goals for investigators working at the interface of biomedical imaging and tumor modeling: (1) the need to construct models amenable to the incorporation of imaging data, (2) the construction of a framework, informed by clinical and cancer biology, that allows for
The majority of existing biomathematical and biophysical models of tumor growth and treatment response are not of the form that are readily amenable to incorporating imaging data; more specifically, the models are not of the kind that can readily be populated by data that can be measured in an intact system at multiple time points with reasonable spatial resolution in 3D. Frequently, the models have a wide variety of parameters that must be estimated or assumed based on measurements from disparate sources. Conversely, the utility of imaging data is that it can be acquired on a patient specific basis so that models can be initialized and constrained by an individual patient’s characteristics. As the majority of current models are not constructed with this in mind, this is a fundamental—though artificial—barrier to progress on integrating imaging data in tumor growth models. Thus, going forward, models need to be constructed with the types of measurements that are available from imaging data in mind. In the author’s view, this should be one of the primary goals for investigators working in this field.
Once tumor growth models are built that are readily populated by the imaging data described above in Section
As imaging data is acquired with minimal interaction with the system under investigation, measurements made early in an experiment can lead to predictions that can then be tested later in the experiment. More specifically, in the context of pre-clinical rodent tumor studies, imaging measurements could be made at baseline (i.e., after an implanted tumor reaches a particular size but before therapy commences), then at a second time point early in the course of treatment. These two data sets could then be used to initialize the model(s) to determine the optimal treatment approach as outlined in the previous two paragraphs. The optimized therapy could then be given to the (say) mouse and the disease progression/regression tracked. In this way the predictions of the imaging based model can be directly compared to experiment to determine their
After reviewing a representative sampling of the emerging, quantitative
The author would like to thank the National Cancer Institute for support through 1U01CA142565, R01 CA138599, 1P50 098131, P30 CA68485, and R25CA092043, R25CA136440 and the Kleberg Foundation for their generous support of our imaging program. Additionally, the author would like to thank Drs. Vito Quaranta, M.D. Mike Miga, Ph.D. Nkiruka Atuegwu, Ph.D., Jared Weis, Ph.D., and Mr. David Hormuth for many informative and engaging discussions on the topic of integrating mathematical modeling and biomedical imaging.