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Representative results from scenario 1. The central eight tumor slices from patient 1 are used to demonstrate the model calibration results using the best model as determined by the AIC. The measured and model determined sum of the enhancing and non-enhancing regions (i.e., ϕE and ϕN) are shown in left and middle columns, respectively, whereas the voxel-wise comparison between the model and measured tumor distributions are shown in the right column. In general, a high level of agreement and correlation resulted in low voxel level errors (CCCs greater than 0.68). MATLAB R2019b (Mathworks, Natick, MA) was used for producing individual figures, images, and graphs. Adobe Photoshop 2020 (Adobe, San Jose, CA) was used to arrange individual panels, draw schematics, and add text.
Source publication
High-grade gliomas are an aggressive and invasive malignancy which are susceptible to treatment resistance due to heterogeneity in intratumoral properties such as cell proliferation and density and perfusion. Non-invasive imaging approaches can measure these properties, which can then be used to calibrate patient-specific mathematical models of tum...
Citations
... A wide variety of model types has been utilized to capture the complexities of GBM, ranging from ordinary differential equations (ODEs) [16][17][18] to partial differential equations (PDEs) [19][20][21], stochastic differential equations (SDEs) [22], cellular automata (CA) [23][24][25], and evolutionary game theory models (EGTs) [26]. Through these diverse methods, mathematical models have successfully addressed various aspects of GBM, shedding light on critical points such as the tumor's response to chemo-and radiotherapy [27][28][29][30][31][32][33][34][35][36][37][38], or other types of treatments [39][40][41][42][43], the development of resistances [44][45][46], as well as phenotypic changes induced by treatments [47][48][49]. Moreover, mathematical models have been used to identify biomarkers with prognostic value [50][51][52][53][54]. ...
... Previous mathematical studies about therapies for brain cancer can be found in [2,24,32,48,87,92,93,111,134,[138][139][140], which however are grounded on ordinary differential equations and kinetic equations, not discussing the interplay with mechanics. Recent investigations by Hormuth et al. [89,91] considered the effect of Von Mises stress in a model for brain tumour growth and response to chemoradiation. Notably, the stress was used to exponentially dampen the motility coefficient of tumour cells, while the mechanical constitutive equations remained relatively simplified, and fluid stresses were not considered. ...
... The proposed framework also incorpo-rates a simple model to simulate therapeutic protocols, which is a relevant feature towards the development of improved treatments for patients. Differently from recent works that introduced mechanical effects into chemoradiation models for brain tumours [89][90][91], our framework accounts for nonlinear elasticity of both the tumour and the host tissue, with constitutive equations motivated by experimental data [22], and incorporates both solid and fluid stresses. Furthermore, the coupling between mechanics and growth is derived via well-established Continuum Mechanics methods in a physically motivated way. ...
... Finally, we note that, by defining the diffusion tensor weighted by φ , we can effectively account for variations in intratumoral cellularity, which in turn influence the Apparent Diffusion Coefficient (ADC), a metric that reflects the magnitude of diffusion [8,89,105]. Permeability tensor K. ...
Malignant brain tumours represent a significant medical challenge due to their aggressive nature and unpredictable locations. The growth of a brain tumour can result in a mass effect, causing compression and displacement of the surrounding healthy brain tissue and possibly leading to severe neurological complications. In this paper, we propose a multiphase mechanical model for brain tumour growth that quantifies deformations and solid stresses caused by the expanding tumour mass and incorporates anisotropic growth influenced by brain fibres. We employ a sharp interface model to simulate localised, non-invasive solid brain tumours, which are those responsible for substantial mechanical impact on the surrounding healthy tissue. By using patient-specific imaging data, we create realistic three-dimensional brain geometries and accurately represent ventricular shapes, to evaluate how the growing mass may compress and deform the cerebral ventricles. Another relevant feature of our model is the ability to simulate therapeutic protocols, facilitating the evaluation of treatment efficacy and guiding the development of personalized therapies for individual patients. Overall, our model allows to make a step towards a deeper analysis of the complex interactions between brain tumours and their environment, with a particular focus on the impact of a growing cancer on healthy tissue, ventricular compression, and therapeutic treatment.
... These measurements enable the calibration of model parameters, the definition of the tumour and host tissue geometry, and the initialisation of model variables (e.g., those describing tumour and other cell species as well as key substances controlling their dynamics) [37,41,55]. Towards these ends, data are usually required at two or more time points during the monitoring of the disease before, during and after treatment [29,40,41,52]. The collection of these data over time contributes to early characterise crucial clinical endpoints (e.g., progression to higher-risk disease, treatment failure, survival) by both analysing the relative change of these measurements [25,49] as well as by using them to obtain more accurate tumour forecasts [29,40,41,52]. ...
... Towards these ends, data are usually required at two or more time points during the monitoring of the disease before, during and after treatment [29,40,41,52]. The collection of these data over time contributes to early characterise crucial clinical endpoints (e.g., progression to higher-risk disease, treatment failure, survival) by both analysing the relative change of these measurements [25,49] as well as by using them to obtain more accurate tumour forecasts [29,40,41,52]. However, depending on the natural history and specific clinical management protocols for each type of cancer, it may not be possible to obtain tumour measurements until the tumour has developed sufficiently to produce symptoms or be detected with standard-of-care screening methods [12,37,40,41]. ...
... There are computational studies regarding the population dynamics and treatment response based on biological data dedicated to the simulation of drug combinations other than TMZ-DOX against GB progress 28 or to other cancer types [29][30][31] . There is a long list of GB -specific computational models that have been in vivo validated with human or xenograft models based on other form of therapies, such as hypoxia-related death 32,33 , radiotherapy and/or combination therapies [34][35][36] , indicating the wide applicability of such a mechanistic approach. A variety of GB -specific TMZ 37-39 -or DOX-only 31,40,41 simulations also exist. ...
Adjuvant Temozolomide is considered the front-line Glioblastoma chemotherapeutic treatment; yet not all patients respond. Latest trends in clinical trials usually refer to Doxorubicin; yet it can lead to severe side-effects if administered in high doses. While Glioblastoma prognosis remains poor, little is known about the combination of the two chemotherapeutics. Patient-derived spheroids were generated and treated with a range of Temozolomide/Doxorubicin concentrations either as monotherapy or in combination. Optical microscopy was used to monitor the growth pattern and cell death. Based on the monotherapy experiments, we developed a probabilistic mathematical framework in order to describe the drug-induced effect at the single-cell level and simulate drug doses in combination assuming probabilistic independence. Doxorubicin was found to be effective in doses even four orders of magnitude less than Temozolomide in monotherapy. The combination therapy doses tested in vitro were able to lead to irreversible growth inhibition at doses where monotherapy resulted in relapse. In our simulations, we assumed both drugs are anti-mitotic; Temozolomide has a growth-arrest effect, while Doxorubicin is able to cumulatively cause necrosis. Interestingly, under no mechanistic synergy assumption, the in silico predictions underestimate the in vitro results. In silico models allow the exploration of a variety of potential underlying hypotheses. The simulated-biological discrepancy at certain doses indicates a supra-additive response when both drugs are combined. Our results suggest a Temozolomide–Doxorubicin dual chemotherapeutic scheme to both disable proliferation and increase cytotoxicity against Glioblastoma.
... to account for the patient's and treating physician's risk preferences. We illustrate the predictive digital twin using an in silico population of HGG patients that is constructed by pooling clinical data of MRI measurements of HGG cellularity and mechanistic parameter values describing the dynamics of HGG growth and RT response from the literature (Qi et al., 2006;Wang et al., 2009;Hormuth et al., 2021c). We investigate varying levels of total dose to analyze the ensuing suite of therapeutic planning options. ...
... All the parameters necessary to define the ODE model are provided in Table 1. We fix the radiosensitivity parameter ratio to αRT /βRT = 10 ( Rockne et al., 2009) and the surviving fraction resulting from chemotherapy to S C = 0.82 (Hormuth et al., 2021c). The system of ODEs in Equation (4) is solved via a forward Euler scheme with sufficiently small time step size of 0.2 days to ensure numerical stability. ...
... In the HGG setting, these observational data can be acquired non-invasively using MRI. Recently, MRI data have been used to calibrate computational models and obtain tumor forecasts (Hormuth et al., 2015(Hormuth et al., , 2020(Hormuth et al., , 2021c after appropriate post-processing. In particular, the MRI data needs to be post-processed to extract the total tumor burden as a cell count, such that it relates to the state variable N of the ODE model in Equation (4). ...
We develop a methodology to create data-driven predictive digital twins for optimal risk-aware clinical decision-making. We illustrate the methodology as an enabler for an anticipatory personalized treatment that accounts for uncertainties in the underlying tumor biology in high-grade gliomas, where heterogeneity in the response to standard-of-care (SOC) radiotherapy contributes to sub-optimal patient outcomes. The digital twin is initialized through prior distributions derived from population-level clinical data in the literature for a mechanistic model's parameters. Then the digital twin is personalized using Bayesian model calibration for assimilating patient-specific magnetic resonance imaging data. The calibrated digital twin is used to propose optimal radiotherapy treatment regimens by solving a multi-objective risk-based optimization under uncertainty problem. The solution leads to a suite of patient-specific optimal radiotherapy treatment regimens exhibiting varying levels of trade-off between the two competing clinical objectives: (i) maximizing tumor control (characterized by minimizing the risk of tumor volume growth) and (ii) minimizing the toxicity from radiotherapy. The proposed digital twin framework is illustrated by generating an in silico cohort of 100 patients with high-grade glioma growth and response properties typically observed in the literature. For the same total radiation dose as the SOC, the personalized treatment regimens lead to median increase in tumor time to progression of around six days. Alternatively, for the same level of tumor control as the SOC, the digital twin provides optimal treatment options that lead to a median reduction in radiation dose by 16.7% (10 Gy) compared to SOC total dose of 60 Gy. The range of optimal solutions also provide options with increased doses for patients with aggressive cancer, where SOC does not lead to sufficient tumor control.
... The most common modeling framework for these studies is the reaction-diffusion model, which simplifies GBM behavior to infiltration (via a diffusion term) and proliferation (via a reaction term). While this approach has been widely applied to modeling external beam radiation, [12][13][14], there is only minimal applications to radionuclide therapy in the preclinical setting [15], and no examples in the clinical setting. The present study serves as an initial application of predicting the spatio-temporal response of GBM to high dose, continuous radiation treatments in patients. ...
... Unfortunately, we are currently limited to employing M0 for the LOOCV due to the small number of parameters included in this model. While calibrating proliferation on a local basis (i.e., at the voxel level) has shown great accuracy when predicting the growth of GBM (see ref. [14], as well as in the present effort), there is currently not an established method to assign locally distributed parameters a priori. Additionally, ten patients is a small sample size to statistically validate the found parameter distributions. ...
Rhenium-186 (¹⁸⁶Re) labeled nanoliposome (RNL) therapy for recurrent glioblastoma patients has shown promise to improve outcomes by locally delivering radiation to affected areas. To optimize the delivery of RNL, we have developed a framework to predict patient-specific response to RNL using image-guided mathematical models.
Methods
We calibrated a family of reaction-diffusion type models with multi-modality imaging data from ten patients (NCR01906385) to predict the spatio-temporal dynamics of each patient’s tumor. The data consisted of longitudinal magnetic resonance imaging (MRI) and single photon emission computed tomography (SPECT) to estimate tumor burden and local RNL activity, respectively. The optimal model from the family was selected and used to predict future growth. A simplified version of the model was used in a leave-one-out analysis to predict the development of an individual patient’s tumor, based on cohort parameters.
Results
Across the cohort, predictions using patient-specific parameters with the selected model were able to achieve Spearman correlation coefficients (SCC) of 0.98 and 0.93 for tumor volume and total cell number, respectively, when compared to the measured data. Predictions utilizing the leave-one-out method achieved SCCs of 0.89 and 0.88 for volume and total cell number across the population, respectively.
Conclusion
We have shown that patient-specific calibrations of a biology-based mathematical model can be used to make early predictions of response to RNL therapy. Furthermore, the leave-one-out framework indicates that radiation doses determined by SPECT can be used to assign model parameters to make predictions directly following the conclusion of RNL treatment.
Statement of Significance
This manuscript explores the application of computational models to predict response to radionuclide therapy in glioblastoma. There are few, to our knowledge, examples of mathematical models used in clinical radionuclide therapy. We have tested a family of models to determine the applicability of different radiation coupling terms for response to the localized radiation delivery. We show that with patient-specific parameter estimation, we can make accurate predictions of future glioblastoma response to the treatment. As a comparison, we have shown that population trends in response can be used to forecast growth from the moment the treatment has been delivered.
In addition to the high simulation and prediction accuracy our modeling methods have achieved, the evaluation of a family of models has given insight into the response dynamics of radionuclide therapy. These dynamics, while different than we had initially hypothesized, should encourage future imaging studies involving high dosage radiation treatments, with specific emphasis on the local immune and vascular response.
... In the past, the tumor growth prediction problem has been approached through mathematical models from biological principles, such as ones based on the reaction-diffusion (RD) equation [5]- [7]. A few recent RD-based methods [8]- [10] have introduced different approaches to adjust the model's parameters towards precise personalizing brain tumor modeling. However, the success of these approaches depends on the quality of the chosen forward model, which always includes simplifying assumptions compared to the actual biological processes [11]. ...
Diffuse gliomas are malignant brain tumors that grow widespread through the brain. The complex interactions between neoplastic cells and normal tissue, as well as the treatment-induced changes often encountered, make glioma tumor growth modeling challenging. In this paper, we present a novel end-to-end network capable of generating future tumor masks and realistic MRIs of how the tumor will look at any future time points for different treatment plans. Our model is built upon cutting-edge diffusion probabilistic models and deep-segmentation neural networks. We extended a diffusion model to include sequential multi-parametric MRI and treatment information as conditioning input to guide the generative diffusion process. This allows us to estimate tumor growth at any given time point. We trained the model using real-world postoperative longitudinal MRI data with glioma tumor growth trajectories represented as tumor segmentation maps over time. The model has demonstrated promising performance across a range of tasks, including the generation of high-quality synthetic MRIs with tumor masks, time-series tumor segmentations, and uncertainty estimation. Combined with the treatment-aware generated MRIs, the tumor growth predictions with uncertainty estimates can provide useful information for clinical decision-making.
... There are three main paradigms to describe tumor growth and therapeutic response with PDE models. First, advection-diffusion-reaction models are usually posed in terms of tumor cell density [157,57,89,35,154,6] or tumor cell volumetric fraction [139,79]. The formulation of these models is obtained by combining two types of phenomena: (i) a mobility mechanism, which is generally represented by a flux featuring diffusive and advective processes within an elastic [157,57,89,35] or poroelastic medium [6,139,79], and (ii) a collection of reaction terms representing local tumor cell mechanisms, such as proliferation, death, metabolism, therapeutic response, and phenotypic change of tumor cells [165,86,93]. ...
... First, advection-diffusion-reaction models are usually posed in terms of tumor cell density [157,57,89,35,154,6] or tumor cell volumetric fraction [139,79]. The formulation of these models is obtained by combining two types of phenomena: (i) a mobility mechanism, which is generally represented by a flux featuring diffusive and advective processes within an elastic [157,57,89,35] or poroelastic medium [6,139,79], and (ii) a collection of reaction terms representing local tumor cell mechanisms, such as proliferation, death, metabolism, therapeutic response, and phenotypic change of tumor cells [165,86,93]. Second, phase-field models are governed by the minimization of a functional featuring all types of energy interacting in the tumor growth problem (e.g., internal energy, kinetic energy, interfacial energy between tumor and healthy tissues, elastic energy), wherein the tumor is represented by a variable identifying the regions occupied by cancerous tissue [94,33,45,86,160,153]. ...
... According to the description of the three types of PDE models provided above, their formulation can accommodate the time-resolved terms appearing on the right-hand side of ODE models into reaction terms with a spatiotemporal definition. This feature enables the representation of intratumoral heterogeneities, for example, using proliferation maps calibrated from longitudinal imaging data [157,57,154], adjusting growth in terms of the local availability of nutrients or the local vascularization [122,6,94,33,59,139], and characterizing therapeutic response based on the local tumor burden [165,35,157,57], nutrient availability [6,120], tumor-supporting vasculature [60,57,136], and drug concentration in the tissue [157,142]. The description of these phenomena driving tumor growth can be further enriched by extending a baseline PDE model to include other key spatiotemporal mechanisms represented by additional PDEs, such as the effect of tumor-driven mechani-cal deformation of the host tissue and the effect of mechanical stress on tumor dynamics [57,157,94,135,6,154], nutrient and drug transport in the tissue [142,94,45,6], the production of key substances (e.g., angiogenic factors, matrix-degrading enzymes or biomarkers) [94,136,153,142,45], blood flow in the native vascular network [160,45,79], and the development of angiogenesis [160,79,59]. ...
Despite the remarkable advances in cancer diagnosis, treatment, and management that have occurred over the past decade, malignant tumors remain a major public health problem. Further progress in combating cancer may be enabled by personalizing the delivery of therapies according to the predicted response for each individual patient. The design of personalized therapies requires patient-specific information integrated into an appropriate mathematical model of tumor response. A fundamental barrier to realizing this paradigm is the current lack of a rigorous, yet practical, mathematical theory of tumor initiation, development, invasion, and response to therapy. In this review, we begin by providing an overview of different approaches to modeling tumor growth and treatment, including mechanistic as well as data-driven models based on ``big data" and artificial intelligence. Next, we present illustrative examples of mathematical models manifesting their utility and discussing the limitations of stand-alone mechanistic and data-driven models. We further discuss the potential of mechanistic models for not only predicting, but also optimizing response to therapy on a patient-specific basis. We then discuss current efforts and future possibilities to integrate mechanistic and data-driven models. We conclude by proposing five fundamental challenges that must be addressed to fully realize personalized care for cancer patients driven by computational models.
... The risk-based optimization formulation allows one to account for the patient's and treating physician's risk preferences. We illustrate the predictive digital twin using an in silico population of HGG patients that is constructed by pooling clinical data of MRI measurements of HGG cellularity and mechanistic parameter values describing the dynamics of HGG growth and RT response from the literature [75,53,24]. We investigate varying levels of total dose to analyze the ensuing suite of therapeutic planning options. ...
... All the parameters necessary to define the ODE model are provided in Table 1. We fix the radiosensitivity parameter ratio to αRT /βRT = 10 [61] and the surviving fraction resulting from chemotherapy to S C = 0.82 [24]. The system of ODEs in Eq. (4) is solved via a forward Euler scheme with sufficiently small time step size of 0.2 days to ensure numerical stability. ...
... In the HGG setting, these observational data can be acquired non-invasively using MRI. Recently, MRI data have been used to calibrate computational models and obtain tumor forecasts [27,26,25,24] after appropriate post-processing. In particular, the MRI data needs to be post-processed to extract the total tumor burden as a cell count, such that it relates to the state variable N of the ODE model in Eq. (4). ...
We develop a methodology to create data-driven predictive digital twins for optimal risk-aware clinical decision-making. We illustrate the methodology as an enabler for an anticipatory personalized treatment that accounts for uncertainties in the underlying tumor biology in high-grade gliomas, where heterogeneity in the response to standard-of-care (SOC) radiotherapy contributes to sub-optimal patient outcomes. The digital twin is initialized through prior distributions derived from population-level clinical data in the literature for a mechanistic model's parameters. Then the digital twin is personalized using Bayesian model calibration for assimilating patient-specific magnetic resonance imaging data and used to propose optimal radiotherapy treatment regimens by solving a multi-objective risk-based optimization under uncertainty problem. The solution leads to a suite of patient-specific optimal radiotherapy treatment regimens exhibiting varying levels of trade-off between the two competing clinical objectives: (i) maximizing tumor control (characterized by minimizing the risk of tumor volume growth) and (ii) minimizing the toxicity from radiotherapy. The proposed digital twin framework is illustrated by generating an in silico cohort of 100 patients with high-grade glioma growth and response properties typically observed in the literature. For the same total radiation dose as the SOC, the personalized treatment regimens lead to median increase in tumor time to progression of around six days. Alternatively, for the same level of tumor control as the SOC, the digital twin provides optimal treatment options that lead to a median reduction in radiation dose by 16.7% (10 Gy) compared to SOC total dose of 60 Gy. The range of optimal solutions also provide options with increased doses for patients with aggressive cancer, where SOC does not lead to sufficient tumor control.
... However, tumor sizebased methods cannot definitively establish changes until the patient has received several treatment cycles [23,24] and, since tissue-based biomarkers require an invasive biopsy, they are prone to sampling errors due to tumor heterogeneity [25][26][27]. Alternatively, imaging-based computational tumor forecasting has been actively investigated to obtain early predictions of patient-specific pathological and therapeutic outcomes that can guide clinical decisionmaking for different tumor types [28][29][30][31][32][33][34][35][36][37]. These tumor forecasting methods align with other imaging-based predictive technologies that have been applied to multiple pathologies within the context of computational medicine [38][39][40][41][42][43]. ...
... [30,44,45,[73][74][75]). In paticular, the tumor cell density maps were approximated from the voxel-based ADC values within the tumor ROI [30,34,44,45,76,77], fibroglandular and adipose tissues were segmented based on enhancement in the DCE-MRI data by leveraging a k-means clustering algorithm, and a normalized perfusion map representing the spatial distribution of blood volume fraction within the breast tissue was calculated using the area under the dynamic curve (AUC) for each voxel in the DCE-MRI data [45]. The interested reader is referred to Supplementary Methods S1 and Ref. [45] for further information on MRI data acquisition and preprocessing. ...
... In Eq. (2), D 0 represents the tumor cell diffusivity in the absence of mechanical inhibition, N denotes an empirical coupling constant, and v (x, t) is the von Mises stress, which is a scalar stress metric calculated as where ij (with i, j = 1, 2, 3 ) are the components of the second-order mechanical stress tensor (x, t) . This mechanically coupled approach has been shown to render superior predictions of breast tumor dynamics during NAC with respect to a baseline model without mechanics [73,74], and it has also been adopted in patient-specific, mechanically constrained models of brain and prostate cancer [34,88,89]. ...
Neoadjuvant chemotherapy (NAC) is a standard-of-care treatment for locally advanced triple negative breast cancer (TNBC) before surgery. The early assessment of TNBC response to NAC would enable an oncologist to adapt the therapeutic plan of a non-responding patient, thereby improving treatment outcomes while preventing unnecessary toxicities. To this end, a promising approach consists of obtaining in silico personalized forecasts of tumor response to NAC via computer simulation of mechanistic models constrained with patient-specific magnetic resonance imaging (MRI) data acquired early during NAC. Here, we present a new mechanistic model of TNBC growth and response to NAC, including an explicit description of drug pharmacodynamics and pharmacokinetics. As longitudinal in vivo MRI data for model calibration is limited, we perform a sensitivity analysis to identify the model mechanisms driving the response to two NAC drug combinations: doxorubicin with cyclophosphamide, and paclitaxel with carboplatin. The model parameter space is constructed by combining patient-specific MRI-based in silico parameter estimates and in vitro measurements of pharmacodynamic parameters obtained using time-resolved microscopy assays of several TNBC lines. The sensitivity analysis is run in two MRI-based scenarios corresponding to a well-perfused and a poorly perfused tumor. Out of the 15 parameters considered herein, only the baseline tumor cell net proliferation rate along with the maximum concentrations and effects of doxorubicin, carboplatin, and paclitaxel exhibit a relevant impact on model forecasts (total effect index, ST\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{T}$$\end{document}0.1). These results dramatically limit the number of parameters that require in vivo MRI-constrained calibration, thereby facilitating the clinical application of our model.
