Contrastive image adaptation for acquisition shift reduction in medical imaging.

The domain shift, or acquisition shift in medical imaging, is responsible for potentially harmful differences between development and deployment conditions of medical image analysis techniques. There is a growing need in the community for advanced methods that could mitigate this issue better than conventional approaches. In this paper, we consider configurations in which we can expose a learning-based pixel level adaptor to a large variability of unlabeled images during its training, i.e. sufficient to span the acquisition shift expected during the training or testing of a downstream task model. We leverage the ability of convolutional architectures to efficiently learn domain-agnostic features and train a many-to-one unsupervised mapping between a source collection of heterogeneous images from multiple unknown domains subjected to the acquisition shift and a homogeneous subset of this source set of lower cardinality, potentially constituted of a single image. To this end, we propose a new cycle-free image-to-image architecture based on a combination of three loss functions : a contrastive PatchNCE loss, an adversarial loss and an edge preserving loss allowing for rich domain adaptation to the target image even under strong domain imbalance and low data regimes. Experiments support the interest of the proposed contrastive image adaptation approach for the regularization of downstream deep supervised segmentation and cross-modality synthesis models.

UroPredict: Machine learning model on real-world data for prediction of kidney cancer recurrence (UroCCR-120).

Renal cell carcinoma (RCC) is most often diagnosed at a localized stage, where surgery is the standard of care. Existing prognostic scores provide moderate predictive performance, leading to challenges in establishing follow-up recommendations after surgery and in selecting patients who could benefit from adjuvant therapy. In this study, we developed a model for individual postoperative disease-free survival (DFS) prediction using machine learning (ML) on real-world prospective data. Using the French kidney cancer research network database, UroCCR, we analyzed a cohort of surgically treated RCC patients. Participating sites were randomly assigned to either the training or testing cohort, and several ML models were trained on the training dataset. The predictive performance of the best ML model was then evaluated on the test dataset and compared with the usual risk scores. In total, 3372 patients were included, with a median follow-up of 30 months. The best results in predicting DFS were achieved using Cox PH models that included 24 variables, resulting in an iAUC of 0.81 [IC95% 0.77-0.85]. The ML model surpassed the predictive performance of the most commonly used risk scores while handling incomplete data in predictors. Lastly, patients were stratified into four prognostic groups with good discrimination (iAUC = 0.79 [IC95% 0.74-0.83]). Our study suggests that applying ML to real-world prospective data from patients undergoing surgery for localized or locally advanced RCC can provide accurate individual DFS prediction, outperforming traditional prognostic scores.

A CT-Based Clinical, Radiological and Radiomic Machine Learning Model for Predicting Malignancy of Solid Renal Tumors (UroCCR-75).

BACKGROUND: Differentiating benign from malignant renal tumors is important for patient management, and it may be improved by quantitative CT features analysis including radiomic. PURPOSE: This study aimed to compare performances of machine learning models using bio-clinical, conventional radiologic and 3D-radiomic features for the differentiation of benign and malignant solid renal tumors using pre-operative multiphasic contrast-enhanced CT examinations. MATERIALS AND METHODS: A unicentric retrospective analysis of prospectively acquired data from a national kidney cancer database was conducted between January 2016 and December 2020. Histologic findings were obtained by robotic-assisted partial nephrectomy. Lesion images were semi-automatically segmented, allowing for a 3D-radiomic features extraction in the nephrographic phase. Conventional radiologic parameters such as shape, content and enhancement were combined in the analysis. Biological and clinical features were obtained from the national database. Eight machine learning (ML) models were trained and validated using a ten-fold cross-validation. Predictive performances were evaluated comparing sensitivity, specificity, accuracy and AUC. RESULTS: A total of 122 patients with 132 renal lesions, including 111 renal cell carcinomas (RCCs) (111/132, 84%) and 21 benign tumors (21/132, 16%), were evaluated (58 +/- 14 years, men 74%). Unilaterality (100/111, 90% vs. 13/21, 62%; p = 0.02), necrosis (81/111, 73% vs. 8/21, 38%; p = 0.02), lower values of tumor/cortex ratio at portal time (0.61 vs. 0.74, p = 0.01) and higher variation of tumor/cortex ratio between arterial and portal times (0.22 vs. 0.05, p = 0.008) were associated with malignancy. A total of 35 radiomics features were selected, and "intensity mean value" was associated with RCCs in multivariate analysis (OR = 0.99). After ten-fold cross-validation, a C5.0Tree model was retained for its predictive performances, yielding a sensitivity of 95%, specificity of 42%, accuracy of 87% and AUC of 0.74. CONCLUSION: Our machine learning-based model combining clinical, radiologic and radiomics features from multiphasic contrast-enhanced CT scans may help differentiate benign from malignant solid renal tumors.

Machine-learning approach for prediction of pT3a upstaging and outcomes of localized renal cell carcinoma (UroCCR-15).

OBJECTIVES: To assess the impact of pathological upstaging from clinically localized to locally advanced pT3a on survival in patients with renal cell carcinoma (RCC), as well as the oncological safety of various surgical approaches in this setting, and to develop a machine-learning-based, contemporary, clinically relevant model for individual preoperative prediction of pT3a upstaging. MATERIALS AND METHODS: Clinical data from patients treated with either partial nephrectomy (PN) or radical nephrectomy (RN) for cT1/cT2a RCC from 2000 to 2019, included in the French multi-institutional kidney cancer database UroCCR, were retrospectively analysed. Seven machine-learning algorithms were applied to the cohort after a training/testing split to develop a predictive model for upstaging to pT3a. Survival curves for disease-free survival (DFS) and overall survival (OS) rates were compared between PN and RN after G-computation for pT3a tumours. RESULTS: A total of 4395 patients were included, among whom 667 patients (15%, 337 PN and 330 RN) had a pT3a-upstaged RCC. The UroCCR-15 predictive model presented an area under the receiver-operating characteristic curve of 0.77. Survival analysis after adjustment for confounders showed no difference in DFS or OS for PN vs RN in pT3a tumours (DFS: hazard ratio [HR] 1.08, P = 0.7; OS: HR 1.03, P > 0.9). CONCLUSIONS: Our study shows that machine-learning technology can play a useful role in the evaluation and prognosis of upstaged RCC. In the context of incidental upstaging, PN does not compromise oncological outcomes, even for large tumour sizes.

A Model-Strengthened Imaging Biomarker for Survival Prediction in EGFR-Mutated Non-small-cell Lung Carcinoma Patients Treated with Tyrosine Kinase Inhibitors.

Non-small-cell lung carcinoma is a frequent type of lung cancer with a bad prognosis. Depending on the stage and genomics, several therapeutical approaches are used. Tyrosine Kinase Inhibitors (TKI) may be successful for a time in the treatment of EGFR-mutated non-small cells lung carcinoma. Our objective is here to introduce a survival assessment as their efficacy in the long run is challenging to evaluate. The study includes 17 patients diagnosed with EGFR-mutated non-small cell lung cancer and exposed to an EGFR-targeting TKI with 3 computed tomography (CT) scans of the primary tumor (one before the TKI introduction and two after). An imaging biomarker based on evolution of texture heterogeneity between the first and the third exams is derived and computed from a mathematical model and patient data. Defining the overall survival as the time between the introduction of the TKI treatment and the patient death, we obtain a statistically significant correlation between the overall survival and our imaging marker ([Formula: see text]). Using the ROC curve, the patients are separated into two populations and the comparison of the survival curves is statistically significant ([Formula: see text]). The baseline exam seems to have a significant role in the prediction of response to TKI treatment. More precisely, our imaging biomarker defined using only the CT scan before the TKI introduction allows to determine a first classification of the population which is improved over time using the imaging marker as soon as more CT scans are available. This exploratory study leads us to think that it is possible to obtain a survival assessment using only few CT scans of the primary tumor.

Role of 3D volume growth rate for drug activity evaluation in meningioma clinical trials: the example of the CEVOREM study.

BACKGROUND: We aimed to improve the assessment of the drug activity in meningioma clinical trials based on the study of the 3D volume growth rate (3DVGR) in a series of aggressive meningiomas. We secondarily aimed to correlate 3DVGR study with patient outcome. METHODS: We performed a post hoc analysis based on volume data and 3DVGR extracted from CEVOREM study including 18 patients with 32 recurrent high-grade meningiomas and treated with everolimus and octreotide. The joint latent class model was used to classify tumor 3DVGR undertreatment. RESULTS: Class 1 includes lesions responding to treatment with decrease in volume in the first 3 months, and then a stabilization thereafter (9.5% of tumors) (mean pretreatment 3DVGR = 6.13%/month; mean undertreatment 3DVGR = -18.7%/month within 3 first months and -0.14%/month after the 3 first months). Class 2 includes lesions considered as stable or with a slight increase in volume undertreatment (65.5%) (mean pretreatment 3DVGR = 6.09%/month; undertreatment 3DVGR = -0.09% within the first 3 months). Class 3 includes lesions without 3DVGR decrease (25%) (mean pretreatment 3DVGR = 46.9%/month; mean undertreatment 3DVGR = 19.2%/month within the first 3 months). Patients with class 3 lesions had a significantly worse progression-free survival (PFS) rate than class 1 and 2 ones. CONCLUSIONS: Tumor 3DVGR could be helpful to detect early signal of drugs antitumoral activity or nonactivity. This volume response classification could help in the assessment of drug activity in tumors with mostly volume stabilization and rare response as aggressive meningiomas even with a low number of patients in complement to 6 months PFS.

Spatial mechanistic modeling for prediction of the growth of asymptomatic meningiomas.

BACKGROUND AND OBJECTIVE: Mathematical modeling of tumor growth draws interest from the medical community as they have the potential to improve patients' care and the use of public health resources. The main objectives of this work are to model the growth of meningiomas - slow-growing benign tumors requiring extended imaging follow-up - and to predict tumor volume and shape at a later desired time using only two times examinations. METHODS: We develop two variants of a 3D partial differential system of equations (PDE) which yield after a spatial integration systems of ordinary differential equations (ODE) that relate tumor volume with time. Estimation of models parameters is a crucial step to obtain a personalized model for a patient that can be used for descriptive or predictive purposes. As PDE and ODE systems share the same parameters, they are both estimated by fitting the ODE systems to the tumor volumes obtained from MRI examinations acquired at different times. A population approach allows to compensate for sparse sampling times and measurement uncertainties by constraining the variability of the parameters in the population. RESULTS: Description capabilities of the models are investigated in 39 patients with benign asymptomatic meningiomas who had had at least three surveillance MRI examinations. The two models can fit to the data accurately and more realistically than a naive linear regression. Prediction performances are validated for 33 patients using a population approach. Mean relative errors in volume predictions are less than 10% with ODE systems versus 12.5% with the naive linear model using only two times examinations. Concerning the shape, the mean Sørensen-Dice coefficients are 85% with the PDE systems in a subset of 10 representative patients. CONCLUSIONS: Our strategy - based on personalization of mathematical model - provides a good insight on meningioma growth and may help decide whether to extend the follow-up or to treat the tumor.

A prospective comparison of dynamic contrast-enhanced MRI and 51Cr-EDTA clearance for glomerular filtration rate measurement in 42 kidney transplant recipients.

OBJECTIVES: To evaluate the performance of dynamic contrast-enhanced MRI measurement of glomerular filtration rate (GFR) compared with the reference standard technique of urinary clearance of 51Cr-EDTA. PATIENTS AND METHODS: All kidney transplant recipients (KTRs) with an indication for non-urgent contrast-enhanced MRI at our institution were prospectively included between 2008 and 2012. Renographies were acquired by low-dose dynamic contrast-enhanced MRI (DCE-MRI) then fitted with a two-compartment pharmacokinetic model. MR-GFR was compared with reference isotopic measurements using Bland-Altman diagrams, intraclass correlation coefficient (ICC) and concordance rates. RESULTS: Forty-two KTRs (mean age 51.5 years, 26-74) were analyzed. Mean estimated GFR was 48.5 ± 27 mL/min/1.73m2 (24-178 mL/min). The mean bias was +13.2 mL/min (6.4-20.0, +36.9%) ranging from -31.0 mL/min (-41.7%) to +101.4 mL/min (+89.2%) with a large variability (standard-deviation: 22.3 mL/min; limits of agreement: [-30.6 (-43.3--18.9); +57.0 (45.3-68.7)]). The ICC was 0.32 (0.02-0.56) and the concordance rate was 28.6% (14.9-42.2). CONCLUSIONS: The large variability of MR-GFR compared with the reference technique precludes its use in KTRs, whose anatomical peculiarities make standardization of arterial input function (AIF) difficult.

Experimental estimation of stored stress within spherical microtissues : What can and cannot be inferred from cutting experiments.

Biological tissues accumulate mechanical stress during their growth. The mere measurement of the stored stress is not an easy task. We address here the spherical case and our experiments consist in performing an incision of a spherical microtissue (tumor spheroid) grown in vitro. On the theoretical part we derive a compatibility condition on the stored stress in spherical symmetry, which imposes a relation between the circumferential and radial stored stress. The numerical implementation uses the hyperelastic model of Ciarlet and Geymonat. A parametric study is performed to assess the influence of each parameter on the shape of the domain after the incision. As a conclusion, the total radial stored stress can be confidently estimated from the measurement of the opening after incision. We validate the approach with experimental data.

Genetic analysis of adults heterozygous for ALPL mutations.

Hypophosphatasia (HPP) is a rare inherited metabolic bone disease due to a deficiency of the tissue nonspecific alkaline phosphatase isoenzyme (TNSALP) encoded by the ALPL gene. Patients have consistently low serum alkaline phosphatase (AP), so that this parameter is a good hallmark of the disease. Adult HPP is heterogeneous, and some patients present only mild nonpathognomonic symptoms which are also common in the general population such as joint pain, osteomalacia and osteopenia, chondrocalcinosis, arthropathy and musculoskeletal pain. Adult HPP may be recessively or dominantly inherited; the latter case is assumed to be due to the dominant negative effect (DNE) of missense mutations derived from the functional homodimeric structure of TNSALP. However, there is no biological argument excluding the possibility of other causes of dominant HPP. Rheumatologists and endocrinologists are increasingly solicited for patients with low AP and nonpathognomonic symptoms of HPP. Many of these patients are heterozygous for an ALPL mutation and a challenging question is to determine if these symptoms, which are also common in the general population, are attributable to their heterozygous ALPL mutation or not. In an attempt to address this question, we reviewed a cohort of 61 adult patients heterozygous for an ALPL mutation. Mutations were distinguished according to their statistical likelihood to show a DNE. One-half of the patients carried mutations predicted with no DNE and were slightly less severely affected by the age of onset, serum AP activity and history of fractures. We hypothesized that these mutations result in another mechanism of dominance or are recessive alleles. To identify other genetic factors that could trigger the disease phenotype in heterozygotes for potential recessive mutations, we examined the next-generation sequencing results of 32 of these patients for a panel of 12 genes involved in the differential diagnosis of HPP or candidate modifier genes of HPP. The heterozygous genotype G/C of the COL1A2 coding SNP rs42524 c.1645C > G (p.Pro549Ala) was associated with the severity of the phenotype in patients carrying mutations with a DNE whereas the homozygous genotype G/G was over-represented in patients carrying mutations without a DNE, suggesting a possible role of this variant in the disease phenotype. These preliminary results support COL1A2 as a modifier gene of HPP and suggest that a significant proportion of adult heterozygotes for ALPL mutations may have unspecific symptoms not attributable to their heterozygosity.

Tumor growth model of ductal carcinoma: from in situ phase to stroma invasion.

This paper aims at modeling breast cancer transition from the in situ stage -when the tumor is confined to the duct- to the invasive phase. Such a transition occurs thanks to the degradation of the duct membrane under the action of specific enzymes so-called matrix metalloproteinases (MMPs). The model consists of advection-reaction equations that hold in the duct and in the surrounding tissue, in order to describe the proliferation and the necrosis of the cancer cells in each subdomain. The divergence of the velocity is given by the increase of the cell densities. Darcy law is imposed in order to close the system. The key-point of the modeling lies in the description of the transmission conditions across the duct. Nonlinear Kedem-Katchalsky transmission conditions across the membrane describe the discontinuity of the pressure as a linear function of the flux. These transmission conditions make it possible to describe the transition from the in situ stage to the invasive phase at the macroscopic level. More precisely, the membrane permeability increases with respect to the local concentration of MMPs. The cancer cells are no more confined to the duct and the tumor invades the surrounding tissue. The model is enriched by the description of nutrients concentration, tumor necrosis factors, and MMPs production. The mathematical model is implemented in a 3D C++-code, which is based on well-adapted finite difference schemes on Cartesian grid. The membrane interface is described by a level-set, and the transmission conditions are precisely approached at the second order thanks to well-suited sharp stencils. Our continuous approach provides new significant insights in the macroscopic modeling of the breast cancer phase transition, due to the membrane degradation by MMP enzymes.

Spatial modelling of tumour drug resistance: the case of GIST liver metastases.

This work is devoted to modelling gastrointestinal stromal tumour metastases to the liver, their growth and resistance to therapies. More precisely, resistance to two standard treatments based on tyrosine kinase inhibitors (imatinib and sunitinib) is observed clinically. Using observations from medical images (CT scans), we build a spatial model consisting in a set of non-linear partial differential equations. After calibration of its parameters with clinical data, this model reproduces qualitatively and quantitatively the spatial tumour evolution of one specific patient. Important features of the growth such as the appearance of spatial heterogeneities and the therapeutical failures may be explained by our model. We then investigate numerically the possibility of optimizing the treatment in terms of progression-free survival time and minimum tumour size reachable by varying the dose of the first treatment. We find that according to our model, the progression-free survival time reaches a plateau with respect to this dose. We also demonstrate numerically that the spatial structure of the tumour may provide much more insights on the cancer cell activities than the standard RECIST criteria, which only consists in the measurement of the tumour diameter. Finally, we discuss on the non-predictivity of the model using only CT scans, in the sense that the early behaviour of the lesion is not sufficient to predict the response to the treatment.

Computational Trials: Unraveling Motility Phenotypes, Progression Patterns, and Treatment Options for Glioblastoma Multiforme.

Glioblastoma multiforme is a malignant brain tumor with poor prognosis and high morbidity due to its invasiveness. Hypoxia-driven motility and concentration-driven motility are two mechanisms of glioblastoma multiforme invasion in the brain. The use of anti-angiogenic drugs has uncovered new progression patterns of glioblastoma multiforme associated with significant differences in overall survival. Here, we apply a mathematical model of glioblastoma multiforme growth and invasion in humans and design computational trials using agents that target angiogenesis, tumor replication rates, or motility. The findings link highly-dispersive, moderately-dispersive, and hypoxia-driven tumors to the patterns observed in glioblastoma multiforme treated by anti-angiogenesis, consisting of progression by Expanding FLAIR, Expanding FLAIR + Necrosis, and Expanding Necrosis, respectively. Furthermore, replication rate-reducing strategies (e.g. Tumor Treating Fields) appear to be effective in highly-dispersive and moderately-dispersive tumors but not in hypoxia-driven tumors. The latter may respond to motility-reducing agents. In a population computational trial, with all three phenotypes, a correlation was observed between the efficacy of the rate-reducing agent and the prolongation of overall survival times. This research highlights the potential applications of computational trials and supports new hypotheses on glioblastoma multiforme phenotypes and treatment options.

Computational Modelling of Metastasis Development in Renal Cell Carcinoma.

The biology of the metastatic colonization process remains a poorly understood phenomenon. To improve our knowledge of its dynamics, we conducted a modelling study based on multi-modal data from an orthotopic murine experimental system of metastatic renal cell carcinoma. The standard theory of metastatic colonization usually assumes that secondary tumours, once established at a distant site, grow independently from each other and from the primary tumour. Using a mathematical model that translates this assumption into equations, we challenged this theory against our data that included: 1) dynamics of primary tumour cells in the kidney and metastatic cells in the lungs, retrieved by green fluorescent protein tracking, and 2) magnetic resonance images (MRI) informing on the number and size of macroscopic lesions. Critically, when calibrated on the growth of the primary tumour and total metastatic burden, the predicted theoretical size distributions were not in agreement with the MRI observations. Moreover, tumour expansion only based on proliferation was not able to explain the volume increase of the metastatic lesions. These findings strongly suggested rejection of the standard theory, demonstrating that the time development of the size distribution of metastases could not be explained by independent growth of metastatic foci. This led us to investigate the effect of spatial interactions between merging metastatic tumours on the dynamics of the global metastatic burden. We derived a mathematical model of spatial tumour growth, confronted it with experimental data of single metastatic tumour growth, and used it to provide insights on the dynamics of multiple tumours growing in close vicinity. Together, our results have implications for theories of the metastatic process and suggest that global dynamics of metastasis development is dependent on spatial interactions between metastatic lesions.

Effects of anti-angiogenesis on glioblastoma growth and migration: model to clinical predictions.

Glioblastoma multiforme (GBM) causes significant neurological morbidity and short survival times. Brain invasion by GBM is associated with poor prognosis. Recent clinical trials of bevacizumab in newly-diagnosed GBM found no beneficial effects on overall survival times; however, the baseline health-related quality of life and performance status were maintained longer in the bevacizumab group and the glucocorticoid requirement was lower. Here, we construct a clinical-scale model of GBM whose predictions uncover a new pattern of recurrence in 11/70 bevacizumab-treated patients. The findings support an exception to the Folkman hypothesis: GBM grows in the absence of angiogenesis by a cycle of proliferation and brain invasion that expands necrosis. Furthermore, necrosis is positively correlated with brain invasion in 26 newly-diagnosed GBM. The unintuitive results explain the unusual clinical effects of bevacizumab and suggest new hypotheses on the dynamic clinical effects of migration by active transport, a mechanism of hypoxia-driven brain invasion.

Patient specific image driven evaluation of the aggressiveness of metastases to the lung.

Metastases to the lung are a therapeutic challenge because some are fast-evolving while others evolve slowly. Any insight that can be provided for which nodule has to be treated first would help clinicians. In this work, we evaluate the aggressiveness but also the response to treatment of these nodules using a calibrated mathematical model. This model is a macroscopic model describing tumoral growth through a set of nonlinear partial differential equations. It has to be calibrated to a specific patient and a specific nodule using a temporal sequence of CT scans. To this end, a new optimization technique based on a reduced order method is developed. Finally, results on two clinical cases are presented that give satisfactory numerical prognosis of the evolution of a nodule during different phases: growth, treatment and post-treatment relapse.

A multilayer grow-or-go model for GBM: effects of invasive cells and anti-angiogenesis on growth.

The recent use of anti-angiogenesis (AA) drugs for the treatment of glioblastoma multiforme (GBM) has uncovered unusual tumor responses. Here, we derive a new mathematical model that takes into account the ability of proliferative cells to become invasive under hypoxic conditions; model simulations generate the multilayer structure of GBM, namely proliferation, brain invasion, and necrosis. The model is able to replicate and justify the clinical observation of rebound growth when AA therapy is discontinued in some patients. The model is interrogated to derive fundamental insights int cancer biology and on the clinical and biological effects of AA drugs. Invasive cells promote tumor growth, which in the long run exceeds the effects of angiogenesis alone. Furthermore, AA drugs increase the fraction of invasive cells in the tumor, which explain progression by fluid-attenuated inversion recovery (FLAIR) signal and the rebound tumor growth when AA is discontinued.

Modeling of the migration of endothelial cells on bioactive micropatterned polymers.

In this paper, a macroscopic model describing endothelial cell migration on bioactive micropatterned polymers is presented. It is based on a system of partial differential equations of Patlak-Keller-Segel type that describes the evolution of the cell densities. The model is studied mathematically and numerically. We prove existence and uniqueness results of the solution to the differential system. We also show that fundamental physical properties such as mass conservation, positivity and boundedness of the solution are satisfied. The numerical study allows us to show that the modeling results are in good agreement with the experiments.

A pharmacologically based multiscale mathematical model of angiogenesis and its use in investigating the efficacy of a new cancer treatment strategy.

Tumor angiogenesis is the process by which new blood vessels are formed and enhance the oxygenation and growth of tumors. As angiogenesis is recognized as being a critical event in cancer development, considerable efforts have been made to identify inhibitors of this process. Cytostatic treatments that target the molecular events of the angiogenesis process have been developed, and have met with some success. However, it is usually difficult to preclinically assess the effectiveness of targeted therapies, and apparently promising compounds sometimes fail in clinical trials. We have developed a multiscale mathematical model of angiogenesis and tumor growth. At the molecular level, the model focuses on molecular competition between pro- and anti-angiogenic substances modeled on the basis of pharmacological laws. At the tissue scale, the model uses partial differential equations to describe the spatio-temporal changes in cancer cells during three stages of the cell cycle, as well as those of the endothelial cells that constitute the blood vessel walls. This model is used to qualitatively assess how efficient endostatin gene therapy is. Endostatin is an anti-angiogenic endogenous substance. The gene therapy entails overexpressing endostatin in the tumor and in the surrounding tissue. Simulations show that there is a critical treatment dose below which increasing the duration of treatment leads to a loss of efficacy. This theoretical model may be useful to evaluate the efficacy of therapies targeting angiogenesis, and could therefore contribute to designing prospective clinical trials.