Parameter non-identifiability of the Gyllenberg-Webb ODE model

Parameter estimation and mathematical modeling for the quantitative description of therapy failure due to drug resistance in gastrointestinal stromal tumor metastasis to the liver

In this work we develop a general mathematical model and devise a practical identifiability approach for gastrointestinal stromal tumor (GIST) metastasis to the liver, with the aim of quantitatively describing therapy failure due to drug resistance. To this end, we have modeled metastatic growth and therapy failure produced by resistance to two standard treatments based on tyrosine kinase inhibitors (Imatinib and Sunitinib) that have been observed clinically in patients with GIST metastasis to the liver. The parameter identification problem is difficult to solve, since there are no general results on this issue for models based on ordinary differential equations (ODE) like the ones studied here. We propose a general modeling framework based on ODE for GIST metastatic growth and therapy failure due to drug resistance and analyzed five different model variants, using medical image observations (CT scans) from patients that exhibit drug resistance. The associated parameter estimation problem was solved using the Nelder-Mead simplex algorithm, by adding a regularization term to the objective function to address model instability, and assessing the agreement of either an absolute or proportional error in the objective function. We compared the goodness of fit to data for the proposed model variants, as well as evaluated both error forms in order to improve parameter estimation results. From the model variants analyzed, we identified the one that provides the best fit to all the available patient data sets, as well as the best assumption in computing the objective function (absolute or proportional error). This is the first work that reports mathematical models capable of capturing and quantitatively describing therapy failure due to drug resistance based on clinical images in a patient-specific manner.

Loci That Control Nonlinear, Interdependent Responses to Combinations of Drought and Nitrogen Limitation

Crop improvement must accelerate to feed an increasing human population in the face of environmental changes. Including anticipated climatic changes with genetic architecture in breeding programs could better optimize improvement strategies. Combinations of drought and nitrogen limitation already occur world-wide. We therefore analyzed the genetic architecture underlying the response of Zea mays to combinations of water and nitrogen stresses. Recombinant inbreds were subjected to nine combinations of the two stresses using an optimized response surface design, and their growth was measured. Three-dimensional response surfaces were fit globally and to each polymorphic allele to determine which genetic markers were associated with different response surfaces. Three quantitative trait loci that produced nonlinear surfaces were mapped. To better understand the physiology of the response, we developed a model that reproduced the shapes of the surfaces, their most characteristic feature. The model contains two components that each combine the nitrogen and water inputs. The relative weighting of the two components and the inputs is governed by five parameters, and each QTL affects all five parameters.

We estimated the model’s parameter values for the experimental surfaces using a mesh of points that covered the surfaces’ most distinctive regions. Surfaces computed using these values reproduced the experimental surfaces well, as judged by three different criteria at the mesh points. The modeling and shape comparison techniques used here can be extended to other complex, high-dimensional, nonlinear phenotypes. We encourage the application of our findings and methods to experiments that mix crop protection measures, stresses, or both, on elite and landrace germplasm.