Accurate state estimation from uncertain data and models: an application of data assimilation to mathematical models of human brain tumors

Biology Direct

Eric J Kostelich²  Yang Kuang²  Joshua M McDaniel²  Nina Z Moore¹  Nikolay L Martirosyan¹  Mark C Preul¹ 

[1] Barrow Neurological Institute, St. Joseph's Hospital and Medical Center, 350 W. Thomas Road, Phoenix, AZ 85013 USA

[2] School of Mathematical & Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804 USA

关键词: glioblastoma multiforme;    mathematical models;    data assimiliation;    State estimation;   

Others  :  797036
DOI  :  10.1186/1745-6150-6-64

 received in 2011-07-04, accepted in 2011-12-21,  发布年份 2011

【 摘 要 】

Background

Data assimilation refers to methods for updating the state vector (initial condition) of a complex spatiotemporal model (such as a numerical weather model) by combining new observations with one or more prior forecasts. We consider the potential feasibility of this approach for making short-term (60-day) forecasts of the growth and spread of a malignant brain cancer (glioblastoma multiforme) in individual patient cases, where the observations are synthetic magnetic resonance images of a hypothetical tumor.

Results

We apply a modern state estimation algorithm (the Local Ensemble Transform Kalman Filter), previously developed for numerical weather prediction, to two different mathematical models of glioblastoma, taking into account likely errors in model parameters and measurement uncertainties in magnetic resonance imaging. The filter can accurately shadow the growth of a representative synthetic tumor for 360 days (six 60-day forecast/update cycles) in the presence of a moderate degree of systematic model error and measurement noise.

Conclusions

The mathematical methodology described here may prove useful for other modeling efforts in biology and oncology. An accurate forecast system for glioblastoma may prove useful in clinical settings for treatment planning and patient counseling.

Reviewers

This article was reviewed by Anthony Almudevar, Tomas Radivoyevitch, and Kristin Swanson (nominated by Georg Luebeck).

【 授权许可】

   
2011 Kostelich et al; licensee BioMed Central Ltd.

【 图 表 】

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【 参考文献 】

• [1]Gossman A, Helbich TH, Kuriyama N, Ostrowitzki S, Roberts TP, Shames DM, van Bruggen N, Wendland MF, Israel MA, Brasch RC: Dynamic contrast-enhanced magnetic resonance imaging as a surrogate marker of tumor response to anti-angiogenic therapy in a xenograft model of glioblastoma multiforme. J Magn Reson Imaging 2002, 15:233-240.
• [2]Rijpkema M, Kaanders JH, Joosten FB, van der Kogel AJ, Heerschap A: Method for quantitative mapping of dynamic MRI contrast agent uptake in human tumors. J Magn Reson Imaging 2002, 14:457-463.
• [3]Norden AD, Wen PY: Glioma therapy in adults. Neurologist 2006, 12:;279-292.
• [4]Amberger VR, Hensel T, Ogata N, Schwab ME: Spreading and migration of human glioma and rat C6 cells on central nervous system myelin in vitro is correlated with tumor malignancy and involves a metalloproteolytic activity. Cancer Res 1998, 58:149-158.
• [5]Demuth T, Berens ME: Molecular mechanisms of glioma cell migration and invasion. J Neurooncol 2004, 70:217-228.
• [6]Eikenberry SE, Sankar T, Preul MC, Kostelich EJ, Thalhauser CJ, Kuang Y: Virtual glioblastoma: growth, migration and treatment in a three-dimensional mathematical model. Cell Prolif 2009, 42:511-528.
• [7]Swanson KR, Alvord EC Jr, Murray JD: A quantitative model of differential motility of gliomas in white and grey matter. Cell Prolif 2000, 33:317-329.
• [8]Swanson KR, C B, Murray JD, Alvord EC Jr: Virtual and real brain tumors: using mathematical modeling to quantify glioma growth and invasion. J Neurol Sci 2003, 216:1-10.
• [9]Swanson KR, Rostomily RC, Alvord EC Jr: A mathematical modeling tool for predicting survival of individual patients following resection of glioblastoma: A proof of principle. Brit J Cancer 2008, 98:113-119.
• [10]BrainWeb: Simulated Brain Database [http://www.bic.mni.mcgill.ca/brainweb] webcite
• [11]Talairach J, Tournoux P: Co-Planar Stereotaxic Atlas of the Human Brain: 3-D Proportional System: An Approach to Cerebral Imaging. New York: Thieme Medical Publishers; 1988.
• [12]Harpold HLP, Alvord EC Jr, R SK: Evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol 2007, 66:1-9.
• [13]Lorenz EN: Deterministic non-periodic flow. J Atmos Sci 1963, 20:130-141.
• [14]Lorenz EN: A study of the predictability of a 28-variable atmospheric model. Tellus 1965, 17:321-333.
• [15]Kalnay E: Atmospheric Modeling, Data Assimilation, and Predictability. Cambridge, UK: Cambridge University Press; 2003.
• [16]Horton JR: An Introduction to Dynamic Meteorology, 4th ed. Amsterdam: Elsevier Academic Press; 2004.
• [17]Hunt BR, Kostelich EJ, Szunyogh I: Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter. Physica D 2007, 230:112-126.
• [18]Ott E, Hunt BR, Szunyogh I, Zimin AV, Kostelich EJ, Corazza M, Kalnay E, Patil DJ, Yorke JA: A local ensemble Kalman filter for atmospheric data assimilation. Tellus A 2004, 56:415-428.
• [19]Evensen G: Data Assimilation: The Ensemble Kalman Filter. Berlin: Springer-Verlag; 2007.
• [20]Julier SJ, Uhlmann JK, Durrant-Whyte HF: A new approach for filtering nonlinear systems. Proc of the American Control Conference 1995, 3:1628-1632. IEEE
• [21]Szunyogh I, Kostelich EJ, Gyarmati G, Patil DJ, Hunt BR, Kalnay E, Ott E, Yorke JA: Assessing a local ensemble Kalman filter: Perfect model experiments with the NCEP global model. Tellus A 2005, 57:528-545.
• [22]Szunyogh I, Kostelich EJ, Gyarmati G, Kalnay E, Hunt BR, Ott E, Satterfield E, Yorke JA: A local ensemble Kalman filter data assimilation system for the NCEP global model. Tellus A 2008, 60:113-130.
• [23]Patil DJ, Hunt BR, Kalnay E, Yorke JA, Ott E: Local low dimensionality of atmospheric dynamics. Phys Rev Lett 2001, 86:5878-5881.
• [24]Pérez-García VM, Calvo GF, Belmonte-Beitia J, Diego D, Pérez-Romasanta L: Bright solitary waves in malignant gliomas. Phys Rev E 2011, 84:021921.
• [25]Hoffman RN, Ponte RM, Kostelich EJ, Blumberg A, Szunyogh I, Vinogradov SV, Henderson JM: A simulation study using a local ensemble transform Kalman filter for data assimilation in New York Harbor. J Atmos Ocean Tech 2008, 25:1638-1656.
• [26]Hoffman MJ, Greybush SJ, Wilson RJ, Gyarmati G, Hoffman RN, Kalnay E, Ide K, Kostelich EJ, Miyoshi T, Szunyogh I: An ensemble Kalman filter data assimilation system for the Martian atmosphere: Implementation and simulation experiments. Icarus 2010, 209:470-481.
• [27]Wang X, Bishop CH, Julier SJ: Which is better, an ensemble of positive-negative pairs or a centered spherical simplex ensemble? Mon Wea Rev 2004, 132:1590-1605.
• [28]Beliën ATJ, Paganetti PA, Schwab ME: Membrane-type 1 matrix metalloprotease (MT1-MMP) enables invasive migration of glioma cells in central nervous system white matter. J Cell Biol 1999, 144:373-384.
• [29]Beadle C, Assanah MC, Monzo P, Vallee R, Rosenfeld SS, Canoli P: The role of myosin II in glioma invasion of the brain. Mol Biol Cell 2008, 19:3357-3368.
• [30]Montana V, Sontheimer H: Bradykinin promotes the chemotactic invasion of primary brain tumor. J Neurosci 2011, 31:4858-4867.
• [31]Edelman LB, Eddy JA, Price ND: In silico models of cancer. WIREs Syst Biol Med 2010, 2:438-459.
• [32]Deisboeck TS, Zhang L, Yoon J, Costa J: In silico cancer modeling: Is it ready for prime time? Nat Rev Clin Oncol 2009, 6:34-42.
• [33]Bearer EL, Lowengrub JS, Frieboes HB, Chuang YL, Jin F, Mand Ferrari WiseMS, Agus DB, Cristini V: Multiparameter computational modeling of tumor invasion. Cancer Res 2009, 69:4493-4501.
• [34]Tsai MH, Cook JA, Chandramouli CV, DeGraff W, Yan H, Zhao S, Coleman CN, Mitchell JB, Chuang EY: Gene expression profiling of breast, prostate, and glioma cells following single versus fractionated doses of radiation. Cancer Res 2007, 67:3845-3852.
• [35]Tracqui P, Cruywagen GC, Woodward DE, Bartoo GT, Murray JD, Alvord ECJ: A mathematical model of glioma growth: the effect of chemotherapy on spatio-temporal growth. Cell Prolif 1995, 28:17-31.
• [36]Piette C, Deprez M, Roger T, Noël A, Foidart JM, Munaut C: The dexamethasone-induced inhibition of proliferation, migration, and invasion in glioma cell lines is antagonized by macrophage migration inhibitory factor (MIF) and can be enhanced by specific MIF inhibitors. J Biol Chem 2009, 284:32483-32492.
• [37]Roniotis A, Marias K, Sakkalis V, Zervakis M: Diffusive modeling of glioma evolution: A review. J Biomed Sci Engr 2010, 3:501-508.
• [38]Hogea C, Davatzikos C, Biros G: Modeling glioma growth and mass effect in 3D MR images of the brain. Med Image Comput Comput Assist Interv 2007, 10:642-650.
• [39]Clatz O, Sermesant M, Bondiau PY, Delingette H, Warfield SK, Malandian G, Ayache N: Realistic simulation of the 3d growth of brain tumors in MR images coupling diffusion with biomechanical deformation. IEEE Trans Med Imaging 2005, 24:1334-1346.
• [40]Baek SJ, Hunt BR, Kalnay E, Ott E, Szunyogh I: Local ensemble Kalman filtering in the presence of model bias. Tellus A 2006, 58:293-306.
• [41]Zupanski D, Zupanski M: Model error estimation employing an ensemble data assimilation approach. Mon Wea Rev 2006, 134:1337-1354.
• [42]Orrell D: Ensemble forecasting in a system with model error. J Atmos Sci 2005, 62:1652-1659.
• [43]Emblem KE, Zoellner FG, Tennoe B, Nedregaard B, Nome T, Due-Tonnesson P, Hald JK, Schreie D, Bjornerud A: Predictive modeling in glioma grading from MR perfusion images using support vector machines. Magn Reson Med 2008, 60:945-952.
• [44]Vannucci RC, Barron TF, Lerro D, Antón SC, Vannucii SJ: Craniometric measures during development using MRI. Neuroimage 2011, 56:1855-1864.
• [45]Ruoslahti E: Brain extracellular matrix. Glycobiology 1996, 6:489-492.
• [46]Bonneh-Barkay D, Wiley CA: Brain extracellular matrix in neurodegeneration. Brain Pathol 2009, 19:573-585.
• [47]Bauer R, Ratzinger S, Wales L, Bosserhoff A, Senner V, Grifka J, Grässel S: Inhibition of collagen XVI expression reduces glioma cell invasiveness. Cell Physiol Biochem 2011, 27:217-226.
• [48]Rong Y, Durden DL, Van Meir EG, Brat DJ: "Pseudopalisading" necrosis in glioblastoma: A familiar morphologic feature that links vascular pathology, hypoxia, and angiogenesis. J Neuropathol Exp Neurol 2006, 65:529-539.