{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:47:21Z","timestamp":1760150841877,"version":"build-2065373602"},"reference-count":112,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,1,20]],"date-time":"2022-01-20T00:00:00Z","timestamp":1642636800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100005722","name":"Ludwig-Maximilians-Universit\u00e4t M\u00fcnchen","doi-asserted-by":"publisher","award":["1"],"award-info":[{"award-number":["1"]}],"id":[{"id":"10.13039\/501100005722","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Background: For the kinetic models used in contrast-based medical imaging, the assignment of the arterial input function named AIF is essential for the estimation of the physiological parameters of the tissue via solving an optimization problem. Objective: In the current study, we estimate the AIF relayed on the modified maximum entropy method. The effectiveness of several numerical methods to determine kinetic parameters and the AIF is evaluated\u2014in situations where enough information about the AIF is not available. The purpose of this study is to identify an appropriate method for estimating this function. Materials and Methods: The modified algorithm is a mixture of the maximum entropy approach with an optimization method, named the teaching-learning method. In here, we applied this algorithm in a Bayesian framework to estimate the kinetic parameters when specifying the unique form of the AIF by the maximum entropy method. We assessed the proficiency of the proposed method for assigning the kinetic parameters in the dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI), when determining AIF with some other parameter-estimation methods and a standard fixed AIF method. A previously analyzed dataset consisting of contrast agent concentrations in tissue and plasma was used. Results and Conclusions: We compared the accuracy of the results for the estimated parameters obtained from the MMEM with those of the empirical method, maximum likelihood method, moment matching (\u201cmethod of moments\u201d), the least-square method, the modified maximum likelihood approach, and our previous work. Since the current algorithm does not have the problem of starting point in the parameter estimation phase, it could find the best and nearest model to the empirical model of data, and therefore, the results indicated the Weibull distribution as an appropriate and robust AIF and also illustrated the power and effectiveness of the proposed method to estimate the kinetic parameters.<\/jats:p>","DOI":"10.3390\/e24020155","type":"journal-article","created":{"date-parts":[[2022,1,20]],"date-time":"2022-01-20T22:40:20Z","timestamp":1642718420000},"page":"155","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Modified Maximum Entropy Method and Estimating the AIF via DCE-MRI Data Analysis"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3207-3090","authenticated-orcid":false,"given":"Zahra","family":"Amini Farsani","sequence":"first","affiliation":[{"name":"Statistics Department, School of Science, Lorestan University, Khorramabad 68151-44316, Iran"},{"name":"Bayesian Imaging and Spatial Statistics Group, Institute of Statistics, Ludwig-Maximilians-Universit\u00e4t M\u00fcnchen, Ludwigstra\u00dfe 33, 80539 Munich, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2195-8130","authenticated-orcid":false,"given":"Volker J.","family":"Schmid","sequence":"additional","affiliation":[{"name":"Bayesian Imaging and Spatial Statistics Group, Institute of Statistics, Ludwig-Maximilians-Universit\u00e4t M\u00fcnchen, Ludwigstra\u00dfe 33, 80539 Munich, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,1,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"620","DOI":"10.1103\/PhysRev.106.620","article-title":"Information Theory and Statistical Mechanics","volume":"106","author":"Jaynes","year":"1957","journal-title":"Phys. Rev."},{"key":"ref_2","first-page":"2069","article-title":"Maximum Entropy Copulas","volume":"1305","author":"Pougaza","year":"2011","journal-title":"AIP Conf. Proc."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1217","DOI":"10.1016\/j.jmva.2007.08.004","article-title":"Multivariate maximum entropy identification, transformation, and dependence","volume":"99","author":"Ebrahimi","year":"2008","journal-title":"J. Multivar. Anal."},{"key":"ref_4","unstructured":"Thomas, A., and Cover, T.M. (2006). Elements of Information Theory, John Wiley."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Cofr\u00e9, R., Herzog, R., Corcoran, D., and Rosas, F.E. (2019). A comparison of the maximum entropy principle across biological spatial scales. Entropy, 21.","DOI":"10.20944\/preprints201907.0240.v1"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Jaynes, E.T. (2003). Probability Theory: The Logic of Science, Cambridge University Press.","DOI":"10.1017\/CBO9780511790423"},{"key":"ref_7","unstructured":"Ozer, H.G. (2008). Residue Associations in Protein Family Alignments. [Ph.D. Thesis, The Ohio State University]."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"078102","DOI":"10.1103\/PhysRevLett.100.078102","article-title":"Maximum entropy approach for deducing amino acid interactions in proteins","volume":"100","author":"Seno","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1073\/pnas.0805923106","article-title":"Identification of direct residue contacts in protein\u2013protein interaction by message passing","volume":"106","author":"Weigt","year":"2009","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"3445","DOI":"10.1021\/ct300112v","article-title":"On the use of experimental observations to bias simulated ensembles","volume":"8","author":"Pitera","year":"2012","journal-title":"J. Chem. Theory Comput."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1607","DOI":"10.1016\/j.cell.2012.04.012","article-title":"Three-dimensional structures of membrane proteins from genomic sequencing","volume":"149","author":"Hopf","year":"2012","journal-title":"Cell"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"02B616","DOI":"10.1063\/1.4792208","article-title":"On the statistical equivalence of restrained-ensemble simulations with the maximum entropy method","volume":"138","author":"Roux","year":"2013","journal-title":"J. Chem. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"03B603","DOI":"10.1063\/1.4793625","article-title":"Molecular dynamics simulations with replica-averaged structural restraints generate structural ensembles according to the maximum entropy principle","volume":"138","author":"Cavalli","year":"2013","journal-title":"J. Chem. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1007\/s12210-020-00909-7","article-title":"Does maximal entropy production play a role in the evolution of biological complexity? A biological point of view","volume":"31","author":"Jennings","year":"2020","journal-title":"Rendiconti Lincei Scienze Fisiche e Naturali"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"012707","DOI":"10.1103\/PhysRevE.87.012707","article-title":"Improved contact prediction in proteins: Using pseudolikelihoods to infer Potts models","volume":"87","author":"Ekeberg","year":"2013","journal-title":"Phys. Rev. E"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Boomsma, W., Ferkinghoff-Borg, J., and Lindorff-Larsen, K. (2014). Combining experiments and simulations using the maximum entropy principle. PLoS Comput. Biol., 10.","DOI":"10.1371\/journal.pcbi.1003406"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"6062","DOI":"10.1073\/pnas.1506257112","article-title":"Topology, structures, and energy landscapes of human chromosomes","volume":"112","author":"Zhang","year":"2015","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Cesari, A., Rei\u00dfer, S., and Bussi, G. (2018). Using the maximum entropy principle to combine simulations and solution experiments. Computation, 6.","DOI":"10.3390\/computation6010015"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Farr\u00e9, P., and Emberly, E. (2018). A maximum-entropy model for predicting chromatin contacts. PLoS Comput. Biol., 14.","DOI":"10.1371\/journal.pcbi.1005956"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"707","DOI":"10.1093\/bioinformatics\/16.8.707","article-title":"Genetic network inference: From co-expression clustering to reverse engineering","volume":"16","author":"Liang","year":"2000","journal-title":"Bioinformatics"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"19033","DOI":"10.1073\/pnas.0609152103","article-title":"Using the principle of entropy maximization to infer genetic interaction networks from gene expression patterns","volume":"103","author":"Lezon","year":"2006","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"12494","DOI":"10.1073\/pnas.0902237106","article-title":"Maximum-entropy network analysis reveals a role for tumor necrosis factor in peripheral nerve development and function","volume":"106","author":"Dhadialla","year":"2009","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"10324","DOI":"10.1073\/pnas.1005283107","article-title":"Information-theoretic analysis of phenotype changes in early stages of carcinogenesis","volume":"107","author":"Remacle","year":"2010","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Sanguinetti, G., and Huynh-Thu, V.A. (2019). Gene regulatory network inference: An introductory survey. Gene Regulatory Networks, Springer.","DOI":"10.1007\/978-1-4939-8882-2"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Locasale, J.W., and Wolf-Yadlin, A. (2009). Maximum entropy reconstructions of dynamic signaling networks from quantitative proteomics data. PLoS ONE, 4.","DOI":"10.1371\/journal.pone.0006522"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"6112","DOI":"10.1073\/pnas.1001149107","article-title":"Maximal entropy inference of oncogenicity from phosphorylation signaling","volume":"107","author":"Graeber","year":"2010","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1089\/cmb.2012.0241","article-title":"Reconstructing Boolean models of signaling","volume":"20","author":"Sharan","year":"2013","journal-title":"J. Comput. Biol."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1007","DOI":"10.1038\/nature04701","article-title":"Weak pairwise correlations imply strongly correlated network states in a neural population","volume":"440","author":"Schneidman","year":"2006","journal-title":"Nature"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"8254","DOI":"10.1523\/JNEUROSCI.1282-06.2006","article-title":"The structure of multi-neuron firing patterns in primate retina","volume":"26","author":"Shlens","year":"2006","journal-title":"J. Neurosci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2278","DOI":"10.1093\/bioinformatics\/btz925","article-title":"MPF\u2013BML: A standalone GUI-based package for maximum entropy model inference","volume":"36","author":"Quadeer","year":"2020","journal-title":"Bioinformatics"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1523\/JNEUROSCI.3359-07.2008","article-title":"A maximum entropy model applied to spatial and temporal correlations from cortical networks in vitro","volume":"28","author":"Tang","year":"2008","journal-title":"J. Neurosci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"14058","DOI":"10.1073\/pnas.0906705106","article-title":"Neuronal couplings between retinal ganglion cells inferred by efficient inverse statistical physics methods","volume":"106","author":"Cocco","year":"2009","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Roudi, Y., Nirenberg, S., and Latham, P.E. (2009). Pairwise maximum entropy models for studying large biological systems: When they can work and when they ca not. PLoS Comput. Biol., 5.","DOI":"10.1371\/journal.pcbi.1000380"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"14419","DOI":"10.1073\/pnas.1004906107","article-title":"Optimal population coding by noisy spiking neurons","volume":"107","author":"Prentice","year":"2010","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1038\/nature09178","article-title":"Sparse coding and high-order correlations in fine-scale cortical networks","volume":"466","author":"Ohiorhenuan","year":"2010","journal-title":"Nature"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"89","DOI":"10.3390\/e12010089","article-title":"Maximum entropy approaches to living neural networks","volume":"12","author":"Yeh","year":"2010","journal-title":"Entropy"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Granot-Atedgi, E., Tka\u010dik, G., Segev, R., and Schneidman, E. (2013). Stimulus-dependent maximum entropy models of neural population codes. PLoS Comput. Biol., 9.","DOI":"10.1371\/journal.pcbi.1002922"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"P03011","DOI":"10.1088\/1742-5468\/2013\/03\/P03011","article-title":"The simplest maximum entropy model for collective behavior in a neural network","volume":"2013","author":"Marre","year":"2013","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"042321","DOI":"10.1103\/PhysRevE.95.042321","article-title":"Random versus maximum entropy models of neural population activity","volume":"95","author":"Ferrari","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Rostami, V., Mana, P.P., Gr\u00fcn, S., and Helias, M. (2017). Bistability, non-ergodicity, and inhibition in pairwise maximum-entropy models. PLoS Comput. Biol., 13.","DOI":"10.1371\/journal.pcbi.1005762"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Nghiem, T.A., Tele\u0144czuk, B., Marre, O., Destexhe, A., and Ferrari, U. (2018). Maximum entropy models reveal the correlation structure in cortical neural activity during wakefulness and sleep. bioRxiv, 243857.","DOI":"10.1101\/243857"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1089\/1066527041410418","article-title":"Maximum entropy modeling of short sequence motifs with applications to RNA splicing signals","volume":"11","author":"Yeo","year":"2004","journal-title":"J. Comput. Biol."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"5405","DOI":"10.1073\/pnas.1001705107","article-title":"Maximum entropy models for antibody diversity","volume":"107","author":"Mora","year":"2010","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Santolini, M., Mora, T., and Hakim, V. (2014). A general pairwise interaction model provides an accurate description of in vivo transcription factor binding sites. PLoS ONE, 9.","DOI":"10.1371\/journal.pone.0099015"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"2172","DOI":"10.1093\/bib\/bbaa041","article-title":"DNA sequence symmetries from randomness: The origin of the Chargaff\u2019s second parity rule","volume":"22","author":"Fariselli","year":"2020","journal-title":"Brief. Bioinform."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Fernandez-de Cossio-Diaz, J., and Mulet, R. (2019). Maximum entropy and population heterogeneity in continuous cell cultures. PLoS Comput. Biol., 15.","DOI":"10.1371\/journal.pcbi.1006823"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"995","DOI":"10.1111\/j.1365-246X.2007.03530.x","article-title":"Maximum entropy regularization of the geomagnetic core field inverse problem","volume":"171","author":"Jackson","year":"2007","journal-title":"Geophys. J. Int."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"e00596","DOI":"10.1016\/j.heliyon.2018.e00596","article-title":"An introduction to the maximum entropy approach and its application to inference problems in biology","volume":"4","year":"2018","journal-title":"Heliyon"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"124301","DOI":"10.1118\/1.4898202","article-title":"Models and methods for analyzing DCE-MRI: A review","volume":"41","author":"Khalifa","year":"2014","journal-title":"Med. Phys."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1593\/tlo.13922","article-title":"Dynamic contrast-enhanced magnetic resonance imaging in prostate cancer clinical trials: Potential roles and possible pitfalls","volume":"7","author":"Fennessy","year":"2014","journal-title":"Transl. Oncol."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1593\/tlo.13838","article-title":"Variations of dynamic contrast-enhanced magnetic resonance imaging in evaluation of breast cancer therapy response: A multicenter data analysis challenge","volume":"7","author":"Huang","year":"2014","journal-title":"Transl. Oncol."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1016\/j.tranon.2016.03.005","article-title":"Hypo-vascular liver metastases treated with transarterial chemoembolization: Assessment of early response by volumetric contrast-enhanced and diffusion-weighted magnetic resonance imaging","volume":"9","author":"Sobhani","year":"2016","journal-title":"Transl. Oncol."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"699","DOI":"10.1016\/j.tranon.2019.02.005","article-title":"Diffusion-weighted magnetic resonance imaging is useful for the response evaluation of chemotherapy and\/or radiotherapy to recurrent lesions of lung cancer","volume":"12","author":"Usuda","year":"2019","journal-title":"Transl. Oncol."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1593\/tlo.12319","article-title":"Mapping tumor hypoxia in vivo using pattern recognition of dynamic contrast-enhanced MRI data","volume":"5","author":"Stoyanova","year":"2012","journal-title":"Transl. Oncol."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"1627","DOI":"10.1109\/TMI.2006.884210","article-title":"Bayesian methods for pharmacokinetic models in dynamic contrast-enhanced magnetic resonance imaging","volume":"25","author":"Schmid","year":"2006","journal-title":"IEEE Trans. Med. Imaging"},{"key":"ref_56","first-page":"223","article-title":"Estimating kinetic parameters from dynamic contrast-enhanced T1-weighted MRI of a diffusable tracer: Standardized quantities and symbols","volume":"10","author":"Tofts","year":"1999","journal-title":"J. Magn. Reson. Imaging Off. J. Int. Soc. Magn. Reson. Med."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"103634","DOI":"10.1016\/j.compbiomed.2020.103634","article-title":"DCE-MRI pharmacokinetic parameter maps for cervical carcinoma prediction","volume":"118","author":"Shao","year":"2020","journal-title":"Comput. Biol. Med."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1002\/mp.13885","article-title":"Tracer kinetic models as temporal constraints during brain tumor DCE-MRI reconstruction","volume":"47","author":"Lingala","year":"2020","journal-title":"Med. Phys."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"3447","DOI":"10.1002\/mp.14222","article-title":"Estimation of pharmacokinetic parameters from DCE-MRI by extracting long and short time-dependent features using an LSTM network","volume":"47","author":"Zou","year":"2020","journal-title":"Med. Phys."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"101690","DOI":"10.1016\/j.media.2020.101690","article-title":"Stochastic Gradient Langevin dynamics for joint parameterization of tracer kinetic models, input functions, and T1 relaxation-times from undersampled k-space DCE-MRI","volume":"62","author":"Dikaios","year":"2020","journal-title":"Med. Image Anal."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1002\/mrm.1910170208","article-title":"Measurement of the blood-brain barrier permeability and leakage space using dynamic MR imaging. 1. Fundamental concepts","volume":"17","author":"Tofts","year":"1991","journal-title":"Magn. Reson. Med."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"174","DOI":"10.1002\/mrm.1910240119","article-title":"Measurement of blood-brain barrier permeability using dynamic Gd-DTPA scanning\u2014A comparison of methods","volume":"24","author":"Larsson","year":"1992","journal-title":"Magn. Reson. Med."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"420","DOI":"10.1002\/mrm.20161","article-title":"Microcirculation and microvasculature in breast tumors: Pharmacokinetic analysis of dynamic MR image series","volume":"52","author":"Brix","year":"2004","journal-title":"Magn. Reson. Med."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"041003","DOI":"10.1115\/1.4026108","article-title":"Cerebral Blood Flow in a Healthy Circle of Willis and Two Intracranial Aneurysms: Computational Fluid Dynamics Versus Four-Dimensional Phase-Contrast Magnetic Resonance Imaging","volume":"15","author":"Berg","year":"2014","journal-title":"ASME J. Biomech. Eng."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"2393","DOI":"10.1088\/0031-9155\/52\/9\/005","article-title":"Bayesian estimation of pharmacokinetic parameters for DCE-MRI with a robust treatment of enhancement onset time","volume":"52","author":"Orton","year":"2007","journal-title":"Phys. Med. Biol."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"989","DOI":"10.1016\/j.media.2014.05.001","article-title":"Direct parametric reconstruction from undersampled (k, t)-space data in dynamic contrast enhanced MRI","volume":"18","author":"Dikaios","year":"2014","journal-title":"Med. Image Anal."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1007\/BF02353787","article-title":"Fitting nonlinear regression models with correlated errors to individual pharmacodynamic data using SAS software","volume":"23","author":"Bender","year":"1995","journal-title":"J. Pharmacokinet. Biopharm."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"1073","DOI":"10.1002\/jmri.20898","article-title":"T1 measurement of flowing blood and arterial input function determination for quantitative 3D T1-weighted DCE-MRI","volume":"25","author":"Cheng","year":"2007","journal-title":"J. Magn. Reson. Imaging JMRI"},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"291","DOI":"10.4329\/wjr.v4.i7.291","article-title":"Impact of the arterial input function on microvascularization parameter measurements using dynamic contrast-enhanced ultrasonography","volume":"4","author":"Gauthier","year":"2012","journal-title":"World J. Radiol."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"736","DOI":"10.1002\/jmri.21489","article-title":"Investigation and optimization of parameter accuracy in dynamic contrast-enhanced MRI","volume":"28","author":"Cheng","year":"2008","journal-title":"J. Magn. Reson. Imaging"},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1016\/j.mri.2014.10.004","article-title":"Simulating the effect of input errors on the accuracy of Tofts\u2019 pharmacokinetic model parameters","volume":"33","author":"Lavini","year":"2015","journal-title":"Magn. Reson. Imaging"},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"e241","DOI":"10.1016\/j.acra.2018.10.018","article-title":"Selection of fitting model and arterial input function for repeatability in dynamic contrast-enhanced prostate MRI","volume":"26","author":"Peled","year":"2019","journal-title":"Acad. Radiol."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"56","DOI":"10.18383\/j.tom.2015.00184","article-title":"The impact of arterial input function determination variations on prostate dynamic contrast-enhanced magnetic resonance imaging pharmacokinetic modeling: A multicenter data analysis challenge","volume":"2","author":"Huang","year":"2016","journal-title":"Tomography"},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"99","DOI":"10.18383\/j.tom.2018.00027","article-title":"The impact of arterial input function determination variations on prostate dynamic contrast-enhanced magnetic resonance imaging pharmacokinetic modeling: A multicenter data analysis challenge, part II","volume":"5","author":"Huang","year":"2019","journal-title":"Tomography"},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/j.mri.2017.04.006","article-title":"Effects of arterial input function selection on kinetic parameters in brain dynamic contrast-enhanced MRI","volume":"40","author":"Keil","year":"2017","journal-title":"Magn. Reson. Imaging"},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"993","DOI":"10.1002\/mrm.21066","article-title":"Experimentally-derived functional form for a population-averaged high-temporal-resolution arterial input function for dynamic contrast-enhanced MRI","volume":"56","author":"Parker","year":"2006","journal-title":"Magn. Reson. Med. Off. J. Int. Soc. Magn. Reson. Med."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"1991","DOI":"10.1007\/s00330-015-4012-9","article-title":"Assessment of repeatability and treatment response in early phase clinical trials using DCE-MRI: Comparison of parametric analysis using MR-and CT-derived arterial input functions","volume":"26","author":"Rata","year":"2016","journal-title":"Eur. Radiol."},{"key":"ref_78","first-page":"457","article-title":"Method for quantitative mapping of dynamic MRI contrast agent uptake in human tumors","volume":"14","author":"Rijpkema","year":"2001","journal-title":"J. Magn. Reson. Imaging Off. J. Int. Soc. Magn. Reson. Med."},{"key":"ref_79","first-page":"791","article-title":"Scan-rescan variability in perfusion assessment of tumors in MRI using both model and data-derived arterial input functions","volume":"28","author":"Ashton","year":"2008","journal-title":"J. Magn. Reson. Imaging Off. J. Int. Soc. Magn. Reson. Med."},{"key":"ref_80","first-page":"167","article-title":"Pharmokinetics of Gd-DTPA\/Dimeglumine after intravenous injection into healthy volunteers","volume":"16","author":"Weinmann","year":"1984","journal-title":"Physiol. Chem. Phys. Med. NMR"},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1002\/mrm.1910360209","article-title":"Measurement of the Arterial Concentration of Gd-DTPA Using MRI: A step toward Quantitative Perfusion Imaging","volume":"36","author":"Rostrup","year":"1996","journal-title":"Magn. Reson. Med."},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"461","DOI":"10.3414\/ME17-01-0027","article-title":"Maximum Entropy Approach in Dynamic Contrast-Enhanced Magnetic Resonance Imaging","volume":"56","author":"Farsani","year":"2017","journal-title":"Methods Inf. Med."},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1016\/j.cad.2010.12.015","article-title":"Teaching\u2013learning-based optimization: A novel method for constrained mechanical design optimization problems","volume":"43","author":"Rao","year":"2011","journal-title":"Comput.-Aided Des."},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"366","DOI":"10.1016\/j.neucom.2018.06.076","article-title":"A survey of teaching\u2013learning-based optimization","volume":"335","author":"Zou","year":"2019","journal-title":"Neurocomputing"},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"564","DOI":"10.1002\/jmri.1880070318","article-title":"Probing tumor microvascularity by measurement, analysis and display of contrast agent uptake kinetics","volume":"7","author":"Parker","year":"1997","journal-title":"J. Magn. Reson. Imaging"},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"621","DOI":"10.1148\/rg.262045187","article-title":"Magnetic resonance imaging workbench: Analysis and visualization of dynamic contrast-enhanced MR imaging data","volume":"26","author":"Collins","year":"2006","journal-title":"Radiographics"},{"key":"ref_87","doi-asserted-by":"crossref","unstructured":"Jackson, A., Parker, G.J.M., and Buckley, D.L. (2005). Measuring Contrast Agent Concentration in T1-Weighted Dynamic Contrast-Enhanced MRI. Dynamic Contrast-Enhanced Magntic Resoncance Imaging in Oncology, Springer. Chapter 5.","DOI":"10.1007\/b137553"},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"1065","DOI":"10.1214\/aoms\/1177704472","article-title":"On estimation of a probability density function and mode","volume":"33","author":"Parzen","year":"1962","journal-title":"Ann. Math. Stat."},{"key":"ref_89","first-page":"509","article-title":"Functional tumor imaging withdynamic contrast-enhanced magnetic resonance imaging","volume":"17","author":"Choyke","year":"2003","journal-title":"Magn. Reson. Med."},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"858","DOI":"10.1002\/mrm.20022","article-title":"Efficient method for calculating kinetic parameters using T1-weighted dynamic contrast-enhanced magnetic resonance imaging","volume":"51","author":"Murase","year":"2004","journal-title":"Magn. Reson. Med."},{"key":"ref_91","unstructured":"Mohammad-Djafari, A. (2004, January 1\u20133). Bayesian Image Processing. Proceedings of the Fifth International Conference on Modelling, Computation and Optimization in Information Systems and Management Sciences (MCO 2004), Metz, France."},{"key":"ref_92","unstructured":"Mohammad-Djafari, A., and Demoment, G. (1990, January 3\u20136). Estimating priors in maximum entropy image processing. Proceedings of the International Conference on Acoustics, Speech, and Signal Processing, Albuquerque, NM, USA."},{"key":"ref_93","doi-asserted-by":"crossref","unstructured":"Mohammad-Djafari, A. (1996). A full Bayesian approach for inverse problems. Maximum Entropy and Bayesian Methods, Springer.","DOI":"10.1007\/978-94-011-5430-7_16"},{"key":"ref_94","unstructured":"Hadamard, J. (1932). Le Probleme de Cauchy et les \u00c9quations aux D\u00e9riv\u00e9es Partielles Lin\u00e9aires Hyperboliques, Paris Russian Translation."},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1016\/0041-5553(67)90117-6","article-title":"Solution of the Fredholm equation of the first kind in a statistical ensemble of smooth functions","volume":"7","author":"Turchin","year":"1967","journal-title":"USSR Comput. Math. Math. Phys."},{"key":"ref_96","doi-asserted-by":"crossref","unstructured":"Denisova, N. (2019). Bayesian maximum-a posteriori approach with global and local regularization to image reconstruction problem in medical emission tomography. Entropy, 21.","DOI":"10.3390\/e21111108"},{"key":"ref_97","doi-asserted-by":"crossref","unstructured":"Sparavigna, A.C. (2019). Entropy in image analysis. Entropy, 21.","DOI":"10.3390\/e21050502"},{"key":"ref_98","doi-asserted-by":"crossref","unstructured":"Skilling, J. (1988). The axioms of maximum entropy. Maximum-Entropy and Bayesian Methods in Science and Engineering, Springer.","DOI":"10.1007\/978-94-015-7860-8"},{"key":"ref_99","doi-asserted-by":"crossref","first-page":"729","DOI":"10.1016\/0895-7177(89)90358-0","article-title":"An Algorithm for Maximum Entropy Image Reconstruction form Noisy Data","volume":"12","author":"Elfving","year":"1989","journal-title":"Mathl. Comput. Model."},{"key":"ref_100","doi-asserted-by":"crossref","first-page":"1140","DOI":"10.1016\/j.enconman.2006.10.004","article-title":"Wind energy analysis based on maximum entropy principle (MEP)-type distribution function","volume":"48","author":"Akpinar","year":"2007","journal-title":"Energy Convers. Manag."},{"key":"ref_101","unstructured":"Casella, G., and Berger, R. (2002). Statistical Inference 2, Duxbury."},{"key":"ref_102","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.ijepes.2013.02.023","article-title":"Optimal distributed generation location and size using a modified teaching-learning based optimization algorithm","volume":"50","author":"Mena","year":"2013","journal-title":"Int. J. Electr. Power Energy Syst."},{"key":"ref_103","first-page":"621","article-title":"Estimation of parameters in the weibdl distribution","volume":"9","author":"Bain","year":"1967","journal-title":"Technometrics"},{"key":"ref_104","first-page":"132","article-title":"The estimation of the parameters of the Weibull wind speed distribution for wind energy utilization purposes","volume":"3","author":"Stevens","year":"1979","journal-title":"Wind Eng."},{"key":"ref_105","doi-asserted-by":"crossref","first-page":"350","DOI":"10.1175\/1520-0450(1978)017<0350:MFEWSF>2.0.CO;2","article-title":"Methods for estimating wind speed frequency distributions","volume":"17","author":"Justus","year":"1978","journal-title":"J. Appl. Meteorol."},{"key":"ref_106","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/j.enconman.2010.06.015","article-title":"Probability distributions for offshore wind speeds","volume":"52","author":"Morgan","year":"2011","journal-title":"Energy Convers. Manag."},{"key":"ref_107","doi-asserted-by":"crossref","first-page":"1761","DOI":"10.1016\/j.enconman.2009.03.020","article-title":"A new method to estimate Weibull parameters for wind energy applications","volume":"50","author":"Dinler","year":"2009","journal-title":"Energy Convers. Manag."},{"key":"ref_108","doi-asserted-by":"crossref","first-page":"976","DOI":"10.1016\/j.egypro.2015.11.596","article-title":"Comparative study of five methods to estimate Weibull parameters for wind speed on Phangan Island, Thailand","volume":"79","author":"Werapun","year":"2015","journal-title":"Energy Procedia"},{"key":"ref_109","doi-asserted-by":"crossref","first-page":"931","DOI":"10.1016\/j.apenergy.2013.07.040","article-title":"Study on the Maximum Entropy Principle applied to the annual wind speed probability distribution: A case study for observations of intertidal zone anemometer towers of Rudong in East China Sea","volume":"114","author":"Zhang","year":"2014","journal-title":"Appl. Energy"},{"key":"ref_110","doi-asserted-by":"crossref","first-page":"5516","DOI":"10.1007\/s11661-017-4294-4","article-title":"Weibull modulus estimated by the non-linear least squares method: A solution to deviation occurring in traditional Weibull estimation","volume":"48","author":"Li","year":"2017","journal-title":"Metall. Mater. Trans. A"},{"key":"ref_111","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/S0167-6105(99)00122-1","article-title":"Modern estimation of the parameters of the Weibull wind speed distribution for wind energy analysis","volume":"85","author":"Seguro","year":"2000","journal-title":"J. Wind. Eng. Ind. Aerodyn."},{"key":"ref_112","doi-asserted-by":"crossref","first-page":"867","DOI":"10.1016\/S0167-6105(00)00088-X","article-title":"\u201cDiscussion on modern estimation of the parameters of the Weibull wind speed distribution for wind speed energy analysis\u201d by J.V. Seguro, T.W. Lambert","volume":"89","author":"Cook","year":"2001","journal-title":"J. Wind. Eng. Ind. Aerodyn."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/2\/155\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:04:26Z","timestamp":1760133866000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/2\/155"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,20]]},"references-count":112,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2022,2]]}},"alternative-id":["e24020155"],"URL":"https:\/\/doi.org\/10.3390\/e24020155","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2022,1,20]]}}}