A Spatial Statistical Analysis of Tumor Growth

Cressie, Noel; Hulting, Frederick L.

doi:10.2307/2290257

Cited by 10 publications

(7 citation statements)

References 39 publications

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“…By restricting the events to X t , the spatial point process is equivalently Poisson with intensity λ t+1 in X t . Equation ( 7) defines a SVAR model that can exhibit both growth and recession; in [11], it was used to characterize growth of breast cancer cells on a glass slide photographed 72 hours apart, and its hitting function was derived. On a transformed space, Z t+1 was assumed to be a disk of random radius R t+1 and, based on the hitting function, method-of-moments estimators (and standard errors) were found for λ t+1 , E(R t+1 ), and var(R t+1 ); t = 1, 2, .…”

Section: And λ(•) Is the Intensity Function In Units Of 1/(d-dimensional Volume)mentioning

confidence: 99%

“…We also investigate the predictive nature of the model by simulating realizations of the SVAR model at the next time point and forecasting the pixelwise probability of being a concentrated-precipitation pixel at that time point. This model was adapted from [11], where the growth of in vitro breast-cancer cells was modeled over a 72-hour period. However, in the case of concentrated precipitation, the rapid movement of precipitation fronts means that a different model and different estimation techniques are needed.…”

Section: Introductionmentioning

confidence: 99%

“…Definitions of random sets and Boolean models can be found in [2,10,29,43,48,51]. Many authors use convex random sets, Boolean models, and hitting functions for the analysis of tumor growth and biological data; some important references are [11,14,18,30]. Recently, growth models have also been considered in the meteorological field for the analysis of storm cells and rainfall, such as in [27] and [28], where a methodology based on nonoverlapping random disk models is used in a multistage hierarchical Bayesian context.…”

Section: Introductionmentioning

confidence: 99%

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Dynamical random-set modeling of concentrated precipitation in North America

Cressie

Assunção

Holan

et al. 2012

Statistics and Its Interface

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In order to study climate at scales where policy decisions can be made, regional climate models (RCMs) have been developed with much finer resolution (∼50 km) than the ∼500 km resolution of atmosphere-ocean general circulation models (AOGCMs). The North American Regional Climate Change Assessment Program (NARCCAP) is an international program that provides 50-km resolution climate output for the United States, Canada, and northern Mexico. In Phase I, there are six RCMs, from which we choose one to illustrate our methodology. The RCMs are updated every 3 hours and contain a number of variables, including temperature, precipitation, wind speed, wind direction, and air pressure; output is available from the years 1968-2000 and from the years 2038-2070. Precipitation is of particular interest to climate scientists, but it can be difficult to study because of its patchy nature: At hourly-up-to-monthly time scales, there are generally many zeroes over the precipitation field. In this research, we study sets of concentrated precipitation (i.e., the union of RCM pixels whose precipitation is above a given threshold), where we are interested in the way these sets evolve from one 3-hour period to the next. Assuming the sets are a realization of a time series of random sets, we are able to build dynamical models for the passage of rainfall fronts over 1-2 days. The dynamics are characterized by a growth/recession model for a time series of random sets, with several parameters that control how the concentrated precipitation changes over time.

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Section: And λ(•) Is the Intensity Function In Units Of 1/(d-dimensional Volume)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Dynamical random-set modeling of concentrated precipitation in North America

Cressie

Assunção

Holan

et al. 2012

Statistics and Its Interface

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“…Here, B(x, r) is a circular disc with centre x and radius r. In this model, t i is thought of as a time point of outburst and x i as the location of the outburst in the tumour, say. A closely related discrete-time Markov growth model has been discussed in detail in Cressie and Hulting (1992). This model can be characterized as a sequence of Boolean models,…”

Section: Related Growth Modelsmentioning

confidence: 99%

Lévy-based growth models

Jónsdóttir¹,

Schmiegel²,

Jensen³

2008

Bernoulli

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In the present paper, we give a condensed review, for the nonspecialist reader, of a new modelling framework for spatio-temporal processes, based on Lévy theory. We show the potential of the approach in stochastic geometry and spatial statistics by studying Lévy-based growth modelling of planar objects. The growth models considered are spatio-temporal stochastic processes on the circle. As a by product, flexible new models for space-time covariance functions on the circle are provided. An application of the Lévy-based growth models to tumour growth is discussed. This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2008, Vol. 14, No. 1, 62-90. This reprint differs from the original in pagination and typographic detail.

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“…Furthermore, t is an independent and uniform random variable on [0,2n] and addition of indices is understood as modulo 271. This set was used in Stoyan & Lippmann (1993) for modelling a figure which was before studied by Cressie & Hulting (1992): the irregular area covered by a human breast cancer grown on a flat dish covered with a nutrient medium. The Brownian circle is an alternative to Cressie's cancer model, being easier to simulate and producing figures more similar to the data, but offering no biological explanation.…”

Section: Models For Random Compact Setsmentioning

confidence: 99%

Random Sets: Models and Statistics

Stoyan

1998

Int Statistical Rev

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This paper surveys aspects of the theory of random closed sets, focussing on issues of practical and current interest. First, some historical remarks on this part of probability theory are made, where the important role of Georges Matheron is emphasized. Then, fundamental characteristics of the distribution of random closed sets are introduced. The very important Boolean model serves as an example for discussing mathematical and statistical problems. A number of further models is then considered, namely excursion sets of random fields, the system of edges of the Poisson Voronoi tessellation and various random systems of non-overlapping spheres. Finally, some ideas of particle statistics are presented, including some models of random compact sets.

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A Spatial Statistical Analysis of Tumor Growth

Cited by 10 publications

References 39 publications

Dynamical random-set modeling of concentrated precipitation in North America

Dynamical random-set modeling of concentrated precipitation in North America

Lévy-based growth models

Random Sets: Models and Statistics

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