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Some expressions and notations related to Equations 1 and 2 were presented incorrectly. The correct text and equations are below.The coefficients (β) in Cox's regression model are estimated by maximizing the partial likelihood function subject to a constraint on the L1-norm of the coefficients. The lasso estimator (β ) maximizes the objective function given below:Here l(β) is the log partial likelihood in the Cox model; for the exact form of this function, see ref. 41. The tuning parameter, λ in Equation 1, was chosen by 10-fold cross-validation. For the implementation, we used the R package "glmnet" (39).PROVAR was defined for each of the 222 TCGA samples as the sum of the estimated coefficients multiplied by protein expression levels, as shown below. Here i represents patients (i = 1,...,222), j represents proteins with nonzero coefficients (j = 1, ..., m), β j is the lasso coefficient of the jth protein marker, and X ij is the expression level of the jth protein for the ith patient. (Equation 2)The JCI regrets the error.

[1] Fluid flow through porous media, and the thermal, electrical, and acoustic properties of these materials, is largely controlled by the geometry and topology (GT) of the pore system, which can be considered as a network. Network extraction techniques have been applied in many research fields, including shape representation, pattern recognition, and artificial intelligence. However, the set of algorithms presented here significantly improves the efficiency of common thinning algorithms by introducing a sufficiency condition based on the idea of a simple set. This paper describes an efficient and accurate algorithm for extracting the geometrical/topological network that represents the pore structure of a porous medium, referred to as the GT network. The accurate medial axis and the specific GT description of the network are achieved by applying symmetrical and interval strategies during the erosion step in the image processing. The GT network extraction algorithm presented here involves a number of steps, including (1) calculation of the three-dimensional Euclidean distance map; (2) clustering of voxels; (3) extraction of the network of the pore space; (4) partitioning of the pore space; and (5) computation of shape factors. The focus of this paper is mainly on the thinning method that underpins points 1-3. The paper is primarily a method description, but we illustrate the functionality of the technique by extracting a pore scale GT network from microcomputer tomography images of three sandstones.

The creation of a 3D pore-scale model of a porous medium is often an essential step in quantitatively characterising the medium and predicting its transport properties. Here we describe a new stochastic pore space reconstruction approach that uses thin section images as its main input. The approach involves using a third-order Markov mesh where we introduce a new algorithm that creates the reconstruction in a single scan, thus overcoming the computational issues normally associated with Markov chain methods. The technique is capable of generating realistic pore architecture models (PAMs), and examples are presented for a range of fairly homogenous rock samples as well as for one heterogeneous soil sample. We then apply a Lattice-Boltzmann (LB) scheme to calculate the permeabilities of the PAMs, which in all cases closely match the measured values of the original samples. We also develop a set of software methods -referred to as pore analysis tools (PATs) -to quantitatively analyse the reconstructed pore systems. These tools reveal the pore connectivity and pore size distribution, from which we can simulate the mercury injection process, which in turn reproduces the measured curves very closely. Analysis of the topological descriptors reveals that a connectivity function based on the specific Euler number may serve as a simple predictor of the threshold pressure for geo-materials.

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