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Abstract BackgroundDelta radiomics is a high‐throughput computational technique used to describe quantitative changes in serial, time‐series imaging by considering the relative change in radiomic features of images extracted at two distinct time points. Recent work has demonstrated a lack of prognostic signal of radiomic features extracted using this technique. We hypothesize that this lack of signal is due to the fundamental assumptions made when extracting features via delta radiomics, and that other methods should be investigated. PurposeThe purpose of this work was to show a proof‐of‐concept of a new radiomics paradigm for sparse, time‐series imaging data, where features are extracted from a spatial‐temporal manifold modeling the time evolution between images, and to assess the prognostic value on patients with oropharyngeal cancer (OPC). MethodsTo accomplish this, we developed an algorithm to mathematically describe the relationship between two images acquired at time and . These images serve as boundary conditions of a partial differential equation describing the transition from one image to the other. To solve this equation, we propagate the position and momentum of each voxel according to Fokker–Planck dynamics (i.e., a technique common in statistical mechanics). This transformation is driven by an underlying potential force uniquely determined by the equilibrium image. The solution generates a spatial‐temporal manifold (3 spatial dimensions + time) from which we define dynamic radiomic features. First, our approach was numerically verified by stochastically sampling dynamic Gaussian processes of monotonically decreasing noise. The transformation from high to low noise was compared between our Fokker–Planck estimation and simulated ground‐truth. To demonstrate feasibility and clinical impact, we applied our approach to¹⁸F‐FDG‐PET images to estimate early metabolic response of patients (n = 57) undergoing definitive (chemo)radiation for OPC. Images were acquired pre‐treatment and 2‐weeks intra‐treatment (after 20 Gy). Dynamic radiomic features capturing changes in texture and morphology were then extracted. Patients were partitioned into two groups based on similar dynamic radiomic feature expression via k‐means clustering and compared by Kaplan–Meier analyses with log‐rank tests (p < 0.05). These results were compared to conventional delta radiomics to test the added value of our approach. ResultsNumerical results confirmed our technique can recover image noise characteristics given sparse input data as boundary conditions. Our technique was able to model tumor shrinkage and metabolic response. While no delta radiomics features proved prognostic, Kaplan–Meier analyses identified nine significant dynamic radiomic features. The most significant feature was Gray‐Level‐Size‐Zone‐Matrix gray‐level variance (p = 0.011), which demonstrated prognostic improvement over its corresponding delta radiomic feature (p = 0.722). ConclusionsWe developed, verified, and demonstrated the prognostic value of a novel, physics‐based radiomics approach over conventional delta radiomics via data assimilation of quantitative imaging and differential equations. 

Abstract BackgroundDue to intrinsic differences in data formatting, data structure, and underlying semantic information, the integration of imaging data with clinical data can be non‐trivial. Optimal integration requires robust data fusion, that is, the process of integrating multiple data sources to produce more useful information than captured by individual data sources. Here, we introduce the concept offusion qualityfor deep learning problems involving imaging and clinical data. We first provide a general theoretical framework and numerical validation of our technique. To demonstrate real‐world applicability, we then apply our technique to optimize the fusion of CT imaging and hepatic blood markers to estimate portal venous hypertension, which is linked to prognosis in patients with cirrhosis of the liver. PurposeTo develop a measurement method of optimal data fusion quality deep learning problems utilizing both imaging data and clinical data. MethodsOur approach is based on modeling the fully connected layer (FCL) of a convolutional neural network (CNN) as a potential function, whose distribution takes the form of the classical Gibbs measure. The features of the FCL are then modeled as random variables governed by state functions, which are interpreted as the different data sources to be fused. The probability density of each source, relative to the probability density of the FCL, represents a quantitative measure of source‐bias. To minimize this source‐bias and optimize CNN performance, we implement a vector‐growing encoding scheme called positional encoding, where low‐dimensional clinical data are transcribed into a rich feature space that complements high‐dimensional imaging features. We first provide a numerical validation of our approach based on simulated Gaussian processes. We then applied our approach to patient data, where we optimized the fusion of CT images with blood markers to predict portal venous hypertension in patients with cirrhosis of the liver. This patient study was based on a modified ResNet‐152 model that incorporates both images and blood markers as input. These two data sources were processed in parallel, fused into a single FCL, and optimized based on our fusion quality framework. ResultsNumerical validation of our approach confirmed that the probability density function of a fused feature space converges to a source‐specific probability density function when source data are improperly fused. Our numerical results demonstrate that this phenomenon can be quantified as a measure of fusion quality. On patient data, the fused model consisting of both imaging data and positionally encoded blood markers at the theoretically optimal fusion quality metric achieved an AUC of 0.74 and an accuracy of 0.71. This model was statistically better than the imaging‐only model (AUC = 0.60; accuracy = 0.62), the blood marker‐only model (AUC = 0.58; accuracy = 0.60), and a variety of purposely sub‐optimized fusion models (AUC = 0.61–0.70; accuracy = 0.58–0.69). ConclusionsWe introduced the concept of data fusion quality for multi‐source deep learning problems involving both imaging and clinical data. We provided a theoretical framework, numerical validation, and real‐world application in abdominal radiology. Our data suggests that CT imaging and hepatic blood markers provide complementary diagnostic information when appropriately fused. 

In this paper we derive the best constant for the following $L^{\mathrm{\infty <\#comment/>}}$-type Gagliardo-Nirenberg interpolation inequality $\Vert <\#comment/>u{\Vert <\#comment/>}_{L^{\mathrm{\infty <\#comment/>}}}\le <\#comment/>C_{q,\mathrm{\infty <\#comment/>},p}\Vert <\#comment/>u{\Vert <\#comment/>}_{L^{q+1}}^{1-<\#comment/>\mathrm{\theta <\#comment/>}}\Vert <\#comment/>\mathrm{\nabla <\#comment/>}u{\Vert <\#comment/>}_{L^p}^{\mathrm{\theta <\#comment/>}},\mathrm{\theta <\#comment/>}=\frac{pd}{dp+(p-<\#comment/>d)(q+1)},$where parameters $q$and $p$satisfy the conditions $p>d\ge <\#comment/>1$, $q\ge <\#comment/>0$. The best constant $C_{q,\mathrm{\infty <\#comment/>},p}$is given by $C_{q,\mathrm{\infty <\#comment/>},p}={\mathrm{\theta <\#comment/>}}^{-<\#comment/>\frac{\mathrm{\theta <\#comment/>}}{p}}(1-<\#comment/>\mathrm{\theta <\#comment/>}{)}^{\frac{\mathrm{\theta <\#comment/>}}{p}}{M}_c^{-<\#comment/>\frac{\mathrm{\theta <\#comment/>}}{d}},M_c≔{\int <\#comment/>}_{{\mathbb{R}}^d}{u}_{c,\mathrm{\infty <\#comment/>}}^{q+1}dx,$where $u_{c,\mathrm{\infty <\#comment/>}}$is the unique radial non-increasing solution to a generalized Lane-Emden equation. The case of equality holds when $u=A{u}_{c,\mathrm{\infty <\#comment/>}}\left(\mathrm{\lambda <\#comment/>}\right(x-<\#comment/>x_0\left)\right)$for any real numbers $A$, $\mathrm{\lambda <\#comment/>}>0$and $x_0\in <\#comment/>{\mathbb{R}}^d$. In fact, the generalized Lane-Emden equation in ${\mathbb{R}}^d$contains a delta function as a source and it is a Thomas-Fermi type equation. For $q=0$or $d=1$, $u_{c,\mathrm{\infty <\#comment/>}}$have closed form solutions expressed in terms of the incomplete Beta functions. Moreover, we show that $u_{c,m}\to <\#comment/>u_{c,\mathrm{\infty <\#comment/>}}$and $C_{q,m,p}\to <\#comment/>C_{q,\mathrm{\infty <\#comment/>},p}$as $m\to <\#comment/>+\mathrm{\infty <\#comment/>}$for $d=1$, where $u_{c,m}$and $C_{q,m,p}$are the function achieving equality and the best constant of $L^m$-type Gagliardo-Nirenberg interpolation inequality, respectively. 

We propose a high-order stochastic–statistical moment closure model for efficient ensemble prediction of leading-order statistical moments and probability density functions in multiscale complex turbulent systems. The statistical moment equations are closed by a precise calibration of the high-order feedbacks using ensemble solutions of the consistent stochastic equations, suitable for modeling complex phenomena including non-Gaussian statistics and extreme events. To address challenges associated with closely coupled spatiotemporal scales in turbulent states and expensive large ensemble simulation for high-dimensional systems, we introduce efficient computational strategies using the random batch method (RBM). This approach significantly reduces the required ensemble size while accurately capturing essential high-order structures. Only a small batch of small-scale fluctuation modes is used for each time update of the samples, and exact convergence to the full model statistics is ensured through frequent resampling of the batches during time evolution. Furthermore, we develop a reduced-order model to handle systems with really high dimensions by linking the large number of small-scale fluctuation modes to ensemble samples of dominant leading modes. The effectiveness of the proposed models is validated by numerical experiments on the one-layer and two-layer Lorenz ‘96 systems, which exhibit representative chaotic features and various statistical regimes. The full and reduced-order RBM models demonstrate uniformly high skill in capturing the time evolution of crucial leading-order statistics, non-Gaussian probability distributions, while achieving significantly lower computational cost compared to direct Monte-Carlo approaches. The models provide effective tools for a wide range of real-world applications in prediction, uncertainty quantification, and data assimilation.