Mathematical modeling of mass and energy transport for thermoembolization.

Background: Thermoembolization presents a unique treatment alternative for patients diagnosed with hepatocellular carcinoma. The approach delivers a reagent that undergoes an exothermic chemical reaction and combines the benefits of embolic as well as thermal- and chemical-ablative therapy modalities. The target tissue and vascular bed are subjected to simultaneous hyperthermia, ischemia, and chemical denaturation in a single procedure. To guide optimal delivery, we developed a mathematical model for understanding the competing diffusive and convective effects observed in thermoembolization delivery protocols.Methods: A mixture theory formulation was used to mathematically model thermoembolization as chemically reacting transport of an electrophile, dichloroacetyl chloride (DCACl), within porous living tissue. Mass and energy transport of each relevant constituent are considered. Specifically, DCACl is injected into the vessels and exothermically reacts with water in the blood or tissue to form dichloroacetic acid and hydrochloric acid. Neutralization reactions are assumed instantaneous in this approach. We validated the mathematical model predictions of temperature using MR thermometry of the thermoembolization procedure performed in ex vivo kidney.Results: Mathematical modeling predictions of tissue death were highly dependent on the vascular geometry, injection pressure, and intrinsic amount of exothermic energy released from the chemical species, and were able to recapitulate the temperature distributions observed in MR thermometry.Conclusion: These efforts present a first step toward formalizing a mathematical model for thermoembolization and are promising for providing insight for delivery protocol optimization. While our approach captured the observed experimental temperature measurements, larger-scale experimental validation is needed to prioritize additional model complexity and fidelity.

Finding representative electrocardiogram beat morphologies with CUR.

In this paper, we use the CUR matrix factorization as a means of dimension reduction to identify important subsequences in electrocardiogram (ECG) time series. As opposed to other factorizations typically used in dimension reduction that characterize data in terms of abstract representatives (for example, an orthogonal basis), the CUR factorization describes the data in terms of actual instances within the original data set. Therefore, the CUR characterization can be directly related back to the clinical setting. We apply CUR to a synthetic ECG data set as well as to data from the MIT-BIH Arrhythmia, MGH-MF, and Incart databases using the discrete empirical interpolation method (DEIM) and an incremental QR factorization. In doing so, we demonstrate that CUR is able to identify beat morphologies that are representative of the data set, including rare-occurring beat events, providing a robust summarization of the ECG data. We also see that using CUR-selected beats to label the remaining unselected beats via 1-nearest neighbor classification produces results comparable to those presented in other works. While the electrocardiogram is of particular interest here, this work demonstrates the utility of CUR in detecting representative subsequences in quasiperiodic physiological time series.

Comparison of reduced models for blood flow using Runge-Kutta discontinuous Galerkin methods.

One-dimensional blood flow models take the general form of nonlinear hyperbolic systems but differ in their formulation. One class of models considers the physically conserved quantities of mass and momentum, while another class describes mass and velocity. Further, the averaging process employed in the model derivation requires the specification of the axial velocity profile; this choice differentiates models within each class. Discrepancies among differing models have yet to be investigated. In this paper, we comment on some theoretical differences among models and systematically compare them for physiologically relevant vessel parameters, network topology, and boundary data. In particular, the effect of the velocity profile is investigated in the cases of both smooth and discontinuous solutions, and a recommendation for a physiological model is provided. The models are discretized by a class of Runge-Kutta discontinuous Galerkin methods.

Predicting the effects of surgically determined parameters on exercise tolerance in Fontan patients.

The Fontan physiology is a surgically created circulation for patients with a single functioning ventricle. Patients with this circulation tend to have lower exercise tolerance compared to those with a normal circulation. Important computational and experimental work has been done to investigate this reduction in exercise tolerance. However, there are few systematic modeling studies that focus on the effect of several surgically determined parameters within the same framework. We propose a mathematical model to describe the Fontan circulation under exercise. We then formulate a heuristic based on clinical data from Fontan patients to estimate exercise tolerance. The model is used to investigate the effect of three important surgically determined parameters on exercise tolerance: the systemic arterial compliance, the systemic-venous to pulmonary-venous fenestration, and the resistance of the total cavopulmonary connection.

1D Thermoembolization Model Using CT Imaging Data for Porcine Liver.

Objective: Innovative therapies such as thermoembolization are expected to play an important role in improving care for patients with diseases such as hepatocellular carcinoma. Thermoembolization is a minimally invasive strategy that combines thermal ablation and embolization in a single procedure. This approach exploits an exothermic chemical reaction that occurs when an acid chloride is delivered via an endovascular route. However, comprehension of the complexities of the biophysics of thermoembolization is challenging. Mathematical models can aid in understanding such complex processes and assisting clinicians in making informed decisions. In this study, we used a Hagen-Poiseuille 1D blood flow model to predict the mass transport and possible embolization locations in a porcine hepatic artery. Method: The 1D flow model was used on imaging data of in-vivo embolization imaging data of three pigs. The hydrolysis time constant of acid chloride chemical reaction was optimized for each pig, and LOOCV method was used to test the model's predictive ability. Conclusion: This basic model provided a balanced accuracy rate of 66.8% for identifying the possible locations of damage in the hepatic artery. Use of the model provides an initial understanding of the vascular transport phenomena that are predicted to occur as a result of thermoembolization.

PocketNet: A Smaller Neural Network for Medical Image Analysis.

Medical imaging deep learning models are often large and complex, requiring specialized hardware to train and evaluate these models. To address such issues, we propose the PocketNet paradigm to reduce the size of deep learning models by throttling the growth of the number of channels in convolutional neural networks. We demonstrate that, for a range of segmentation and classification tasks, PocketNet architectures produce results comparable to that of conventional neural networks while reducing the number of parameters by multiple orders of magnitude, using up to 90% less GPU memory, and speeding up training times by up to 40%, thereby allowing such models to be trained and deployed in resource-constrained settings.

An Extended DEIM Algorithm for Subset Selection and Class Identification.

The discrete empirical interpolation method (DEIM) has been shown to be a viable index-selection technique for identifying representative subsets in data. Having gained some popularity in reducing dimensionality of physical models involving differential equations, its use in subset-/pattern-identification tasks is not yet broadly known within the machine learning community. While it has much to offer as is, the DEIM algorithm is limited in that the number of selected indices cannot exceed the rank of the corresponding data matrix. Although this is not an issue for many data sets, there are cases in which the number of classes represented in a given data set is greater than the rank of the data matrix; in such cases, it is impossible for the standard DEIM algorithm to identify all classes. To overcome this issue, we present a novel extension of DEIM, called E-DEIM. With the proposed algorithm, we also provide some theoretical results for using extensions of DEIM to form the CUR matrix factorization in identifying both rows and columns to approximate the original data matrix. Results from applying variations of E-DEIM to two different data sets indicate that the presented extension can indeed allow for the identification of additional classes along with those selected by standard DEIM. In addition, comparing these results to those of some more familiar methods demonstrates that the proposed deterministic E-DEIM approach including coherence performs comparably to or better than the other evaluated methods and should be considered in future class-identification tasks.

Identification of Kernels in a Convolutional Neural Network: Connections Between Level Set Equation and Deep Learning for Image Segmentation.

Medical image segmentation remains a difficult, time-consuming task; currently, liver segmentation from abdominal CT scans is often done by hand, requiring too much time to construct patient-specific treatment models for hepatocellular carcinoma. Image segmentation techniques, such as level set methods and convolutional neural networks (CNN), rely on a series of convolutions and nonlinearities to construct image features: neural networks that use strictly mean-zero finite difference stencils as convolution kernels can be treated as upwind discretizations of differential equations. If this relationship can be made explicit, one gains the ability to analyze CNN using the language of numerical analysis, thereby providing a well-established framework for proving properties such as stability and approximation accuracy. We test this relationship by constructing a level set network, a type of CNN whose architecture describes the expansion of level sets; forward-propagation through a level set network is equivalent to solving the level set equation; the level set network achieves comparable segmentation accuracy to solving the level set equation, while not obtaining the accuracy of a common CNN architecture. We therefore analyze which convolution filters are present in a standard CNN, to see whether finite difference stencils are learned during training; we observe certain patterns that form at certain layers in the network, where the learned CNN kernels depart from known convolution kernels used to solve the level set equation.

A simple mathematical model of signaling resulting from the binding of lipopolysaccharide with Toll-like receptor 4 demonstrates inherent preconditioning behavior.

The complex biology of Gram-negative bacterial lipopolysaccharide (LPS) is central to the acute inflammatory response in sepsis and related diseases. Repeated treatment with LPS can lead to desensitization or enhancement of subsequent responses both in vitro and in vivo (a phenomenon known as preconditioning). Previous computational studies have demonstrated a role for anti-inflammatory influences in this process (J. Day, J. Rubin, Y. Vodovotz, C.C. Chow, A. Reynolds, G. Clermont, A reduced mathematical model of the acute inflammatory response: II. Capturing scenarios of repeated endotoxin administration. J. Theor. Biol. 242 (2006) 237). Since LPS signals via Toll-like receptor 4 (TLR4), we created a simple mathematical model in order to address the role of this receptor in both the normal and preconditioned response to LPS. We created a non-linear system of ordinary differential equations, consisting of free LPS, free TLR4, bound complex LPS-TLR4, and an intracellular signaling cascade (lumped into a single variable). We simulate the effects of preconditioning by small and large repeated doses of LPS on the system, varying the timing of the doses as well as the rate of expression of TLR4. Our simulations suggest that a simplified model of LPS/TLR4 signaling can account for complex preconditioning phenomena without invoking a specific signaling inhibition mechanism, but rather based on the dynamics of the signaling response itself, as well as the timing and magnitude of the LPS stimuli.

An implicit discontinuous Galerkin method for modeling acute edema and resuscitation in the small intestine.

Edema, also termed oedema, is a generalized medical condition associated with an abnormal aggregation of fluid in a tissue matrix. In the intestine, excessive edema can lead to serious health complications associated with reduced motility. A $7.5\%$ solution of hypertonic saline (HS) has been hypothesized as an effective means to reduce the effects of edema following surgery or injury. However, detailed clinical edema experiments can be difficult to implement, or costly, in practice. In this manuscript we introduce an implicit in time discontinuous Galerkin method with novel adaptations for modeling edema in the 3D layered physiology of the intestine. The model improves over early work via inclusion of the tissue intrinsic storage coefficient, and the effects of Starling overestimation for high venous pressures. Validation against a recent clinical experiment in HS resuscitation of acute edema is presented; the results support the clinical hypothesis that 7.5% HS solution may be effective in the resuscitation of acute edema formation. New results include an improved view into the effects of resuscitation on the hydrostatic pressure profile of edematous rats, effects on lumenal volume attenuation, relative fluid gain and an estimation of the impacts of both acute edema and resuscitation on intestinal motility.

An energy-based equilibrium contact angle boundary condition on jagged surfaces for phase-field methods.

We consider an energy-based boundary condition to impose an equilibrium wetting angle for the Cahn-Hilliard-Navier-Stokes phase-field model on voxel-set-type computational domains. These domains typically stem from µCT (micro computed tomography) imaging of porous rock and approximate a (on µm scale) smooth domain with a certain resolution. Planar surfaces that are perpendicular to the main axes are naturally approximated by a layer of voxels. However, planar surfaces in any other directions and curved surfaces yield a jagged/topologically rough surface approximation by voxels. For the standard Cahn-Hilliard formulation, where the contact angle between the diffuse interface and the domain boundary (fluid-solid interface/wall) is 90°, jagged surfaces have no impact on the contact angle. However, a prescribed contact angle smaller or larger than 90° on jagged voxel surfaces is amplified. As a remedy, we propose the introduction of surface energy correction factors for each fluid-solid voxel face that counterbalance the difference of the voxel-set surface area with the underlying smooth one. The discretization of the model equations is performed with the discontinuous Galerkin method. However, the presented semi-analytical approach of correcting the surface energy is equally applicable to other direct numerical methods such as finite elements, finite volumes, or finite differences, since the correction factors appear in the strong formulation of the model.

One-dimensional elastic continuum model of enterocyte layer migration.

Necrotizing enterocolitis is the leading cause of death from gastrointestinal disease in preterm infants. It results from an injury to the mucosal lining of the intestine, leading to translocation of bacteria and endotoxin into the circulation. Intestinal mucosal defects are repaired by the process of intestinal restitution, during which enterocytes migrate from healthy areas to sites of injury. In this article, we develop a mathematical model of migration of enterocytes during experimental necrotizing enterocolitis. The model is based on a novel assumption of elastic deformation of the cell layer and incorporates the following effects: i), mobility promoting force due to lamellipod formation, ii), mobility impeding adhesion to the cell matrix, and iii), enterocyte proliferation. Our model successfully reproduces the behavior observed for enterocyte migration on glass coverslips, namely the dependence of migration speed on the distance from the wound edge, and the finite propagation distance in the absence of proliferation that results in an occasional failure to close the wound. It also qualitatively reproduces the dependence of migration speed on integrin concentration. The model is applicable to the closure of a wound with a linear edge and, after calibration with experimental data, could be used to predict the effect of chemical agents on mobility, adhesion, and proliferation of enterocytes.

Cardiovascular mechanics in the early stages of pulmonary hypertension: a computational study.

We formulate and study a new mathematical model of pulmonary hypertension. Based on principles of fluid and elastic dynamics, we introduce a model that quantifies the stiffening of pulmonary vasculature (arteries and arterioles) to reproduce the hemodynamics of the pulmonary system, including physiologically consistent dependence between compliance and resistance. This pulmonary model is embedded in a closed-loop network of the major vessels in the body, approximated as one-dimensional elastic tubes, and zero-dimensional models for the heart and other organs. Increasingly severe pulmonary hypertension is modeled in the context of two extreme scenarios: (1) no cardiac compensation and (2) compensation to achieve constant cardiac output. Simulations from the computational model are used to estimate cardiac workload, as well as pressure and flow traces at several locations. We also quantify the sensitivity of several diagnostic indicators to the progression of pulmonary arterial stiffening. Simulation results indicate that pulmonary pulse pressure, pulmonary vascular compliance, pulmonary RC time, luminal distensibility of the pulmonary artery, and pulmonary vascular impedance are much better suited to detect the early stages of pulmonary hypertension than mean pulmonary arterial pressure and pulmonary vascular resistance, which are conventionally employed as diagnostic indicators for this disease.

A computational study of the Fontan circulation with fenestration or hepatic vein exclusion.

Fontan patients may undergo additional surgical modifications to mitigate complications like protein-losing enteropathy, liver cirrhosis, and other issues in their splanchnic circulation. Recent case reports show promise for several types of modifications, but the subtle effects of these surgeries on the circulation are not well understood. In this paper, we employ mathematical modeling of blood flow to systematically quantify the impact of these surgical changes on extracardiac Fontan hemodynamics. We investigate two modifications: (1) the fenestrated Fontan and (2) the Fontan with hepatic vein exclusion. Closed-loop hemodynamic models are used, which consist of one-dimensional networks for the major vessels and zero-dimensional models for the heart and organ beds. Numerical results suggest the hepatic vein exclusion has the greatest overall impact on the hemodynamics, followed by the largest sized fenestration. In particular, the hepatic vein exclusion drastically lowers portal venous pressure while the fenestration decreases pulmonary artery pressure. Both modifications increase flow to the intestines, a finding consistent with their utility in clinical practice for combating complications in the splanchnic circulation.

A comprehensive mathematical framework for modeling intestinal smooth muscle cell contraction with applications to intestinal edema.

The contraction of intestinal smooth muscle cells (ISMCs) involves many coordinated biochemical and mechanical processes. In this work, we present a framework for modeling ISMC contractility that begins with chemical models of calcium dynamics, continues with myosin light chain phosphorylation and force generation, and ends with a cell model of the ISMC undergoing contraction-relaxation. The motivation for developing this comprehensive framework is to study the effects of edema (excess fluid build-up in the muscle tissue) on ISMC contractility. The hypothesis is that more fluid equates to dilution of an external stimulis, eventually leading to reduced contractility. We compare our results to experimental data collected from normal versus edematous intestinal muscle tissue.

Numerical Method of Characteristics for One-Dimensional Blood Flow.

Mathematical modeling at the level of the full cardiovascular system requires the numerical approximation of solutions to a one-dimensional nonlinear hyperbolic system describing flow in a single vessel. This model is often simulated by computationally intensive methods like finite elements and discontinuous Galerkin, while some recent applications require more efficient approaches (e.g. for real-time clinical decision support, phenomena occurring over multiple cardiac cycles, iterative solutions to optimization/inverse problems, and uncertainty quantification). Further, the high speed of pressure waves in blood vessels greatly restricts the time step needed for stability in explicit schemes. We address both cost and stability by presenting an efficient and unconditionally stable method for approximating solutions to diagonal nonlinear hyperbolic systems. Theoretical analysis of the algorithm is given along with a comparison of our method to a discontinuous Galerkin implementation. Lastly, we demonstrate the utility of the proposed method by implementing it on small and large arterial networks of vessels whose elastic and geometrical parameters are physiologically relevant.

A mathematical model of intestinal oedema formation.

Intestinal oedema is a medical condition referring to the build-up of excess fluid in the interstitial spaces of the intestinal wall tissue. Intestinal oedema is known to produce a decrease in intestinal transit caused by a decrease in smooth muscle contractility, which can lead to numerous medical problems for the patient. Interstitial volume regulation has thus far been modelled with ordinary differential equations, or with a partial differential equation system where volume changes depend only on the current pressure and not on updated tissue stress. In this work, we present a computational, partial differential equation model of intestinal oedema formation that overcomes the limitations of past work to present a comprehensive model of the phenomenon. This model includes mass and momentum balance equations which give a time evolution of the interstitial pressure, intestinal volume changes and stress. The model also accounts for the spatially varying mechanical properties of the intestinal tissue and the inhomogeneous distribution of fluid-leaking capillaries that create oedema. The intestinal wall is modelled as a multi-layered, deforming, poroelastic medium, and the system of equations is solved using a discontinuous Galerkin method. To validate the model, simulation results are compared with results from four experimental scenarios. A sensitivity analysis is also provided. The model is able to capture the final submucosal interstitial pressure and total fluid volume change for all four experimental cases, and provide further insight into the distribution of these quantities across the intestinal wall.

Ecodynamics of PAHs at a peri-urban site of the French Mediterranean Sea.

The PAH contamination level and biochemical composition of sinking particles and surficial sediments (0-0.5 cm layer) were assessed at a rural coastal site in the northwestern Mediterranean Sea. Surficial sediment contamination (≈20 ng g(-1)) was considerably lower than at other Mediterranean sites, yet particles collected in sediment traps had 6-8 times more PAH. Contaminated particles were mostly marine in origin. Temporal variation of contamination levels correlated with organic content of the particles, but some of the observed variability could be attributed to seasonal changes in pyrolytic PAH production. Sinking organic particles were potentially as readily digestible as surficial sediments for prospective consumers however, transfer of PAHs along the benthic food chain is probably enhanced because of the particles' higher nutritional value.

Agent-based model of inflammation and wound healing: insights into diabetic foot ulcer pathology and the role of transforming growth factor-beta1.

Inflammation and wound healing are inextricably linked and complex processes, and are deranged in the setting of chronic, nonhealing diabetic foot ulcers (DFU). An ideal therapy for DFU should both suppress excessive inflammation while enhancing healing. We reasoned that biological simulation would clarify mechanisms and help refine therapeutic approaches to DFU. We developed an agent-based model (ABM) capable of reproducing qualitatively much of the literature data on skin wound healing, including changes in relevant cell populations (macrophages, neutrophils, fibroblasts) and their key effector cytokines (tumor necrosis factor-alpha [TNF], interleukin [IL]-1beta, IL-10, and transforming growth factor [TGF]-beta1). In this simulation, a normal healing response results in tissue damage that first increases (due to wound-induced inflammation) and then decreases as the collagen levels increase. Studies by others suggest that diabetes and DFU are characterized by elevated TNF and reduced TGF-beta1, although which of these changes is a cause and which one is an effect is unclear. Accordingly, we simulated the genesis of DFU in two ways, either by (1) increasing the rate of TNF production fourfold or (2) by decreasing the rate of TGF-beta1 production 67% based on prior literature. Both manipulations resulted in increased inflammation (elevated neutrophils, TNF, and tissue damage) and delayed healing (reduced TGF-beta1 and collagen). Our ABM reproduced the therapeutic effect of platelet-derived growth factor/platelet releasate treatment as well as DFU debridement. We next simulated the expected effect of administering (1) a neutralizing anti-TNF antibody, (2) an agent that would increase the activation of endogenous latent TGF-beta1, or (3) latent TGF-beta1 (which has a longer half-life than active TGF-beta1), and found that these therapies would have similar effects regardless of the initial assumption of the derangement that underlies DFU (elevated TNF vs. reduced TGF-beta1). In silico methods may elucidate mechanisms of and suggest therapies for aberrant skin healing.