Continuous Partial Differential Equations Research Articles

The impact of multiple stenosis and aneurysms on arterial diseases: A cardiovascular study

The comparative effect of serial stenosis and aneurysms arteries on blood flow is examined to identify atherosclerotic diseases. The finite element approach has been used to solve the continuity, momentum, and Oldroyd-B partial differential equations to analyze the blood flow. Newtonian and non-Newtonian both cases are taken for the viscoelastic response of blood. In this study, the impact of multiple stenotic and aneurysmal arteries on blood flow have been studied to determine the severity of atherosclerosis diseases through the analysis of blood behavior. The novel aspect of the study is its assessment of the severity of atherosclerotic disorders for the occurrence of serial stenosis and aneurysm simultaneously in the blood vessel wall in each of the four cases. The maximum abnormal arterial blood flow effect is found for the presence of serial stenoses compared to aneurysms which refers to the severity of atherosclerosis. At the hub of stenosis, the blood velocity magnitude and wall shear stress (WSS) are higher, whereas the arterial wall normal gradient values are lower. For all cases, the contrary results are observed at the hub of the aneurysmal model. The blood flow has been affected significantly by the increases in Reynolds number for both models. The influence of stenotic and aneurysmal arteries on blood flow is graphically illustrated in terms of the velocity profile, pressure distribution, and WSS. Medical experts may use this study's findings to assess the severity of cardiovascular diseases.

Numerical and statistical analysis of unsteady hydromagnetic hybrid nanofluid flow in a porous medium over an inclined disk with dissipative heat transfer

AbstractThe present analysis deals with the three‐dimensional radiative, convective hybrid nanofluid flow over a rotating and stretching inclined disk under the action of the applied magnetic field, Joule's heating, and viscous dissipation effects. The mathematical model considers nanoparticles graphene oxide (GO) and molybdenum disulfide (MoS) suspended in water as the base fluid within a Darcy medium. The cylindrical coordinate system is used to express continuity, momentum, and energy partial differential equations, out of which momentum and energy partial differential equations are transformed into ordinary differential equations using a suitable transformation method under the boundary‐layer approximation. The transformed equations are further solved using the spectral quasi‐linearization method. Graphical and numerical data are presented to investigate the behavior of velocity and temperature under various parameters and shape factors. The nature of fluid flow at the boundary wall is determined by tables of skin friction and the Nusselt number. Statistical analysis of the parameters for axial and tangential skin friction and the Nusselt number is performed using the quadratic regression model.

Soliton approximation in continuum models of leader-follower behavior.

Complex biological processes involve collective behavior of entities (bacteria, cells, animals) over many length and time scales and can be described by discrete models that track individuals or by continuum models involving densities and fields. We consider hybrid stochastic agent-based models of branching morphogenesis and angiogenesis (new blood vessel creation from preexisting vasculature), which treat cells as individuals that are guided by underlying continuous chemical and/or mechanical fields. In these descriptions, leader (tip) cells emerge from existing branches and follower (stalk) cells build the new sprout in their wake. Vessel branching and fusion (anastomosis) occur as a result of tip and stalk cell dynamics. Coarse graining these hybrid models in appropriate limits produces continuum partial differential equations(PDEs) for endothelial cell densities that are more analytically tractable. While these models differ in nonlinearity, they produce similar equationsat leading order when chemotaxis is dominant. We analyze this leading order system in a simple quasi-one-dimensional geometry and show that the numerical solution of the leading order PDE is well described by a soliton wave that evolves from vessel to source. This wave is an attractor for intermediate times until it arrives at the hypoxic region releasing the growth factor. The mathematical techniques used here thus identify common features of discrete and continuum approaches and provide insight into general biological mechanisms governing their collective dynamics.

A Numerical Assessment of Time-Dependent Magneto-Convective Thermal-Material Transfer over a Vertical Permeable Plate

The objective of this work is to investigate the influences of thermal radiation, heat generation, and buoyancy force on the time-dependent boundary layer (BL) flow across a vertical permeable plate. The fluid is unsteady, incompressible, viscous, and electrically insulating. The heat transfer mechanism happens due to free convection. The nondimensional partial differential equations of continuity, momentum, energy, and concentration are discussed using appropriate transformations. The impressions of thermal radiation and buoyancy forces are exposed in the energy and momentum equation, respectively. For numerical model, a set of nonlinear dimensionless partial differential equations can be solved using an explicit finite difference approach. The stability and convergence analyses are also established to complete the formulation of the model. The thermophysical effects of entering physical parameters on the flow, thermal, and material fields are analyzed. The variations in local and average skin friction, material, and heat transfer rates are also discussed for the physical interest. The analysis of the obtained findings is shown graphically, and relevant parameters pointedly prejudice the flow field. Studio Developer FORTRAN 6.2 and Tecplot 10.0 are applied to simulate the schematic model equations and graphical presentation numerically. The intensifying values of the magnetic field are affected decreasingly in the flow field. The temperature profiles decrease within the BL to increase the value of radiation parameters. The present study is on the consequences for petroleum engineering, agriculture engineering, extraction, purification processes, nuclear power plants, gas turbines, etc. To see the rationality of the present research, we compare these results and the results available in the literature with outstanding compatibility.

Optimum Volume Fraction and Inlet Temperature of an Ideal Nanoparticle for Enhanced Oil Recovery by Nanofluid Flooding in a Porous Medium

Nowadays, oil companies employ nanofluid flooding to increase oil production from oil reservoirs. Herein the present work, a multiphase flow in porous media was used to simulate oil extraction from a three-dimensional porous medium filled with oil. Interestingly, the finite element method was used to solve the nonlinear partial differential equations of continuity, energy, Darcy’s law, and the transport of nanoparticles (NPs). The proposed model used nanofluids (NFs) empirical formulas for density and viscosity on NF and oil relative permeabilities and NP transport equations. The NPs thermophysical properties have been investigated and compared with their oil recovery factor (ORF) to determine the highest ORF. Different NPs (SiO2, CuO, and Al2O3) were used as the first parameter, keeping all parameters constant. The simulation was run three times for the injected fluid using the various NPs to compare the effects on enhanced oil recovery. The second parameter, volume fraction (VF), has been modeled six times (0.5, 1, 2, 3, 4, and 5%), with all other parameters held constant. The third parameter, the injected NF inlet temperature (293.15–403.15 K), was simulated assuming that all other parameters are kept constant. The energy equation was applied to choose the inlet temperature that fits the optimum NP and VF to determine the highest ORF. Findings indicated that SiO2 shows the best ORF compared to the other NPs. Remarkably, SiO2 has the lowest density and highest thermal capacity. The optimum VF of SiO2 was 4%, increasing the ORF but reduced when the VF was higher than 4%. The ORF was improved when the viscosity and density of the oil decreased by increasing the injected inlet temperature. Furthermore, the results indicated that the highest ORF of 37% was obtained at 353.15 K when SiO2 was used at a VF of 4%. At the same time, the lowest recovery is obtained when a volume of 5% was used at 403.15 K.

Flow and Heat Transfer Analysis on Reiner-Philippoff Fluid Flow over a Stretching Sheet in the Presence of First and Second Order Velocity Slip and Temperature Jump Effects

Most of the fluid used in industrial application (i.e. Oils and gas industry, food manufacturing, lubrication and biomedical) do not conform to the Newtonian postulate. In contrast to the Newtonian fluid, the viscosity of the fluid can change when under force to either more liquid or more solid and dependent on shear rate history. This behaviour of fluids is commonly known as non-Newtonian fluid. The non-Newtonian fluid is so widespread in nature and technology resulting in very high interest of investigating among scientist. The Reiner-Philippoff fluid is one of the types of non-Newtonian fluid models that exhibiting the dilatant, pseudoplastic and Newtonian behaviors. Hence, this study is devoted to analyze the flow and heat transfer of Reiner-Philippoff fluid with the presence of first and second order velocity slip together with the temperature jump effects over a stretching sheet. Partial differential equations of continuity, momentum and energy equations were transformed into the similarity equations. The obtained equations were then solved via bvp4c function in MATLAB software. For the validation purpose, the present model and its numerical solution were compared with previous established solutions under limiting case where the present model is condensed to be identical with the reported model and turn to be in very strong agreement. The consequences of pertinent parameters on fluid’s characteristics are analyzed in details through the plotted graphic visuals and tabular form.

Numerical Assessment of Nanofluid Natural Convection Using Local RBF Method Coupled with an Artificial Compressibility Model

In this paper, natural heat convection inside square and equilateral triangular cavities was studied using a meshless method based on collocation local radial basis function (RBF). The nanofluids used were Cu-water or -water mixture with nanoparticle volume fractions range of . A system of continuity, momentum, and energy partial differential equations was used in modeling the flow and temperature behavior of the fluids. Partial derivatives in the governing equations were approximated using the RBF method. The artificial compressibility model was implemented to overcome the pressure velocity coupling problem that occurs in such equations. The main goal of this work was to present a simple and efficient method to deal with complex geometries for a variety of problem conditions. To assess the accuracy of the proposed method, several test cases of natural convection in square and triangular cavities were selected. For Rayleigh numbers ranging from to , a validation test of natural convection of Cu-water in a square cavity was used. The numerical investigation was then extended to Rayleigh number , as well as -water nanofluid with a volume fraction range of . In a second investigation, the same nanofluids were used in a triangular cavity with varying volume fractions to test the proposed meshless approach on non-rectangular geometries. The numerical results appear to be in agreement with those from earlier investigations. Furthermore, the suggested meshless method was found to be stable and accurate, demonstrating that it may be a viable alternative for solving natural heat transfer equations of nanofluids in enclosures with irregular geometries.

On a diffusive bacteriophage dynamical model for bacterial infections

Bacteriophages or phages are viruses that infect bacteria and are increasingly used to control bacterial infections. We develop a reaction–diffusion model coupling the interactive dynamic of phages and bacteria with an epidemiological bacteria-borne disease model. For the submodel without phage absorption, the basic reproduction number [Formula: see text] is computed. The disease-free equilibrium (DFE) is shown to be globally asymptotically stable whenever [Formula: see text] is less than one, while a unique globally asymptotically endemic equilibrium is proven whenever [Formula: see text] exceeds one. In the presence of phage absorption, the above stated classical condition based on [Formula: see text], as the average number of secondary human infections produced by susceptible/lysogen bacteria during their entire lifespan, is no longer sufficient to guarantee the global stability of the DFE. We thus derive an additional threshold [Formula: see text], which is the average offspring number of lysogen bacteria produced by one infected human during the phage–bacteria interactions, and prove that the DFE is globally asymptotically stable whenever both [Formula: see text] and [Formula: see text] are under unity, and infections persist uniformly whenever [Formula: see text] is greater than one. Finally, the discrete counterpart of the continuous partial differential equation model is derived by constructing a nonstandard finite difference scheme which is dynamically consistent. This consistency is shown by constructing suitable discrete Lyapunov functionals thanks to which the global stability results for the continuous model are replicated. This scheme is implemented in MatLab platform and used to assess the impact of spatial distribution of phages, on the dynamic of bacterial infections.

A Continuation Method for Large-Scale Modeling and Control: From ODEs to PDE, a Round Trip

In this article, we present a continuation method, which transforms spatially distributed ordinary differential equation (ODE) systems into a continuous partial differential equation (PDE). We show that this continuation can be performed for both linear and nonlinear systems, including multidimensional, space-varying, and time-varying systems. When applied to a large-scale network, the continuation provides a PDE describing the evolution of a continuous-state approximation that respects the spatial structure of the original ODE. Our method is illustrated by multiple examples, including transport equations, Kuramoto equations, and heat diffusion equations. As a main example, we perform the continuation of a Newtonian system of interacting particles and obtain the Euler equations for compressible fluids, thereby providing an original solution to Hilbert’s sixth problem. Finally, we leverage our derivation of Euler equations to solve a control problem multiagent systems, by designing a nonlinear control algorithm for robot formation based on its continuous approximation.

Continuum soft tissue models from upscaling of arrays of hyperelastic cells

Constitutive models for soft tissue mechanics are typically constructed by fitting phenomenological models to experimental measurements. However, a significant challenge is to rationally construct soft tissue models that encode the properties of the constituent cells and their extracellular matrix. This work presents a framework to derive multiscale soft tissue models that incorporate properties of individual cells without assuming homogeneity or periodicity at the cell level. We consider a viscoelastic model for each cell (which can deform, grow and divide), that we couple to form a network description of a one-dimensional line of cells. We use a discrete-to-continuum approach to form (nonlinear) continuum partial differential equation models for the tissue. These models elucidate the contrasting role of the two forms of dissipation: substrate dissipation localizes the deformation to the neighbourhood of the free boundary and inhibits morphoelastic growth, whereas internal cell dissipation promotes spatial uniformity and does not influence the elongation length. Furthermore, cell division is shown to increase the rate of elongation of the array compared with growth alone, provided the substrate dissipation is proportional to the cell surface area.

Comparative Mathematical Study of Blood Flow Through Stenotic and Aneurysmatic Artery with the Presence and Absence of Blood Clots

Numerical predictions of blood flow and hemodynamic properties through stenosis and aneurysm artery have been studied in the presence of blood clots at the constricted area. The finite element method has been used to solve the partial differential equations of continuity, momentum, Oldroyd-B, and bioheat transport in cartesian coordinates systems. The present investigation carries the potential to compute blood velocity, pressure, and drag coefficients with significance at the throat of stenosis and aneurysm. The models have also been employed to study simulation, blood clots, and hemodynamic characteristics for all modifications. The impact of shearthinning on blood flow is intensified compared to the viscoelastic properties. It is found that the maximum effect of the drag coefficient is visible at the hub of stenotic for nonclotting models. The highest pressure and the lowest velocity are gained for the presence of blood clots at the constraint area. The impact of stenosis and aneurysm artery, drag coefficient, and blood clots on blood flow is the main physical outcome that may be reported in medical science to identify atherosclerosis diseases. The quantitative analysis has been completed numerically with the physiological significance of hemodynamic factors of blood flow which shows the validity of the present model.

Continuum Covariance Propagation for Understanding Variance Loss in Advective Systems

Motivated by the spurious variance loss encountered during covariance propagation in atmospheric and other large-scale data assimilation systems, we consider the problem for state dynamics governed by the continuity and related hyperbolic partial differential equations. This loss of variance is often attributed to reduced-rank representations of the covariance matrix, as in ensemble methods for example, or else to the use of dissipative numerical methods. Through a combination of analytical work and numerical experiments, we demonstrate that significant variance loss, as well as gain, typically occurs during covariance propagation, even at full rank. The cause of this unusual behavior is a discontinuous change in the continuum covariance dynamics as correlation lengths become small, for instance in the vicinity of sharp gradients in the velocity field. This discontinuity in the covariance dynamics arises from hyperbolicity: the diagonal of the kernel of the covariance operator is a characteristic surface for advective dynamics. Our numerical experiments demonstrate that standard numerical methods for evolving the state are not adequate for propagating the covariance, because they do not capture the discontinuity in the continuum covariance dynamics as correlations lengths tend to zero. Our analytical and numerical results demonstrate in the context of mass conservation that this leads to significant, spurious variance loss in regions of mass convergence and gain in regions of mass divergence. The results suggest that developing local covariance propagation methods designed specifically to capture covariance evolution near the diagonal may prove a useful alternative to current methods of covariance propagation.

F-actin bending facilitates net actomyosin contraction By inhibiting expansion with plus-end-located myosin motors

Contraction of actomyosin networks underpins important cellular processes including motility and division. The mechanical origin of actomyosin contraction is not fully-understood. We investigate whether contraction arises on the scale of individual filaments, without needing to invoke network-scale interactions. We derive discrete force-balance and continuum partial differential equations for two symmetric, semi-flexible actin filaments with an attached myosin motor. Assuming the system exists within a homogeneous background material, our method enables computation of the stress tensor, providing a measure of contractility. After deriving the model, we use a combination of asymptotic analysis and numerical solutions to show how F-actin bending facilitates contraction on the scale of two filaments. Rigid filaments exhibit polarity-reversal symmetry as the motor travels from the minus to plus-ends, such that contractile and expansive components cancel. Filament bending induces a geometric asymmetry that brings the filaments closer to parallel as a myosin motor approaches their plus-ends, decreasing the effective spring force opposing motor motion. The reduced spring force enables the motor to move faster close to filament plus-ends, which reduces expansive stress and gives rise to net contraction. Bending-induced geometric asymmetry provides both new understanding of actomyosin contraction mechanics, and a hypothesis that can be tested in experiments.

Continuous Limit, Rational Solutions, and Asymptotic State Analysis for the Generalized Toda Lattice Equation Associated with 3 × 3 Lax Pair

Discrete integrable nonlinear differential difference equations (NDDEs) have various mathematical structures and properties, such as Lax pair, infinitely many conservation laws, Hamiltonian structure, and different kinds of symmetries, including Lie point symmetry, generalized Lie bäcklund symmetry, and master symmetry. Symmetry is one of the very effective methods used to study the exact solutions and integrability of NDDEs. The Toda lattice equation is a famous example of NDDEs, which may be used to simulate the motions of particles in lattices. In this paper, we investigated the generalized Toda lattice equation related to 3×3 matrix linear spectral problem. This discrete equation is related to continuous linear and nonlinear partial differential equations under the continuous limit. Based on the known 3×3 Lax pair of this equation, the discrete generalized (m,3N−m)-fold Darboux transformation was constructed for the first time and extended from the 2×2 Lax pair to the 3×3 Lax pair to give its rational solutions. Furthermore, the limit states of such rational solutions are discussed via the asymptotic analysis technique. Finally, the exponential–rational mixed solutions of the generalized Toda lattice equation are obtained in the form of determinants.

Duel Solutions in Hiemenz Flow of an Electro-Conductive Viscous Nanofluid Containing Elliptic Single-/Multi-Wall Carbon Nanotubes With Magnetic Induction Effects

Abstract Modern magnetic nanomaterials are increasingly embracing new technologies including smart coatings, intelligent lubricants, and functional working fluids in energy systems. Motivated by studying the manufacturing magnetofluid dynamics of electroconductive viscous nanofluids, in this work, we analyzed the magnetohydrodynamics (MHD) convection flow and heat transfer of an incompressible viscous nanofluid containing carbon nanotubes (CNTs) past a stretching sheet. Magnetic induction effects are included. Similarity solutions are derived where possible in addition to dual branch solutions. Both single-wall carbon nanotubes (SWCNTs) and multi-wall carbon nanotubes (MWCNTs) are considered taking water and kerosene oil as base fluids. The governing continuity, momentum, magnetic induction, and heat conservation partial differential equations are converted to coupled, nonlinear systems of ordinary differential equations via similarity transformations. The emerging control parameters are shown to be Prandtl number (Pr), nanoparticle volume fraction parameter (φ), inverse magnetic Prandtl number (λ), magnetic body force parameter (β) and stretching rate parameter (A), and the type of carbon nanotube. Numerical solutions to the ordinary differential boundary value problem are conducted with the efficient bvp4c solver in matlab. Validation with earlier studies is included. Computations of reduced skin friction and reduced wall heat transfer rate (Nusselt number) are also comprised in order to identify the critical parameter values for the existence of dual solutions (upper and lower branch) for velocity, temperature, and induced magnetic field functions. Dual solutions are shown to exist for some cases studied. The simulations indicate that when the stretching rate ratio parameter is less than 1, SWCNT nanofluids exhibit higher velocity than MWCNT nanofluids with increasing magnetic parameters for water- and kerosene-oil-based CNT nanofluids. Generally, SWCNT nanofluids achieve enhanced heat transfer performance compared to MWCNT nanofluids. Water-based CNT nanofluids also attain greater flow acceleration compared with kerosene-oil-based CNT nanofluids.

Significance low oscillating magnetic field and Hall current in the nano-ferrofluid flow past a rotating stretchable disk

The present investigation involves the Hall current effects past a low oscillating stretchable rotating disk with Joule heating and the viscous dissipation impacts on a Ferro-nanofluid flow. The entropy generation analysis is carried out to study the impact of rotational viscosity by applying a low oscillating magnetic field. The model gives the continuity, momentum, temperature, magnetization, and rotational partial differential equations. These equations are transformed into the ODEs and solved by using bvp4c MATLAB. The graphical representation of arising parameters such as effective magnetization and nanoparticle concentration on thermal profile, velocity profile, and rate of disorder along with Bejan number is presented. Drag force and the heat transfer rate are given in the tabular form. It is comprehended that for increasing nanoparticle volume fraction and magnetization parameter, the radial, and tangential velocity reduce while thermal profile surges. The comparison of present results for radial and axial velocity profiles with the existing literature shows approximately the same results.

Analysis of finite-volume discrete adjoint fields for two-dimensional compressible Euler flows

This work deals with a number of questions related to the discrete and continuous adjoint fields associated with the compressible Euler equations and classical aerodynamic functions. The consistency of the discrete adjoint equations with the corresponding continuous adjoint partial differential equations is one of them. It has been established or at least discussed only for some numerical schemes and a contribution of this article is to give the adjoint consistency conditions for the 2-D Jameson-Schmidt-Turkel scheme in cell-centered finite-volume formulation. The consistency issue is also studied here from a new heuristic point of view by discretizing the continuous adjoint equation for the discrete flow and adjoint fields. Both points of view prove to provide useful information. Besides, it has been often noted that discrete or continuous inviscid lift and drag adjoint exhibit numerical divergence close to the wall and stagnation streamline for a wide range of subsonic and transonic flow conditions. This is analyzed here using the physical source term perturbation method introduced in reference [1]. With this point of view, the fourth physical source term of [1] appears to be the only one responsible for this behavior. It is also demonstrated that the numerical divergence of the adjoint variables corresponds to the response of the flow to the convected increment of stagnation pressure and diminution of entropy created at the source and the resulting change in lift and drag.

Analysis of Vapor Condensation Process From Moist Air.(Dept.M)

Condensation process, from moist airflow over flat plate, is investigated theoretically and experimentally. This process has many important engineering applications, which are concerning with moist air treatment. In The theoretical analysis of the present work, the flow is assumed to be laminar, steady and of constant physical properties. The process is described by continuity, momentum, energy and concentration partial differential equations expressed in Cartesian coordinates system. Due to the nature of the studied problem and with proper transformation of the problem dependent and independent variables, these governing equations are transformed to set of ordinary differential equations. Accordingly, one can obtain similar solutions of the problem, which are obtained by application of the well-known numerical method Rung-Kutta accompanied with shooting method. According to this solution, the values of Nusselt and Sherwood numbers, for all values of Reynolds number are obtained. The test section, of the experimental work, is a rectangular duct with cooled bottom. The moist air is forced to flow inside the duct, where some of water is condensed over its lower surface. The values of dry bulb temperature and wet bulb temperature are measured through out the flow field for different values of flow velocity. In accordance heat and mass transfer coefficients are estimated for different operating conditions. A correlation formula of the problem parameters is proposed. According to the obtained experimental results, the average heat and mass transfer coefficients increases by increasing velocity up to certain value of it then decreases for more increase of velocity. Comparisons are made between present obtained experimental results and those results obtained in previous work, exhibit good agreement in the studied region.

Natural Convection Air cooling of Electronic Components in Partially top vented Enclosures.(Dept.M)

The study of natural convection from a heat source located at the bottom of partially top vented enclosure is presented. Air cooling for electronic component situated at the bottom of a top vented enclosure is studied theoretically. The flow is assumed to be laminar, steady and of constant physical properties. The process is described by continuity, momentum and energy partial differential equations, which are be expressed in Cartesian coordinates system. Due to the nature of the studied problem and with proper transformation of the problem dependent and independent variables, these governing equations are transformed to a set of dimensionless partial differential equations. This set of differential equations is transformed to set of difference equations by the implantation of finite difference technique. Accordingly, one can obtain the solutions for this problem, which are obtained by application of the well known Gauss Siedel iteration method. A computer program is developed to solve the present proposed mathematical model. According to this solution, the values of Nusselt number, for different values of Rayleigh number are obtained. Comparisons between the present obtained results and those results obtained in previous theoretical results are performed for parametric variations of the vent opening size and shape. Also a correlation for Nusselt number as a function of Rayleigh number and equivalent diameter of the vent is proposed.

Integrating Quantitative Assays with Biologically Based Mathematical Modeling for Predictive Oncology.

SummaryWe provide an overview on the use of biological assays to calibrate and initialize mechanism-based models of cancer phenomena. Although artificial intelligence methods currently dominate the landscape in computational oncology, mathematical models that seek to explicitly incorporate biological mechanisms into their formalism are of increasing interest. These models can guide experimental design and provide insights into the underlying mechanisms of cancer progression. Historically, these models have included a myriad of parameters that have been difficult to quantify in biologically relevant systems, limiting their practical insights. Recently, however, there has been much interest calibrating biologically based models with the quantitative measurements available from (for example) RNA sequencing, time-resolved microscopy, and in vivo imaging. In this contribution, we summarize how a variety of experimental methods quantify tumor characteristics from the molecular to tissue scales and describe how such data can be directly integrated with mechanism-based models to improve predictions of tumor growth and treatment response.