Collective cell migration in single and dual cell layers

Collective cell migration plays a substantial role in maintaining the cohesion of epithelial cell layers, in wound healing, and in embryonic development. We extend a previously developed one-dimensional continuum mechanical model of cell layer migration based on an assumption of elastic deformation of the cell layer to incorporate stretch-dependent proliferation, which leads to a generalized Stefan problem for the density of the layer. The resulting partial differential equation system is solved numerically using an adaptive finite difference method and similarity solutions are studied analytically. We show the existence of traveling wave solutions with constant wave speed for a large class of constitutive equations for the dependence of proliferation on stretch. We then extend the corresponding two-dimensional model of cell migration to incorporate two adhering cell layers. A numerical method to solve the model equations is based on a level set method for free boundary problems with a domain decomposition method to account for where the migrating cells in each layer are located. We apply the model to experimental migration of epithelial and mesenchymal cell layers during gastrulation, an early phase of development, in animal cap explants of Xenopus laevis embryos to analyze the mechanical properties of each cell layer. Understanding the mechanics of collective cell migration during embryonic development will aid in developing tools to perturb pathological cases such as during wound healing and to aid in the prediction and early detection of birth defects

Modeling the Wound healing in Necrotizing Enterocolitis and Diabetic Foot Ulcer

In my thesis, I present three different models for the wound healing in Necrotizing Enterocolitis and Diabetic Foot Ulcer. (I) NEC results after an injury to the mucosal lining of the intestine, leading to translocation of bacteria and endotoxin. Intestinal mucosal defects are repaired by the process of intestinal restitution, during which enterocytes migrate from healthy areas to sites of injury. To model the migration of enterocytes, first we formulate a one-dimensional mathematical model based on the assumption of elastic deformation of the cell layer. Then we extend the model into a two-dimension space and the resulting moving boundary problem is solved by using modified Finite Element Method. (II) Diabetic foot ulcers (DFU) are caused by both vascular and neurologic complications of diabetes, in combination with persistent opportunistic infections and deficient wound healing. We develop an Agent-based computational model to simulate its inflammation and the resolution of the inflammatory response in its wound healing process

Continuum Models of Collective Migration in Living Tissues

This dissertation investigates the physical mechanics of collective cell migration in monolayers of epithelial cells. Coordinated cell motion underlies a number of biological processes, including wound healing, morphogenesis and cancer metastasis, and is controlled by the interplay of single cell motility, cell-cell adhesions, cell-substrate interaction, and cell contractility modulated by the acto-myosin cytoskeleton. Here we examine the competing roles of these mechanisms via a continuum model of a tissue as an active elastic medium, where mechanical deformations are coupled to and feed back onto chemical signaling. We begin in Chapter 1 with a brief review of cell migration at both the single-cell and many-cell levels, and of the experimental tools used to probe the mechanical properties of cells and tissues. In Chapter 2 we formulate our minimal continuum model of a tissue as an overdamped active elastic medium on a frictional substrate. The model couples mechanical deformations in the tissue to myosin-based contractile activity and to cell polarization. Two new ingredients of our model are: (i) a feedback between the on-off dynamics of myosin motors and the active contractile stresses they induce in the tissue, and (ii) the coupling of cell directed motion or polarization to tissue strain. In the following two chapters we employ this model to describe collective cell dynamics in expanding (Chapter 3) and confined (Chapter 4) tissues and compare with experiments. In expanding monolayers, as realized for instance in wound healing assays where an initially confined tissue is allowed to expand freely on a substrate, our model reproduces the propagating waves of mechanical stress observed in experiments and believed to play a key role in controlling the transmission of information across the tissue and mediating coordinated cell motion. Combining analytical and numerical work we construct a phase diagram that identifies various dynamical regimes in terms of single-cell properties, such as contractility and stiffness. In Chapter 4, we use our model to describe collective dynamics of cells confined to a circular geometry. In this case the propagating waves are replaced by standing sloshing waves guided by both contractility and polarization. The work on confined tissues was carried out in collaboration with the experimental group of Jeff Fredberg at the Harvard School of Public Health. By combining theory and experiment we can provide a quantitative understanding of how contractility and polarization regulate the mechanics of the tissue by renormalizing the tissue elastic moduli and controlling the frequency of oscillatory modes

Modeling the extracellular matrix in cell migration and morphogenesis:a guide for the curious biologist

The extracellular matrix (ECM) is a highly complex structure through which biochemical and mechanical signals are transmitted. In processes of cell migration, the ECM also acts as a scaffold, providing structural support to cells as well as points of potential attachment. Although the ECM is a well-studied structure, its role in many biological processes remains difficult to investigate comprehensively due to its complexity and structural variation within an organism. In tandem with experiments, mathematical models are helpful in refining and testing hypotheses, generating predictions, and exploring conditions outside the scope of experiments. Such models can be combined and calibrated with in vivo and in vitro data to identify critical cell-ECM interactions that drive developmental and homeostatic processes, or the progression of diseases. In this review, we focus on mathematical and computational models of the ECM in processes such as cell migration including cancer metastasis, and in tissue structure and morphogenesis. By highlighting the predictive power of these models, we aim to help bridge the gap between experimental and computational approaches to studying the ECM and to provide guidance on selecting an appropriate model framework to complement corresponding experimental studies

Mechanical characterization of cervical tissue

A multi-scale constitutive model for the nonpregnant cervical tissue is presented. The mechanical response of the cervix is described by a model which takes into account material properties at different structural hierarchies of tissue through a multi-scale coupling scheme. The model introduces the deformation mechanisms of collagen fibrils at the microscale into a macroscopic continuum description of the mechanical behavior of tissue. The mechanical behavior of the cervix is governed by the directional structures in the collagen fiber architecture. The prefer- entially aligned fibers are responsible for the typical anisotropic behavior to the material and the solid matrix (ground substance) originates its incompressible response. The model assumes uncoupled contributions of the matrix and collagen fibers. The matrix is modeled as a simple isotropic material. On the other hand, results from a constitutive model of randomly crimped collagen fibers are used to modeled the fibrous part, and a parameter to quantify the stochastic dispersion of the collagen orientation is introduced. The collective mechanical behavior of collagen fibers is presented in terms of an explicit expression for the strain-energy function (SEF). And at the macro-scale, the constitutive response of the cervical tissue is formulated by homogenizing a fiber-reinforced material. Non-destructive evaluation using ultrasonic signals is a well-established method to obtain physically relevant mechanical parameters. This work aims to understand the ultrasonic transmission through soft tissues, in order to develop a useful tool to quantify mechanical parameters, which may be applied as a future diagnosis method. To this end, experimental ultrasound measurements were carried out in soft tissue samples, as well as simulations by finite difference time-domain method. Finally, a comparative study between experimental and simulated signals is presented. Results show the ability to describe the mechanical behavior of the cervical tissue like a fiber reinforced material, and that the ultrasonic wave propagation phenomena can be exploited to reconstruct the mechanical properties of soft tissues, and thus to diagnose pathologies that manifest by tissue consistency changes.Se presenta un modelo constitutivo multi-escala para el tejido cervical de mujeres no embarazadas. La respuesta mecánica del cuello del útero se describe por un modelo que tiene en cuenta las propiedades del material en las diferentes jerarquías estructurales del tejido a través de un esquema de acoplamiento multi-escala. El comportamiento macromecánico del tejido introduce los mecanismos de deformación de la fibras de colágeno que ocurren en la escala microscópica. Las direcciones preferentes de las fibras de colágeno rigen el comportamiento mecánico del cervix, creando el comportamiento anisotrópico típico del tejido, siendo la matriz (sustancia fundamental) la responsable de su respuesta incompresible. El modelo supone contribuciones desacoplados para la matriz y las fibras de colágeno. La matriz se modela como un material isotrópico sencillo. Por otro lado, se utiliza un modelo constitutivo de fibras onduladas de colágeno para la parte fibrosa, donde se introduce un parámetro para cuantificar la dispersión estocástica en la orientación de las fibras. El comportamiento colectivo de las fibras de colágeno se presenta en términos de potencial de energía de deformación (SEF). La respuesta constitutiva del tejido cervical en la macro escala se formula para la homogeneización de un material reforzado con fibras. La evaluación no destructiva utilizando señales ultrasónicas es un método reconocido para obtener parámetros mecánicos físicamente pertinentes. Este trabajo tiene tiene como objetivo comprender la transmisión de ultrasonidos a través de los tejidos blandos, para conseguir una herramienta útil para cuantificar los parámetros mecánicos y convertirse en la base de un futuro método de diagnóstico. Se han realizado medidas experimentales con ultrasonidos y una simulación por el método de las diferencias finitas en muestras de tejido blando. Finalmente, se presenta un estudio comparativo entre las señales experimentales y simuladas. Los resultados muestran la capacidad de describir el comportamiento mecánico del tejido del cuello uterino como un material reforzado con fibras, y que los fenómenos de propagación de ondas ultrasónicas pueden ser explotados para reconstruir las propiedades mecánicas de los tejidos blandos por los que viajan, y por lo tanto, como herramienta para el diagnóstico de patologías que se manifiestan por cambios en la consistencia de los tejidos.Universidad de Granada. Departamento de Mecánica de Estructuras e Ingeniería Hidráulica. Máster Universitario en Estructuras, curso 2010-2011This work has been supported by the Ministry of Science and Innovation of Spain through FPI grant BES-2011-044970 within Proyect number DPI2010-17065 (MICINN)

Mechanics of Growing Tissues: A Continuum Description Approach

During development, higher organisms grow from a single fertilized egg cell to the adult animal. The many processes that lead to the eventual shape of the developed organism are subsumed as morphogenesis, which notably involves the growth of tissues by repeated rounds of cell division. Whereas coordinated tissue growth is a prerequisite for animal development, excessive cell division in adult animals is the key ingredient to cancer. In this thesis, we investigate the collective organization of cells by cell division and cell death. The multicellular dynamics of growing tissues is influenced by mechanical conditions and can give rise to cell rearrangements and movements. We develop a continuum description of tissue dynamics, which describes the stress distribution and the cell flow field on large scales. Cell division and apoptosis introduce stress sources that, in general, are anisotropic. By combining cell number balance with dynamic equations for the stress source, we show that the tissue effectively behaves as a viscoelastic fluid with a relaxation time set by the rates of division and apoptosis. If the tissue is confined in a fixed volume, it reaches a homeostatic state in which division and apoptosis balance. In this state, cells undergo a diffusive random motion driven by the stochasticity of division and apoptosis. We calculate the effective diffusion coefficient as a function of the tissue parameters and compare our results concerning both diffusion and viscosity to simulations of multicellular systems. Introducing a second material component that accounts for the extracellular fluid, we show that a finite permeability of the tissue gives rise to additional mechanical effects. In the limit of long times, the mechanical response of the tissue to external perturbations is confined to a region of which the size depends on the ratio of tissue viscosity and cell-fluid friction. The two-component description furthermore allows to clearly distinguish the different contributions to the isotropic part of the mechanical stress, i.e., the fluid pressure and the stress exerted by cells. Last but not least, we study the propagation of an interface between two different cell populations within a tissue driven by differences in the mechanical control of the rates of cell division and apoptosis. Combining simple analytical limits and numerical simulations, we distinguish two different modes of propagation of the more proliferative population: a diffusive regime in which relative fluxes dominate the expansion, and a propulsive regime in which the proliferation gives rise to dominating convective flows.Die Entwicklung höherer Organismen beginnt mit einer einzelnen befruchteten Eizelle und endet beim erwachsenen Tier. Die vielen Prozesse, die zur endgültigen Form des entwickelten Organismus führen, werden als Morphogenese zusammengefasst; diese umfasst insbesondere das Wachstum von Geweben durch wiederholte Zellteilungszyklen. Während koordiniertes Gewebewachstum eine Voraussetzung normaler Entwicklung ist, führt übermäßige, unkontrollierte Zellteilung letztlich zu Krebs. In dieser Arbeit untersuchen wir den Einfluss von Zellteilung und Zelltod auf die Organisation von Zellen in Geweben. Die Dynamik wachsender Gewebe wird durch mechanische Bedingungen beeinflusst, die u.a.~Anlass zu Zellbewegungen sein können. Wir entwickeln eine Kontinuumsbeschreibung der Gewebedynamik, die die mechanischen Spannungen und das Zellströmungsfeld auf großen Skalen beschreibt. Zellteilung und Apoptose wirken als Spannungsquellen, die in der Regel anisotrop sind. Indem wir die Erhaltungsgleichung für die Zellanzahldichte mit dynamischen Gleichungen für die Spannungsquellen kombinieren, zeigen wir, dass sich das Gewebe effektiv wie eine viskoelastische Flüssigkeit verhält, deren Relaxationszeit von Zellteilungs- und Apoptose-Raten abhängt. Wenn das Gewebe in einem gegebenen Volumen eingeschlossen ist, erreicht es einen homöostatischen Zustand, in dem Zellteilung und der Apoptose im Gleichgewicht sind. In diesem Zustand unterliegen die Zellen einer diffusiven Bewegung aufgrund der Stochastizität von Zellteilung und Apoptose. Wir berechnen den effektiven Diffusionskoeffizienten als Funktion der Gewebeparameter und vergleichen unsere Ergebnisse sowohl hinsichtlich der Diffusion und als auch der Viskosität mit numerischen Simulationen solcher vielzelliger Systeme. Die Berücksichtigung der extrazellulären Flüssigkeit als einer zweiten Materialkomponente erlaubt uns zu zeigen, dass eine endliche Permeabilität des Gewebes zusätzliche mechanische Effekte bedingt. Auf langer Zeitskalen bleibt die mechanische Reaktion des Gewebes auf externe Störungen auf einen Bereich beschränkt, dessen Größe vom Verhältnis der Gewebeviskosität zum Permeabilitätskoeffizienten abhängt. Die Zweikomponenten-Beschreibung erlaubt darüber hinaus eine klare Unterscheidung der verschiedenen Beiträge zum isotropen Teil der mechanischen Spannung, d.h., des hydrodynamischen und des von Zellen ausgeübten Drucks. Zuletzt untersuchen wir die Dynamik einer Grenzfläche zwischen zwei verschiedenen Zellpopulationen innerhalb eines Gewebes, die durch Unterschiede in der mechanischen Kontrolle der effektiven Zellteilungsraten angetrieben wird. Mithilfe der Kombination einfacher analytischer Grenzfälle und numerischer Simulationen zeigen wir, dass zwei unterschiedliche Ausbreitungsmodi unterschieden werden können: ein diffusives Regime, in dem relative Flüsse die Expansion der stärker wachsenden Zellpopulation dominieren, sowie ein Regime, in dem die Grenzfläche durch konvektive Strömungen angetrieben wird.Les organismes supérieurs se développent à partir d\'une seule cellule fécondée jusqu\'à l\'animal adulte. Les nombreux processus qui conduisent à la forme finale de l\'organisme sont connus sous le nom de morphogenèse, qui comprend notamment la croissance des tissus par des cycles répétés de division cellulaire. Alors que la croissance coordonnée des tissus est une condition nécessaire au développement des animaux, la division cellulaire excessive chez les animaux adultes est l\'ingrédient clé du cancer. Dans cette thèse, nous étudions l\'organisation collective des cellules par division et mort cellulaire. La dynamique multicellulaire des tissus en croissance est influencée par des conditions mécaniques et peut donner lieu à des réarrangements ainsi qu\'à des mouvements cellulaires. Nous élaborons une description continue de la dynamique des tissus qui décrit la distribution des contraintes et le champ d\'écoulement des cellules sur de grandes échelles. La division cellulaire et l\'apoptose introduisent des sources de contraintes qui, en général, sont anisotropes. En combinant l\'équation de conservation du nombre de cellules avec des équations dynamiques des sources de contraintes, nous montrons que le tissu se comporte de manière effective comme un fluide viscoélastique avec un temps de relaxation fixé par les taux de division et d\'apoptose. Si le tissu est confiné dans un volume donné, il atteint un état homéostatique dans lequel division et apoptose s\'équilibrent. Dans cet état, les cellules subissent un mouvement diffusif aléatoire dû à la stochasticité de la division et de l\'apoptose. Nous calculons le coefficient de diffusion effectif en fonction des paramètres du tissu et comparons nos résultats concernant à la fois la diffusion et la viscosité à des simulations numériques de tels systèmes multicellulaires. En introduisant un deuxième composant qui représente le liquide extracellulaire, nous montrons qu\'une perméabilité finie du tissu donne lieu à des effets mécaniques supplémentaires. Dans la limite des temps longs, la réponse mécanique du tissu à des perturbations extérieures est confinée à une région dont la taille dépend du rapport entre la viscosité tissulaire et le coefficient de frottement entre les cellules et le liquide extracellulaire. La description à deux composants permet en outre de distinguer clairement les différentes contributions à la partie isotrope de la contrainte mécanique, c\'est-à-dire la pression du fluide et la contrainte exercée par les cellules. Finalement, nous étudions la propagation d\'une interface entre deux populations de cellules différentes, due à des différences dans le contrôle mécanique des taux de division et de mort cellulaire. En combinant de simples limites analytiques et des simulations numériques, nous distinguons deux modes de propagation différents de la population cellulaire la plus proliférante : un régime diffusif dans lequel les flux relatifs dominent l\'expansion, et un régime de propulsion dans lequel la prolifération domine et entraine des flux convectifs

Computational modeling of epithelial wound healing: Short and long term chemo-mechanical mechanisms

During the lifetime of all living multicellular organisms, wounds in their tissues are frequently observed. Thecapability of closing those gaps is fundamental for a healthy development. If done deficiently, many diseasesmay occur from simple inflammation to tumor formation. The wound healing process in epithelial tissueoccurs in three different stages. The first one is the assembly of a supra-cellular actomyosin cable and itsmigration towards the wound edge, triggered by biochemical processes in which calcium plays a distinctiverole. How this process is orchestrated following damage remains unclear. Later, after its positioning, thecable contracts driving the tissue towards the gap and reducing the wound area. Finally, cell migrationtowards the interior of the wound ends up sealing the tissue. In this work, we make use of a mechanicalcontinuum model for the first two stages in order to developed and 2D finite element simulations within amonolithically fully implicit implementation. The model for the actomyosin cable formation involves thecoupling of transient calcium ions transport, with actin fibers and myosin motors recruitment and non-linearmechanical response of the tissue. The contraction stage, the active deformation of the previously formedactomyosin cable is taken into account. The relative motion of the myosin motors over the actin filaments ismodeled so there exists an active tissue contraction in the direction of those fibers. Upon implementation,the model is capable of performing a wide range of biophysical situations reported experimentally, as wedemonstrate in our numerical results. We have been able to rationalize through computational mechanicsthe firing of calcium in the wound right after damage infliction as well as the consequent formation of actinring, reproducing nicely what has been reported in biological literature. Thereafter, the numerical modelof acto-myosin contraction, fully integrated with the non-linear mechanics of the problem, correlates withthe mechanics of wound closure at the actin-ring contraction stage. More importantly, the approach is thefirst of its kind in the modeling of epithelial and embryonic cell layers, where a wide number of complexmechanics has been integrated and solved though computational methods in engineering. We believe thatthe simulations will help to unravel new insights in open questions of developmental biology.Peer ReviewedPostprint (author's final draft

On the theory of cell migration: durotaxis and chemotaxis

Cell migration is a fundamental element in a variety of physiological and pathological processes. Alteration of its regulatory mechanisms leads to loss of cellular adhesion and increased motility, which are critical steps in the initial stages of metastasis, before a malignant cell colonizes a distant tissue or organ. Consequently, cell migration has become the focus of intensive experimental and theoretical studies; however the understanding of many of its mechanism remains elusive. Cell migration is the result of a periodic sequence of protrusion, adhesion remodeling and contraction stages that leads to directed movement of cells towards external stimuli. The spatio-temporal coordination of these processes depends on the di erential activation of the signaling networks that regulate them at specific subcellular locations. Particularly, proteins from the family of small RhoGTPases play a central role in establishing cell polarization, setting the direction of migration, regulating the formation of adhesion sites and the generation of the forces that drive motion. Theoretical models based on an independent description of these processes have a limited capacity to predict cellular behavior observed in vitro, since their functionality depends intrinsically on the cross-regulation between their signaling pathways. This thesis presents a model of cell migration that integrates a description of force generation and cell deformation, adhesion site dynamics and RhoGTPases activation. The cell is modeled as a viscoelastic body capable of developing active traction and protrusion forces. The magnitude of stresses is determined by the activation level of the RhoGTPases, whose distribution in the cell body is described by a set of reaction-di usion equations. Adhesion sites are modeled as punctual clusters of transmembrane receptors that dynamically bind and unbind the extracellular matrix depending on the force transmitted to them and the distance with ligands on the substrate. Onthe theoretical level, the major findings concern the relationship between the topology of a crosstalk scheme and the properties, as defined in [1], inherited by the associated reaction network as a gradient sensing and regulatory system: persistent and transient polarization triggered by external gradients, adaptation to uniform stimulus, reversible polarization, multi-stimuli response and amplification. This leads to models that remain functional against the biological diversity associated to di erent cell types and matches the observed cell behaviour in Chemotaxis essays [2, 3, 4, 5]: the capacity of cells to amplify gradients, polarize without featuring Turing patterns of activation, and switch the polarization axis and the direction of migration after the source of the external stimulus is changed. The RhoGTPase model, derived on theoretical premises, challenges a long held view on the mechanisms of RhoGTPase crosstalk and suggests that the role of GDIs, GEFs and GAPs has to be revised. Recent experimental evidence supports this idea[6]. In addition, the model allows to recapitulate a continuous transition between the tear-like shape adopted by neutrophiles and the fan-like shape of keratocytes during migration [7] by varying the relative magnitudes of protrusion and contraction forces or, alternatively, the strength of RhoGTPase Crosstalk. The second mechanism represents a novel explanation of the di erent morphologies observed in migrating cells. Di erences in RhoGTPase crosstalk strength could be mediated by di erences between the activity or concentration of GEFs, GAPs and GDIs in di erent cell types; an idea that can be explored experimentally. On cell mechanosensing, a new hypothesis based on a simple physical principle is proposed as the mechanism that might explain the universal preference of cells (bar neurons) to migrate along sti ness gradients. The theory provides a simple unifying explanation to a number of recent observations on force development and growth in real time at cell Focal adhesions [8, 9, 10, 11]. The apparently conflicting results have been attributed to the di erences in experimental set-ups and cell types used, and have fueled a longstanding controversy on how cells prove the mechanical properties of the extra-cellular matrix. The predictions of the theory recapitulate these experimental observations, and its founding hypothesis can be tested experimentally. This hypothesis directly suggests the mechanism that could explain the preference of cells to migrate along sti ness gradients, and for the first time, a plausible biological function for its existence. This phenomenon is known as Durotaxis, and its abnormal regulation has been associated to the malignant behaviour of cancer cells. &nbsp

Forces and Flow of Contractile Networks

Biological cells use contractile networks of cross-linked semiflexible biopolymers, the so-called actin cytoskeleton, to control their shapes and to probe the mechanical properties of their environment. These processes are essential for cell survival and function. In this thesis we present a general framework to model two-dimensional contractile networks embedded in either two- or three-dimensional space. A surface representation with triangles and edges allows us to explicitly address the heterogeneity of biopolymer networks. In adherent cells, thick polymer bundles called stress fibers strongly influence cellular mechanics. We establish methods to assess their contribution to traction force generation, intracellular force balance, and intracellular flow from experimental data. Further, we develop a theory for the excitable nature of the cell cortex, which is a thin polymer layer lining the inner side of the cell membrane, and show how it is related to global cell shape changes

Partial Differential Equation Models of Collective Migration During Wound Healing

This dissertation is concerned with the derivation, analysis, and parameter inference of mathematical models of the collective migration of epithelial cells. During the wound healing process, epidermal keratinocytes collectively migrate from the wound edge into the wound area as a means to re-establish the outermost layer of skin. This migration into the wound is stimulated by the presence of epidermal growth factor. Accordingly, this dissertation focuses on the migratory response of epidermal keratinocytes in response to this growth factor. Such studies will suggest suitable clinical treatments to consider for chronic wounds and invasive cancers.We begin with a study into the role of cell-cell adhesions on keratinocyte migration during wound healing. We use an inverse problem methodology in combination with model validation to show that cells use these connections to promote migration by pulling on their follower cells as they migrate into the wound. We next derive a biochemically-structured version of Fisher's Equation that provides a framework to study how patterns of biochemical activation influence migration into the wound. We prove the existence of a self-similar traveling wave solution. In considering a more complicated scenario where cell migration depends on biochemical activity levels, we show numerically that the threshold parameter where all cells in the population become activated yields the simulations that migrate farthest into the wound. Lastly, we consider the role of numerical error on an inverse problem methodology. The numerical approximation of a cost function is dominated by either numerical or experimental error in computations, which leads to different rates of convergence as numerical resolution increases. We use residual analysis to derive an autocorrelative statistical model for cases where numerical error is the main source of error for first order schemes. This autocorrelative statistical model can correct confidence interval computation for these methods and hence improve uncertainty quantification