[Show abstract][Hide abstract] ABSTRACT: In this paper we investigate the limit behavior of the solution to
quasi-static Biot's equations in thin poroelastic plates as the thickness tends
to zero. We choose Terzaghi's time corresponding to the plate thickness and
obtain the strong convergence of the three-dimensional solid displacement,
fluid pressure and total poroelastic stress to the solution of the new class of
plate equations. In the new equations the in-plane stretching is described by
the 2D Navier's linear elasticity equations, with elastic moduli depending on
Gassmann's and Biot's coefficients. The bending equation is coupled with the
pressure equation and it contains the bending moment due to the variation in
pore pressure across the plate thickness. The pressure equation is parabolic
only in the vertical direction. As additional terms it contains the time
derivative of the in-plane Laplacean of the vertical deflection of the plate
and of the the elastic in-plane compression term.
Archive for Rational Mechanics and Analysis 11/2014; · 2.02 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Myelodysplastic syndromes (MDS) are triggered by an aberrant hematopoietic stem cell (HSC). It is, however, unclear how this clone interferes with physiologic blood formation. In this study, we followed the hypothesis that the MDS clone impinges on feedback signals for self-renewal and differentiation and thereby suppresses normal hematopoiesis. Based on the theory that the MDS clone affects feedback signals for self-renewal and differentiation and hence suppresses normal hematopoiesis, we have developed a mathematical model to simulate different modifications in MDS-initiating cells and systemic feedback signals during disease development. These simulations revealed that the disease initiating cells must have higher self-renewal rates than normal HSCs to outcompete normal hematopoiesis. We assumed that self-renewal is the default pathway of stem and progenitor cells which is down-regulated by an increasing number of primitive cells in the bone marrow niche - including the premature MDS cells. Furthermore, the proliferative signal is up-regulated by cytopenia. Overall, our model is compatible with clinically observed MDS development, even though a single mutation scenario is unlikely for real disease progression which is usually associated with complex clonal hierarchy. For experimental validation of systemic feedback signals, we analyzed the impact of MDS patient derived serum on hematopoietic progenitor cells in vitro: in fact, MDS serum slightly increased proliferation, whereas maintenance of primitive phenotype was reduced. However, MDS serum did not significantly affect colony forming unit (CFU) frequencies indicating that regulation of self-renewal may involve local signals from the niche. Taken together, we suggest that initial mutations in MDS particularly favor aberrant high self-renewal rates. Accumulation of primitive MDS cells in the bone marrow then interferes with feedback signals for normal hematopoiesis - which then results in cytopenia.
[Show abstract][Hide abstract] ABSTRACT: Deregulation of signaling pathways and subsequent abnormal interactions of downstream genes very often results in carcinogenesis. In this paper, we propose a two-compartment model describing intricate dynamics of the target genes of the Wnt signaling pathway in medulloblastoma. The system of nine nonlinear ordinary differential equations accounts for the formation and dissociation of complexes as well as for the transcription, translation and transport between the cytoplasm and nucleus. We focus on the interplay between MYC and SGK1 (serum and glucocorticoid-inducible kinase 1), which are the products of Wnt/β-catenin signaling pathway, and GSK3β (glycogen synthase kinase). Numerical simulations of the model solutions yield a better understanding of the process and indicate the importance of the SGK1 gene in the development of medulloblastoma, which has been confirmed in our recent experiments. The model is calibrated based on the gene expression microarray data for two types of medulloblastoma, characterized by monosomy and trisomy of chromosome 6q to highlight the difference between diagnoses.
Journal of Theoretical Biology 03/2014; · 2.35 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In the adult hippocampus, neurogenesis-the process of generating mature granule cells from adult neural stem cells-occurs throughout the entire lifetime. In order to investigate the involved regulatory mechanisms, knockout (KO) experiments, which modify the dynamic behaviour of this process, were conducted in the past. Evaluating these KOs is a non-trivial task owing to the complicated nature of the hippocampal neurogenic niche. In this study, we model neurogenesis as a multicompartmental system of ordinary differential equations based on experimental data. To analyse the results of KO experiments, we investigate how changes of cell properties, reflected by model parameters, influence the dynamics of cell counts and of the experimentally observed counts of cells labelled by the cell division marker bromodeoxyuridine (BrdU). We find that changing cell proliferation rates or the fraction of self-renewal, reflecting the balance between symmetric and asymmetric cell divisions, may result in multiple time phases in the response of the system, such as an initial increase in cell counts followed by a decrease. Furthermore, these phases may be qualitatively different in cells at different differentiation stages and even between mitotically labelled cells and all cells existing in the system.
Journal of The Royal Society Interface 02/2014; 11(94):20140144. · 3.86 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Recent experimental evidence suggests that acute myeloid leukaemias may originate from multiple clones of malignant cells. Nevertheless, it is not known how the observed clones may differ with respect to cell properties, such as proliferation and self-renewal. There are scarcely any data on how these cell properties change due to chemotherapy and relapse. We propose a new mathematical model to investigate the impact of cell properties on the multi-clonal composition of leukaemias. Model results imply that enhanced self-renewal may be a key mechanism in the clonal selection process. Simulations suggest that fast proliferating and highly self-renewing cells dominate at primary diagnosis, while relapse following therapy-induced remission is triggered mostly by highly self-renewing but slowly proliferating cells. Comparison of simulation results to patient data demonstrates that the proposed model is consistent with clinically observed dynamics based on a clonal selection process.
Journal of The Royal Society Interface 02/2014; 11(94):20140079. · 3.86 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: It is generally accepted that the effective velocity of a viscous flow over a
porous bed satisfies the Beavers-Joseph slip law. To the contrary, in the case
of a forced infiltration of a viscous fluid into a porous medium the interface
law has been a subject of controversy. In this paper, we prove rigorously that
the effective interface conditions are:
(i) the continuity of the normal effective velocities;
(ii) zero Darcy's pressure and
(iii) a given slip velocity.
The effective tangential slip velocity is calculated from the boundary layer
and depends only on the pore geometry. In the next order of approximation, we
derive a pressure slip law. An independent confirmation of the analytical
results using direct numerical simulation of the flow at the microscopic level
is given, as well.
Computer Methods in Applied Mechanics and Engineering 01/2014; · 2.63 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Self-renewal is a constitutive property of stem cells. Testing the cancer
stem cell hypothesis requires investigation of the impact of self-renewal on
cancer expansion. To understand better this impact, we propose a mathematical
model describing dynamics of a continuum of cell clones structured by the
self-renewal potential. The model is an extension of the finite
multi-compartment models of interactions between normal and cancer cells in
acute leukemias. It takes a form of a system of integro-differential equations
with a nonlinear and nonlocal coupling, which describes regulatory feedback
loops in cell proliferation and differentiation process. We show that such
coupling leads to mass concentration in points corresponding to maximum of the
self-renewal potential and the model solutions tend asymptotically to a linear
combination of Dirac measures. Furthermore, using a Lyapunov function
constructed for a finite dimensional counterpart of the model, we prove that
the total mass of the solution converges to a globally stable equilibrium.
Additionally, we show stability of the model in space of positive Radon
measures equipped with flat metric. The analytical results are illustrated by
[Show abstract][Hide abstract] ABSTRACT: Hematopoiesis is a complex and strongly regulated process. In case of regenerative pressure, efficient recovery of blood cell counts is crucial for survival of an individual. We propose a quantitative mathematical model of white blood cell formation based on the following cell parameters: (1) proliferation rate, (2) self-renewal, and (3) cell death. Simulating this model we assess the change of these parameters under regenerative pressure. The proposed model allows to quantitatively describe the impact of these cell parameters on engraftment time after stem cell transplantation. Results indicate that enhanced self-renewal during the posttransplant period is crucial for efficient regeneration of blood cell counts while constant or reduced self-renewal leads to delayed recovery or graft failure. Increased cell death in the posttransplant period has a similar impact. In contrast, reduced proliferation or pre-homing cell death causes only mild delays in blood cell recovery which can be compensated sufficiently by increasing the dose of transplanted cells.
Advances in Experimental Medicine and Biology 01/2014; 844:347-367. · 2.01 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Motivated by recent experimental findings, we propose a novel mechanism of embryonic pattern formation based on coupling of tissue curvature with diffusive signaling by a chemical factor. We derive a new mathematical model using energy minimization approach and show that the model generates a variety of morphogen and curvature patterns agreeing with experimentally observed structures. The mechanism proposed transcends the classical Turing concept which requires interactions between two morphogens with a significantly different diffusivity. Our studies show how biomechanical forces may replace the elusive long-range inhibitor and lead to formation of stable spatially heterogeneous structures without existence of chemical prepatterns. We propose new experimental approaches to decisively test our central hypothesis that tissue curvature and morphogen expression are coupled in a positive feedback loop.
PLoS ONE 12/2013; 8(12):e82617. · 3.53 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Coupling diffusion process of signaling molecules with nonlinear interactions
of intracellular processes and cellular growth/transformation leads to a system
of reaction-diffusion equations coupled with ordinary differential equations
(diffusion-ODE models), which differ from the usual reaction-diffusion systems.
One of the mechanisms of pattern formation in such systems is based on the
existence of multiple steady states and hysteresis in the ODE subsystem.
Diffusion tries to average different states and is the cause of spatio-temporal
patterns. In this paper we provide a systematic description of stationary
solutions of such systems, having the form of transition or boundary layers.
The solutions are discontinuous in the case of non-diffusing variables whose
quasi-stationary dynamics exhibit hysteresis. The considered model is motivated
by biological applications and elucidates a possible mechanism of formation of
patterns with sharp transitions.
[Show abstract][Hide abstract] ABSTRACT: It is known that the number of transplanted cells has a significant impact on the outcome after SCT. We identify issues that cannot be addressed by conventional analysis of clinical trials and ask whether it is possible to develop a refined analysis to conclude about the outcome of individual patients given clinical trial results. To accomplish this, we propose an interdisciplinary approach based on mathematical modeling. We devise and calibrate a mathematical model of short-term reconstitution and simulate treatment of large patient groups with random interindividual variation. Relating model simulations to clinical data allows quantifying the effect of transplant size on reconstitution time in the terms of patient populations and individual patients. The model confirms the existence of lower bounds on cell dose necessary for secure and efficient reconstitution but suggests that for some patient subpopulations higher thresholds might be appropriate. Simulations demonstrate that relative time gain because of increased cell dose is an 'interpersonally stable' parameter, in other words that slowly engrafting patients profit more from transplant enlargements than average cases. We propose a simple mathematical formula to approximate the effect of changes of transplant size on reconstitution time.Bone Marrow Transplantation advance online publication, 23 September 2013; doi:10.1038/bmt.2013.138.
Bone marrow transplantation 09/2013; · 3.00 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Recently developed theoretical framework for analysis of structured
population dynamics in the spaces of nonnegative Radon measures with a suitable
metric provides a rigorous tool to study numerical schemes based on particle
methods. The approach is based on the idea of tracing growth and transport of
measures which approximate the solution of original partial differential
equation. In this paper we present analytical and numerical study of two
versions of Escalator Boxcar Train (EBT) algorithm which has been widely
applied in theoretical biology, and compare it to the recently developed
split-up algorithm. The novelty of this paper is in showing well-posedness and
convergence rates of the schemes using the concept of semiflows on metric
spaces. Theoretical results are validated by numerical simulations of test
cases, in which distances between simulated and exact solutions are computed
using flat metric.
[Show abstract][Hide abstract] ABSTRACT: We present a global stability analysis of two-compartment models of a hierarchical cell production system with a nonlinear regulatory feedback loop. The models describe cell differentiation processes with the stem cell division rate or the self-renewal fraction regulated by the number of mature cells. The two-compartment systems constitute a basic version of the multicompartment models proposed recently by Marciniak-Czochra and collaborators () to investigate the dynamics of the hematopoietic system. Using global stability analysis, we compare different regulatory mechanisms. For both models, we show that there exists a unique positive equilibrium that is globally asymptotically stable if and only if the respective reproduction numbers exceed one. The proof is based on constructing Lyapunov functions, which are appropriate to handle the specific nonlinearities of the model. Additionally, we propose a new model to test biological hypothesis on the regulation of the fraction of differentiating cells. We show that such regulatory mechanism is incapable of maintaining homeostasis and leads to unbounded cell growth. Potential biological implications are discussed.
[Show abstract][Hide abstract] ABSTRACT: We analyze an integro-differential initial value problem obtained as a
shadow-type limit from a system of reaction-diffusion-ODE equations modeling
pattern formation. We show that nonlocal terms in such models induce a
destabilization of stationary solutions, which may lead to a blowup of
spatially inhomogeneous solutions, either in finite or infinite time.
[Show abstract][Hide abstract] ABSTRACT: We present modeling of an incompressible viscous flow through a fracture
adjacent to a porous medium. We consider a fast stationary flow, predominantly
tangential to the porous medium. Slow flow in such setting can be described by
the Beavers-Joseph-Saffman slip. For fast flows, a nonlinear filtration law in
the porous medium and a non- linear interface law are expected. In this paper
we rigorously derive a quadratic effective slip interface law which holds for a
range of Reynolds numbers and fracture widths. The porous medium flow is
described by the Darcys law. The result shows that the interface slip law can
be nonlinear, independently of the regime for the bulk flow. Since most of the
interface and boundary slip laws are obtained via upscaling of complex systems,
the result indicates that studying the inviscid limits for the Navier-Stokes
equations with linear slip law at the boundary should be rethought.
Discrete and Continuous Dynamical Systems - Series S 03/2013;
[Show abstract][Hide abstract] ABSTRACT: We explore a mechanism of pattern formation arising in processes
described by a system of a single reaction-diffusion equation coupled
with ordinary differential equations. Such systems of equations arise
from the modeling of interactions between cellular processes and
diffusing growth factors. We focused on the model of early
carcinogenesis proposed by Marciniak-Czochra and Kimmel, which is an
example of a wider class of pattern formation models with an
autocatalytic non-diffusing component. We present a numerical study
showing emergence of periodic and irregular spike patterns due to
diffusion-driven instability. To control the accuracy of simulations, we
develop a numerical code based on the finite element method and adaptive
mesh. Simulations, supplemented by numerical analysis, indicate a novel
pattern formation phenomenon based on the emergence of nonstationary
structures tending asymptotically to the sum of Dirac deltas.
Mathematical Methods in the Applied Sciences 03/2013; 37(9). · 0.78 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: It is generally accepted that the effective velocity of a viscous flow over a
porous bed satisfies the Beavers-Joseph slip law. To the contrary, interface
law for the effective stress has been a subject of controversy. Recently, a
pressure jump interface law has been rigorously derived by Marciniak-Czochra
and Mikeli\'c. In this paper, we provide a confirmation of the analytical
result using direct numerical simulation of the flow at the microscopic level.
Journal of Fluid Mechanics 01/2013; 732:510-536. · 2.29 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The aim of this paper is to contribute to understanding of the pattern
formation phenomenon in reaction-diffusion equations coupled with ordinary
differential equations. Such systems of equations arise, for example, from the
modeling of interactions between proliferating cells and diffusing growth
factors. We focus on stability of solutions to an initial-boundary value
problem for a system consisting of a single reaction-diffusion equation coupled
with an ordinary differential equation. We show that such systems exhibit
diffusion-driven instabilities (Turing instability) under the condition of
autocatalysis of non-diffusing component. Nevertheless, there exist no stable
Turing patterns, i.e. all continuous spatially heterogenous stationary
solutions are unstable. In addition, we formulate instability conditions for
discontinuous patterns for a class of nonlinearities.
[Show abstract][Hide abstract] ABSTRACT: In this article, we propose a new notion of multiscale convergence, called ‘three-scale’, which aims to give a topological framework in which to assess complex processes occurring at three different scales or levels within a heterogeneous medium. This generalizes and extends the notion of two-scale convergence, a well-established concept that is now commonly used for obtaining an averaged, asymptotic value (homogenization) of processes that exist on two different spatial scales. The well-posedness of this new concept is justified via a compactness theorem which ensures that all bounded sequences in L 2(Ω) are relative compact with respect to the three-scale convergence. This is taken further by giving a boundedness characterization of three-scale convergent sequences and is then continued with the introduction of the notion of ‘strong three-scale convergence’ whose well-posedness is also discussed. Finally, the three-scale convergence of the gradients is established.