CTW: Molecular to Systems Physiology

(May 5,2014 - May 9,2014 )

Organizers


Daniel Beard
Department of Physiology, Medical College of Wisconsin
Laura Ellwein
Mathematics, Virginia Commonwealth University
Mette Olufsen
Department of Mathematics, North Carolina State University

More than a decade after the completion of the Human Genome Project, our ability to predict important high-level phenotypes from molecular information at the cellular level remains woefully inadequate. Statistical mapping between variants identified by genome -wide association studies and complex traits such as hypertension do not effectively explain the range of phenotypes in the population, nor do they provide useful predictions of disease risk. In short, the standard machinery of statistical genetics has fallen short as a tool to understand complex disease. This provides the opportunity and motivation for a more comprehensive approach to the grand challenge of understanding the mechanistic relationships between high-level phenotypes and molecular information. 

Multi-scale simulation of physiological systems represents a powerful vehicle for linking multiple levels of causality. Mathematical modeling in combination with high-performance computing and high-resolution data has led to tremendously sophisticated and reliable multi-scale multi-physics based simulations of certain physiological systems. In particular, system dynamics from the cellular to the system levels have long been studied using mathematical modeling, for example, computer models of the heart. Yet such dynamics models rarely make any use of data gathered at the molecular level, and therefore cannot capitalize on the emerging availability of patient data collected at multiple scales (e.g. genome information). This workshop will discuss the state-of-the-art mathematical techniques (and outstanding needs) for effectively synthesizing data ranging from genomic through molecular and organ up to the system level with multi-scale computational techniques. Efforts will be focused on addressing how models can be adapted to couple data measured at different scales and from different species, yet belong to the same physiological system. This question will be studied within the respiratory, cardiovascular, and renal systems. We expect that it is possible to extract common features from these systems, and that techniques applied will have applicability outside the systems studied. 

This workshop will bring together domain experts from physiology, mathematics, and statistics. Physiologists and statisticians will help identify key data sets of interest and address questions related to uncertainty in data sampling, including discussion of known variation within species, and between in-vivo and in-vitro sampling. Mathematicians will bring expertise in modeling, model reduction, and solving inverse problems. The aim will be to discuss ways to combine data from multiple sources and scales with relevant models to predict patient specific responses. New techniques that have shown promise for solving these types of problems include reformulation of models using techniques from algebra, uncertainty quantification, parameter estimation, and networks. This diverse group of researchers will have potential to generate new projects and ideas for linking statistical and physics-based techniques for building multi-scale mathematical models that incorporate physiological data from multiple sources and scales, which may eventually elucidate relationships between phenotypes and the underlying physiology.

Accepted Speakers

Kellie Archer
Biostatistics, Virginia Commonwealth University
Julia Arciero
Mathematics, Indiana University--Purdue University
Jerry Batszel
Institute of Mathematics, University of Graz, Austria
Daniel Beard
Department of Physiology, Medical College of Wisconsin
Daniela Calvetti
Mathematics, Applied Mathematics and Statistics, Case Western Reserve University
Brian Carlson
Physiology, Medical College of Wisconsin
Michael Chappell
Warwick Engineering in Biomedicine, University of Warwick
Naomi Chesler
Biomedical Engineering, University of Wisconsin
Gheorghe Craciun
Mathematics and Biomolecular Chemistry, University of Wisconsin-Madison
Laura Ellwein
Mathematics, Virginia Commonwealth University
Alberto Figueroa
Department of Biomedical Engineering, King's College
John Gennari
Biomedical Informatics and Medical Education, University of Washington
Leif Hellevik
Structural Engimeering, Norwegian University of Science and Technology
Nick Hill
School of Mathematics and Statistics, University of Glasgow
Alison Hu
Biomedical Engineering, University of Southern California
Ghassan Kassab
Biomedical Engineering&MedicineCardiology , University of California, Irvine
Anita Layton
Mathematics, Duke University
Adam Mahdi
Mathematics, North Carolina State University
Robert Moss
Mathematics, Duke University
Mette Olufsen
Department of Mathematics, North Carolina State University
Johnny Ottesen
Department of Mathematics and Physics, Roskilde University Center
Amina Qutub
Bioengineering, Rice University
Michael Reed
Mathematics, Duke University
John Schild
Department of Biomedical Engineering, Neuroscience, Indiana University--Purdue University
Santiago Schnell
Department of Molecular & Integrative Biology, University of Michigan Medical School
Christian Schulze
Medicine/Cardiology, Columbia University
Hien Tran
Mathematics, North Carolina State University
Frans van de Vosse
Department of Biomedical Engineering, Technische Universiteit Eindhoven
Alessandro Veneziani
Department of Mathematics and Computer Science, Emory University
Carsten Wiuf
Department of Mathematical Sciences, University of Copenhagen
Monday, May 5, 2014
Time Session
08:00 AM

Shuttle to MBI

08:15 AM
09:00 AM

Breakfast

09:00 AM
09:15 AM

Welcome to MBI - Marty Golubitsky

09:15 AM
09:35 AM

Introduction by Organizers

09:45 AM
10:20 AM
Santiago Schnell - On the mechanisms of sensing unfolded protein in the endoplasmic reticulum.

One of the main functions of the endoplasmic reticulum (ER) is to serve as the cell protein-folding factory. The ER is responsible for the synthesis, folding, assembly and modification of one third of the eukaryotic proteome. Proteins enter the ER as unfolded polypeptide chains with variable fluxes depending on the physiological state of the cell. A sudden increase in the demand for a protein or the disruption of a folding reaction causes an imbalance between protein-folding load and capacity of the ER, which can lead to the accumulation of unfolded protein in the ER lumen. The ER protein balance is regulated by several signaling pathways, which are collectively termed the unfolded protein response. The unfolded protein response is activated by three transducers, which are enzymes whose oligomerization-induced activation is linked to perturbed protein folding in the ER. Three model mechanisms have been proposed for how these enzymes sense the unfolded protein load in the ER lumen: (i) direct recognition, (ii) indirect recognition and (iii) hybrid recognition. We developed detailed reaction mechanisms for each model and analyzed their dynamical behavior. We found that some of these mechanisms have serious discrepancies with the experimental data. We suggest a set of experiments that have not been yet carried out to test a detailed novel model mechanism of protein load sensing in the ER lumen, which explains current experimental findings. Our new model could provide new insights into the mechanisms of protein homeostasis in the ER.

10:30 AM
11:05 AM
Amina Qutub - Molecular Signatures of Cells during Hypoxic-Stimulated Tissue Growth

Oxygen is fundamental to life on Earth. In diseases affecting the vasculature including cancer and neurodegenerative diseases, abberrant hypoxic response is a critical part of the disease. Limited oxygen can lead to more aggressive tumors or determine our susceptibility to dementia. On the other hand, appropriate manipulation of proteins involved in cellular hypoxic response can help restore blood vessels and regenerate tissues. A challenge lies in understanding the complex cellular response to hypoxia both across different diseases and between patients with the detail needed to develop effective therapies. In this presentation, I will share how we are developing and integrating methods in multiscale modeling, machine learning, molecular biology, and microscopy image analysis to tackle the challenge of interpreting how changes at the molecular level affect cellular response and multicellular dynamics. My lab’s goal is to use computational systems biology methods to understand – and ultimately control –biological response to oxygen across scales.

11:10 AM
12:00 PM

Discussion: Integrating multiple-scale biophysical processes (Discussion Leader: Daniela Calvetti)

12:00 PM
01:30 PM

Lunch Break

01:30 PM
02:05 PM
Gheorghe Craciun - Persistence, Permanence, and Global Stability in Biological Interaction Networks

Complex interaction networks are present in all areas of biology, and manifest themselves at very different spatial and temporal scales. Persistence, permanence and global stability are emergent properties of complex networks, and play key roles in the dynamics of living systems.

Mathematically, a dynamical system is called persistent if, for all positive solutions, no variable approaches zero. In addition, for a permanent system, all variables are uniformly bounded. We describe criteria for persistence and permanence of solutions, and for global convergence of solutions to an unique equilibrium, in a manner that is robust with respect to initial conditions and parameter values.

A thorough understanding of these properties will allow for a better understanding of essential biological processes, such as homeostasis and adaptability.

02:15 PM
02:50 PM
Michael Reed - How Mathematicians can Contribute to Genomic Medicine

Mathematical models of physiological processes allow one to study the homeostatic mechanisms that keep important phenotypic variables within certain normal ranges. When these variables leave the homeostatic range often disease processes ensue. From the models one can derive surfaces that show the relationship between genetic polymorphisms and particularly important phenotypic variables. Known gene polymorphisms correspond to particular points on the surface, some of which are located near the edge of the homeostatic region. The purpose of medical advice tailored to the patient’s genotype is to suggest dietary changes or exercise changes that move the patient back towards the middle of the homeostatic region.

03:00 PM
03:35 PM
Anita Layton, Robert Moss - Assessment of Renal Autoregulatory Mechanisms

A mathematical model of renal hemodynamics is used to assess the individual contributions of the tubuloglomerular feedback (TGF) mechanism and the myogenic response to glomerular filtration rate regulation in the rat kidney. The model represents an afferent arteriole segment, glomerular filtration, and a short loop of Henle. The afferent arteriole model exhibits myogenic response, which is activated by hydrostatic pressure variations to induce changes in membrane potential and vascular muscle tone. The tubule model predicts tubular fluid and Cl- transport. Macula densa Cl- concentration is sensed as the signal for TGF, which acts to constrict or dilate the afferent arteriole. With this configuration, the model afferent arteriole maintains stable glomerular filtration rate within a physiologic range of perfusion pressure (80-180 mmHg). The contribution of TGF to overall autoregulation is significant only within a narrow band of perfusion pressure values (80-110 mmHg). Model simulations of ramp-like perfusion pressure perturbations agree well with findings by Flemming et al. (J Am Soc Nephrol 12:2253-2262, 2001), which indicate that changes in vascular conductance is markedly sensitive to pressure velocity. That asymmetric response is attributed to the rate-dependent kinetics of the myogenic mechanism. Moreover, simulations of renal autoregulation in diabetes mellitus predict that, due to the impairment of the voltage-gated Ca2+ channels of the afferent arteriole smooth muscle cells, the perfusion pressure range in which SNGFR remains stable is reduced by ~70%, and that TGF gain is reduced by nearly 40%, consistent with experimental findings.

04:00 PM
05:00 PM
Alberto Figueroa - Recent Advances in 3D Blood flow Simulation: From Parameter Estimation Methods to Clinical Applications

In this talk we will give an overview of a series of methods for 3D blood flow modeling, ranging from Kalman filtering techniques for automatic outflow and material parameter estimation to baroreflex model for automatic control of blood pressure. We will also discuss recent progress made on the validation of CFD predictions of pressure gradients in coarctation patients at rest and stress using clinical pressure data.

05:00 PM
06:00 PM

Reception in MBI lounge

06:00 PM

Shuttle pick-up from MBI

Tuesday, May 6, 2014
Time Session
08:00 AM

Shuttle to MBI

08:15 AM
09:00 AM

Breakfast

09:00 AM
09:35 AM
Hien Tran
09:45 AM
10:20 AM
Daniela Calvetti
10:30 AM
11:05 AM
Leif Hellevik
11:10 AM
12:00 PM

Discussion: Practical approaches to multi-scale physiology modeling (Discussion Leader: Alberto Figueroa)

12:00 PM
01:30 PM

Lunch Break

01:30 PM
02:05 PM
Carsten Wiuf - Model reduction is biochemical reaction networks

In many situations we apply simplified models to complex dynamical systems, either because we are unaware of what the 'correct' model should look like, or because the 'correct' model is too complex to handle statistically/mathematically. In this talk, I will discuss model reduction for stochastic as well as deterministic biochemical reaction networks. In particular, I will focus on reduction by elimination of intermediate species, transient species that typically are consumed at a faster rate than non-intermediates and provide a number of results concerning equilibrium dynamics as well as non-equilibrium dynamics.

02:15 PM
02:50 PM
Michael Chappell
02:50 PM
03:15 PM

Break

03:15 PM
03:50 PM
Kellie Archer - Ordinal Response Models for Modeling Longitudinal High-Dimensional Genomic Feature Data

Ordinal scales are commonly used to measure health status and disease related outcomes. Notable examples include cancer staging, histopathological classification, adverse event rating, and severity of illness. In addition, repeated measurements are common in clinical practice for tracking and monitoring the progression of complex diseases. Classical likelihood-based ordinal modeling methods have contributed to the analysis of data in which the response categories are ordered and the number of covariates (p) is smaller than the sample size (n). With the emergence of genomic technologies being increasingly applied to identify molecular markers associated with complex disease phenotypes and outcomes, many research studies now include high dimensional feature data where p >> n, so that traditional methods cannot be applied. To fill this void we have developed an innovative penalized random coefficient ordinal response model for classifying and predicting disease progression along with time. Specifically our method extends the Generalized Monotone Incremental Forward Stagewise method (Hastie et al, 2007) to the ordinal response setting in combination with classical mixed effects modeling methods. We demonstrate our method using data from the Inflammation and the Host Response to Injury study in which Affymetrix gene expression profiles and Marshall Multiple Organ Dysfunction Score on six body systems were longitudinally collected at hospitalization day 1 up to day 30 in 169 patients.

04:00 PM
04:35 PM
John Gennari, John Gennari - Codeword annotation for sharing and merging physiological models

Hunter and Bassingthwaithe define the Physiome as a set of multiscale, interacting mathematical models of physiology. Although available model repositories are an initial step toward this vision, it is a critical next step to develop computer-readable annotation for connecting codewords across models. Current hand-crafted model-building methods must be formalized and standardized to better support knowledge interaction and sharing. In particular, we argue for semantic annotations as a way of communicating the biophysical meaning of individual model codewords. Once annotated in a computable format, we can automatically find and connect models based on the annotation semantics of the biological entities and physiological properties.

In this talk, we present our approach to semantic annotation, using standard bio-ontology terms to relate physiological properties (e.g. pressure), to anatomical entities (e.g. blood). In turn, we use these annotations to semi-automatically find relevant models from repositories, and ultimately merge those models where appropriate. We present our results with SemGen, a prototype tool, for both building annotations and merging models, even across different modeling languages. If successful, our approach to develop interacting model repositories could accelerate model sharing and integration, and research that depends on the construction of complex models.

04:35 PM
06:00 PM

Poster Session

06:00 PM

Shuttle pick-up from MBI

Wednesday, May 7, 2014
Time Session
08:00 AM

Shuttle to MBI

08:15 AM
09:00 AM

Breakfast

09:00 AM
09:35 AM
Christian Schulze
09:45 AM
10:20 AM

TBD

10:30 AM
11:05 AM
Julia Arciero - Assessing vascular risk factors for glaucoma using a mathematical model of blood flow in the retina

Glaucoma is the second leading cause of blindness in the world and is characterized by progressive retinal ganglion cell death and irreversible visual field loss. Although elevated intraocular pressure has been identified as the primary risk factor for glaucoma and is the main target of glaucoma treatments, several vascular risk factors that lead to impaired retinal blood flow have also been correlated with the progression and incidence of glaucoma. Here, a multi-scale mathematical model is used to investigate the relative contributions of vascular risk factors on flow regulation and tissue oxygenation in the retina. A previously-developed fluid-structure interaction system modeling the central retinal artery is coupled to a vascular wall mechanics model for the vessels of the retinal microcirculation. Under normal conditions, the model predicts a 14% decrease in retinal perfusion if oxygen demand is decreased by 50% and a 33% increase in perfusion if demand is increased by 50%. These responses are impaired significantly if the metabolic or carbon dioxide mechanisms of retinal blood flow autoregulation are impaired. Changes in oxygen saturation levels in the retinal vascular network are also assessed as levels of mean arterial pressure, oxygen demand, and intraocular pressure are varied. Overall, the model results suggest that impaired autoregulation might increase the risk of retinal ischemic damage, as would occur in glaucoma, under conditions of elevated metabolic demand or decreased mean arterial pressure.

11:10 AM
12:00 PM

Discussion: When and how are models from different labs compatible: Is there any value in archiving and disseminating computational models? (Discussion Leader: Klas Petersen)

12:00 PM
01:30 PM

Lunch Break

01:30 PM
02:05 PM
Nick Hill - A structured tree model for the pulmonary circulation

Abstract coming soon.

02:15 PM
02:50 PM
Frans van de Vosse - Personalization of 1D Wave Propagation Models of the Cardiovascular System

One of the main di?culties in the translation of mathematical models to the clinic for supporting clinical decision-making is assessing patient-speci?c values for the model parameters, the boundary and the initial conditions. Measurement modalities or data are not always available for all model parameters. In addition, the precision and accuracy of clinical measurements are hampered by large (biological) variations. Consequently, a balance is needed between the uncertainty resulting from model input parameters and the uncertainty resulting from model assumptions. For this, it is essential to quantify the uncertainty resulting from model input and to determine whether the complexity of the model is su?cient for the application of interest.

The aim of this study is to investigate model personalization (parameter ?xing and prioritization), model output uncertainty, and the number of runs required to reach convergence of their sensitivity estimates (i.e. computational cost) in case of a 1D pulse wave propagation model that was developed to support vascular access surgery planning [1].

The most common and straightforward method is to use crude Monte Carlo simulations in which the model is executed multiple times to estimate the sensitivity indices. This method, however, requires a lot of computational e?ort. Saltelli et al. [2] introduced a method that is computationally less demanding. This makes the method better applicable to computational expensive models or models with many model parameters. However, large computing resources are still required when applying the method to models with many model parameters. Finally, the method of Morris [3] is a global sensitivity analysis that is able to identify the few important model parameters among the many model parameters in the model with a relatively small number of model evaluations.

Our specific aim was to investigate whether model personalization could be performed by ?rst applying the Morris screening method that identi?es the non-important parameters and subsequently applying the Saltelli method to the resulting subset of important parameters. As this is expected to reduce the computational cost of the uncertainty and sensitivity analysis, this might improve clinical applicability. In addition the uncertainty of the model outputs was quantified using the same data that was generated for the sensitivity analysis.

The Saltelli method, which in general requires many model runs, is found to be a robust method for model personalization. Screening for the important parameters using the Morris method is found to work well for the complex cardiovascular wave propagation model for vascular access. The Morris method can therefore be used for parameter ?xing. However, it does not o?er any information in the setting of parameter prioritization, i.e. in identifying which parameters are most rewarding to measure as accurately as possible. The subsets of important parameters identi?ed for the output of interest lead to a significant complexity reduction.

We conclude that for model personalization of complex models it is advised to perform a screening for the important parameters using the method of Morris ?rst, and then perform a variance-based sensitivity analysis on the subset with only important parameters. For this purpose a Saltelli method can be used. Alternative and more computationally e?cient estimation methods not presented in this study are stochastic collocation methods based on polynomial chaos expansion.

[1]W. Huberts, C de Jonge, W.P.M. van der Linden, M.A Inda, J.H.M. Tordoir, F.N. van de Vosse, and E.M.H. Bosboom. A sensitivity analysis of a personalized pulse wave propagation model for arteriovenous ?stula surgery. Part A: Identi?cation of most in?uential model parameters. Med Eng Phys., 35(6):810–26, 2013.

[2]A. Saltelli. Making best use of model evaluations to compute sensitivity indices. Comp Phys Comm, 145:280–297, 2002.

[3]M.D. Morris. Factorial sampling plans for preliminary computational experiments. Technometrics, 33(2):161–174, 1991.

03:00 PM
03:15 PM

Break

03:15 PM
03:50 PM
Laura Ellwein
04:00 PM
04:35 PM
Alessandro Veneziani
04:35 PM
04:45 PM

Discussion/Day's Wrap-up [Dan Beard]

04:45 PM

Shuttle pick-up from MBI

Thursday, May 8, 2014
Time Session
08:00 AM

Shuttle to MBI

08:15 AM
09:00 AM

Breakfast

09:00 AM
09:35 AM
Mette Olufsen
09:45 AM
10:20 AM
John Schild
10:30 AM
11:05 AM
Johnny Ottesen - Patient specific modelling of the endocrine HPA-axis and its relation to depression: Ultradian and circadian oscillations

Depression is a widely spread disease: In the Western world approximately 10% of the population experience severe depression at least once in their lifetime and many more experience a mild form of depression. We establish a statistical significant correlation between depression and a recently defined index characterising the hypothalamus-pituitary-adrenal (HPA) axis. The relation supports the common belief that depression is caused by malfunctions in the HPA-axis. We suggest a novel model capable of showing both circadian as well as ultradian oscillations of the hormone concentrations related to the HPA-axis. The fast ultradian rhythm is generated in the hippocampus whereas the slower circadian rhythm is caused by the circadian clock. We show that these patterns fit data from 29 subjects. We demonstrate that patient-specific modelling is capable of making more precise diagnostics and offers a tool for individual treatment plans and more effective design of pharmaceutical molecules as a consequence. Three parameters related to depression are identified by non-linear mixed effects modelling and statistical hypothesis testing. These parameters represent underlying physiological mechanisms controlling the average levels as well as the ultradian frequency and amplitudes of the hormones ACTH and cortisol. The results are promising since they offer an exact aetiology for depression going from molecular level to systems physiology.

11:10 AM
12:00 PM

Discussion: What can/should models be used for? To fit data? To make discoveries? To cure disease? (Discussion Leader: Stig Omholt)

12:00 PM
01:30 PM

Lunch Break

01:30 PM
02:05 PM
Jerry Batszel
02:15 PM
02:50 PM
Brian Carlson
02:50 PM
03:15 PM

Break

03:15 PM
03:50 PM
Alison Hu - Modeling autonomic and metabolic dysfunction in sleep-disordered breathing using PNEUMA

There is increasing recognition that sleep-disordered breathing (SDB), which is quite prevalent in obese subjects, can play an independent role in facilitating the development of autonomic and metabolic dysfunction. These abnormalities can lead to the emergence of metabolic syndrome, and subsequently with disease progression, to overt Type 2 diabetes (T2DM). The causal pathways linking SDB to T2DM remain controversial and relatively unexplored. We are developing a large-scale simulation model that would enable competing hypotheses of these causal pathways to be tested at the organ systems level. Our current efforts are based on an integrative model of respiratory, cardiovascular and sleep state control (“PNEUMA”) that was developed by us to characterize the underlying mechanisms that lead to SDB and to determine the effects of SDB on autonomic control of the cardiovascular system and sleep-wake control. We have extended PNEUMA by incorporating a metabolic component, representing the regulation of glucose, insulin, glucagon and free fatty acids using a multi-compartment model. An additional feature is the incorporation of the dynamics of beta-cell regulation. Changes in sympathetic output from the cardiorespiratory portion of PNEUMA, as well as changes in sleep-wake state, lead to changes in epinephrine output and blood flow to the tissues, in turn affecting the metabolism of glucose, insulin and FFA. “Metabolic feedback” takes the form of changes in insulin level, which lead to changes in sympathetic tone through stimulation of the alpha-sympathetic receptors. Consistent with clinical observations, the model predicts that increased severity of sleep apnea, as reflected in an increase in apnea-hypopnea index, leads to higher levels of fasting plasma insulin. Ongoing efforts are aimed at incorporating biological and biochemical processes that occur at the cellular or sub-cellular level, that would enable PNEUMA to simulate disease progression.

04:00 PM
04:35 PM
Adam Mahdi
04:35 PM
04:45 PM

Discussion/Day's Wrap-up [Laura Ellwein]

04:45 PM

Shuttle pick-up from MBI

05:00 PM
06:00 PM

Cash Bar

06:00 PM
08:00 PM

Banquet in the Fusion Room at Crowne Plaza

Friday, May 9, 2014
Time Session
08:00 AM

Shuttle to MBI

08:15 AM
09:00 AM

Breakfast

09:00 AM
09:35 AM
Daniel Beard - Multi-scale modular modeling of cardiovascular function to probe the etiology of complex cardiovascular disease

It is increasingly recognized that multifactorial diseases arise from interaction between genetic and environmental factors, and physiological systems. Examples of particular relevance to human health include the major health burdens that we face: cardiovascular disease and heart failure; metabolic syndrome and type 2 diabetes; and cancer. In all of these examples, acute and chronic (mal)adaptions of specific molecular mechanism and pathways in disease states occur against a background of physiological regulation. Since processes involved in complex disease operate in the context of physiological regulatory mechanisms, an understanding of a disease process builds upon an understanding of the associated physiological systems.

The Virtual Physiological Rat (VPR) is a multi-national research program combining model-driven experiments and experimentally validated multi-scale models to develop theoretical and computational framework explaining: (1.) the long-term regulation of arterial pressure; and (2.) the etiology and sequelae of hypertensive heart disease, spanning molecular genetic to whole-body function. Recent results elucidating novel hypotheses for the mechanisms underlying primary hypertension and the role of metabolic alterations in heart failure will we presented.

09:45 AM
10:20 AM
Naomi Chesler - Toward more comprehensive and data-driven mathematical models of the heart and circulations

According to Claude Bernard, “the application of mathematics to natural phenomena is the aim of all science, because the expression of the laws of phenomena should always be mathematical.” While much progress has been made in understanding natural phenomena since 1865 when Bernard made this statement and developing mathematical models of these phenomena, much work remains to be done. Whether these models range from the genome to the whole body or are more focused on a particular length-scale, time-scale and organ system, development and validation of physiological, mathematical models still require close collaboration between the theoretician and the experimentalist.

An achievable goal in mathematical modeling today is a model of the cardiovascular system that describes the ejection of blood from the heart, from cross-bridge cycling dynamics to ventricular contraction; incorporates the anatomy, morphometry and biomechanics of the pulmonary and systemic circulations; and is able to connect these systems into one integrated system dependent on and responsible for oxygen delivery, waste removal, and homeostasis. In this presentation, I will share my perspective as an experimentalist. In particular, I will show a set of experimental data that are being used to validate a mathematical model of the heart, pulmonary and systemic circulations and preliminary modeling results. I will also present a vision for more in-depth experimental work that will enable development and validation of a more detailed model with shorter length scales, smaller time scales and better integration between the organ systems with the eventual and lofty goal of the application of mathematics to all cardiovascular phenomena.

10:30 AM
10:45 AM

Break

10:45 AM
11:20 AM
11:30 AM
12:05 PM
Ghassan Kassab
12:05 PM
12:30 PM

Discussion and wrap-up [Mette Olufsen]

12:30 PM

Shuttle pick-up from MBI

Name Affiliation
Archer, Kellie kjarcher@vcu.edu Biostatistics, Virginia Commonwealth University
Arciero, Julia jarciero@math.iupui.edu Mathematics, Indiana University--Purdue University
arthur, bright kiddie58@ymail.com Biological Sciences, Philander Smith College
Batszel, Jerry Jerry_batzel@uni-graz.at Institute of Mathematics, University of Graz, Austria
Battista, Christina cbattis2@ncsu.edu Department of Mathematics, North Carolina State University
Beard, Daniel beardda@gmail.com Department of Physiology, Medical College of Wisconsin
Bilinsky, Lydia Lydia.Bilinsky@asu.edu Mathematics, Duke University
Calvetti, Daniela daniela.calvetti@case.edu Mathematics, Applied Mathematics and Statistics, Case Western Reserve University
Cao, Yang ycao@cs.vt.edu Computer Science, Virginia Tech
Carlson, Brian becarlson@mcw.edu Physiology, Medical College of Wisconsin
Chappell, Michael M.J.Chappell@warwick.ac.uk Warwick Engineering in Biomedicine, University of Warwick
Chesler, Naomi chesler@engr.wisc.edu Biomedical Engineering, University of Wisconsin
Cook, Daniel dCook@uw.edu
Craciun, Gheorghe craciun@math.wisc.edu Mathematics and Biomolecular Chemistry, University of Wisconsin-Madison
Ellwein, Laura laura.ellwein@gmail.com Mathematics, Virginia Commonwealth University
Figueroa, C. Alberto alberto.figueroa@kcl.ac.uk Department of Biomedical Engineering, King's College
Ford Versypt, Ashlee ashleefv@mit.edu Chemical Engineering, Massachusetts Institute of Technology
Frisbee, Jefferson jfrisbee@hsc.wvu.edu Physiology and Pharmacology, West Virginia University
Fry, Brendan yrfnadnerb@hotmail.com Mathematics, Duke University
Gennari, John gennari@uw.edu Biomedical Informatics and Medical Education, University of Washington
Hellevik, Leif Rune leif.r.hellevik@ntnu.no Structural Engimeering, Norwegian University of Science and Technology
Hill, Nicholas Nicholas.Hill@maths.gla.ac.uk School of Mathematics and Statistics, University of Glasgow
Hu, Wen-Hsin wenhsin@usc.edu Biomedical Engineering, University of Southern California
Joo, Jaewook jjoo1@utk.edu Physics, University of Tennessee
Kassab, Ghassan gkassab@iupui.edu Biomedical Engineering&MedicineCardiology , University of California, Irvine
Kim, Jae Kyoung kim.5052@mbi.osu.edu Mathematical Biosciences Institute, The Ohio State University
Layton, Anita alayton@math.duke.edu Mathematics, Duke University
Lee, Namyong nlee@mnsu.edu Department of Mathematics, Minnesota State University
Lee, Pilhwa pilee@med.umich.edu Department of Molecular and Integrative Physiology, University of Michigan
Mahdi, Adam adam.mahdi@gmail.com Mathematics, North Carolina State University
Makrides, Elizabeth elizabeth_makrides@brown.edu Division of Applied Mathematics, Brown University
Malka, Roy Roy_Malka@hms.harvard.edu Systems Biology, Harvard Medical School
Moss, Robert robm@math.duke.edu Mathematics, Duke University
Novak, Vera vnovak@caregroup.org
Olsen, Christian chaarga@ncsu.edu Biomathematics, North Carolina State University
Olufsen, Mette msolufse@ncsu.edu Department of Mathematics, North Carolina State University
Ottesen, Johnny Johnny@ruc.dk Department of Mathematics and Physics, Roskilde University Center
Pantea, Casian cpantea@math.wvu.edu Mathematics, West Virginia University
Pettersen, Klas klas.pettersen@umb.no Centre for Molecular Medicine Norway, University of Oslo
Qutub, Amina aminaq@rice.edu Bioengineering, Rice University
Reed, Michael reed@math.duke.edu Mathematics, Duke University
Schild, John jschild@iupui.edu Department of Biomedical Engineering, Neuroscience, Indiana University--Purdue University
Schnell, Santiago schnells@umich.edu Department of Molecular & Integrative Biology, University of Michigan Medical School
Schulze, Christian pcs2121@columbia.edu Medicine/Cardiology, Columbia University
Sturdy, Jacob jsturdy@ncsu.edu Mathematics, North Carolina State University
Temamogullari, Nihal ezgitamam@yahoo.com Mathematics, Duke University
Tran, Hien tran@ncsu.edu Mathematics, North Carolina State University
van de Vosse, Frans F.N.v.d.Vosse@tue.nl Department of Biomedical Engineering, Technische Universiteit Eindhoven
Veneziani, Alessandro ale@mathcs.emory.edu Department of Mathematics and Computer Science, Emory University
Vik, Jon Olav jonovik@gmail.com Department of Animal and Aquacultural Sciences, Norwegian University of Life Sciences
Wiuf, Carsten wiuf@math.ku.dk Department of Mathematical Sciences, University of Copenhagen
Zuhr, Erica ezuhr@highpoint.edu Mathematics, High Point University
Ordinal Response Models for Modeling Longitudinal High-Dimensional Genomic Feature Data

Ordinal scales are commonly used to measure health status and disease related outcomes. Notable examples include cancer staging, histopathological classification, adverse event rating, and severity of illness. In addition, repeated measurements are common in clinical practice for tracking and monitoring the progression of complex diseases. Classical likelihood-based ordinal modeling methods have contributed to the analysis of data in which the response categories are ordered and the number of covariates (p) is smaller than the sample size (n). With the emergence of genomic technologies being increasingly applied to identify molecular markers associated with complex disease phenotypes and outcomes, many research studies now include high dimensional feature data where p >> n, so that traditional methods cannot be applied. To fill this void we have developed an innovative penalized random coefficient ordinal response model for classifying and predicting disease progression along with time. Specifically our method extends the Generalized Monotone Incremental Forward Stagewise method (Hastie et al, 2007) to the ordinal response setting in combination with classical mixed effects modeling methods. We demonstrate our method using data from the Inflammation and the Host Response to Injury study in which Affymetrix gene expression profiles and Marshall Multiple Organ Dysfunction Score on six body systems were longitudinally collected at hospitalization day 1 up to day 30 in 169 patients.

Assessing vascular risk factors for glaucoma using a mathematical model of blood flow in the retina

Glaucoma is the second leading cause of blindness in the world and is characterized by progressive retinal ganglion cell death and irreversible visual field loss. Although elevated intraocular pressure has been identified as the primary risk factor for glaucoma and is the main target of glaucoma treatments, several vascular risk factors that lead to impaired retinal blood flow have also been correlated with the progression and incidence of glaucoma. Here, a multi-scale mathematical model is used to investigate the relative contributions of vascular risk factors on flow regulation and tissue oxygenation in the retina. A previously-developed fluid-structure interaction system modeling the central retinal artery is coupled to a vascular wall mechanics model for the vessels of the retinal microcirculation. Under normal conditions, the model predicts a 14% decrease in retinal perfusion if oxygen demand is decreased by 50% and a 33% increase in perfusion if demand is increased by 50%. These responses are impaired significantly if the metabolic or carbon dioxide mechanisms of retinal blood flow autoregulation are impaired. Changes in oxygen saturation levels in the retinal vascular network are also assessed as levels of mean arterial pressure, oxygen demand, and intraocular pressure are varied. Overall, the model results suggest that impaired autoregulation might increase the risk of retinal ischemic damage, as would occur in glaucoma, under conditions of elevated metabolic demand or decreased mean arterial pressure.

Multi-scale modular modeling of cardiovascular function to probe the etiology of complex cardiovascular disease

It is increasingly recognized that multifactorial diseases arise from interaction between genetic and environmental factors, and physiological systems. Examples of particular relevance to human health include the major health burdens that we face: cardiovascular disease and heart failure; metabolic syndrome and type 2 diabetes; and cancer. In all of these examples, acute and chronic (mal)adaptions of specific molecular mechanism and pathways in disease states occur against a background of physiological regulation. Since processes involved in complex disease operate in the context of physiological regulatory mechanisms, an understanding of a disease process builds upon an understanding of the associated physiological systems.

The Virtual Physiological Rat (VPR) is a multi-national research program combining model-driven experiments and experimentally validated multi-scale models to develop theoretical and computational framework explaining: (1.) the long-term regulation of arterial pressure; and (2.) the etiology and sequelae of hypertensive heart disease, spanning molecular genetic to whole-body function. Recent results elucidating novel hypotheses for the mechanisms underlying primary hypertension and the role of metabolic alterations in heart failure will we presented.

Toward more comprehensive and data-driven mathematical models of the heart and circulations

According to Claude Bernard, “the application of mathematics to natural phenomena is the aim of all science, because the expression of the laws of phenomena should always be mathematical.” While much progress has been made in understanding natural phenomena since 1865 when Bernard made this statement and developing mathematical models of these phenomena, much work remains to be done. Whether these models range from the genome to the whole body or are more focused on a particular length-scale, time-scale and organ system, development and validation of physiological, mathematical models still require close collaboration between the theoretician and the experimentalist.

An achievable goal in mathematical modeling today is a model of the cardiovascular system that describes the ejection of blood from the heart, from cross-bridge cycling dynamics to ventricular contraction; incorporates the anatomy, morphometry and biomechanics of the pulmonary and systemic circulations; and is able to connect these systems into one integrated system dependent on and responsible for oxygen delivery, waste removal, and homeostasis. In this presentation, I will share my perspective as an experimentalist. In particular, I will show a set of experimental data that are being used to validate a mathematical model of the heart, pulmonary and systemic circulations and preliminary modeling results. I will also present a vision for more in-depth experimental work that will enable development and validation of a more detailed model with shorter length scales, smaller time scales and better integration between the organ systems with the eventual and lofty goal of the application of mathematics to all cardiovascular phenomena.

Codeword annotation for sharing and merging physiological models

Hunter and Bassingthwaithe define the Physiome as a set of multiscale, interacting mathematical models of physiology. Although available model repositories are an initial step toward this vision, it is a critical next step to develop computer-readable annotation for connecting codewords across models. Current hand-crafted model-building methods must be formalized and standardized to better support knowledge interaction and sharing. In particular, we argue for semantic annotations as a way of communicating the biophysical meaning of individual model codewords. Once annotated in a computable format, we can automatically find and connect models based on the annotation semantics of the biological entities and physiological properties.

In this talk, we present our approach to semantic annotation, using standard bio-ontology terms to relate physiological properties (e.g. pressure), to anatomical entities (e.g. blood). In turn, we use these annotations to semi-automatically find relevant models from repositories, and ultimately merge those models where appropriate. We present our results with SemGen, a prototype tool, for both building annotations and merging models, even across different modeling languages. If successful, our approach to develop interacting model repositories could accelerate model sharing and integration, and research that depends on the construction of complex models.

Persistence, Permanence, and Global Stability in Biological Interaction Networks

Complex interaction networks are present in all areas of biology, and manifest themselves at very different spatial and temporal scales. Persistence, permanence and global stability are emergent properties of complex networks, and play key roles in the dynamics of living systems.

Mathematically, a dynamical system is called persistent if, for all positive solutions, no variable approaches zero. In addition, for a permanent system, all variables are uniformly bounded. We describe criteria for persistence and permanence of solutions, and for global convergence of solutions to an unique equilibrium, in a manner that is robust with respect to initial conditions and parameter values.

A thorough understanding of these properties will allow for a better understanding of essential biological processes, such as homeostasis and adaptability.

Recent Advances in 3D Blood flow Simulation: From Parameter Estimation Methods to Clinical Applications

In this talk we will give an overview of a series of methods for 3D blood flow modeling, ranging from Kalman filtering techniques for automatic outflow and material parameter estimation to baroreflex model for automatic control of blood pressure. We will also discuss recent progress made on the validation of CFD predictions of pressure gradients in coarctation patients at rest and stress using clinical pressure data.

Codeword annotation for sharing and merging physiological models

Hunter and Bassingthwaithe define the Physiome as a set of multiscale, interacting mathematical models of physiology. Although available model repositories are an initial step toward this vision, it is a critical next step to develop computer-readable annotation for connecting codewords across models. Current hand-crafted model-building methods must be formalized and standardized to better support knowledge interaction and sharing. In particular, we argue for semantic annotations as a way of communicating the biophysical meaning of individual model codewords. Once annotated in a computable format, we can automatically find and connect models based on the annotation semantics of the biological entities and physiological properties.

In this talk, we present our approach to semantic annotation, using standard bio-ontology terms to relate physiological properties (e.g. pressure), to anatomical entities (e.g. blood). In turn, we use these annotations to semi-automatically find relevant models from repositories, and ultimately merge those models where appropriate. We present our results with SemGen, a prototype tool, for both building annotations and merging models, even across different modeling languages. If successful, our approach to develop interacting model repositories could accelerate model sharing and integration, and research that depends on the construction of complex models.

Codeword annotation for sharing and merging physiological models

Hunter and Bassingthwaithe define the Physiome as a set of multiscale, interacting mathematical models of physiology. Although available model repositories are an initial step toward this vision, it is a critical next step to develop computer-readable annotation for connecting codewords across models. Current hand-crafted model-building methods must be formalized and standardized to better support knowledge interaction and sharing. In particular, we argue for semantic annotations as a way of communicating the biophysical meaning of individual model codewords. Once annotated in a computable format, we can automatically find and connect models based on the annotation semantics of the biological entities and physiological properties.

In this talk, we present our approach to semantic annotation, using standard bio-ontology terms to relate physiological properties (e.g. pressure), to anatomical entities (e.g. blood). In turn, we use these annotations to semi-automatically find relevant models from repositories, and ultimately merge those models where appropriate. We present our results with SemGen, a prototype tool, for both building annotations and merging models, even across different modeling languages. If successful, our approach to develop interacting model repositories could accelerate model sharing and integration, and research that depends on the construction of complex models.

A structured tree model for the pulmonary circulation

Abstract coming soon.

Modeling autonomic and metabolic dysfunction in sleep-disordered breathing using PNEUMA

There is increasing recognition that sleep-disordered breathing (SDB), which is quite prevalent in obese subjects, can play an independent role in facilitating the development of autonomic and metabolic dysfunction. These abnormalities can lead to the emergence of metabolic syndrome, and subsequently with disease progression, to overt Type 2 diabetes (T2DM). The causal pathways linking SDB to T2DM remain controversial and relatively unexplored. We are developing a large-scale simulation model that would enable competing hypotheses of these causal pathways to be tested at the organ systems level. Our current efforts are based on an integrative model of respiratory, cardiovascular and sleep state control (“PNEUMA”) that was developed by us to characterize the underlying mechanisms that lead to SDB and to determine the effects of SDB on autonomic control of the cardiovascular system and sleep-wake control. We have extended PNEUMA by incorporating a metabolic component, representing the regulation of glucose, insulin, glucagon and free fatty acids using a multi-compartment model. An additional feature is the incorporation of the dynamics of beta-cell regulation. Changes in sympathetic output from the cardiorespiratory portion of PNEUMA, as well as changes in sleep-wake state, lead to changes in epinephrine output and blood flow to the tissues, in turn affecting the metabolism of glucose, insulin and FFA. “Metabolic feedback” takes the form of changes in insulin level, which lead to changes in sympathetic tone through stimulation of the alpha-sympathetic receptors. Consistent with clinical observations, the model predicts that increased severity of sleep apnea, as reflected in an increase in apnea-hypopnea index, leads to higher levels of fasting plasma insulin. Ongoing efforts are aimed at incorporating biological and biochemical processes that occur at the cellular or sub-cellular level, that would enable PNEUMA to simulate disease progression.

Assessment of Renal Autoregulatory Mechanisms

A mathematical model of renal hemodynamics is used to assess the individual contributions of the tubuloglomerular feedback (TGF) mechanism and the myogenic response to glomerular filtration rate regulation in the rat kidney. The model represents an afferent arteriole segment, glomerular filtration, and a short loop of Henle. The afferent arteriole model exhibits myogenic response, which is activated by hydrostatic pressure variations to induce changes in membrane potential and vascular muscle tone. The tubule model predicts tubular fluid and Cl- transport. Macula densa Cl- concentration is sensed as the signal for TGF, which acts to constrict or dilate the afferent arteriole. With this configuration, the model afferent arteriole maintains stable glomerular filtration rate within a physiologic range of perfusion pressure (80-180 mmHg). The contribution of TGF to overall autoregulation is significant only within a narrow band of perfusion pressure values (80-110 mmHg). Model simulations of ramp-like perfusion pressure perturbations agree well with findings by Flemming et al. (J Am Soc Nephrol 12:2253-2262, 2001), which indicate that changes in vascular conductance is markedly sensitive to pressure velocity. That asymmetric response is attributed to the rate-dependent kinetics of the myogenic mechanism. Moreover, simulations of renal autoregulation in diabetes mellitus predict that, due to the impairment of the voltage-gated Ca2+ channels of the afferent arteriole smooth muscle cells, the perfusion pressure range in which SNGFR remains stable is reduced by ~70%, and that TGF gain is reduced by nearly 40%, consistent with experimental findings.

Assessment of Renal Autoregulatory Mechanisms

A mathematical model of renal hemodynamics is used to assess the individual contributions of the tubuloglomerular feedback (TGF) mechanism and the myogenic response to glomerular filtration rate regulation in the rat kidney. The model represents an afferent arteriole segment, glomerular filtration, and a short loop of Henle. The afferent arteriole model exhibits myogenic response, which is activated by hydrostatic pressure variations to induce changes in membrane potential and vascular muscle tone. The tubule model predicts tubular fluid and Cl- transport. Macula densa Cl- concentration is sensed as the signal for TGF, which acts to constrict or dilate the afferent arteriole. With this configuration, the model afferent arteriole maintains stable glomerular filtration rate within a physiologic range of perfusion pressure (80-180 mmHg). The contribution of TGF to overall autoregulation is significant only within a narrow band of perfusion pressure values (80-110 mmHg). Model simulations of ramp-like perfusion pressure perturbations agree well with findings by Flemming et al. (J Am Soc Nephrol 12:2253-2262, 2001), which indicate that changes in vascular conductance is markedly sensitive to pressure velocity. That asymmetric response is attributed to the rate-dependent kinetics of the myogenic mechanism. Moreover, simulations of renal autoregulation in diabetes mellitus predict that, due to the impairment of the voltage-gated Ca2+ channels of the afferent arteriole smooth muscle cells, the perfusion pressure range in which SNGFR remains stable is reduced by ~70%, and that TGF gain is reduced by nearly 40%, consistent with experimental findings.

Patient specific modelling of the endocrine HPA-axis and its relation to depression: Ultradian and circadian oscillations

Depression is a widely spread disease: In the Western world approximately 10% of the population experience severe depression at least once in their lifetime and many more experience a mild form of depression. We establish a statistical significant correlation between depression and a recently defined index characterising the hypothalamus-pituitary-adrenal (HPA) axis. The relation supports the common belief that depression is caused by malfunctions in the HPA-axis. We suggest a novel model capable of showing both circadian as well as ultradian oscillations of the hormone concentrations related to the HPA-axis. The fast ultradian rhythm is generated in the hippocampus whereas the slower circadian rhythm is caused by the circadian clock. We show that these patterns fit data from 29 subjects. We demonstrate that patient-specific modelling is capable of making more precise diagnostics and offers a tool for individual treatment plans and more effective design of pharmaceutical molecules as a consequence. Three parameters related to depression are identified by non-linear mixed effects modelling and statistical hypothesis testing. These parameters represent underlying physiological mechanisms controlling the average levels as well as the ultradian frequency and amplitudes of the hormones ACTH and cortisol. The results are promising since they offer an exact aetiology for depression going from molecular level to systems physiology.

Molecular Signatures of Cells during Hypoxic-Stimulated Tissue Growth

Oxygen is fundamental to life on Earth. In diseases affecting the vasculature including cancer and neurodegenerative diseases, abberrant hypoxic response is a critical part of the disease. Limited oxygen can lead to more aggressive tumors or determine our susceptibility to dementia. On the other hand, appropriate manipulation of proteins involved in cellular hypoxic response can help restore blood vessels and regenerate tissues. A challenge lies in understanding the complex cellular response to hypoxia both across different diseases and between patients with the detail needed to develop effective therapies. In this presentation, I will share how we are developing and integrating methods in multiscale modeling, machine learning, molecular biology, and microscopy image analysis to tackle the challenge of interpreting how changes at the molecular level affect cellular response and multicellular dynamics. My lab’s goal is to use computational systems biology methods to understand – and ultimately control –biological response to oxygen across scales.

How Mathematicians can Contribute to Genomic Medicine

Mathematical models of physiological processes allow one to study the homeostatic mechanisms that keep important phenotypic variables within certain normal ranges. When these variables leave the homeostatic range often disease processes ensue. From the models one can derive surfaces that show the relationship between genetic polymorphisms and particularly important phenotypic variables. Known gene polymorphisms correspond to particular points on the surface, some of which are located near the edge of the homeostatic region. The purpose of medical advice tailored to the patient’s genotype is to suggest dietary changes or exercise changes that move the patient back towards the middle of the homeostatic region.

On the mechanisms of sensing unfolded protein in the endoplasmic reticulum.

One of the main functions of the endoplasmic reticulum (ER) is to serve as the cell protein-folding factory. The ER is responsible for the synthesis, folding, assembly and modification of one third of the eukaryotic proteome. Proteins enter the ER as unfolded polypeptide chains with variable fluxes depending on the physiological state of the cell. A sudden increase in the demand for a protein or the disruption of a folding reaction causes an imbalance between protein-folding load and capacity of the ER, which can lead to the accumulation of unfolded protein in the ER lumen. The ER protein balance is regulated by several signaling pathways, which are collectively termed the unfolded protein response. The unfolded protein response is activated by three transducers, which are enzymes whose oligomerization-induced activation is linked to perturbed protein folding in the ER. Three model mechanisms have been proposed for how these enzymes sense the unfolded protein load in the ER lumen: (i) direct recognition, (ii) indirect recognition and (iii) hybrid recognition. We developed detailed reaction mechanisms for each model and analyzed their dynamical behavior. We found that some of these mechanisms have serious discrepancies with the experimental data. We suggest a set of experiments that have not been yet carried out to test a detailed novel model mechanism of protein load sensing in the ER lumen, which explains current experimental findings. Our new model could provide new insights into the mechanisms of protein homeostasis in the ER.

Personalization of 1D Wave Propagation Models of the Cardiovascular System

One of the main di?culties in the translation of mathematical models to the clinic for supporting clinical decision-making is assessing patient-speci?c values for the model parameters, the boundary and the initial conditions. Measurement modalities or data are not always available for all model parameters. In addition, the precision and accuracy of clinical measurements are hampered by large (biological) variations. Consequently, a balance is needed between the uncertainty resulting from model input parameters and the uncertainty resulting from model assumptions. For this, it is essential to quantify the uncertainty resulting from model input and to determine whether the complexity of the model is su?cient for the application of interest.

The aim of this study is to investigate model personalization (parameter ?xing and prioritization), model output uncertainty, and the number of runs required to reach convergence of their sensitivity estimates (i.e. computational cost) in case of a 1D pulse wave propagation model that was developed to support vascular access surgery planning [1].

The most common and straightforward method is to use crude Monte Carlo simulations in which the model is executed multiple times to estimate the sensitivity indices. This method, however, requires a lot of computational e?ort. Saltelli et al. [2] introduced a method that is computationally less demanding. This makes the method better applicable to computational expensive models or models with many model parameters. However, large computing resources are still required when applying the method to models with many model parameters. Finally, the method of Morris [3] is a global sensitivity analysis that is able to identify the few important model parameters among the many model parameters in the model with a relatively small number of model evaluations.

Our specific aim was to investigate whether model personalization could be performed by ?rst applying the Morris screening method that identi?es the non-important parameters and subsequently applying the Saltelli method to the resulting subset of important parameters. As this is expected to reduce the computational cost of the uncertainty and sensitivity analysis, this might improve clinical applicability. In addition the uncertainty of the model outputs was quantified using the same data that was generated for the sensitivity analysis.

The Saltelli method, which in general requires many model runs, is found to be a robust method for model personalization. Screening for the important parameters using the Morris method is found to work well for the complex cardiovascular wave propagation model for vascular access. The Morris method can therefore be used for parameter ?xing. However, it does not o?er any information in the setting of parameter prioritization, i.e. in identifying which parameters are most rewarding to measure as accurately as possible. The subsets of important parameters identi?ed for the output of interest lead to a significant complexity reduction.

We conclude that for model personalization of complex models it is advised to perform a screening for the important parameters using the method of Morris ?rst, and then perform a variance-based sensitivity analysis on the subset with only important parameters. For this purpose a Saltelli method can be used. Alternative and more computationally e?cient estimation methods not presented in this study are stochastic collocation methods based on polynomial chaos expansion.

[1]W. Huberts, C de Jonge, W.P.M. van der Linden, M.A Inda, J.H.M. Tordoir, F.N. van de Vosse, and E.M.H. Bosboom. A sensitivity analysis of a personalized pulse wave propagation model for arteriovenous ?stula surgery. Part A: Identi?cation of most in?uential model parameters. Med Eng Phys., 35(6):810–26, 2013.

[2]A. Saltelli. Making best use of model evaluations to compute sensitivity indices. Comp Phys Comm, 145:280–297, 2002.

[3]M.D. Morris. Factorial sampling plans for preliminary computational experiments. Technometrics, 33(2):161–174, 1991.

Model reduction is biochemical reaction networks

In many situations we apply simplified models to complex dynamical systems, either because we are unaware of what the 'correct' model should look like, or because the 'correct' model is too complex to handle statistically/mathematically. In this talk, I will discuss model reduction for stochastic as well as deterministic biochemical reaction networks. In particular, I will focus on reduction by elimination of intermediate species, transient species that typically are consumed at a faster rate than non-intermediates and provide a number of results concerning equilibrium dynamics as well as non-equilibrium dynamics.

Posters


The validity of quasi steady-state approximations in discrete stochastic simulations

In biochemical networks, reactions often occur on disparate timescales and can be characterized as either “fast” or “slow.” The quasi-steady state approximation (QSSA) utilizes timescale separation to project models of biochemical networks onto lower-dimensional slow manifolds. As a result, fast elementary reactions are not modeled explicitly, and their effect is captured by non-elementary reaction rate functions (e.g. Hill functions). The accuracy of the QSSA applied to deterministic systems depends on how well timescales are separated. Recently, it has been proposed to use the non-elementary rate functions obtained via the deterministic QSSA to define propensity functions in stochastic simulations of biochemical networks. In this approach, termed the stochastic QSSA, fast reactions that are part of non-elementary reactions are not simulated, greatly reducing computation time. However, it is unclear when the stochastic QSSA provides an accurate approximation of the original stochastic simulation. We show that, unlike the deterministic QSSA, the validity of the stochastic QSSA does not follow from timescale separation alone, but also depends on the sensitivity of the non-elementary reaction rate functions to changes in the slow species. The stochastic QSSA becomes more accurate when this sensitivity is small. Different types of QSSAs result in non-elementary functions with different sensitivities, and the total QSSA results in less sensitive functions than the standard or the pre-factor QSSA. We prove that, as a result, the stochastic QSSA becomes more accurate when non-elementary reaction functions are obtained using the total QSSA. Our work provides a novel condition for the validity of the QSSA in stochastic simulations of biochemical reaction networks with disparate timescales.