Mathematical models for kidney function focusing on clinical interest.

We give an overview of mathematical models of renal physiology and anatomy with the clinician in mind. Beyond the past focus on issues of local transport mechanisms along the nephron and the urine concentrating mechanism, recent models have brought insight into difficult problems such as renal ischemia (oxygen and CO2 diffusion in the medulla) or calcium and potassium homeostasis. They have also provided revealing 3D reconstructions of the full trajectories of families of nephrons and collecting ducts through cortex and medulla. The recent appearance of sophisticated whole-kidney models representing nephrons and their associated renal vasculature promises more realistic simulation of renal pathologies and pharmacological treatments in the foreseeable future.

Mathematical Model of Ammonia Handling in the Rat Renal Medulla.

The kidney is one of the main organs that produces ammonia and release it into the circulation. Under normal conditions, between 30 and 50% of the ammonia produced in the kidney is excreted in the urine, the rest being absorbed into the systemic circulation via the renal vein. In acidosis and in some pathological conditions, the proportion of urinary excretion can increase to 70% of the ammonia produced in the kidney. Mechanisms regulating the balance between urinary excretion and renal vein release are not fully understood. We developed a mathematical model that reflects current thinking about renal ammonia handling in order to investigate the role of each tubular segment and identify some of the components which might control this balance. The model treats the movements of water, sodium chloride, urea, NH3 and [Formula: see text], and non-reabsorbable solute in an idealized renal medulla of the rat at steady state. A parameter study was performed to identify the transport parameters and microenvironmental conditions that most affect the rate of urinary ammonia excretion. Our results suggest that urinary ammonia excretion is mainly determined by those parameters that affect ammonia recycling in the loops of Henle. In particular, our results suggest a critical role for interstitial pH in the outer medulla and for luminal pH along the inner medullary collecting ducts.

Implementation of a model of bodily fluids regulation.

The classic model of blood pressure regulation by Guyton et al. (Annu Rev Physiol 34:13-46, 1972a; Ann Biomed Eng 1:254-281, 1972b) set a new standard for quantitative exploration of physiological function and led to important new insights, some of which still remain the focus of debate, such as whether the kidney plays the primary role in the genesis of hypertension (Montani et al. in Exp Physiol 24:41-54, 2009a; Exp Physiol 94:382-388, 2009b; Osborn et al. in Exp Physiol 94:389-396, 2009a; Exp Physiol 94:388-389, 2009b). Key to the success of this model was the fact that the authors made the computer code (in FORTRAN) freely available and eventually provided a convivial user interface for exploration of model behavior on early microcomputers (Montani et al. in Int J Bio-med Comput 24:41-54, 1989). Ikeda et al. (Ann Biomed Eng 7:135-166, 1979) developed an offshoot of the Guyton model targeting especially the regulation of body fluids and acid-base balance; their model provides extended renal and respiratory functions and would be a good basis for further extensions. In the interest of providing a simple, useable version of Ikeda et al.'s model and to facilitate further such extensions, we present a practical implementation of the model of Ikeda et al. (Ann Biomed Eng 7:135-166, 1979), using the ODE solver Berkeley Madonna.

Hormonal regulation of salt and water excretion: a mathematical model of whole kidney function and pressure natriuresis.

We present a lumped-nephron model that explicitly represents the main features of the underlying physiology, incorporating the major hormonal regulatory effects on both tubular and vascular function, and that accurately simulates hormonal regulation of renal salt and water excretion. This is the first model to explicitly couple glomerulovascular and medullary dynamics, and it is much more detailed in structure than existing whole organ models and renal portions of multiorgan models. In contrast to previous medullary models, which have only considered the antidiuretic state, our model is able to regulate water and sodium excretion over a variety of experimental conditions in good agreement with data from experimental studies of the rat. Since the properties of the vasculature and epithelia are explicitly represented, they can be altered to simulate pathophysiological conditions and pharmacological interventions. The model serves as an appropriate starting point for simulations of physiological, pathophysiological, and pharmacological renal conditions and for exploring the relationship between the extrarenal environment and renal excretory function in physiological and pathophysiological contexts.

A model of calcium transport along the rat nephron.

We developed a mathematical model of calcium (Ca(2+)) transport along the rat nephron to investigate the factors that promote hypercalciuria. The model is an extension of the flat medullary model of Hervy and Thomas (Am J Physiol Renal Physiol 284: F65-F81, 2003). It explicitly represents all the nephron segments beyond the proximal tubules and distinguishes between superficial and deep nephrons. It solves dynamic conservation equations to determine NaCl, urea, and Ca(2+) concentration profiles in tubules, vasa recta, and the interstitium. Calcium is known to be reabsorbed passively in the thick ascending limbs and actively in the distal convoluted (DCT) and connecting (CNT) tubules. Our model predicts that the passive diffusion of Ca(2+) from the vasa recta and loops of Henle generates a significant axial Ca(2+) concentration gradient in the medullary interstitium. In the base case, the urinary Ca(2+) concentration and fractional excretion are predicted as 2.7 mM and 0.32%, respectively. Urinary Ca(2+) excretion is found to be strongly modulated by water and NaCl reabsorption along the nephron. Our simulations also suggest that Ca(2+) molar flow and concentration profiles differ significantly between superficial and deep nephrons, such that the latter deliver less Ca(2+) to the collecting duct. Finally, our results suggest that the DCT and CNT can act to counteract upstream variations in Ca(2+) transport but not always sufficiently to prevent hypercalciuria.

A vision and strategy for the virtual physiological human: 2012 update.

European funding under Framework 7 (FP7) for the virtual physiological human (VPH) project has been in place now for 5 years. The VPH Network of Excellence (NoE) has been set up to help develop common standards, open source software, freely accessible data and model repositories, and various training and dissemination activities for the project. It is also working to coordinate the many clinically targeted projects that have been funded under the FP7 calls. An initial vision for the VPH was defined by the FP6 STEP project in 2006. In 2010, we wrote an assessment of the accomplishments of the first two years of the VPH in which we considered the biomedical science, healthcare and information and communications technology challenges facing the project (Hunter et al. 2010 Phil. Trans. R. Soc. A 368, 2595-2614 (doi:10.1098/rsta.2010.0048)). We proposed that a not-for-profit professional umbrella organization, the VPH Institute, should be established as a means of sustaining the VPH vision beyond the time-frame of the NoE. Here, we update and extend this assessment and in particular address the following issues raised in response to Hunter et al.: (i) a vision for the VPH updated in the light of progress made so far, (ii) biomedical science and healthcare challenges that the VPH initiative can address while also providing innovation opportunities for the European industry, and (iii) external changes needed in regulatory policy and business models to realize the full potential that the VPH has to offer to industry, clinics and society generally.

Virtual patients and sensitivity analysis of the Guyton model of blood pressure regulation: towards individualized models of whole-body physiology.

Mathematical models that integrate multi-scale physiological data can offer insight into physiological and pathophysiological function, and may eventually assist in individualized predictive medicine. We present a methodology for performing systematic analyses of multi-parameter interactions in such complex, multi-scale models. Human physiology models are often based on or inspired by Arthur Guyton's whole-body circulatory regulation model. Despite the significance of this model, it has not been the subject of a systematic and comprehensive sensitivity study. Therefore, we use this model as a case study for our methodology. Our analysis of the Guyton model reveals how the multitude of model parameters combine to affect the model dynamics, and how interesting combinations of parameters may be identified. It also includes a "virtual population" from which "virtual individuals" can be chosen, on the basis of exhibiting conditions similar to those of a real-world patient. This lays the groundwork for using the Guyton model for in silico exploration of pathophysiological states and treatment strategies. The results presented here illustrate several potential uses for the entire dataset of sensitivity results and the "virtual individuals" that we have generated, which are included in the supplementary material. More generally, the presented methodology is applicable to modern, more complex multi-scale physiological models.

Integration of detailed modules in a core model of body fluid homeostasis and blood pressure regulation.

This paper presents a contribution to the definition of the interfaces required to perform heterogeneous model integration in the context of integrative physiology. A formalization of the model integration problem is proposed and a coupling method is presented. The extension of the classic Guyton model, a multi-organ, integrated systems model of blood pressure regulation, is used as an example of the application of the proposed method. To this end, the Guyton model has been restructured, extensive sensitivity analyses have been performed, and appropriate transformations have been applied to replace a subset of its constituting modules by integrating a pulsatile heart and an updated representation of the renin-angiotensin system. Simulation results of the extended integrated model are presented and the impacts of their integration within the original model are evaluated.

The global risk approach should be better applied in French hypertensive patients: a comparison between simulation and observation studies.

BACKGROUND: The prediction of the public health impact of a preventive strategy provides valuable support for decision-making. International guidelines for hypertension management have introduced the level of absolute cardiovascular risk in the definition of the treatment target population. The public health impact of implementing such a recommendation has not been measured. METHODOLOGY/PRINCIPAL FINDINGS: We assessed the efficiency of three treatment scenarios according to historical and current versions of practice guidelines on a Realistic Virtual Population representative of the French population aged from 35 to 64 years: 1) BP≥160/95 mm Hg; 2) BP≥140/90 mm Hg and 3) BP≥140/90 mm Hg plus increased CVD risk. We compared the eligibility following the ESC guidelines with the recently observed proportion of treated amongst hypertensive individuals reported by the Etude Nationale Nutrition Santé survey. Lowering the threshold to define hypertension multiplied by 2.5 the number of eligible individuals. Applying the cardiovascular risk rule reduced this number significantly: less than 1/4 of hypertensive women under 55 years and less than 1/3 of hypertensive men below 45 years of age. This was the most efficient strategy. Compared to the simulated guidelines application, men of all ages were undertreated (between 32 and 60%), as were women over 55 years (70%). By contrast, younger women were over-treated (over 200%). CONCLUSION: The global CVD risk approach to decide for treatment is more efficient than the simple blood pressure level. However, lack of screening rather than guideline application seems to explain the low prescription rates among hypertensive individuals in France. Multidimensional analyses required to obtain these results are possible only through databases at the individual level: realistic virtual populations should become the gold standard for assessing the impact of public health policies at the national level.

The Virtual Kidney: an eScience interface and Grid portal.

The Virtual Kidney uses a web interface and distributed computing to provide experimental scientists and analysts with access to computational simulations and knowledge databases hosted in geographically separated laboratories. Users can explore a variety of complex models without requiring the specific programming environment in which applications have been developed. This initiative exploits high-bandwidth communication networks for collaborative research and for shared access to knowledge resources. The Virtual Kidney has been developed within a specialist community of renal scientists but is transferable to other areas of research requiring interaction between published literature and databases, theoretical models and simulations and the formulation of effective experimental designs. A web-based three-dimensional interface provides access to experimental data, a parameter database and mathematical models. A multi-scale kidney reconstruction includes blood vessels and serially sectioned nephrons. Selection of structures provides links to the database, returning parameter values and extracts from the literature. Models are run locally or remotely with a Grid resource broker managing scheduling, monitoring and visualization of simulation results and application, credential and resource allocation. Simulation results are viewed graphically or as scaled colour gradients on the Virtual Kidney structures, allowing visual and quantitative appreciation of the effects of simulated parameter changes.

SAPHIR: a physiome core model of body fluid homeostasis and blood pressure regulation.

We present the current state of the development of the SAPHIR project (a Systems Approach for PHysiological Integration of Renal, cardiac and respiratory function). The aim is to provide an open-source multi-resolution modelling environment that will permit, at a practical level, a plug-and-play construction of integrated systems models using lumped-parameter components at the organ/tissue level while also allowing focus on cellular- or molecular-level detailed sub-models embedded in the larger core model. Thus, an in silico exploration of gene-to-organ-to-organism scenarios will be possible, while keeping computation time manageable. As a first prototype implementation in this environment, we describe a core model of human physiology targeting the short- and long-term regulation of blood pressure, body fluids and homeostasis of the major solutes. In tandem with the development of the core models, the project involves database implementation and ontology development.

Robert Rosen in the age of systems biology.

The widespread use of the term Systems Biology (SB) signals a welcome recognition that organisms must be understood as integrated systems. Although just what this is taken to mean varies from one group to another, it generally implies a focus on biological functions and processes rather than on biological parts and a reliance on mathematical modeling to arrive at an understanding of these biological processes based on biological observations or measurements. SB, thus, falls directly in the line of reflection carried out by Robert Rosen throughout his work. In the present article, we briefly introduce the various currents of SB and then point out several ways Rosen's work can be used to avoid certain pitfalls associated with the use of dynamical systems models for the study of complex systems, as well as to inspire a productive path forward based on loosely organized cooperation among dispersed laboratories.

QxDB: a generic database to support mathematical modelling in biology.

QxDB (quantitative x-modelling database) is a web-based generic database package designed especially to house quantitative and structural information. Its development was motivated by the need for centralized access to such results for development of mathematical models, but its usefulness extends to the general research community of both modellers and experimentalists. Written in PHP (Hyper Preprocessor) and MYSQL, the database is easily adapted to new fields of research and ported to Apache-based web servers. Unlike most existing databases, experimental and observational results curated in QxDB are supplemented by comments from the experts who contribute input to the database, giving their evaluations of experimental techniques, breadth of validity of results, experimental conditions, and the like, thus providing the visitor with a basis for gauging the quality (or appropriateness) of each item for his/her needs. QxDB can be easily customized by adapting the contents of the database table containing the descriptors that characterize each data record according to an informal ontology of the research domain. We will illustrate this adaptability of QxDB by presenting two examples, the first dealing with modelling in oncology and the second with mechanical properties of cells and tissues.

The microbial experience of environmental phosphate fluctuations. An essay on the possibility of putting intentions into cell biochemistry.

We present a model of microbial information processing that contains characteristic features of the phenomenon of physiological adaptation. The backbone of the model is the "adaptive event" in which energy-converting subsystems of the cell interact with the changing environment. In this process, the subsystems pass, via an adaptive operation mode, from one adapted state to the next. An adaptive operation mode takes place when an adapted state is disturbed by an environmental alteration. These two manifestations of an adaptive event were differently treated in the simulation, based on an application of linear irreversible thermodynamics to the energy transduction of adaptive subsystems. In adapted states, the conductivity coefficients of the flow-force relationships employed remained constant, whereas during an adaptive operation mode, these coefficients were altered in a directional manner during the simulation. An example dealing with the complex relationship between phosphate uptake and cyanobacterial growth is given. In this example, the simulation of adapted states of two subsystems of the incorporating machinery, namely the phosphate carrier in the cell membrane and the F-ATPase in the thylakoid membrane, was in accordance with the measured uptake kinetics, and when fixed, predetermined conductivity coefficients were used. In the adaptive operation mode, however, the simulated behavior was in agreement with experimental observations when the program was able to "interpret" its own performance in the light of environmental phosphate fluctuations, experienced by the cell in the past, and to reconstruct the two subsystems according to this interpretation. Via transitions between adapted states and adaptive modes, information is transferred from one adaptive event to the next: the latter "inherits" the results of former interpretations. By appropriating them selectively, it is entering into a future in which its own interpretation is passed on to the following adaptive event. The model is discussed with respect to the concept of autopoiesis.

Expression of the human erythroid Rh glycoprotein (RhAG) enhances both NH3 and NH4+ transport in HeLa cells.

The erythroid Rh-associated glycoprotein (RhAG) is strictly required for the expression of the Rh blood group antigens carried by Rh (D,CE) proteins. A biological function for RhAG in ammonium transport has been suggested by its ability to improve survival of an ammonium-uptake-deficient yeast. We investigated the function of RhAG by studying the entry of NH3/NH4+ in HeLa cells transiently expressing the green fluorescent protein (GFP)-RhAG fusion protein and using a fluorescent proton probe to measure intracellular pH (pHi). Under experimental conditions that reduce the intrinsic Na/H exchanger activity, exposure of control cells to a 10 mM NH4Cl- containing solution induces the classic pHi response profile of cells having a high permeability to NH3 (PNH3) but relatively low permeability to NH4+ (PNH4). In contrast, under the same conditions, the pHi profile of cells expressing RhAG clearly indicated an increased PNH4, as evidenced by secondary reacidification during NH4Cl exposure and a pHi undershoot below the initial resting value upon its removal. Measurements of pHi during methylammonium exposure showed that RhAG expression enhances the influx of both the unprotonated and ionic forms of methylammonium. Using a mathematical model to adjust passive permeabilities for a fit to the pHi profiles, we found that RhAG expression resulted in a threefold increase of PNH4 and a twofold increase of PNH3. Our results are the first evidence that the human erythroid RhAG increases the transport of both NH3 and NH4+.

Inner medullary lactate production and urine-concentrating mechanism: a flat medullary model.

We used a mathematical model to explore the possibility that metabolic production of net osmoles in the renal inner medulla (IM) may participate in the urine-concentrating mechanism. Anaerobic glycolysis (AG) is an important source of energy for cells of the IM, because this region of the kidney is hypoxic. AG is also a source of net osmoles, because it splits each glucose into two lactate molecules, which are not metabolized within the IM. Furthermore, these sugars exert their full osmotic effect across the epithelia of the thin descending limb of Henle's loop and the collecting duct, so they are apt to fulfill the external osmole role previously attributed to interstitial urea (whose role is compromised by the high urea permeability of long descending limbs). The present simulations show that physiological levels of IM glycolytic lactate production could suffice to significantly amplify the IM accumulation of NaCl. The model predicts that for this to be effective, IM lactate recycling must be efficient, which requires high lactate permeability of descending vasa recta and reduced IM blood flow during antidiuresis, two conditions that are probably fulfilled under normal circumstances. The simulations also suggest that the resulting IM osmotic gradient is virtually insensitive to the urea permeability of long descending limbs, thus lifting a longstanding paradox, and that this high urea permeability may serve for independent regulation of urea balance.