Conceived and designed the experiments: RM TG FG SRT. Performed the experiments: RM TG SRT. Analyzed the data: RM TG SRT. Contributed reagents/materials/analysis tools: RM TG NL SRT. Wrote the paper: RM TG IM NL FG SRT.
The authors have declared that no competing interests exist.
Mathematical models that integrate multiscale 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 multiparameter interactions in such complex, multiscale models. Human physiology models are often based on or inspired by Arthur Guyton's wholebody 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 realworld patient. This lays the groundwork for using the Guyton model for
As our understanding of the human body at all scales increases, the construction of a “Virtual Physiological Human” is becoming more feasible and will be an important step towards individualized diagnosis and treatment. As computational models increase in complexity to reflect this growth in understanding, analysis of these models becomes ever more complex. We present a methodology for systematically analysing the interactions between parameters and outputs of such complicated models, using the Guyton model of circulatory regulation as a case study. This model remains a landmark achievement that contributed to the development of our current understanding of blood pressure control, and we present the first comprehensive sensitivity analysis of this model. Effects of varying each parameter are explored over randomized simulations; our analysis demonstrates how to use these results to infer relationships between model parameters and the predicted physiological behaviour. Understanding these relationships is of the utmost importance for developing an optimal treatment strategy for individual patients. These results provide new insight into the multilevel interactions in the cardiovascularrenal system and will be useful to researchers wishing to use the model in pathophysiological or pharmacological settings. This methodology is applicable to current and future physiological models of arbitrary complexity.
Global initiatives such as the IUPS Physiome project
In short, computational models that integrate physiological data from multiple scales (both physical and temporal) provide a framework for understanding the maintenance of biological entities under physiological and pathological conditions. One significant application for such models is
Many challenges must be overcome before a truly integrative model of human physiology can be constructed
A number of models have already been used to develop insight into aspects of human physiology
In previous work, the Guyton models were modularized and reimplemented in Fortran, C++ (M2SL
The analysis and results presented here arose naturally from this body of work. Our motivation was to develop a methodology for systematically exploring the ramified implications of multiparameter interactions in multiscale physiological models. We present such a methodology, which incorporates the
The results provide valuable information about the interdependencies of parameter effects on the model outputs, thus providing direction for future physiologicallyapplicable sensitivity studies of the effects of changes to multiple parameters. These results also lay the groundwork for the use of multiparameter models such as the Guyton model in systematic
An additional outcome is the production of a
Note that beyond the methodology itself, the results presented in this manuscript also serve to demonstrate some of the uses to which the complete set of elementary effects and virtual individuals may be applied. We provide tables of all of the resulting output in the supplementary material (
The Guyton model comprises

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The influence
For any point
Each elementary effect was calculated
Given a set of values for a single elementary effect
For each random input vector
The parameter under investigation (
Throughout the simulations, a number of output variables were monitored to ensure that they remained within physiological bounds (i. e., that the virtual individuals remained “alive”, see
Parameter  Minimum  Maximum  Unit 
GFR  0.015  –  L/min 
CNA  120  160  mEq/L 
CKE  2.5  8  mEq/L 
HM  24  80  – 
MAP  50  200  mmHg 
HR  20  200 

Since the system is highly nonlinear, the effects of a perturbation in the parameter

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This entailed
The results presented here are intended as a demonstration of the analyses that are possible with the complete set of simulation results, which are given in the supplementary material, namely: means and deviations of each elementary effect at each time
The distribution of mean arterial pressure (MAP) in the virtual population is shown in
The vertical line marks the threshold of hypertension (
Population  Criteria  Size  Fraction 
Normotensive 

135,263  35% 
Hypertensive  248,737  65% 
Also shown in
The analysis of the
Definitions of all model 96 parameters and 276 variables are tabulated
Given our interest in the development of hypertension, we focus the discussion here on variables directly related to blood pressure. For example,
The single largest effect on all three variables is that of HYL (the quantity of interstitial hyaluronic acid), which affects the tissue hydrostatic and osmotic pressures. This effect is only observed
The effects on mean arterial pressure (MAP), cardiac output (QAO) and rate of urine production (VUD) are plotted as
Consider the elementary effects on MAP at time
At time
Some of these effects wane over time (ANCSNS, ANUM, ANY and VV9), while the remaining effects become stronger over time (AARK, CPR, EARK, GFLC) and exhibit the largest steadystate elementary effects on MAP. Other parameters also exhibit significant elementary effects over the longer timescales: HM6 (erythropoietic limiter), LPPR (rate of liver protein production), NID (rate of sodium intake) and RNAUGN (basal renal autoregulation feedback multiplier).
Thus, as the model approaches the steadystate following a perturbation, the effects of hormones such as angiotensin are reduced, whilst properties that directly affect glomerular filtration exhibit the largest elementary effects. As one would expect, an increase in AARK or a decrease in EARK (in isolation) results in a permanent increase in the mean arterial pressure, due to a decreased filtration rate. A decrease in glomerular permeability (GFLC) produces a similar change for the same reason. The other major longterm effects (
The only observable effects on QAO at time
A number of parameters exhibit significant effects at time
The major longterm effects on cardiac output (
The nature of the steadystate effects on QAO differ from those on MAP. Several parameters cause permanent changes in both QAO and MAP, but with smaller effects on QAO: AARK (
The elementary effects on VUD at time
Several parameters that exhibited no elementary effect at
By time
The nature of the steadystate effects on VUD (
The elementary effects on NOD (sodium excretion, not shown here) were found to be nearidentical to those on VUD (urine production) at all times
We now demonstrate how the results of the sensitivity analysis can be used to determine which parameters influence an elementary effect. These are the parameters that are most likely to be of interest when investigating the effects of multiparameter perturbations. By identifying such parameters with this method, the number of multiparameter combinations under consideration can be greatly reduced, somewhat mitigating the combinatorial growth in parameters combinations as the number of model parameters increases. This information is of particular use when trying to regulate some physiological function (e. g., pharmacological applications).
As illustrated in
The correlations and partial correlations (controlling for HYL) between the variables (MAP, QAO and VUD) and the model parameters differ by less than
Correlations (
The significant parameters that have been identified by this correlation analysis are all related to vessel and interstitial oncotic pressures, which explains the nature of their influence on the elementary effect of HYL. Hyaluronic acid plays a large role in determining the hydrostatic and oncotic pressure of the tissue gel in the Guyton model, and this effect is a function of the amount of hyaluronic acid (HYL) in the tissue and the interstitial fluid volume (VTS).
This analysis also demonstrates that the Guyton model fails to account for other physiological effects of hyaluronic acid, such as its role in water and solute balance in the inner medulla
A limitation of the sensitivity analysis presented here is that only a single parameter was perturbed during each simulation. However since, for each parameter, this was done with thousands of randomized sets of values for all of the remaining parameters, we demonstrate that the results of our analysis can inform the selection of interesting/relevant multiparameter perturbations, greatly reducing the computational cost of exhaustively searching all possible multiparameter perturbations.
Parameter interactions, which are evidenced by large variances, are more prevalent at the shortest timescales (
The parameters that demonstrated the largest elementary effects on multiple output variables at the steadystate (
A perturbation in any of these parameters changes the steadystate variable values. The importance of these parameters reflects the role of the kidney in longterm blood pressure autoregulation in both the Guyton model and human physiology
Correlations were calculated between each parameter and each output variable at each time
Correlations are shown for mean arterial pressure (MAP), cardiac output (QAO) and rate of urine production (VUD), where
Consider the correlations with MAP; the mosthighly correlated parameters (
In contrast, the parameters mosthighly correlated with QAO (
The parameters mosthighly correlated with VUD (
One parameter, CPR, is notable for being highly correlated with all three output variables MAP, QAO and VUD. In particular, CPR has a correlation of
The virtual individuals were divided into normotensive and hypertensive subpopulations based on their mean arterial pressure, as illustrated in
Probability densities are shown for AAR (the afferent arteriolar resistance), POR (the reference value of capillary
However, obvious differences were observed for very few parameters, all of which had already been highlighted in the sensitivity and correlation analyses.
Correlations between parameters and variables were then compared between the two populations; some results are shown in
For a given variable, the correlations with each parameter are plotted against the xaxis for the normotensive population, and against the yaxis for the hypertensive population (only correlations
The correlations with MAP in the hypertensive population are systematically larger than those in the normotensive population (
When correlations with blood volume are considered (
The correlations with urine production (
The large virtual population that has been assembled here (
The virtual population was divided in two: a randomlychosen training set
This classifier was then evaluated on the evaluation set (i. e., the rest of the virtual population), shown in
Each classifier (binomial GLM) was fitted to a random
But no matter how accurately this classifier can predict hypertension in the virtual population, one should not conclude that it will be of practical use for predicting hypertension in realworld patients. The classifier is a function of
Of the parameters listed in
Classifier  
Parameter  Optimal  Renal+Liver  Liver  Renal 
(intercept)  −32.66  −12.6066  −5.3854  0.92989 
A3K  1.073e5  
AARK  16.95  11.9073  4.63921  
ALDMM  −0.1565  
AMCSNS  0.5476  
AMNAM  0.2681  
ANCSNS  2.312  
ANMAM  2.599  
ANMEM  −1.069  
ANMSLT  0.2056  
ANPXAF  0.5081  
ANUM  0.2067  
AUMK1  1.836  
AUTOSN  −0.9326  
CPR  0.3703  0.2589  0.1310  
DIURET  0.1422  
DTNAR  0.5744  
EARK  −7.800  −5.7318  −2.56488  
GFLC  −307.4  −235.5629  −115.02363  
HM6  3.451e3  
HSL  1.315  
HSR  1.374  
LPPR  26.90  18.9798  9.7504  
NID  4.322  2.7301  −0.2501  0.07619 
PCR  2.188e2  
RNAUGN  −1.645  
RTPPR  −0.1121  
SR  −0.3817  
SRK2  3.059e5  
TENSTC  −4.460  
VV9  −1.035 
Name  Description  Unit  GLM 
CPR  plasma protein concentration for protein destruction  g/L  Liver 
AARK  basic afferent arteriolar resistance  mmHg min/L  Renal 
EARK  basic efferent arteriolar resistance  mmHg min/L  Renal 
LPPR  rate of liver protein production  g/min  Liver 
GFLC  glomerular filtration coefficient  L/min/mmHg  Renal 
NID  rate of sodium intake  mEq/min  Both 
The Guyton model was constructed and refined over many years, and has been validated against a large amount of experimental data
Here we present a brief comparison of the Guyton model to the human renal/body fluid model of Uttamsingh et al.
These simulations reproduced the conditions shown in
Larger variation between the two models is observed when aldosterone is increased fourfold, in order to simulate the administration of deoxycorticosterone acetate (DOCA), a mineralocorticoid with similar effects to those of aldosterone
This simulation reproduced the conditions shown in
The differences highlighted here between the Guyton model and the model of Uttamsingh et al. are certainly due in part to the lower level of detail in the renal portion of the Guyton model, but the Guyton model also includes a more complete cardiovascular model, which would necessarily alter the dynamics produced in response to a chronic increase in aldosterone load. Thus, these observations
In our analysis we perturbed a single parameter in each simulaton (although each parameter was perturbed 1000 times, each simulation with a different set of randomlyselected parameter values). Perturbation of multiple parameters would yield a wealth of additional information, but without any guidance the only recourse would be to exhaustively search every combination of
Given the population of virtual individuals that was presented here, an obvious and desirable application is to draw comparisons between subsets of this population and a given realworld patient. That is, given some observations of a realworld patient, we can select those virtual individuals who best match these observations and see whether one can draw conclusions about the condition of the realworld patient based on the longterm dynamics of the selected virtual individuals.
Beyond using virtual populations merely as a reference for the current and ongoing condition of realworld patients
Development of chronic diseases such as cardiovascular disease is a complex process that involves environmental and cultural factors shared by the individuals living in the same geographical area, as well as ageing, genetic and disease determinants. Hunter et al.
Researchers of the BIMBO project have defined a modeling approach to estimate the public health impact, in terms of the reduction in the number of cardiovascular deaths (CVD), of administering blood pressure lowering drugs to a virtual population of patients
The work presented here illustrates the value of using population information to predict the success of treatment strategies, whilst also moving towards a more ambitious goal: taking into account the individual genetic backgrounds and pathophysiological profiles. This would contribute to the delivery of individualized healthcare, by optimizing the impact of treatments for both the individual patient and at the population level. Future challenges include the development of more sophisticated effect models
The implications of pharmacogenetic parameters on drug efficacy have been explored in the context of diuretic treatment for blood pressure
With regard to the diagnosis and treatment of hypertension, a practical model would predict the effects of the various diuretics and other drugs that are commonly administered to ameliorate hypertension. This would allow the model predictions to be directly compared to clinical studies such as INDANA
We have presented a sensitivity analysis of the Guyton model of human physiology (1992 version), which examined the elementary effects of each parameter over a range of timescales and the correlations between model parameters and key output variables. We also demonstrated how interesting multiparameter combinations can be identified, and how this can highlight shortcomings in the model.
A pool of
Work is currently underway on comparing these results to realworld patient data from clinical studies of the effect of Avastin on hypertension in cancer patients
The methodology we have presented here and applied to the Guyton model is generic in that it can be applied to any mathematical model of sufficient complexity. As physiological models encompass larger and larger scales, both spatially and temporally, this methodology should prove beneficial in elucidating the subtle interactions between model parameters in these complex models.
Such an effort is required to evaluate the clinical suitability of using the Guyton model to assist in providing individualized predictive medicine, as per the goals of both the IUPS Physiome and the Virtual Physiological Human projects.
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Portions of this work were presented in abstract form at the following meetings: VPH 2010, Sept. 28–29, 2010, Brussels, Belgium; Experimental Biology 2011, April 9–13, 2011, Washington D.C., USA; Printemps de Cardiologie 2011, May 12–14, 2011, Lyon, France; ECMTB2011, June 28–July 2, 2011, Kraków, Poland; MEFANET 5th Annual Meeting, November 24–25, 2011, Brno, Czech Republic. We thank JeanPierre Boissel (Novadiscovery cofounder & CSO) for discussions on physiological modelling, including the Effect Model law.