Published online before print September 19, 2005, doi: 10.1161/01.HYP.0000184390.27545.40
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(Hypertension. 2005;46:652.)
© 2005 American Heart Association, Inc.
Editorial Commentaries |
Modeling the Vasculature
A Judicious Approach?
From the Department of Pharmacology, University of Aarhus, Denmark.
Correspondence to Prof. Michael J. Mulvany, Department of Pharmacology, University of Aarhus, University Park 240, 8000 Aarhus C, Denmark, Tel. +45 8942 1711, Fax +45 8612 8804, mm{at}farm.au.dk
Although William Hazlitt believed that "rules and models destroy genius and art,"1 it was the firm belief of Cecil Murray that "physiological organization, like gravitation, is a stubborn fact."2 Thus, Murray maintained that "organization is a legitimate field for scientific inquiry and not an affair of reflective judgment," and that one of the objectives of vascular science should therefore be to uncover the laws that underlie the remarkable similarities of the vascular architecture in many species. In this issue, Pries et al3 present an update of their previous models of the vasculature,4,5 showing that with appropriate coefficients, a comparatively small number of parameters can simulate real vascular trees. The meticulousness and sophistication of their work is not in doubt, but does it advance our understanding of why the vasculature is organized the way it is?
There are various approaches to modeling. One approach, such as the classic cardiovascular model of Guyton,6 is to identify as many components of the system as possible and to insert known characteristics of these and their known interconnections, and then to observe how the model operates.7 Such an approach can lead to new insights as to the parameters of importance, in this case, the apparent role of pressure-natriuresis. Another approach is to find simple relationships that explain a variety of phenomena, as did Watson and Crick regarding the structure of DNA.8 A third approach is to use a known physical principle to explain a complex system, and this was the approach of Murray.2,9 In all cases, these approaches lead to hypotheses that can be tested.
It is self-evident that development of the vasculature must depend on general mechanisms and not on genetic programming of each vessel. Murray’s approach was to assess what could be the evolutionary pressures that would be of importance in determining these mechanisms and suggested that an important factor would be minimization of energy expenditure. From the concept that the energy cost of sending blood through a vessel was the sum of the frictional loss, and of the energy cost of producing the blood within the vessel, he made the deduction that blood velocity should be proportional to the cube root of the vessel radius, now known as "Murray’s law." This provided prediction for the tapering of the arterial vasculature (aorta and veins were excluded from the analysis). This prediction has, in general, been confirmed in systemic vascular beds with reasonable accuracy,10,11 although more recent studies find exponents below the cube law,12 and there is some evidence that the cube law might be fortuitous.13 Murray’s law also predicted that shear stress should be constant throughout the vasculature, but Pries et al3,5 have not confirmed this, possibly because certain of Murray’s simplifications are incorrect. For example, Murray took no account of the non-Newtonian properties of blood. It is also possible that the failure to confirm the prediction is that the measurements have not been made under resting in vivo conditions (see below).
The work of Pries et al can, to some extent, be seen as a development of Murray’s work in that they attempt to identify relationships between vascular wall components that will result in a known vascular architecture. The basic controlling parameters are proposed to be the blood flow and blood pressure, and thus wall shear stress (increased shear stress leads to increased diameter) and wall circumferential stress (increased circumferential stress leads to wall thickening). They then introduce a third basic parameter: the metabolic demand that is proposed to affect how shear stress controls diameter and how wall stress affects wall thickening. Furthermore, they propose that the effect of wall stress on wall thickening is reduced if the wall is thick, and that a thick wall also limits the effect of shear stress on diameter increase. Also, other assumptions are made to ensure convergence. Finally, account is taken of the physical constraints imposed by vascular geometry and the Laplace relationship.
Pries et al’s starting point is the mapping of a real rat mesenteric vascular network containing 913 vessels with regard to diameter and wall thickness, all measured in vivo in anesthetized animals under conditions of complete vascular relaxation. From this network, blood flow and pressure in each vessel were calculated for given input and output pressures, taking previously used account of the Poiseuille relationship and blood rheology. This allowed calculation of the shear stress and circumferential stress in each vessel. The network was then allowed to adapt according to the model until all parameters converged. Coefficients were adjusted iteratively until reasonable concordance between model and the observed situation was achieved. The authors then show, using in part a simplified 23-vessel network, that raising input pressure results in increased peripheral resistance and structural adaptation similar to that observed in experiments reported with hypertensive rats. Raising metabolic demand (eg, exercise) results in decreased resistance, whereas reduction of vascular response to increased shear stress (eg, endothelial dysfunction) results in raised peripheral resistance. Thus, although it would have been preferable to have performed all these interventions on the full 913-vessel network, it seems that the model reacts in an expected manner.
Pries et al have therefore provided a model based on reasonable assumptions that can reproduce a complex network and that reacts in an expected manner to various interventions. Of the three types of model referred to in my introduction, the model presented by Pries et al most closely resembles the approach of Guyton6 but has the disadvantage that rather than being based on experiments, some of the relationships that have been used are assumed, and the coefficients have been determined by computer fitting. Does this negate the value of the model? Not so in my opinion because it has the strength that it is based on a realistic vascular network with all its intricacy. Therefore, in principle, the model provides the basis for experiments that can test the assumed relationships and determine the coefficients.
A criticism of the model, which is acknowledged by the authors, is that the measured values are made on vessels that are completely relaxed. In practice, the vessels will have tone and be contracted. Thus, the vessel diameters will differ from those used in the model, and the shear and wall stresses that are actually experienced by vessels during adaptation will be substantially different from those that have been calculated. This is perhaps not so important if there is a constant relationship between relaxed diameters and actual diameters (for which there is some indirect evidence for the vasculature as a whole14); but unless this can be demonstrated for all vessels, there must be some concern regarding the basis for the model. In particular, it might be that shear stress was more constant for the in vivo situation (as predicted by Murray) if the smaller vessels have a greater tone than the larger vessels.
Together, Pries et al3 are to be thanked for providing a realistic framework for investigating fundamental relationships of importance in the development of the vasculature. Experiments should now be initiated, perhaps using organ culture,15 to test the proposed relationships and determine the various coefficients. Further development of the model should, I suggest, include vascular tone. It would also be interesting to study the adaptation of the 913-vessel network on the basis of the Murray minimum energy concept.
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Footnotes |
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The opinions expressed in this editorial commentary are not necessarily those of the editors or of the American Heart Association.
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References |
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- Hazlitt W. Thoughts on taste. In: The Edinburgh Magazine. Edinburgh, UK: Archibald Constable and Company; 1818.
- Murray CD. The physiological principle of minimum work. I, The vascular system and the cost of blood volume. Proc Natl Acad Sci U S A. 1926; 12: 207–214.
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- Guyton AC, Coleman TC. Long-term regulation of the circulation: Interrelationships with body fluid volumes; In: Reeve EB, Guyton AC, eds. Long-Term Regulation of the Circulation: Interrelationships with Body Fluid Volumes. Philadelphia, Pa: W.B. Saunders; 1967.
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- Murray CD. The physiological principle of minimum work applied to the angle of branching of arteries. J Gen Physiol. 1926; 9: 835–841.
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[Abstract/Free Full Text] - Christensen KL, Mulvany MJ. Vasodilatation, not hypotension, improves resistance vessel design during treatment of essential hypertension: a literature survey. J Hypertens. 2001; 19: 1001–1006.[CrossRef][Medline] [Order article via Infotrieve]
- Bakker EN, Buus CL, Spaan JA, Perree J, Ganga A, Rolf TM, Sorop O, Bramsen LH, Mulvany MJ, VanBavel E. Small artery remodeling depends on tissue-type transglutaminase. Circ Res. 2005; 96: 119–126.
[Abstract/Free Full Text]
Related Article:
- Remodeling of Blood Vessels: Responses of Diameter and Wall Thickness to Hemodynamic and Metabolic Stimuli
- Axel R. Pries, Bettina Reglin, and Timothy W. Secomb
Hypertension 2005 46: 725-731.[Abstract] [Full Text] [PDF]
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