Modeling of vascular networks

The general subject of this review is the modeling of the microscopic vascular (capillary) networks in animals, with attention directed specifically to mammals. The modeling to be dealt with is that associated with the derivation and discussion of `scaling laws' for these networks for mammals of vastly different size. For reviews of the basic concepts see also Dawson(1991, 2001, 2003). Experimental measurements on the scaling of the vascular networks of mammals have been reported by Schmidt-Nielsen and Pennycuik(1961), Gehr et al.(1981), and Hoppeler et al.(1981).

Scaling laws of the kind to be considered here are important in the general understanding of the vascular and cardiovascular system. If found appropriate,such scaling laws can illustrate that these systems all follow the same general pattern. Hence, the study of these systems on the basis of measurements from, say, the mouse can then reveal important information for the human, as well as other mammals.

The vascular system of mammals consists of two major parts: the systemic part associated with transport of blood to the body and the pulmonary part associated with transport of blood to the lungs. The architecture is basically the same for both parts. The systemic vascular system, for example, consists of the aorta, carrying oxygenated blood directly from the left side of the heart, and the smaller vessels branching from it. The latter lead ultimately to the capillary networks spread throughout the body. These vascular networks consist of very small arteries (the arterioles) that take blood to the capillaries, the capillaries themselves, through which exchange of products with the tissues occurs, and then the very small veins (the venules) that take blood away from the capillaries and ultimately return it to the right side of the heart for circulation in the pulmonary system for discharge of unwanted gaseous products and for recharge of oxygen in the blood. The flow system is illustrated in Fig. 1 (from Dawson, 1991). Attention here will be directed mainly towards the systemic system, although some reference will also be made to the pulmonary system, as appropriate.

The systemic vascular networks of mammals consist of some 10-100 capillaries in each network. In muscle tissue, there are also reserve capillaries that open with exercise and provide increased blood flow, as first shown by Krogh (1920). This matter will be dealt with later; but for now, resting conditions will be assumed.

where ω denotes heart rate, M_b denotes mammal mass (with M_b⁰ denoting no dependence on it),and where the symbol α denotes proportionality.

Direct evidence for the latter relation exists for the pulmonary capillaries (Gehr et al.,1981) and there is no reason to expect differently for the systemic capillaries. Relations (2) and (3) thus provide two relations, backed by measurement, between the four unknown quantities R_C, L_C, N_C and ω. Two additional relations are thus needed to complete the description. These may either be empirical or theoretical. The latter is first considered. The following discussion of these relations is based on earlier work by the author(Dawson, 1991).

assuming all dimensions scale the same, like in fact the dimensions of the heart, which all scale as mammal mass to the power 1/3 (Dawson, 1991, 2001).

Now, cardiac tissue consists mainly of contraction fibers which, when excited, provide the pumping action of the heart. The fibers consist of series connections of small cardiac cells, separated by membranes. Contraction is initiated in the upper heart and spreads over the heart through progressive influx of ions into the cardiac cells, making the fibers contract. The required two relations follow from consideration of the resting heart rate as determined by the diameter of the cardiac fibers, with the latter assumed the same as the linear dimension d defined above; and from consideration of the influx of ions into the fibers.

Experimental evidence for the relation for heart rate has existed for many years (Rihl, 1926; Clark, 1927). More recent work by Holt et al. (1968) provides experimental support for both the heart-rate and cardiac-output relations.

It is worthwhile to note the differences that arise if different values of`b' are used for the exponent in the above contraction-speed relation leading to relation (5). For the lower value of 0.5, the equations predict that heart rate should vary with mammal mass to a power of -4/15, that is, a power of about -0.27; and that the cardiac output should vary with mammal mass to a power of 11/15, that is, a power of about 0.73. For the higher value of 1, the equations predict an exponent value for heart rate of -2/9, that is, a power of -0.22; and that for cardiac output of 7/9, that is, a power of about 0.78. The differences are not especially significant, and for either choice of exponent `b', the relations for heart rate and cardiac output, if determined experimentally, could reasonably be rounded to the exponent values of -1/4 and 3/4, respectively.

The value of 2/3 for the exponent `b' in the contraction-speed relation,noted in connection with relation (5), was chosen earlier by the author(Dawson, 1991) because it led to the relations for cardiac output and heart rate, as generally accepted. Interestingly, if an extreme value for `b' of zero is assumed, the resulting expressions are that heart rate varies as mammal mass to the power -1/3 and that cardiac output varies with mammal mass to the power 2/3. The latter is unlikely, based in part on the work of Holt et al.(1968).

In addition to showing that the scaling laws for the capillary networks, as well as the scaling laws for cardiac output and heart rate, can be determined from fundamental considerations, it has been illustrated here that heart rate varies inversely as the square of the diameter of the cardiac contraction fibers. The latter offers an explanation (and the only one that I am aware of)as to how and why the resting heart rate of, say, the mouse is many times that of the human; that is, about 600 beats min^-1 for mouse and about 60 beats min^-1 for human.

Additional means for obtaining the above scaling laws have been considered earlier and involved use of empirical relations for heart rate (reciprocal 1/4 power relation) and oxygen consumption rate (3/4 power relation), as described by Dawson (2001, 2003). This approach leads to the same scaling laws as relations (8a-c).

It may first be noted that direct experimental measurements are limited regarding the systemic side of the circulation, but the data available are consistent with the theory described here. In particular, there are the counting measurements of Kunkel(1930) for number of nephrons in the kidneys of mammals of various sizes. The nephron is the basic unit in the kidney and consists of a collection of capillary vessels. The number of capillaries per nephron can be expected to be the same for any mammal, and hence a count of nephrons in a kidney can be expected to be proportional to a count of capillaries in the kidney. The measurements of Kunkel (mouse to ox range) were analyzed by Adolph(1949) and shown to obey a power law relation, with mammal mass raised to the power 0.62, which may also be taken as 5/8, in agreement with relation (8c).

There are also measurements of Kunkel(1930) concerning the diameter of renal capsules of the nephrons that indicate variation with mammal mass to a power of essentially 1/12 (Adolph,1949). This variation may be taken as evidence for the scaling relation for relation (8a), since the diameter of the renal capsules can be expected to be proportional to the radius of the contained capillaries within them.

In addition, there are measurements of rate of urine output of resting mammals (mouse to elephant range) that indicate variation with mammal mass to a power of essentially 5/6, as noted by Adolph(1949). Net fluid flow from the capillaries in the kidneys, insofar as scaling is concerned, can be expected to be proportional to the product of capillary number N_Cand capillary length L_C as discussed earlier by the author(Dawson, 1991). Relations(8b,c) provide, in fact, the scaling exponent 5/6.

Finally, on the systemic side, there are measurements of capillary density in muscles by Schmidt-Nielsen and Pennycuik(1961) that presumably apply to the resting state, as considered here. Capillary spacing in resting muscle can be expected to be proportional to capillary radius, and capillary density can therefore be expected to be inversely proportional to the square of capillary radius; that is, as mammal mass to the power -1/6, based on relation(8a). The relation just described was recently confirmed from theoretical considerations (Dawson, 2003). The measurements of Schmidt-Nielsen and Pennycuik (bat to pig range) support this variation as illustrated in Fig. 2 for capillary density in the masseter (jaw) muscle of mammals. Similar agreement exists for measurements of Schmidt-Nielsen and Pennycuik(1961) regarding resting capillary density in the gastrocnemius (leg) muscle of mammals(Dawson, 2003).

For the pulmonary side of the circulation, measurements exist for capillary volume and surface area (Gehr et al.,1981). These data (shrew to cow range) may be used to obtain data for capillary radius and net capillary length(Dawson, 1991). Typical values for the capillary radius are shown in Fig. 3 and support predictions from relation (8a). In average terms,the volume of the capillaries was found by Gehr et al.(1981) to be directly proportional to mammal mass, and their surface area was found proportional to mammal mass raised to about the power 0.93 (average of two sets of data). These results imply that capillary radius R_C and net capillary length N_CL_C scale with mammal mass to powers 0.07 and 0.86, respectively, in good agreement with the theoretical value from relations (8a-c) of 0.08 and 0.83.

The basic scaling relations (8a-c) may also be derived by replacing the two theoretical relations (5) and (7) by the empirical relation for heart rate,and either of the above experimentally confirmed relations for capillary number and capillary radius.

Among the many biological processes involved with the vascular networks,the most important is perhaps the transfer of oxygen from the blood to the surrounding tissues. The resting oxygen consumption rate of mammals (mouse to elephant range) is known to obey an average power-law relation with mammal mass raised to the power 3/4 (Kleiber,1932; Brody and Procter,1932; Brody,1945). This relation applies over a wide range of mammal size,but, insofar as empirical relations are concerned, may be subject to modification for smaller mammals (Bartels,1982; Heusner,1991; Dodds et al.,2001). Interestingly, the careful work of Bishop(1999) on mammals and birds indicates no such deviations.

Moreover, as already known (Dawson,1991) on the basis of measurements by Schmidt-Nielsen and Larimer(1958), the oxygen pressure P₀ in the blood can be considered to be proportional to mammal mass to the power -1/12 (mouse to mule range). This correlation is illustrated in Fig. 4, together with an additional measurement for the elephant(Bartels et al., 1963). With the product N_CL_C varying with mammal mass to the power 5/6, as required by relations (8a-c), the oxygen transfer rate is thus seen to vary with mammal mass to the power 3/4, in accordance with measurements over a broad range on mammal size.

It may be noted (at least within the writer's limited understanding) that`constructal theory', based in part on the assumption that vascular formation is an optimized flow architecture, also provides the power 3/4 for resting oxygen consumption rate (Bejan, 1997, 2000).

A practical application of capillary scaling involves consideration of therapeutic drug concentrations in the blood of small mammals as they relate to humans. A specific example is provided by the chemotherapy drug methotrexate, which is a widely used (non-metabolized) drug in the treatment of cancer. Dedrick et al.(1970) showed that data from various mammals could be consolidated into a broad correlation by plotting the ratio of drug concentration in the blood to initial dose as a function of time since injection, when the latter was divided by mammal mass raised to the power 1/4. Since the blood-circulation time is proportional to mammal mass to the power 1/4 (Dawson, 1991),the resulting time scale is simply proportional to circulation time. It may be anticipated that some refinement could be gained by including the capillary geometry and scaling into the correlation.

The drug is assumed to be injected on the systemic venous side in a relatively short period of time. On passing through a general capillary, the classical view (Guyton, 1971)is that the blood initially expels some water and drug to surrounding tissue(Starling, 1896; Pappenheimer et al., 1951). Near the end of the capillary blood flow, some of the mixed drug is reabsorbed(see Fig. 5; from Dawson, 1991). Of course, in the event of fluctuating osmotic pressure, some capillary exchange, back and forth, could be expected along the entire length of the capillary.

In either case, a simplified process may be assumed for scaling purposes where the entire drug is removed by the capillaries during the initial cycle and some of the drug reabsorbed. In this case, the drug concentration at the injection site, after the initial cycle, will equal the amount reabsorbed. This process would perhaps lead to a reduced concentration at the site,compared with the actual process. However the same percentage difference may be expected (for a non-metabolized drug) for all mammals so that scaling is still possible in the present case.

The concentration of drug, C (in units of mass per volume of blood), in the blood after the first complete cycle of blood circulation in the above-simplified process will be proportional to the net volume of drug reabsorbed per unit volume of blood. Flow into and out of capillaries can be considered for scaling purposes (Dawson,1991) to be proportional to a scale-independent constant and the net capillary length, that is, BN_CL_C,where B denotes the filtration-absorption constant (units of area per time). The concentration of drug reabsorbed over the time, T, for a complete circulation is thus proportional, for scale-independent initial concentration, to the ratio N_CL_CT/V_B,where V_B denotes total blood volume. For any arbitrary initial drug concentration D₀, the ratio is D₀N_CL_C/V_B)T.

where subscripts `H' and `M' denotes values for the human and mammal,respectively.

Fig. 6 illustrates application of this scaling law in projecting measurements from the mouse(mass of 0.022 kg) to the human (mass of 70 kg) for the plasma concentration of methotrexate in the blood as a function of time. The basic measurements for the mouse, as used in developing the results of Fig. 6, are due to Dedrick et al. (1970) and the data for the human are due to Henderson et al.(1965). The solid curve shown in Fig. 6 is a simple`best-fit' power-law expression for comparison purposes.

The agreement of the scaled data from the mouse with that of the human, as shown in Fig. 6, is indeed very good. The factor(M_H/M_M)^1/12 in the scaling relation (13a) represents the effect of capillary process and is equal to 1.96. Without this factor, the predictions from the mouse would thus be reduced by a factor of about 2. Overall, the factor can be seen to improve the predictions over what they would be otherwise.

An interesting aspect of the work in the present discussion is that the scaling laws for the radius and length of the capillaries forming the vascular networks can be determined on the basis of the resting condition of mammals. This aspect has been noted earlier(Dawson, 2003). Since the size of the capillaries cannot change with exercise, there can only be one set of scaling laws for these, and resting conditions provide these laws. Of course,resting conditions also provide the scaling laws for resting heart rate,cardiac output and oxygen consumption rate, as illustrated here.

For a number of years, it was generally thought that all scaling relations associated with the physiological functions of mammals (for example, cardiac output, oxygen consumption rate and heart rate) were the same for both resting and exercise states. This could be the case if such physiological variables increased during exercise by the same factor for all mammals. This would change the proportional factor in the scaling relations, but leave the scaling exponents with body mass unchanged. This is now known not to be the case, as a result of work by Baudinette(1978), Taylor et al. (1981),Weibel et al. (1991) and Bishop (1997, 1999), among others.

In particular, the oxygen consumption rate of mammals (bat to steer range)in strenuous exercise has been shown by Bishop(1999), in careful analysis of existing data, to vary with mammal mass raised to the power 0.88, that is, a power of about 7/8. This result is consistent with earlier work of Weibel et al. (1991) and the more recent work of Weibel et al. (2004). The power 7/8 is also consistent with `constructal theory' and the assumption that flow systems survive by forming easier access to the required flow(Bejan, 2000).

On the capillary scale, allowance appears to be made for exercise by increased activity of the capillary network through openings of reserve capillaries in the muscles (Krogh,1920). This will change the scaling law for operative number of capillaries in the body, but will leave the scaling laws for dimensions of the capillaries unchanged. This view is, perhaps, consistent with the principle of`symmorphosis' (Taylor and Weibel,1981; Weibel et al.,1991; Hoppeler and Weibel,1998) requiring (in simple terms) that nothing more be provided in the design of mammals than that required for their purposes.

This relation corresponds to a `weighted' average for scaling of the number of capillaries in the body. The concept of a weighted average is consistent with recent discussion by Weibel(2002) and work of Darveau et al. (2002) on the rate of oxygen consumption during exercise. When compared with the resting state, with capillary number proportional to mammal mass to the power 5/8, the implication is that the ratio of average capillary number in strenuous exercise to resting number increases in proportion to mammal mass to the power 1/8. Perhaps gravity enters here in that larger mammals have to work harder than small mammals in overcoming their weight (as opposed to their mass) in exercise.

Of course, with strenuous exercise, resting-state similarity in the physiological response of mammals is lost. With fixed scaling of radius and length dimensions of the capillaries, it is impossible to satisfy all the earlier relations required for similar response among resting mammals. In this regard, capillary density in muscles that includes both resting and reserve capillaries (for strenuous exercise) cannot be expected to follow the simple scaling law noted earlier where resting capillary density varies with mammal mass to the power -1/6. This observation is consistent with sophisticated measurements of Hoppeler et al.(1981), which presumably detected both active and inactive capillaries as discussed recently(Dawson, 2003).

In spite of the loss of resting-state similarity in exercise, it is worthwhile to note that cardiac output in strenuous exercise appears to vary in the same way as oxygen consumption rate(Bishop, 1997), namely as mammal mass to the power 7/8. In this case, the pressure drop across capillary networks is still scale invariant like in the resting state, as can be seen from Equation (1). The rate at which energy is dissipated by viscous(frictional) resistance in the capillaries is proportional to the product of pressure drop and cardiac output. Considering all the capillary networks, this can then be seen to be proportional to the rate of oxygen supply to the body and, in particular, also to the heart since heart mass and body mass are proportional. A similar situation exists for the resting state. The ratio of rate of energy dissipation in capillary blood flow to rate of oxygen (fuel)supply to the heart is thus invariant with scale for both exercise and resting, consistent with good design practice.

This last matter may perhaps have implications regarding work by West et al. (1997), Banavar et al.(1999) and Dodds et al.(2001) on the efficiency of the design of the vascular networks of mammals. It may also have relevance in constructing a general basis for similarity in the strenuous-exercise state that provides, for example, the 7/8ths power law for oxygen-consumption rate,like that provided here for the 3/4-law for the resting state.

Modeling of the vascular networks of mammals has been reviewed here, with the expressed goal of discussing relevant scaling laws for resting mammals of vastly different size. Both geometry and process have been considered. Results have been shown to be in good agreement with measurements for a wide range of mammal sizes. A main conclusion to be drawn from this review is that similarity exists for the vascular networks of mammals, as well as for related physiological processes.

Overall, the design of the vascular networks of mammals appears to be the same, with predictable adjustments for scale for resting mammals and some additional adjustment for exercise. An interesting question that presents itself is why the design follows essentially the same pattern for all mammals. The answer would appear to be simply because it is the `best' pattern, all things considered. In this regard, it has been noted that, for both resting and strenuous exercise, the rate of energy dissipation in blood flow in the vascular capillary networks appears to be proportional under change of scale to the oxygen rate supplied to the heart. The ratio of the two is thus invariant under change of scale for both resting and exercise.

I am grateful to the referees for their valuable comments and criticisms of the original manuscript.