Abstract
Kitaoka, Hiroko, and Béla Suki. Branching design of the bronchial tree based on a diameterflow relationship.J. Appl. Physiol. 82(3): 968–976, 1997.—We propose a method for designing the bronchial tree where the branching process is stochastic and the diameter (d) of a branch is determined by its flow rate (Q). We use two principles: the continuum equation for flow division and a powerlaw relationship betweend and Q, given by Q ∼ d ^{n}, where n is the diameter exponent. The value ofn has been suggested to be ∼3. We assume that flow is divided iteratively with a random variable for the flowdivision ratio, defined as the ratio of flow in the branch to that in its parent branch. We show that the cumulative probability distribution function of Q, P(>Q) is proportional to Q^{−1}. We analyzed prior morphometric airway data (O. G. Raabe, H. C. Yeh, H. M. Schum, and R. F. Phalen, Report No. LF53, 1976) and found that the cumulative probability distribution function of diameters, P(>d), is proportional to d ^{−n}, which supports the validity of Q ∼ d ^{n} sinceP(>Q) ∼ Q^{−1}. This allowed us to assign diameters to the segments of the flowbranching pattern. We modeled the bronchial trees of four mammals and found that their statistical features were in good accordance with the morphometric data. We conclude that our design method is appropriate for robust generation of bronchial tree models.
 airway
 diameter exponent
 flow distribution
 fractals
 power law distributions
the function of a ductal structure is to transport fluid to designated areas of an organ. In addition to efficient fluid transport, for large branching ductal systems such as the airway tree, one should also consider the optimal flow distribution at each terminal. What kind of a branching system is required from the point of view of flow distribution at the terminals where a given amount of fluid has to be delivered? The flow rate (Q) distribution is supposed to have a small deviation and be stable against perturbations. Statistical descriptions (19, 28) are necessary to evaluate this aspect of flow distribution. Previous morphometric airway models such as those proposed by Weibel (34) and Horsfield et al. (11, 12) are not adequate for studying the heterogeneity at the terminal ends.
In Weibel’s model (34), each terminal is completely identical. This simplification makes Weibel’s model attractive and easy to handle from the point of view of fluid mechanical computations along the airways. However, its usage is limited because there are no such ideal branching trees and under certain conditions (e.g., brochoconstriction) the assumptions of symmetry and homogeneity cannot be maintained. The airway models proposed by Horsfield et al. (11, 12) are more realistic, because here each terminal has a different pathway, giving rise to a natural asymmetry of the tree. This feature has been exploited to predict the acoustic properties of the airways by Fredberg and Hoenig (3) and more recently has been further developed by Lutchen et al. (21) to investigate lung function (e.g., lung and airway resistances) during heterogeneous constrictions. Nevertheless, Horsfield’s models are selfsimilar in the sense that the branching patterns are completely determined by a rule of branching order, which, in turn, leads to identical diameters (d) of the terminals. In reality, according to the nomenclature of Boyden (2), we can name nearly a hundred proximal branches that may be genetically determined in the human airways. However, these airways constitute <0.5% of the total number of branches down to terminal bronchioles (TB), where no deterministic branching patterns have been found. We therefore realize that a stochastic description of the branching process is necessary, which leads to some heterogeneity ofd and flow distribution at the terminal ends.
An optimal relationship between Q andd was proposed decades ago (7, 17, 23,30) as follows
Here we propose a method for designing branching ductal structures where the branching pattern is stochastic and the d of each branch is determined by its Q through Eq.1 . First, we will describe a stochastic process whereby Q is iteratively divided at each bifurcation. We then examine some general properties of this branching process that can be used to generate the branching pattern or the topology of the tree (seestochastic branching process in a ductal system). Next, we will present evidence of the applicability of the above diameterflow relationship based on our analysis of Raabe’s morphometric data (25) (seedesigning the diameters of the branches). Finally, combining the branching pattern of flow division with Eq. 1 will allow us to design the d of the bronchial tree, and we will present simulation trees for four mammals and compare the results with actual morphometric data.
STOCHASTIC BRANCHING PROCESS IN A DUCTAL SYSTEM
FlowDividing Process Under Continuum Equation
First, we describe how fluid is divided at a bifurcation. We assume that mass is conserved at each bifurcation that is equivalent to the continuum equation for incompressible fluid. When the volume change of the duct during flow is negligible, the Q before branching (Q_{0}) is equal to the sum of the Q of the two daughter branches (Q_{1}and Q_{2}; Q_{1} ≥ Q_{2}).
In a finite structure like a living organ, there are terminal branches where the flowdividing process stops and the fluid is delivered into the terminal units of the organ. In the lung, the TB is defined as a terminal branch of the conductive airway tree (11, 34). Although there is further branching within an acinus, the acinus is defined as the functional unit for gas exchange because the respiratory bronchioles are no longer pure conductive ducts. Accordingly, it is reasonable to assume that there is a threshold flow (Qc) below which there is no more conductive flow division. By definition, Qc provides the maximum Q at the terminal branches. In the following, we will assume that the Q is normalized to unity at the root. Thus, Qc represents the maximum fraction of the total flow that can be delivered to a TB.
The above three rules (continuum equation, flowdividing ratio as a random variable, and the existence of a Qc as a parameter providing the maximum Q at a terminal branch) enable us to create large branching systems simulating the actual branching pattern of the bronchial tree. Specification of r and Qc that are appropriate for various mammals will be described instochastic models of mammalian bronchial trees.
Statistical Characteristics of Flow Distributions
Plotted on a loglog graph, Fig. 1 shows the cumulative distribution of Q, including every branch of a tree.N(≥Q) is the number of branches whose Q is larger than a given value Q. The cumulative probability distribution function P(≥Q) is then given byN(≥Q)/Nt
, whereN
tis the total number of branches in the tree. As can be seen from Fig.1, for Q ≥ Qc, N(≥Q) is proportional to Q^{−1} so that
An interesting implication of Eq. 3 is that when a physical quantity is related to Q according to a power law (e.g., the d of the branch throughEq. 1 ), the probabilitydistribution function of that quantity will also be a power law, as will be discussed in the next section. Since Mandelbrot (22) proposed the concept of fractals, branching structures like the bronchial tree have often been categorized as fractal objects (1, 5, 6, 16, 18, 20, 33). Fractals are selfsimilar objects characterized by powerlaw distributions. Thus, since the results in Fig. 1 are quite independent of the particular distribution of r, the powerlaw distribution of flow demonstrates the general statistical selfsimilar property of branching structures.
DESIGNING THE DIAMETERS OF THE BRANCHES
Having generated branching patterns, we next propose a method to assignd to the individual branches by using the diameterexponent rule of Eq. 1
. Combining Eqs. 1
and
2
, we obtain the following relationship between the d of a bifurcation
In this section, we first reanalyze the morphometric data of four mammalian airways published by Raabe et al. (25) and then examine the validity of Eq. 1 in two ways: by directly using Eq. 4 and by examining the probabilitydistribution function of the diameter exponent. The significance of this is that by establishing the validity of the dQ rule (Eq. 1 ), we will be able to connect the d of a branch to its Q. Thus, we can then simply transform the flowbranching pattern obtained in the previous section to diameterbranching pattern.
Resampling Raabe’s Morphometric Data
The morphometric data of Raabe et al. (25) are based on two human lungs, two dog lungs, one rat lung, and one hamster lung. To every data set, they assigned a minimumd beyond which the measurement was complete. For branches in which d was smaller than the minimum d, the tree was arbitrary. The precision of measurement was 0.1 mm; therefore, we did not use d <0.5 mm because of the large relative measurement errors. We resampled Raabe’s data so as to include only those branches that formed a complete tree with their diameters larger than the minimumd. The trunk of a tree was not limited to the trachea. As long as all branches beyond the minimumd belonging to one trunk were measured, the tree arising from this trunk was included in the statistical analysis. However, to obtain statistically meaningful results, we required that a tree contained at least 400 branches.
We generated nine trees from Raabe’s data (Table1). Six of the trees were from bilateral lungs of six individuals, one tree was from a subsegmental bronchus of a human right upper lobe (HM272), and two trees were obtained from the right apical lobe and the right intermediate lobe of a dog (DM272).
Calculation of the Diameter Exponent
Calculation of the d exponent was performed by solvingEq. 4 numerically at each bifurcation. For the human and dog airway trees, all bifurcations were selected whered _{0}was larger than the minimum d. For rat and hamster airway trees, only those bifurcations were selected whered _{0},d _{1}, andd _{2}were all >0.5 mm. Bifurcations withd _{1}(ord _{2}) ≥d _{0}were excluded from the analysis, since in this case there is no finite positive solution to Eq. 4 .
The calculated values of n are summarized in Table 1. In all cases, the SD were large. Figure2 shows a histogram ofn obtained from the data set HM272. The distribution of n could be approximated with a lognormal distribution not only for HM272 but also in all other cases, similar to previous reports (13, 30). As explained in appendix , the wide distribution of n could primarily be due to the high sensitivity of the calculation ofn from the measured values ofd using Eq.4 . Although airways are not ideally cylindrical,Eq. 4 is based on the assumption of cylindrical configurations; therefore, slight changes in the measured values of d can cause a wide distribution of n.
There was a significant correlation between geometric mean (GM) ofn and the minimumd (correlation coefficient = 0.86,P < 0.01), which is in accord with Horsfield’s report (13). This correlation, however, is a result of the influence of measurement error in d(see appendix ). Also, there was a significant correlation between GM ofn and body mass (correlation coefficient = 0.83, P < 0.01). However, one should not conclude that there are speciesrelated differences in n, because this correlation is also influenced by the correlation between GM ofn and the minimumd. This analysis thus indicates that the diameter exponent rule (Eq. 1 ) may be acceptable in large trees.
ProbabilityDistribution Function of Diameters
The cumulative distributions of diameters,N(≥d ), were examined by counting the number of branches whosed was larger than a given diameterd. When the total number of branches in the tree isNt
, the corresponding probabilitydistribution function,P(≥d) is given byN(≥d )/Nt
. Before the calculation ofP(≥d ), the measured d were first corrected for measurement error by subtracting the absolute error, 0.05 mm, from the values of the d. The reason is that the number of branches with dlarger than the measured value Dshould also include the number of branches whose diameters are betweenD − 0.05 andD. The cumulative distribution ofd showed an apparent inverse power law for all species, i.e., N(≥d ) ∼d
^{−m}. Figure3 shows examples ofN(≥d ) in the two human bronchial trees. The values ofm were obtained from the slope of the regression line (Table 1). However, since the total number of branches,Nt
, was not large enough in all trees to establish reliable statistics of the slopem, mwas further corrected according to the procedure detailed inappendix
. This correction procedure resulted in values of m that were quite close to the GM of n (Table 1). Indeed, it is easy to show that m andn should be the same. FromEqs. 1
and
3
, the probability distribution ofd,P(≥d ), is derived as
Besides Raabe’s actual morphometric data, Fig.4 demonstrates that the cumulative distribution of d both in Weibel’s (34) and Horsfield’s human model (11) complies withEq. 5 . Because Weibel usedEq. 4 withn = 3 for generations between 2 and 10, the slope on a loglog plot in this range is 3.0, with a correlation coefficient of 0.995. Although Horsfield did not use this relationship, the cumulative distribution ofd in his model also showed a good accordance with an inverse power function. The regression lines from the trachea to preterminal bronchioles showed a slope of 3.2 with a correlation coefficient of 0.981 in Weibel’s model, and a slope of 3.1 with a correlation coefficient of 0.988 in Horsfield’s model. In summary, these together provide evidence that the slopem of the distribution of dis statistically equivalent to the slopen obtained from Eq.3 , further supporting the applicability of the diameterflow relationship given by Eq.1 .
The Value of Diameter Exponent
What is the appropriate value for n? Arguments have been offered by several groups (29, 32, 35, 36) to explain the discrepancy between empirical values ofn and the theoretical value 3 for laminar flow. Considering the results of other studies (13, 30) and our analysis of measurement errors in this study (seeappendix ), we adopted a single value of n = 2.8 for modeling the mammalian airways. The case of HM272 in Raabe’s data, a 60yrold human man with unknown smoking history was described as “Tissue section showed emphysematous change, typical of aging.” The GM ofn in this case was slightly higher (3.0 vs. 2.8) than that of the other human, HM283, whose tissue section showed no apparent change. This slight increase ofn appears to be consistent with aging (31).
STOCHASTIC MODELS OF MAMMALIAN BRONCHIAL TREES
Selection of Model Parameters in Designing the Bronchial Tree
In the previous section, we established a method for determining thed of the branches where the Q is known. Combining this with the stochastic method of flow division introduced in stochastic branching process in a ductal system now allows us to design thed of the bronchial tree. There is one deterministic parameter, Qc, and one random variable,r, in our model. We will first discuss how Qc and the probabilitydensitydistribution function ofr should be assigned, and then we will present simulation trees for four mammals based on the actual morphometric data obtained by Raabe et al. (25).
The value of Qc and the distribution ofr should be assigned based on actual morphometric data. Qc can be approximately determined from the number of TB. If, for example, r = 0.5, the branching is symmetric and all TB have the same flow. Recall that the flow at the trunk was assumed to be unity. The total flow at the terminals is the sum of the individual flows at the TB, and hence the product of the number of terminals and the mean Q at TB is equal to 1. Accordingly, Qc is the reciprocal of the number of terminals. If nowr is a random variable with a given mean and SD, Qc will be the maximum flow at TB, and the number of terminal branches will be >1/Qc. Nevertheless, as a first approximation, this argument can still be used to determine the value of Qc. The expected value of r is related to the distributions of the generation numbers and Q of TB. As the expected value of r is closer to 0.5, these distributions become narrower. In the present work, we only used a uniform distribution of r. For example, when we assign Qc = 0.00006 andr is distributed uniformly between 0.2 and 0.5, the distribution of generation numbers of terminal branches is approximately normal, with a mean of 15.9 and SD of 2.0, as shown in Fig. 5. This result agrees well with those derived from morphometric data of human airways reported in the literature (10, 34).
We generated 10 trees with the same Qc and the same range ofr. There were no identical trees; however, the statistical features of these trees were identical. The mean ± SD of the number of terminal branches was 26,431 ± 19. The means and the SD of the generation numbers at the terminal branches were distributed between 15.8 and 16.3 and between 1.9 and 2.2, respectively. The mean Q at the terminal branches were identical, and the coefficient of variation (CV) of the Q ranged from 31 to 34%.
Comparison with Morphometric Data
In Raabe’s data, TBs were reported in one of the humans (HM272), one of the dogs (DM272), and in the rat and the hamster lungs. The terminated branches exactly recognized as TB were assigned a T, and other terminated branches were assigned an F. We only analyzed those which were exactly recognized, and the results are given in Table2. Because the measurement ofd in rat and hamster was completed down to TB as closely as possible, we estimated the total number of TB to be between (T + F) and (T + 1.5 × F), where T and F denote the number of branches marked with a T and an F, respectively. On the other hand, in the human and dog lungs, the total number of TB were not from Raabe’s data but from other reports (8, 11, 14, 35), because the population of TB measured in Raabe’s data was too small. In Table 2,d are normalized with thed of the trachea except for the hamster, where it was normalized with the largest airwayd available.
For each mammal, we assigned values for Qc and the ranges ofr (Table3) based on the number of TB and distribution of the generation numbers of TB, and subsequently we used them to create model trees of the four mammalian bronchial trees. The diameter distributions in these model trees showed good accordance with the morphometric data summarized in Table 2. Additionally, when we changed the value of the d exponent from 2.8 to 3.0 in humans, which corresponded to the case of HM272 in Table 2, the mean d of TB increased by 27%. This is consistent with aging actually observed in that case, as pointed out in designing the diameters of the branches.
DISCUSSION
To design the bronchial tree in this study, we introduced a stochastic flowdividing process instead of using deterministic branching patterns as proposed earlier (11, 12, 34). Although we used a uniform distribution of the flowdividing ratior, there are no direct data to support this assumption. Nevertheless, we can estimate the range ofr in Horsfield’s human and dog models (11, 12) as follows. Horsfield assigned identical Q to all terminal branches in his human model (11); therefore, the value ofr can be calculated at each bifurcation by using Eq. 1 . Although Horsfield did not use Q in his dog model (12), we can calculate the value of r in the same way as in his human model. The histogram of r for Horsfield’s human and dog models is shown in Fig.6. In the human model, the branching pattern between the lobular bronchi and TB is completely symmetric. However, there have been several reports demonstrating that the branching pattern within the secondary lobule is not symmetric (4, 26). When we exclude this part of his model, the mean value ofr was 0.36, almost equal to the mean value of r in our simulation, 0.35. In his dog model, r has a wider distribution than in his human model with a mean of 0.25. We need to point out that this value is exactly the same that we used in our simulation, although we determined rfrom Raabe’s data. The distributions ofr in Fig. 6 are not smooth, most likely due to the rigid branching pattern of the Horsfield model, and hence they appear to be less realistic than the uniform distribution we used.
It seems feasible that the distribution ofr also depends on the Q in the parent branch. We examined the correlation betweenr and the Q of the parent branch in both Horsfield’s models and found no systematic correlation except in the central airways. We also examined the influence of a nonuniform probabilitydensity distribution function ofr on the statistical features of the tree model. For example, we replaced the uniform distribution ofr with a normal distribution truncated at 0 and 0.5. We found that the distributions of generation numbers and Q at the terminal branches were almost completely determined by the expected value of r, rather than the type of the probability distribution. Therefore, we suggest that, for simplicity, the uniform distribution ofr is sufficient to generate large tree models. If a more realistic branching pattern is required in the proximal part of the tree, one can assign deterministic rules tor for the first several generations. This alteration, however, will not change the overall statistical features of the tree because of the small number of such proximal bifurcations.
Several studies have pointed out that the diameters in a branching structure are distributed according to a power law (15, 24). However, to our knowledge, no study has proposed how this distribution could be related to the distribution of Q. Horsfield et al. (13) also analyzed the Raabe data and obtained the value ofn in two different ways. One way was to calculate it based on his own ordering method, and the other way was by using Eq. 4 . One can show that his former method is approximately equal to the method of obtaining a distribution function of d. Values ofn he obtained with this method were similar to our results.
Although Eq. 1
predicts that the value of n is constant in a branching system, the calculated values of nusing Eq. 4
showed a wide distribution (see Fig. 2). We therefore examined the sensitivity ofP(≥d) to fluctuation inn. First, we generated a tree where Q of all branches were assigned. Then, starting from the trunk, we iteratively calculated the diameters at each bifurcation by applying the following equations
We also examined the distribution of airway lengths in Raabe’s data. However, there were no significant powerlaw distributions. Although we tried to extract relationships amongd, lengths, and angles, we did not find any significant correlation that would have allowed us to build deterministic relationships among these quantities. There are two type of angles in Raabe’s reports. The first is the angle of the branch relative to the direction of gravity, and the other is relative to the parent branch. These two angles are not sufficient to reconstruct the threedimensional structure of the trees from Raabe’s data. If the angles had been measured so as to determine the location in the threedimensional space, some significant correlation might have been detected. These are further problems to be investigated for modeling a threedimensional branching structure.
Our model lends itself to immediate investigation of the distribution of d of TB that can then be compared with those derived from morphometric data. Although the model is based on the relationship between d and Q (Eq. 1 ), in its current form, it may not be used to predict Q at TB, because the model does not incorporate the effect of local compliance which determines the regional Q (27). However, from the point of view of designing the bronchial tree, this does not seem to be a serious limitation, because the predictions of the model were in excellent agreement with morphometric data. To predict more precisely the distribution of Q at TB, one would have to know the distribution of the local compliances. There is evidence that parenchymal expansion is quite heterogeneous (27), indicating a wider distribution of local compliances. Presumably, the spatial arrangement of the peripheral airway tree and the spatial distribution of local compliances are closely related. Thus, to take local compliance into account, one would also have to know the relationship between local compliance and the terminal airway structure, which is beyond the scope of the present study. Nevertheless, we note that the model presented here may be directly applicable to designing vascular trees where tissue compliance is less of an issue.
Despite the simplicity of the model, our method of designing the bronchial tree offers two advantages over previous airway tree models. Because there are no assumptions of any kinds of unity at the terminals, we are able to investigate heterogeneity at the terminals. More importantly, our model also enables us a diameterbased analysis. Most methods investigating branching ductal structures have been based on various branching orders (1, 5, 16, 22). When we use a branching order, starting either from the top or the bottom of the tree, we have to transform this quantity into a physical quantity such as the d of the branch. The anatomic structure of the airway wall is, however, more correlated with the diameter than with the branching order, and the most effective regulation of local flow is achieved by changing the diameter of the airway. We may use our model in a statistical sense to relate diameter to various quantities in lung, such as flow or airway resistance, independently from branching order.
In summary, we have presented a stochastic design of the bronchial tree based on the d exponent law. We find that the design is robust against perturbations in its parameters and provides trees that are statistically equivalent to morphometric data. The design principle is simple and will enable us to predict distributions of various quantities related to lung function.
Acknowledgments
We thank Drs. O. G. Raabe and H. C. Yeh and also the Lovelace Foundation for providing us with data, which were obtained from research performed at the Inhalation Toxicology Research Institute supported by the National Institute of Environmental Health Science under an interagency agreement with the US Energy Research and Developmental administration (now the Department of Energy), Contract No. DEAC04–76EV01013.
Footnotes

Address for reprint requests: H. Kitaoka, Takaki Laboratory, Dept. of Mechanical Engineering, Tokyo Univ. of Agriculture and Technology, Konganei, Tokyo 184, Japan.

This research was supported by National Science Foundation Grant BES9503008.
 Copyright © 1997 the American Physiological Society
Appendix
Distribution of Flow in a Branching Tree
There is a simple relationship (which can be symmetric or asymmetric) between the total number (Nt
) of branches in the tree and the number of terminal branches (E) as follows
When the branching pattern is asymmetric, the number of bifurcations where one of the two daughters does not belong to the partial tree cannot be neglected. The total flow coming out of the terminals of the partial tree, Q′, is <1 because of the missing branches. The number of terminal branches is ∼Q′/Q_{a} which is also <N(≥Q_{a})/2. However, the total flow coming out of the partial tree is also smaller than unity by an amount Q" = 1 − Q′. The number of missing daughter branches can then be approximated as the ratio of Q" to the mean flow through the missing branches. The Q of the missing branches are ranged from 0 to Q_{a} with an average of ∼Q_{a}/2. Thus, the number of missing branches is estimated to be 2Q"/Q_{a}. If we now add the missing daughters to the partial tree, this new tree has a complete set of bifurcations, and soEq. EA1
would again be applicable. Accordingly, for this new tree, the total number of branches isN(≥Q_{a}) + 2Q"/Q_{a}. Therefore, Eq. EA1
can now be written as
Appendix
Influence of Measurement Errors on the Calculation of Diameter Exponent
When calculating the d exponent with the use of Eq. 4 , the influence of measurement errors ind is extremely important. In Raabe’s data, the precision of measurement was 0.1 mm. If it were 0.01 mm, there would be 1,000 possible combinations of the measured d of the three branches at a bifurcation. We examined how n is distributed corresponding to all possible combinations. In the following, we defined _{i}(I = 0, 1, or 2, with 0 denoting the parent, and 1 and 2 the daughters) as a measured value ofd andD _{i}as its true value. For example, let us examine the combination of measured values ofd _{0}= 0.8 mm,d _{1}= 0.7 mm, andd _{2}= 0.5 mm. The calculated value of n at this bifurcation is 2.61. IfD _{0}= 0.75 mm,D _{1}= 0.74 mm, andD _{2}= 0.54 mm, the n is 7.90. IfD _{0}= 0.84 mm,D _{1}= 0.65 mm, andD _{2}= 0.45 mm, the n is 1.83. We calculated n for all possible combinations, and obtained the distribution ofn as shown in Fig.7. The distribution ofn is approximately lognormal, having a higher mean value than the original value. When the diameters are larger, the distribution of n is narrower and the mean value of n is closer to the original value, because the relative measurement error is smaller for larger diameters. This is one reason for the significant correlation between the minimum diameter and the mean value ofn in the analysis of Raabe’s data. Another reason is as follows. WhenD _{1}and/orD _{2}is close toD _{0}, the value of n is high. However, such a bifurcation is often excluded when the relative measurement error is large (e.g., a combination ofD _{0}= 0.74 andD _{1}= 0.73, both of which would be measured at 0.7 mm when the precision is 0.1 mm). Therefore, at a smaller diameter, there are less bifurcations with higher values of n, resulting in smaller mean of n in the analyses of Raabe’s data.
Appendix
Correcting the Slope m for Smaller Trees
In practice, when theNt
of branches in a tree is not large enough to neglect the −1 inEq. EA1
, the estimated value ofm should be corrected for it. WhenNt
is smaller, the influence of −1 is bigger and the slope obtained from the loglog plot of flow distribution becomes slightly >1, as shown in Fig. 1. The slope of P(≥Q) on a loglog plot (a) can then be calculated as