During the 50 years since the concept of urban metabolism was proposed, this field of research has evolved slowly. On the basis of an analogy with an organism's metabolism, the concept of urban metabolism has become an effective method to evaluate the flows of energy and materials within an urban system, thereby providing insights into the system's sustainability and the severity of urban problems such as excessive social, community, and household metabolism at scales ranging from global to local. Researchers have improved this approach, evolving from models of linear to cyclic processes and then to network models. Researchers account for flows of energy and materials, ecological footprints, inputs and outputs, and the characteristics of the system's ecological network. However, the practical methods of analysis need to be improved. Future analysis should focus on establishing a multilevel, unified, and standardized system of categories to support the creation of consistent inventory databases; it should also seek to improve the methods used in the analysis to provide standards and guidance that will help governments to achieve sustainable development. Finally, researchers must improve the ability to provide spatially explicit analyses that facilitate the task of applying research results to guide practical decision-support.

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According to Thompson's principle of similarity, the area of an object should be proportional to its length squared. However, leaf area-length data of some plants have been demonstrated not to follow the principle of similarity. We explore the reasons why the leaf area-length allometry deviates from the principle of similarity and also examine whether there is a general model describing the relationship among leaf area, width and length. More than 11,800 leaves from the six classes of woody and herbaceous plants were sampled to check the leaf area-length allometry. Six mathematical models were compared based on root-mean-square error as the measure of goodness-of-fit. The best supported model described a proportional relationship between leaf area and the product of leaf width and length (i.e., the Montgomery model). We found that the extent to which the leaf area-length allometry deviates from the principle of similarity depends upon the variation of the ratio of leaf width to length. Estimates of the parameter of the Montgomery model ranged between 1/2 and π/4 for the six classes of plants. This is a narrower range than imposed by the limits 1/2 (for a triangular leaf with leaf length as its height and leaf width as its base) to π/4 (for an elliptical leaf with leaf length as its major axis and leaf width as its minor axis). The narrow range in practice implies an evolutionary stability for the leaf area of large-leaved plants despite the fact that leaf shapes of these plants are rather different.

The nondestructive measurement of leaf area is important for expediting data acquisition in the field. The Montgomery equation (ME) assumes that leaf area (A) is a proportional function of the product of leaf length (L) and width (W), i.e., A = cLW, where c is called the Montgomery parameter. The ME has been successfully applied to calculate the surface area of many broad-leaved species with simple leaf shapes. However, whether this equation is valid for more complex leaf shapes has not been verified. METHODS: Leaf A, L, and W were measured directly for each of 5601 leaves of 15 vine species, and ME and three other models were used to fit the data. All four models were compared based on their root mean square errors (RMSEs) to determine whether ME provided the best fit. RESULTS: The ME was a reliable method for estimating the A of all 15 species. In addition, the numerical values of 13 of the 15 values of c fell within a previously predicted numerical range (i.e., between 1/2 and π/4). The data show that the numerical values of c are largely affected by the value of W/L, the concavity of the leaf base, and the number of lobes on the lamina. CONCLUSIONS: The Montgomery parameter can reflect the influence of leaf shape on leafarea calculations and can serve as an important tool for nondestructive measurements of leaf area for many broad-leaved species and for the investigation of leaf morphology.