Most physical and other natural systems are complex entities that are composed of a large number of interacting individual elements. It is a surprising fact that they often obey the so-called scaling laws that relate an observable quantity to a measure of the size of the system. Here, we describe the discovery of universal superlinear metabolic scaling laws in human cancers. This dependence underpins increasing tumour aggressiveness, owing to evolutionary dynamics, that leads to an explosive growth as the disease progresses. We validated this dynamic using longitudinal volumetric data of different histologies from large cohorts of patients with cancer. To explain our observations we tested complex, biologically inspired mathematical models that describe the key processes that govern tumour growth. Our models predict that the emergence of superlinear allometric scaling laws is an inherently three-dimensional phenomenon. Moreover, the scaling laws that we identified allowed us to define a set of metabolic metrics with prognostic value, which adds clinical utility to our findings.

大多数物理系统和其他自然系统都是复杂的实体，由大量相互作用的单个元素组成。令人惊讶的事实是，它们经常遵守所谓的缩放定律，该定律将可观察的量与系统大小的度量联系起来。在这里，我们描述了人类癌症中普遍的超线性代谢比例定律的发现。由于进化动力学，这种依赖性增强了肿瘤的侵略性，导致随着疾病的发展爆发性增长。我们使用来自大量癌症患者的不同组织学的纵向体积数据验证了这种动态变化。为了解释我们的观察结果，我们测试了受生物学启发的复杂数学模型，这些模型描述了控制肿瘤生长的关键过程。我们的模型预测超线性异形比例定律的出现是固有的三维现象。此外，我们确定的缩放定律使我们能够定义一组具有预后价值的代谢指标，这为我们的发现增加了临床实用性。