Growth curve analysis of asymptomatic and symptomatic meningiomas

Abstract

Meningiomas sometimes appear to stop growing after attaining a large size. Commonly used exponential growth models do not reflect this phenomenon. We attempted to find the best curve to simulate their growth. Fifty-two patients with meningioma were followed up for 3.1–21.7 years (mean 7.5 years) with four or more imaging studies each. Thirty-one patients had asymptomatic tumors. The other 21 patients with residual or recurrent tumor were followed up after surgery. Time–volume curves for each tumor were plotted. Nonlinear regression analyses were performed against power, exponential, logistic, and Gompertzian curves. Time–volume curves corresponded to the Gompertzian and logistic growth curves better than to power or exponential curves. When simulating time–volume curves with Gompertzian curves, the majority of benign meningiomas began to slow their growth before patient age of 80 years. Twenty-three of 31 asymptomatic meningiomas had already passed the inflection point before diagnosis. In contrast, this happened less frequently in symptomatic tumors. Especially, all six atypical meningiomas continued to grow quasi-exponentially. Sigmoid curves that approach a plateau were better descriptors of the growth of benign meningiomas than were curves of unlimited growth. However, atypical meningiomas were unlikely to slow their growth.

Appendix

Growth curve equations used in this article are shown below, where α, β, and γ are constants, and V(t) is the volume at time t.

$$ {\text{Exponential curve}}: \, V(t) = \alpha \beta^{t} + \gamma $$

Power curve [28] : $V(t)\, = \,\left( {\alpha t\left( {1 - \beta } \right) + \gamma } \right)^{1/1 - \beta }$

The power curves show rapid increase of volume when 0 < β < 1, linear growth when β = 0, and gradual deceleration when β < 0.

$$ {\text{Logistic curve : }}V(t)\, = \;\gamma /\left( {1 + \alpha \exp \left( { - \beta t} \right)} \right) $$

$$ {\text{Gompertzian curve}}: \, V(t)\, = \,\alpha \exp \left( { - \beta \exp \left( { - \gamma t} \right)} \right) $$

$$ {\text{Inflection point}}: \, Ti\, = \,\left( {\ln \left( {\alpha /\beta } \right)} \right)/\beta $$

Ti was described as days after the initial observation time. A negative Ti value indicates that the inflection point was before the initial imaging study. However, if Ti is beyond the observation period, the tumor was still growing quasi-exponentially.