Physically-Based Simulation of Plant Leaf Growth

Computer Animation and Virtual Worlds, vol 15, pages 237--244, 2004.
(Co-authors: Iris R. Wang, Gladimir Baranoski)

A mathematical model is proposed to simulate the growth of a plant leaf. The tissue in the leaf is regarded as a viscous, incompressible fluid whose 2D expansion comes from the non-zero specific growth rate in area. The resulting system of equations are composed of the modified Navier-Stokes equations. A level set method is used to capture the expanding leaf front. Numerical simulations indicate qualitatively that different portions of the leaf expand at different rates, which is consistent with the biological observations in the growth of a plant leaf. Numerical results for the case of the Xanthium leaf growth are also presented.

Validation against experimental data

The spatial and temporal growth of a plant leaf is not uniform in general, and hence a realistic growth model needs to be able to capture these effects. The following tables show the simulated results are in good agreement with the experimental results.

(Note: the "life" of a leaf is measured by the leaf plastochron index (LPI), defined as:

LPI = (log L_n - log 10) / (log L_n - log L_{n+1})

where n denotes the nth oldest leaf of a Xanthium plant and L_n is the length of leaf n (measured in mm).)

LPI Experimental data Simulation result
0.74 0.34 0.61
2.64 6.60 6.20
4.18 10.7 10.3
The absolute growth rates of a Xanthium leaf when LPI=0.74, 2.64, and 4.18.

The absolute growth rates of a Xanthium leaf when LPI=0.74, 2.64, and 4.18.
LPI	Experimental data	Simulation result
0.74	0.34	0.61
2.64	6.60	6.20
4.18	10.7	10.3

LPI Experimental data Simulation result
2.64 1.353 1.353
4.18 1.146 1.164
The length to width ratios of a Xanthium leaf growing from LPI=2.64 to 4.18.

The length to width ratios of a Xanthium leaf growing from LPI=2.64 to 4.18.
LPI	Experimental data	Simulation result
2.64	1.353	1.353
4.18	1.146	1.164

Simulation sequence

Frames of a simulation sequence illustrating the growth of golden pathos, also known as the "devil's ivy". Notice the faster growth rate in width than that in length.

leafgrow.mpeg (size=8Mb)
Animation simulating the growing of a golden pothos leaf for 3 days.