Hmm, no that isn't helpful, in fact it is confusing and somewhat contradictory.
Convention for liquids is an approximation to an exponential model, derived from an emptying rate proportional to the contents, i.e.
dv/dt = -kv
Where v is volume, t is time and k is a constant. This is a simple ODE which is trivially solved analytically
v(t) = v
0 exp(-kt)
Where v
0 is the initial conditions (volume at time t=0). Hence the term exponential model. For a given fixed change in time, ΔT, we get the expression
v(t+ΔT) = v
0 exp(-k(t+ΔT)) = v
0 exp(-kt) exp(-kΔT)
If we look at the ratio of these equations
v(t+ΔT)/v(t) = v
0 exp(-kt) exp(-kΔT) / (v
0 exp(-kt))
You get a whole bunch of cancelling leaving
v(t+ΔT)/v(t) = exp(-kΔT)
If we set the volume ratio to be a fixed value, say 1/2, we can solve for ΔT=T
1/2 and find independence from v
0:
0.5 = exp(-kT
1/2)
Now that is fine for an exponential model. But the authors of the paper above argue for a linear model. A linear model takes the form
dm/dt = -k
where m is mass of solid food, t is time and k is a constant. Solving the ODE:
m(t) = m
0 - kt
Where m
0 are initial conditions. Now expanding this by a fixed chunk of time ΔT
m(t+ΔT) = m
0 - k(t+ΔT)
Computing the ratio to allow T
1/2 to be assessed
m(t+ΔT)/m(t) = [m
0 - k(t+ΔT)] / [m
0 - kt]
Well, that's a mess. Substituting 0.5 for m(t+ΔT)/m(t) and ΔT = T
1/2 we get
0.5 [m
0 - kt] = m
0 - k(t+T
1/2)
0.5m
0 - 0.5kt = m
0 - kt - kT
1/2
0.5m
0 - 0.5kt - m
0 + kt = - kT
1/2
-kT
1/2 = -0.5m
0 + 0.5kt
T
1/2 = 0.5m
0/k - 0.5t
This shows that a linear model is, by rigorous mathematical definition, dependent on the initial mass of solid food. So quite why a paper would firstly endorse a linear model, then insist on independence from the initial mass of the meal, is a bit confusing. It is mathematically inconsistent.
Now it could be that the assumptions made by these authors are different to the assumptions I've made above - but if they are, then the authors are using the technical terminology in a very different way to - well, just about every other scientific field in the world. Which would be surprising. It is also possible I've made an error above. In which case I would invite corrections.
The thing that makes me more confident that my analysis above is correct is that pretty much every forensic text I've seen cautions that gastric emptying of solids is a function of the size of the initial meal. e.g. here, on Moore's analysis of gastric emptying for people eating meals until they felt full:
http://books.google.co.uk/books?id=XyG3802xSdwC&pg=PA37&lpg=PA37
With apologies if the following has typos (I hand copied it):
Other than the fact that the authors should really know grams are a unit of mass, not weight, we find that full meals have a far longer emptying time than the small meals used in scientific research, due to the dependency on initial conditions for the linear model that I outline above. Furthermore, even amongst healthy subjects, long lag times after a full meal can occur. Checking subject 5 from Moore et al 1981 referenced in the above text book, we find they had a T
1/2 of 638 minutes after their full meal - that is over 10 hours