Hyperbolastic Models
Hyperbolastic
Growth Models
Hyperbolastic
growth models are a family of flexible growth models that can predict variety
of growth behaviors for continuous outcomes in many fields of research, for
instance cancer growth, stem cell growth, cranofacial growth and development
and infectious disease outbreak.
We
have developed three models and we call them H1 (generalizes logistic growth
model), H2 (stand alone) and H3 (generalizes Weibull growth model). They are
called “hyperbolastic” because the outcome is a function of inverse hyperbolic
sine function (arcsinh).
H1 is the three parameter
model of the form
where
.
H2 is the three parameter
model of the form
where
.
H3 model is the four
parameter model of the form
where
.
The
H1 function has one more parameter than the classic logistic and Gompertz
functions, but it is more flexible and can fit asymmetric growth patterns as
well as increasing and decreasing growth. The H2 function has the same number
of parameters as H1 and can fit asymmetric curves, but it cannot fit decreasing
growth patterns, so it is less flexible. The H3 function has the same
flexibility as the H1 function at the expense of one more parameter, similar to
the Weibull and Richards equations. Some of the flexibility of the H1, H2 and
H3 functions is illustrated in the figure below (parameter M is held constant
at 100 while other parameters are varied).
a. H1
,
,
,
,
,,
b. H2
,,
,
, ,,
c. H3
,,
,
,
,, 
,,,
These
functions can be easily implemented in SPSS or SAS PROC NLIN (see example SAS
code used to fit Deisboeck et al. cancer volume data) or other readily
available software packages. Using a flexible and highly accurate predictive
model such as hyperbolastic can significantly improve the outcome of a study
and it is the accuracy of a model that determines its utility.
References
Tabatabai
M, Williams DK, Bursac Z. (2005). Hyperbolastic
Growth Models: Theory and Application. Theoretical
Biology and Medical Modeling, 2(14):1-13.
[BioMed Central] [PubMed
Central]
Bursac
Z, Tabatabai M, Williams DK. (2006). Non-linear
Hyperbolastic Growth Models and Applications in Cranofacial and Stem Cell
Growth. 2005 Proceedings of the American Statistical
Association, Biometrics Section [CD-ROM],
Deisboeck TS, Berens ME, Kansal AR et al. (2001). Pattern of
self-organization in tumour systems: complex growth dynamics in a novel brain
tumour spheroid model. Cell Prolif,
34:115-134.
Hypertabastic
Survival Model
We
introduce a new two-parameter continuous probability distribution called hypertabastic
probability distribution. The hypertabastic hazard function can assume a
different variety of shapes. It can be used to analyze biomedical data such as
cancer recurrence time. Based on the hypertabastic distribution, we introduce
the hypertabastic survival model which includes the hypertabastic proportional
hazards model with parametric baseline hazard function, the hypertabastic
accelerated failure model and the hypertabastic proportional odds model.
(Hypertabastic
Distribution) We say a continuous random variable T has a hypertabastic distribution if its cumulative distribution
function is
for
0 for
The
parameters α and β are both positive and 
and 
are hyperbolic secant and hyperbolic cotangent respectively.
We often read as “T is
hypertabastically distributed with parameters α and β “and
write it as H (α, β). The probability density function of T is given by
for
0 for
where
.
(The
Hypertabastic Survival Function) Let T
be a continuous random variable representing the waiting time until the
occurrence of an event. Then the hypertabastic survival function is defined as
where
P (T>t) is the probability that
waiting time exceeds t.
(The
Hypertabastic Hazard Function) The hypertabastic hazard function 
which represents the
instantaneous failure rate at time t
given survival up to time t is defined
as
.
The
cumulative hazard function 
is defined as
.
Then,
the hypertabastic proportional hazards log-likelihood function can be written
as
where
0 if 
is a right censored observation
1 otherwise.
Some
statistical software packages use logarithm of survival time as their survival
time variable in their model fittings. If this is the case, then the following
alternative formula can be used as the proportional hazards log-likelihood
function:
.
For
the right censored data, the hypertabastic accelerated failure time model would
have a log-likelihood function of the form
where
.
Again,
if someone prefers to use the logarithm of time as the model survival time variable,
then the alternative accelerated failure time log-likelihood function is
.
h(t) t
a. 
b. 
h(t) t
c. 
d. 
h(t) t
e. 
f. 
Figure 1. a) Hypertabastic
hazard curve for
; b) Hypertabastic hazard curve for 0
; c) Hypertabastic hazard curve for
; d) Hypertabastic hazard curve for
; e) Hypertabastic hazard curve for
; f) Hypertabastic hazard curve for
.
References
Tabatabai
MA, Bursac Z, Williams DK, Singh KP.
(2007). Hypertabastic Survival Model.
Theoretical Biology and Medical Modeling,
4(40):1-13.
Bursac Z, Tabatabai M, Williams
DK, Singh K. (2007). Hyperbolastic
Models for Survival Data. 2006 Proceedings of the American
Statistical Association, Biometrics Section [CD-ROM],