Gompertz Curve

 

Also sprach Gompertz


How Gompertz invented Gompertz function? I tell you that by interpreting the original article by Gompertz.(p513)


Following data reveal a population reduction in certain town every ten years. (Table 1)



Table 1  Chronological population reduction from the data of the article by Gompertz.


At age of 10 years old, there were 6460 people. At age of 20 years old, the population decreased to 6090 people due to death of some people. At 30 years old, the population were 5642 people. When we transform the data to logarithmic value, the data took a form of geometric sequence (mp^n). (Fig. 1)



Fig. 1 A logarithmic translated data of the population reduction.

A logarithmic translated data of the population reduction revealed that subtraction of F(t+10) from F(t) took a form of geometric sequence (mp^n).


In Fig.1, subtraction of logarithmic population of age 50 from age 40 becomes mp^3. Therefore, it can be expressed as;

F(40)-F(50)=mp^3.

Subtraction of F(t+s) from F(t) becomes;





Fig. 2 The tangent slope of Gompertz function.


When “s”, as a time interval, is comparably small, slope of F(t) becomes;




We can substitute above values as;



(, because both are constants.)


Therefore;



When we integrate above formula;


When t = -(infinity),



(“ln” means logarithmic value.)

Therefore;



Slope of F(t) becomes;


 

Gompertz said that “if the average exhaustions of a man’s power to avoid death were such that at the end of equal infinitely small intervals of time, he lost equal portions of his remaining power to oppose destruction (page518)”.


It means that dF/dt (decline slope) is proportional to lnGmax-lnG(t). (lnGmax-lnG(t) is sum of fatalities.)


*In the original article of Gompertz, p=exp(ks), because p>1. When p<1, p=exp(-ks).

A growth curve has a p value less than 1. Therefore, exp(-ks) is opted in a logarithmic translated growth curve.


*The most popular Gompertz formula is;


We can substitute the values as follows;

y=G(t), a=Gmax, b=exp(-A0/k), c=exp(k), x=t.


Originally, Gompertz function was used for a population reduction curve.

On 1926, 100 years after Gompertz published the original paper, Sewall discovered that Gompertz curve could be used as a growth curve. (Wright, Sewall, book review in Jour. Am. Stat. Assoc., 21, 494, 1926.)

In a growth curve, exp(kt) is substituted to exp(-kt).

Following figure reveals (1) Gompertz’s original curve for population reduction, (2) a growth curve, (3) HCV reduction curve by anti-viral therapy, respectively. (Fig. 3)

Fig. 3 Gompertzian original curve and derivatives.


The graph of


becomes a sigmoid curve. (Fig. 4)


Fig. 4 Sigmoidal Gompertz curve.

*“Gompertz Curve Calculator”, the iPhone application, calculates a non-logarithmic sigmoid curve with 3 point times and values.


When vertical value is transformed to logarithmic value, the graph is not sigmoid.

(Fig. 5)


Fig.5 Logarithmic translated Gompertz curve.


When a doubling time of cancer cells is always constant, the slope of the logarithmic translated data is also constant and the line becomes straight (Skipper’s model).

Whereas, Gompertz curve reveals a decrease of the slope with time (exp(-kt) times). It means an elongation of the doubling time in Gompertz curve.

The smaller the cancer, the shorter the doubling time. The larger the cancer, the longer the doubling time.

Because cancer growth obeys Gompertz model.

 

Keiji Matsui M.D., Ph.D., Kanagawa, Japan