Exponential growth

In 1934 Mottram [4, 5] reported work on the rat tar wart. Tar warts are tar-carcinogen induced neoplasms of the skin starting 75-100 days after the continuous painting of the rat's neck with tar. Histologically, some warts appear benign while others are clearly malignant.

Using the tar wart tumor growth model, Mottram was the first to describe tumor expansion as exponential. Exponentially growing tumors graphically produce straight lines by plotting linear time on the x-axis versus the log (at any base) of either tumor area, tumor cell number, tumor volume or tumor diameter on the y-axis see Figure 2.

Semilog growth plol of » Collins

Ihree bronchogenic lumors reporled by- • Meyer

I no evidence ol Gompertzian slowing) # Schwai

Semilog growth plol of » Collins

Ihree bronchogenic lumors reporled by- • Meyer

I no evidence ol Gompertzian slowing) # Schwai

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Days After First Measurement

0 100 200 300 400 500 600 700 800 900 1000 1100 1200

Days After First Measurement

Figure 2. Friberg, Collins, Spratt, Steel, Schwartz affirmed that in the clinic, an exponential growth pattern adequately described tumor growth for most patients. A semi-log plot of tumor diameter vs. time illustrates the linear relationship characteristic of exponential growth.

Twenty years later Laird [6, 7] reported on the growth of transplanted tumors in the rat. Under her specific laboratory conditions, most tumor growth could be described in terms of the Gompertzian model. Her experiments lead her to accept that for her laboratory model; most transplantable, rapidly growing tumors could be described in Gompertzian terms.

Studies of tumor growth in clinic patients have been described in terms of both exponential and Gompertzian models. Nevertheless, several investigators reported data that was inconsistent with the Gompertzian model for the majority of their patients. These authors engaged routine imaging of both metastatic and primary pulmonary lesions in an attempt to resolve whether exponential growth could be confirmed in the clinic. Friberg, Collins, Spratt, Steel, Schwartz [8, 9, 10, 11, 12,13] affirmed that in the clinic, an exponential growth pattern adequately described tumor growth for most patients Figures 2, 3.

Time (Years)

Growth rales tor 12 cases with primary cancer ol the breasi. Abscissa: lime. Orinate: number ol (umor cells (logarithmic scare). Each curve represents one case. Tbe sow orange line indicaies the period ol observaiion. Blue lines mark ihe mvta&to phase Modrfied Irom Von Fournier et a) Fournier D v. Weber E. Hoeflkno W. el al.. Growth rated 147 mammary carcinomas. Cancer 1980,45:2198±2207.

Figure 3. Von Fournier et al. confirm a straight-line (by semi-log plot) relationship for patients with breast tumors supporting the model of exponential growth model.

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