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Figure 11.11. (continued) (See the comment in parentheses in Figure 11.10.)

Parameter Estimates

Parameter estimates are obtained from data on mid to high grade astrocytomas (glioblastoma multiforme). High grade astrocytomas account for ~ 50% of all astrocytomas, which in turn make up ~ 50% of all primary brain tumours (Alvord and Shaw 1991).

We use the Alvord and Shaw (1991) estimate of one week to 12 months as the doubling times for gliomas. So, for cell proliferation of a high grade astrocytoma, we take a growth rate of p = 0.012/day which corresponds to a 60-day doubling time.

In the last section, we discussed the marked motility of glioma cells in vitro and in vivo in rats and used experimental data of Chicoine and Silbergeld (1995) and Silbergeld and Chicoine (1997) to estimate the parameters in the model.

For the human brain model, we are interested in the variation in parameter values in white and grey matter regions. Practically, the data describing the tumour growth and invasion for a patient are obtained from medical images such as CT scans and MRI (magnetic resonance imaging). As mentioned these scans approximately define the profile of the detectable portion of the tumour. While Figure 11.5 is the idealized form of a CT scan an actual scan is very much less clear. This study originally began with analysis

Figure 11.12. CT scans during the terminal year of a patient with anaplastic astrocytoma who was undergoing chemotherapy and radiation treatment. The image on left was taken approximately 180 days after the image on the right. (Figures from Tracqui et al. 1995)

Figure 11.12. CT scans during the terminal year of a patient with anaplastic astrocytoma who was undergoing chemotherapy and radiation treatment. The image on left was taken approximately 180 days after the image on the right. (Figures from Tracqui et al. 1995)

of serial CT scans during the terminal year of a patient with anaplastic astrocytoma. The patient underwent various chemotherapy and radiation treatments which we discuss in detail in Section 11.10; an example of one of these scans is shown in Figure 11.12.

As in the previous section, we use the Fisher-Kolmogoroff approximation to approximate D in terms of the linear velocity v and proliferation p. We then have Dw = vW,/4p and Dg = v2g/4p with the experimentally observed linear velocities vw and vg, respectively. We now have to determine vw and vg for a given patient. The most accessible information regarding tumour infiltration for a human patient is given by CT, MRI or other imaging scans. We use the CT scans used in the development of the original model (see Figure 11.12) to determine the velocities of the tumour front in grey and white matter. Within the right hemisphere the margin of detectable tumour moved about 1.5 cm in 180 days (Tracqui et al. 1995, Woodward et al. 1996), that is, an average speed of v = 8.0 x 10-3 cm/day. For the growth rate p = 0.012/day, the Fisher-Kolmogoroff approximation then gives the diffusion coefficient D = v2/4p = 1.3 x 10-3 cm2/day. Due to the proximity of the invasion to the deep cerebral nuclei (grey matter in the right hemisphere), we associate this value with grey matter diffusion so vg ~ 8.0 x 10-3 cm/day (Tracqui et al. 1995) and Dg « 1.3 x 10-3 cm2/day. From the CT scans, the speed of advance of the tumour margin across the corpus callosum (white matter) was two to three times as fast as that in (predominately) grey matter so we estimate vw > 2vg « 1.6 x 10-2 cm/day and Dw > 4Dg « 4.2 x 10-3 cm2/day. In our simulations, we generally assumed a five-fold difference in the diffusion coefficient in grey and white matter Dw = 5 Dg.

Table 11.5. Parameter estimates from patient data for a high-grade human glioma. (From Swanson 1999)

Parameter

Symbol

Range of Values

Linear velocity in grey matter

vg

8.0 x 10-3 cm/day

Linear velocity in white matter

Vw

> 1.6 x 10-2 cm/day

Diffusion coefficient in grey matter

Dg

1.3 x 10-3 cm2/day

Diffusion coefficient in white matter

Dw

> 4.2 x 10-3 cm2/day

Tumour doubling time

td

2 months

Net growth rate

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