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Cancerous tumours, or neoplasms, originate from the mutation of one or more cells which usually undergo rapid uncontrolled growth thereby impairing the functioning of normal tissue. There are many different cancers each with their own characteristics. In this chapter we shall only be concerned with brain tumours, and in particular gliomas or glioblastomas, which make up about half of all primary brain tumours diagnosed; they are particularly nasty tumours with a depressingly dismal prognosis for recovery. Gliomas are highly invasive and infiltrate the surrounding tissue. The impressive increased detection capabilities (but as we shall see, still woefully inadequate) in computerized tomography (CT) and magnetic resonance imaging (MRI) over the past 20 years have resulted in earlier detection of glioma tumours. Despite this progress, the benefits of early treatment have been minimal (see, for example, Silbergeld et al. 1991, Alvord and Shaw 1991, Kelley and Hunt 1994, Cook et al. 1995 and Burgess et al. 1997). For example, even with extensive surgical excision well beyond the grossly visible tumour boundary, regeneration near the edge of resection ultimately results in eventually leading to death (see, for example, Matsukado et al. 1961, Kreth et al. 1993, Woodward et al. 1996 and other references there). This failure of resection is analogous to trying to put out a forest fire from behind the advancing front. The action of the fire (tumour growth) is primarily at the periphery.

The brain basically consists of two types of tissue: grey matter and white matter. Grey matter is composed of neuronal and glial cell bodies that control brain activity while the cortex (like the 'bark') is a coat of grey matter that covers the brain. White matter fibre tracts are myelinated neuron axon bundles located throughout the inner regions of the brain. These fibres establish pathways between grey matter regions. The corpus callosum is a thick band of white matter fibres connecting the left and right cerebral hemispheres of the brain. Within each hemisphere, there are several white matter pathways connecting the cortex to the nuclei deep within the brain; see Figure 11.2. Figure 11.3 shows two photographs of a human brain showing grey and white matter distribution and the corpus collosum.

Gliomas are neoplasms of glial cells (neural cells capable of division) that usually occur in the upper cerebral hemisphere but which can be found throughout the brain (Alvord and Shaw 1991). Astrocytomas, originating from an abnormally multiplying astrocyte glial cell, are the most common gliomas. Depending on their aggressiveness (grade), astrocytomas are further divided into several subcategories. Astrocytomas are the least aggressive or lowest grade, anaplastic astrocytomas are the more aggressive or mid-grade and glioblastomas (multiforme) are the most aggressive or highest grade. Tumour grade indicates the level of malignancy and is based on the degree of anaplasia (or deformity in behaviour and form) seen in the cancerous cells under a microscope. Gliomas often contain several different grade cells with the highest or most malignant grade of cells determining the grade, even if most of the tumour is lower grade. There is still no general clinical agreement on the grading.

Generally, the higher-grade cancer cells are more capable of invading normal tissue and so are more malignant. However, even with their invasive abilities, gliomas very rarely metastasize outside the brain.

The prognosis for patients with neoplasms affecting the nervous system depends on many factors. A major element in the prognosis is the quantitative evaluation of the spatiotemporal infiltration of the tumour, taking into account the anatomic site of the tumour as well as the effectiveness of the various treatments.

Since we believe that the modelling developed in this chapter has a practical bearing on patient treatment it is necessary to give more detailed medical information which we believe is an important part of realistic medical modelling.

Difficulties in Treating Brain Tumours

An enormous amount of experimental and some theoretical work has been devoted to trying to understand why gliomas are so difficult to treat. Unlike many other tumours, gliomas can be highly diffuse. Experiments indicate that within 7 days of tumour implantation in a rat brain, glioma cells can be identified throughout the central nervous system (Silbergeld and Chicoine 1997). A locally dense tumour growth remains where the cancerous tissue was initially implanted but there are solitary tumour cells throughout the central nervous system (Silbergeld and Chicoine 1997 and Silbergeld, personal communication, 1998). Most glioma treatments are directed locally to the bulk mass when, in fact, the action of tumour growth and invasion is elsewhere.

There are various, regularly used, treatments for gliomas, mainly chemotherapy, radiation therapy and surgical intervention. Resection, the surgical removal of an accessible tumour, has a wretched history of success. Recurrence of tumour growth at the

Figure 11.2. Two cross-sections of a human brain showing fibrous white matter and the corpus callosum which connects the left and right cerebral hemispheres. (Figure courtesy of Dr. E.C. Alvord, Jr.)

Figure 11.2. Two cross-sections of a human brain showing fibrous white matter and the corpus callosum which connects the left and right cerebral hemispheres. (Figure courtesy of Dr. E.C. Alvord, Jr.)

resection boundary is a well-documented phenomenon in glioma research (see, for example, Silbergeld and Chicoine 1997, Woodward et al. 1996, Kreth et al. 1993, Kelley and Hunt 1994 and other references there). Both experimentalists and theoreticians believe that the distantly invaded cells are responsible for tumour regeneration following surgery (Chicoine and Silbergeld 1995, Silbergeld and Chicoine 1997). Since the density of cancerous cells (remaining after resection) is highest at the resection boundary,

Figure 11.3. Cross-section of the human brain. The cortex consists of grey matter and is connected to other grey matter regions by white matter fibre bundles. The corpus callosum is a white matter tract connecting the left and right cerebral hemispheres. (Figure reproduced with the permission of Professor Paul Pietsch; derived from T.H. Williams, N. Gluhbegovic and J.Y. Jew Virtual Hospital Figure 5-30. http://www.vh.org/Providers/Textbooks/BrainAnatomy/BrainAnatomy.html)

Figure 11.3. Cross-section of the human brain. The cortex consists of grey matter and is connected to other grey matter regions by white matter fibre bundles. The corpus callosum is a white matter tract connecting the left and right cerebral hemispheres. (Figure reproduced with the permission of Professor Paul Pietsch; derived from T.H. Williams, N. Gluhbegovic and J.Y. Jew Virtual Hospital Figure 5-30. http://www.vh.org/Providers/Textbooks/BrainAnatomy/BrainAnatomy.html)

regrowth seems most probable at this location. Alternatively, Silbergeld and Chicoine (1997) suggested, and are presently testing, the hypothesis that damaged brain tissue at the resection site releases cytokines that recruit the diffusely invaded tumour cells. Nevertheless, both explanations are consistent with the argument that the diffuse nature of gliomas is fundamentally responsible for tumour recurrence near the resection boundary. The difference is that the former is a physical model and the later is more biochemical. In this chapter we shall study a model (basically an incredibly simple oneā€”even linear) for resection therapy and show why it generally fails. As we shall show, it increases life expectancy minimally; we compare the results and predictions with patient data.

Chemotherapy essentially uses specialized chemicals to poison the tumour cells. The brain is naturally defended from these and other types of chemicals by the intricate capillary structure of the blood-brain barrier. Water-soluble drugs, ions and proteins cannot permeate the blood-brain barrier but lipid-soluble agents can. Recently, agents have been devised to temporarily disrupt the blood-brain barrier. Many chemothera-peutic treatments are cell-cycle-dependent: the drugs are triggered by certain phases of the cell cycle. Silbergeld and Chicoine (1997) have observed that the motile cells distant from the bulk tumour do not appear to enter mitosis so cell-cycle specific drugs and standard radiation therapy have limited effectiveness. Not only that, gliomas are often heterogeneous tumours. Those drugs that do reach the cancerous cells are hindered by drug resistance commonly associated with cancer cell heterogeneity. While one cell type is responsive to treatment and dies off, other types are waiting to dominate. This phenomenon requires a model which includes cell mutation to drug resistance cells, in other words a polyclonal model. Below we describe and analyze such a model for chemotherapy and again compare the results with patient data.

The biological complexity of gliomas makes treatment a difficult undertaking. For planning effective (or seemingly so) treatment strategies, information regarding the growth rates and invasion characteristics of tumours is crucial. The use of mathematical modelling can help to quantify the effects of resection, chemotherapy and radiation (it tries to kill the tumour cells with radiation) on the growth and diffusion of malignant gliomas. In this chapter we shed some light on certain aspects of brain tumour treatment with the aim of helping to determine better, or even optimal, therapeutic regimes for patients. A major goal is the development of interactive computer models with which the effects of various treatment strategies for specific tumours could be examined. We believe that the work described in this chapter goes some way in achieving this goal. Having said this, however, all of the treatments mentioned above have a very poor record of success. There is a pressing need for a totally different approach to the treatment of gliomas, several of which are currently being investigated.

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