**Mathematical Models Vs. Animal Models**

Irwin D. Bross, Ph.D.

**1. Who is an expert on mathematical models?**

To most people, including most biomedical researchers, mathematical models
are an esoteric, mysterious scientific instrument. In part, this is because
of the mystique that shrouds many mathematical techniques. In part, it is because
the concept and role of a scientific model is poorly understood. However, there
is nothing arcane or mysterious about mathematical models. On careful examination,
they turn out to be common sense reduced to calculation, as is discussed in
some detail in *Scientific Strategies to Save Your Life* (Bross, 1981).

Since biomedical scientists are often allergic to algebra, the mathematics
in models gives them a convenient excuse for not taking the trouble to try
to understand this methodology. Although mathematicians lack this excuse, I
have found that they tend to have an equally poor understanding of the mathematical
models in biomedical research. Mathematicians are likely to be bored with the
models because the mathematics is often so elementary! They are also hampered
by their poor understanding of what science is all about. Some have the mistaken
notion that pure mathematics is science.

For practical purposes, therefore, neither biomedical scientists nor mathematicians
have much advantage over non-technical readers when it comes to understanding
the contrast between mathematical models and animal model systems as scientific
instruments for biomedical research. If anything, the biomedical scientists
and mathematicians are seriously hampered in understanding these matters because
of their preconceptions and prejudices.

Such biases are reinforced by the ego-gratification they provide - for instance
the misconceptions of mathematicians that they are doing science. However,
as the more enlightened philosophers of science have long recognized (Reichenbach,
1949), the logical manipulations in mathematics are entirely different from
the methodological processes of modern empirical sciences. Another problem
is that there are few experts in mathematical modeling, because mathematical
modeling in biomedical research has largely evolved in the second half of the
20th century.

**2. What is a model?**

Before dealing with particular kinds of models, let us start with the question:
What is a model? What is the role of models in science, engineering, and other
human affairs? A model is a simplified representation or replica of the real
thing. The model can itself be a real thing, for example when a mouse in an
experiment plays the role of "little man." However, a model can also
be abstract or conceptual, as when ordinary or technical languages are used
to represent something in the real world. In ordinary language, the simplest
kind of abstract model is called an "analogy."

In logic, argument by analogy has a poor reputation. In more technical languages,
it is customary to refer to an "analogue," but the basic idea is
the same. In my first book (Bross, 1953), Chapter 10 was archly subtitled "Secrets
of a Scientist's Model," and it is still a good introduction to the role
of models in science. It uses model aircraft as an example of physical models.
The model aircraft a boy might make has only a superficial resemblance to the
actual aircraft - in shape or in function or in both. The model aircraft that
engineers test in wind tunnels has more accurate dimensions to scale so that
it can provide useful predictions of the aerodynamic performance of the full
size aircraft. The predictions are a form of argument by analogy and suggest
that, if suitable precautions are taken, it may be possible to have a valid
argument by analogy. Both animal model systems and mathematical models employ
argument by analogy, and both require great care if the argument is to be scientifically
valid.

If valid arguments are feasible, scientific studies of models may offer substantial
advantages over corresponding studies of the real thing, for example human
beings. Models are often simpler to control and manipulate. Ethical questions
may be less troublesome. The sacrifice of animals is taken for granted by animal
researchers whereas the sacrifice of human beings might not be acceptable to
them. In general, conceptual models are not constrained by such ethical problems.
However, ethical problems have only been postponed since they will arise if
the findings from conceptual models are applied to human beings. While models
can be useful in terms of ease and economy in answering scientific questions,
they do not reduce the need for careful and thoughtful consideration of the
risks and benefits in a given application. All decisions or actions in the
real world eventually involve such considerations.

**3. What is a mathematical model?**

A mathematical model is a conceptual model that uses mathematical languages
rather than ordinary languages to represent a particular scientific context.
The scientific context itself would ordinarily be one that exists in the real
world and the model is necessarily a simplified description of the actual context.
Ordinarily a model represents what are believed to be a few crucial features
of this context and achieves simplification by omitting other aspects which
are less important or are irrelevant. The danger, of course, is that something
that is crucial may be left out.

As an example of simplification, consider the physical models of aircraft used
in engineering. The dimensions of the wings would be accurately reproduced
to scale but the color of the wings might not be reproduced. Color is usually
considered irrelevant to the "lift" and other aerodynamic characteristics
that are being predicted. Even here, though, it is possible to conceive of
special circumstances where the color might be relevant (e.g. by affecting
wing temperatures).

To try to dissipate some of the mystique of mathematical models, the strengths
and the weaknesses of this methodology will now be briefly considered. One
obvious strength of mathematical models is that they use mathematical languages
to describe a given scientific context in the real world. This permits a high
degree of precision in the statement of descriptive hypotheses or assumptions.
It also facilitates the logical manipulation of the statements. Finally, it
often provides quantitative (i.e., numerical) conclusions that may not be obtained
if natural languages are used to describe the scientific context.

On the other hand, this strength can also be a weakness. It is just as easy
to make erroneous assumptions in a mathematical language as in ordinary English.
There is nothing magical about mathematical languages: If you can't describe
the biomedical context correctly in plain English, you can't describe it correctly
in a mathematical language either. However in algebraic form the error may
be less apparent.

**4. The nature of mathematical languages**

From linguistics research, it has become clear that logical languages and mathematics
have evolved from ordinary natural languages (Bross, 1975). Galileo, the founder
of modern science, used a language half way between Latin and algebra! It should
not be surprising, then, that the underlying structure of natural and mathematical
languages is quite similar.

One strength of mathematical models is that they permit precise description
of a scientific context when there is precise scientific knowledge of the primary
and pertinent factors in that context. The weakness here is that biomedical
knowledge tends to be incomplete and imprecise. Hence, mathematical languages
may give an illusion of precision when it really is not there.

One key point that is simultaneously a strength and a weakness of manipulations
in mathematical languages is this: Mathematical (and logical) languages are
informationless. In other words, they do not add any scientific information
to the characterization of events in the real world and, ideally, they do not
lose information either. One danger is that statements in a mathematical language
may suggest a false precision (e.g. in carrying more decimal places than are
warranted). This may, in turn, mislead both those who write in mathematical
languages and those who read these languages into thinking that the information
in a given scientific context is more precise than it really is. Sometimes
this false precision may be hidden by a smokescreen - the ubiquitous mathematical
mystique.

Readers who would like a more comprehensive discussion of the role of mathematical
models in biomedical research than can be given in this brief overview can
find such a discussion in Chapter 10 of my more recent textbook (Bross, 1981).
This text works through several specific examples of mathematical models, including
a model which explains how myeloid leukemia in children and adults can be caused
by diagnostic medical X-rays. (Bross, 1981)

**5. How can a mathematical model by useful?**

How can something that is a mathematical abstraction be useful in dealing with
biomedical contexts in the real world? This is not a simple question. Mathematicians,
scientists, and philosophers have used many pages of text trying to explain
how abstract entities can be connected to events in the real world, Insofar
as there is a simple answer to the question it is this: The power of a mathematical
model is not something that is inherent in the model; it comes from the fact
that it is possible to set up a correspondence between language and reality.

In particular, it is possible to coordinate abstract entities in the model
(e.g. what is nowadays called "parameters") with their counterparts
in the real world (e.g., estimates of the parameters based on scientific data
from actual clinical studies). The success or failure of mathematical models
as a scientific tool depends to a large extent on the degree of coordination
that is achieved.

To understand where there are reasonable prospects for the successful application
of mathematical models, it is important to realize that such models do not
create something out of nothing. If there is little knowledge or scientific
data on a subject, there is little chance that a mathematical model will be
of value. Probably the main strength of mathematical models is that they can
apply the most pertinent information from clinical data to assist policy decisions
that affect humans.

Consequently, mathematical models illuminate a medical context by, in effect,
forcing the user to articulate what is known and what is not known. They can
be a powerful exploratory tool because the model can evolve as biomedical knowledge
evolves or more information becomes available. In other words, a mathematical
model can be sufficiently flexible to conform to the shape of the reports from
the real world. This may require complicating the original model or even changing
the original assumptions that were used in framing the model. Thus, a mathematical
model is a scientific theory that can evolve to fit the facts.

**6. The early successes of animal research**

Let us now turn briefly to animal model systems. These are physical models.
The rationale for an animal model system in biomedical research is that the
pattern of events in the model will mimic the corresponding pattern in humans
or human diseases. The U.S. National Institutes of Health has spent huge sums
of money on research that is aimed at finding particular animal model systems
that will imitate specific human diseases.

The enormous prestige of animal model systems among doctors is due to their
early successes with the infectious diseases. One of the greatest animal researchers
was Dr. Paul Ehrlich. His most famous success was the discovery of an effective
chemotherapy for syphilis on the 606th experiment in a series. (Ehrlich, 1960)
However, when Dr. Ehrlich tried to repeat his triumph in experiments on therapy
for cancer, he failed. He was discouraged that his approach to infectious diseases
was not nearly as effective against animal tumors (Rhoads, 1954-55). Nevertheless,
20th century scientists have studied animal tumors extensively, even though
animal experiments have contributed very little to the treatment of cancer
(Reines, 1987).

Why was animal research so successful with the infectious diseases in the 19th
century and such a failure with the chronic diseases in the 20th century? The
answer is that the infectious diseases are caused by an external factor - by
invasion of bacteria and other microorganisms. In contrast, nearly all of the
chronic diseases, including atherosclerotic vessel disease, appear to be caused
by an internal factor - by genetic damage to human cells, also called "mutagenesis." (Ames,
1982)

In order to do medical research on infectious diseases it was necessary to
grow the microorganism that caused the disease. While scientists could grow
some organisms in a glass dish (*in vitro*), others, such as the spirochete
that causes syphilis, could not be grown in this way. Thus, for research on
infectious diseases, animals served as a form of living tissue culture. With
animal reservoirs of microorganisms, researchers could test the efficacy of
chemotherapeutic agents. This is what Dr. Ehrlich was doing when, after 605
failures, he finally found an agent that was effective against spirochetes.

It is important to note that animals were *not* being used as a model
system. Dr. Ehrlich's animals played the same role that tissue culture played
in the development of a polio virus - both were merely a practical way to grow
and to test the infectious agent. What happened in the tissue culture was not
a model of human polio. It was simply a test system for the efficacy of vaccines.

In contrast, genetically defective human cells cause mutagenic diseases. Here
an animal system is useless unless it can provide a realistic model of the
human disease.

In what follows, the focus will be on the chronic diseases that cause the most
morbidity and mortality in the United States. Mathematical models will be compared
to animal models in their ability to provide a reliable guide to the etiology
or treatment of a given human disease.

**7. Development of mathematical models for biomedical research**

The reputation of animal model systems largely reflected the research on infectious
diseases in the late 19th century and early 20th century. Although in principle
the development of mathematical models for medical research might be traced
back to Harvey's proof of the circulation of blood, in practice the use of
mathematical models in biomedical research largely developed in the second
half of the 20th century.

There are three general approaches to mathematical modeling. They are distinguished
by the extent to which the model attempts to describe the underlying factors
that generate the observed clinical data. The first mathematical models did
not attempt to describe these underlying factors. They only tried to describe
the patterns in the observations themselves. Such models can be called "surface" models.
Most of the mathematical models that are currently used are surface models.

Some mathematical models describe a few, but not all, of the key underlying
factors in a scientific context. These can be called "intermediate" models.
Most of the current efforts to go beyond the surface models fall into this
category.

Starting in the 1960's, a few mathematical models were developed in an attempt
to describe how a complex set of observed clinical data was generated by the
underlying factors in the biomedical context. These underlying factors cannot
be directly observed. They can only be inferred from the observed data by using
the mathematical structure of the model. This kind of mathematical model can
be called a "deep" model.

As with any new technology, there have been both successes and failures with
all three types of mathematical models. However, all three kinds of mathematical
models have proved useful in biomedical research. Sometimes the different kinds
of models work together. For example, a successful surface model may lead to
better understanding of the underlying factors, and this may enable construction
of deeper models. Again, deeper models are often integrated with standard statistical
models that are surface models.

The three kinds of mathematical models tend to have somewhat different areas
of application. Thus, when the only purpose of a statistical analysis of a
clinical study is to characterize the observations succinctly and precisely,
a surface model can do the job. Since deeper modeling tends to be a slow and
difficult task, there is no point in doing this job if a standard surface model
can serve the purposes of a clinical investigator. The three kinds of mathematical
models also have some things in common: All three kinds of models may provide
efficient extraction, handling, and summarization of the clinical information.

**8. Surface models**

Surface models are used most commonly. Many such models are available for dealing
with survival data, dosage-response curves, multivariate data, and other standard
biomedical contexts. For examples most of the usual risk/benefit assessments
for drugs employ surface models - whether the data is from clinical or animal
model studies.

The very brief discussion of surface models here reflects the fact that they
are extensively discussed in the standard statistical literature, and interested
readers can consult this literature. The techniques for parameter estimation
that enable surface models to fit the observed facts and other technical aspects
of mathematical modeling tend to apply (with some modifications and extensions)
to all three kinds of models. These are also extensively discussed in the statistical
literature. Since the literature covers intermediate or deep models much less
adequately, the focus here will be on such models.

**9. An intermediate mathematical model**

Intermediate models are much less common, but they do have a fairly wide usage.
One recent example of an intermediate model is a dosage-response model for
the radiological, chemical, or biological agents that cause genetic damage
(mutagens).

This model was originally developed to account for the striking new findings
from a high-tech in vitro system that uses a somatic cell hybrid with a single
human chromosome (Waldren et al., 1986). This important new technology for
low-dose mutagenic risk assessment is far more sensitive and far more accurate
than the usual in vitro systems or the animal model systems that are widely
used by U.S. government agencies. Dr. Charles Waldren and his co-workers have
used their new technique to demonstrate that systems currently used by the
federal agencies may underestimate the actual mutagenic risks by factors of
200 or more (Waldren et al., 1986).

These findings from the somatic cell hybrid system have been recently confirmed
by epidemiological findings. Official data on the survivorship of servicemen
exposed to fallout at nuclear weapons tests show that the risk of cancer from
low levels of ionizing radiation are at least 200 times greater than the official
estimates (Bross and Bross, 1987).

Why is there this enormous error in what is supposed to be a scientific risk
assessment procedure? The main reason is that one or more invalid statistical
procedures have almost always been used to get the official estimates. One
such procedure is called "linear extrapolation from high dose data to
low dose risk estimates." The procedure is based on a surface model that
assumes that the "dosage response curve" is a straight line (for
example, Gross et al., 1959). By using an intermediate model, it is easy to
show that this assumption is wrong for mutagenic risks. The model also shows
that the extrapolated estimates of low-dose risks from high-dose data are going
to be gross underestimates (Bross, 1987a), again by factors of 200 or more.

**10. Common sense reduced to calculation**

It is worthwhile to consider briefly the nature of this intermediate mathematical
model for several reasons. In this instance, it is possible to give the same
general argument in both mathematical language and in ordinary language. Hence
it can be seen that the model is simply common sense reduced to calculation.

In either language, the model provides a better understanding of the nature
of mutagenic diseases. Most physicians and most scientists have a poor understanding
of the mutagenic diseases and how they differ from the more familiar infectious
diseases. This is a major reason for the misconceptions about animal model
systems that are prevalent in the medical and scientific communities.

At issue here is the shape of what is called a "dosage response curve." This
curve is simply a graphical plot in which the horizontal axis shows the dosage
and the vertical axis shows the risk - for example the risk of cancer. Linear
extrapolation assumes that this dosage response curve is a straight line over
the entire dosage range from low doses to high doses. For radiation, the range
of the extrapolation can be from 500 millirem to 500 rem - a factor of 1000.

Linear extrapolation may sometimes be adequate for some of the older, 19th
century poisons, but it fails completely for the newer, 20th century mutagens.
The simple intermediate mathematical model of the dosage response curve for
a single cell provides a formal proof that the dosage response curve for mutagens
starts out as a straight line but then bends off, reaches a maximum, and then
goes down. There is no way that this complicated curve can be adequately approximated
over the range of the extrapolation by a single straight line. A mathematical
proof is given in Chapter 7 of my latest book (Bross, 1988).

By using common sense it is easy to see why the dosage response curve for mutagens
must have a maximum. The key point is that it takes two steps to produce a
cancer (or other mutagenic disease). The first step is the physical production
of genetic damage in a cell. The second step is the biological reproduction
of this genetic damage by cloning. (Greene and Hamaoka, 1987) If the cell has
so much genetic damage that it cannot reproduce, it cannot cause mutagenic
disease. Hence, at high enough dosages the risk must go to zero, and the dosage
response curve cannot possibly go steadily upward as a straight line.

In the first step that is required to cause a clinical cancer, the mutagen,
for example low-level ionizing radiation, must produce genetic damage (i.e.
lesions in the biochemical structure of the DNA of cells). By the laws of physics,
the number of lesions produced by a given radiation dose is simply proportional
to the dosage. This is why risks in the dosage response curve starts out by
increasing linearly.

The second step is the biological cloning of the damage. A single cell can
have very little effect on the economy of the human body. As will be shown
later, a genetically damaged cell must double and redouble more than 30 times
to produce a clinically detectable breast cancer (Bross et al., 1968). For
a cell to reproduce, it must be able to function well. Common sense says that
as the genetic damage in a cell increases the chemical instructions in the
DNA will become increasingly garbled and the viability of the cell is going
to decrease. Thus, the physical and biological forces are countervailing.

**11. Mathematical models vs. animal models, part I: Mutagenic
risks**

There is nothing mysterious about using the intermediate model to prove that
the dosage response curve for mutagens has a maximum. Common sense says that,
if there are countervailing forces, the increase in genetic damage will be
offset by the decrease in cell viability after a certain point. When this happens,
the mutagenic risk will reach a maximum. In other words, the dosage response
curve will stop going up and start going down.

A mathematical model can give us additional useful information, for instance
where, on average, the maximum occurs. As we might guess, dosage of the mutagen
at the maximum produces slightly more than one chemical lesion, i.e., one instance
of genetic damage in the DNA of the cell. It is much easier to prove this rigorously
with a mathematical model. It is also easy to show that if the dosage response
curve has a maximum, then using a straight line from high dose animal or human
data to determine low dose risks is going to produce a gross underestimate
of the low-level risks.

Linear extrapolation is used in almost all official estimates of mutagenic
risks used by the Environmental Protection Agency, the Nuclear Regulatory Commission,
and the Department of Energy for setting "safe" levels of human exposure
to mutagens. (Committee on Pathologic Effects of Atomic Radiation, 1961; Subcommittee
on Hematologic Effects, 1961) When the actual risks are underestimated by a
factor of 200, this means that the federal agencies will permit workers in
nuclear installations and chemical plants, Americans living near chemical or
radiological dumpsites, and patients getting mutagenic medical modalities to
be exposed to dangerously high doses. Many of them will eventually suffer or
die from cancer and other mutagenic diseases because of the excessive exposures
permitted by the government agencies that are supposed to be protecting the
public.

This is the first of three examples in which mathematical models and animal
model systems yield contradictory results. In all three cases, the results
from animal model systems have caused extensive human suffering and death.
In this first case, the facts about low-level mutagenic risks that are given
here are well-known to the government agencies that use the erroneous risk
estimates. Why, then, do these agencies continue to permit these deadly exposures
to radiological and chemical mutagens and to issue false assurances of safety
repeatedly?

Basically, the answer is that it is politically expedient to do so. The scientifically
invalid risk estimates protect mutagenic nuclear, chemical, and biological
technologies, which U.S. Executive Branch agencies are promoting for military
or commercial reasons. Protecting the public health has low priority relative
to political priorities. For the full story, one may read Crimes of Official
Science: A Casebook (Bross, 1988). It suffices to note here that animal research
on mutagenic risks is scientific fraud. They fund fraudulent research generously
because it gives them what they want - gross underestimates of the actual risks
that can be presented to the mass media and to the public "in the name
of science."

**12. The first collaborative clinical trial for breast cancer**

Note that the intermediate model did not attempt to describe the entire, complex
process that leads to a human cancer. Rather, it focused on genetic damage
in a single cell, which is the start of the process. A much deeper model would
bring human host defense systems into the picture. While one deep model for
a host defense system will be discussed, the overall complexities of human
host defense systems are not easily characterized mathematically.

The first successful deep model in biomedical research described the growth
and spread of human breast cancer. This model was developed by the staff of
my Biostatistics Department at Roswell Park Memorial Institute for Cancer Research
(RPMI) in Buffalo, New York in the 1960's. The mathematical development of
this model was largely the work of Dr. Leslie Blumenson.

The data on breast cancer came from hospitals scattered across the country.
At this time, the Director of Roswell Park was Dr. George E. Moore, a truly
remarkable physician and scientist. He had been instrumental in organizing
the first collaborative clinical trial of breast cancer in the United States.
When I became Director of Biostatistics at RPMI in 1959, the responsibility
for the statistical operations of this collaborative study became an important
part of my job.

At first, the Biostatistics Department analyzed the breast cancer data with
standard surface models in the usual way. Almost from the start, however, the
study had some unusual features. To almost everyone's surprise, the initial
study demonstrated that the new modality under test (a mild chemotherapy) had
produced the first major advance in the treatment of breast cancer in more
than 50 years! (Zubrod, 1979)

This unexpected finding posed some challenging puzzles that the surface models
could not solve. One such puzzle that the deep model eventually solved was
the question: Why did surgery succeed in some of the women and fail in the
others?

**13. A deep model for the growth and spread of breast cancer**

It was largely in an attempt to answer this question that a deep model for
the growth and spread of breast cancer was developed. To underlying assumption
of this model was that there were two distinct kinds of breast cancer. Both
kinds looked very much the same to the surgeon and to the pathologist. However,
one kind of breast cancer was a highly malignant form of cancer. The second
kind was barely malignant. It even seems possible that the second type was
a benign tumor that merely looked like a malignant one.

To test this hypothesis, a two-disease mathematical model of breast cancer
was developed. Although the two diseases could not be distinguished clinically,
it was simple enough to set up a model that had a set of parameters which characterized
each of the two types. The first step was to try to fit the observed data using
just one set of parameters - with a one-disease model.

Only when this theory totally failed to represent the actual observations was
a two-disease model tested. In mathematical modeling as in most scientific
theory, what is called "Occam's Razor" or the "Principle of
Simplicity" is used: Never multiply entities without necessity. In other
words, prefer the simplest theory that fits the facts.

The two-disease model fit the facts beautifully. Hence, the mathematical model
could be used to infer that there were two different kinds of breast cancer.
The fitted parameters for the two types showed the enormous difference in malignancy.
Another parameter showed that, roughly speaking, the two types occurred with
approximately equal frequency.

Why did surgery succeed in curing many women with breast cancer but fail with
many others? The model gave a simple answer: The surgery tended to fail for
the very malignant type of breast cancer and to succeed in patients with the
less malignant type.

In practice, the malignancy of a primary breast cancer is determined by pathologists
who look at tissue samples under the microscope. Only relatively gross abnormalities
can be seen under a microscope, so appearances can be deceiving for mutagenic
diseases. However there was one observation of the pathologists that did seem
to distinguish between the two types of cancer:

The presence or absence of cancer cells in the lymphatic nodes. The
first collaborative study used axillary lymph node involvement as a guideline,
and this has been done in most of the later research studies on breast
cancer. This study found that the adjuvant chemotherapy under study,
at a mild dose, seemed to benefit a few patients who had very early cases
of the malignant type (Noer, 1961; Fisher et al., 1968). It was cautiously
given in a low dose that produced little toxicity in the patients. Unfortunately,
as will be seen, this small success has led to the excessive use of highly
toxic modalities in breast cancer and other forms of cancer.

The mathematical details of the deep model and the clinical studies used to
estimate the parameters of the model are given in a series of publications
(Bross et aL, 1968, Blumenson and Bross, 1968, Blumenson and Bross, 1969, Bross
and Blumenson, 1971). Dr. Leslie Blumenson and I collaborated for many years
on deep models. He was usually responsible for the mathematics while I was
usually responsible for the science.

**14. Solving another mystery with a deep mathematical model**

I was so confident of our "two-disease" model for breast cancer that
I predicted in print that a new clinical trial would be a failure (Bross, 1972).
I also predicted that a later study using the drug 5-FU would also be a dismal
failure. There have now been some additional studies, all of which have produced
the results that could be easily predicted from the original two-disease model
(Slack et al, 1969; Medical World News, 1970).

The acid test of a deep model is the same as that of any other scientific theory:
Can it predict things to come? Since all of the predictions from this deep
model have been right on the mark, it has repeatedly passed the acid test over
a 20-year period.

While the full details of this deep model are too complicated to discuss here,
one minor finding of the breast cancer model may give some feel for its power.
Initially, we had no idea what extent of cloning of the original cancer cell
was required for the detection of a solid breast cancer. Solving this mystery
once again shows how mathematical models are simply common sense reduced to
calculation.

The number of cells in a minimal detectable tumor can be determined, in essence,
by simply dividing the minimal detectable tumor volume by the cell volume.
However, there are some tricky mathematical questions about "close packing" of
cells that Dr. Blumenson worked out. Fortunately, this merely introduces a
multiplicative factor. Writing the number of cells in the minimal detectable
tumor as a power of 2 tells us the number of "doubling times" required
to clone this many cells. The doubling times for each kind of tumor are parameters
of the model and can be estimated from the data. Simple arithmetic tells us
that more than 30 doubling-times are required for a genetically damaged cell
to clone into a clinically detectable tumor. If the doubling time of a tumor
is six months, it would take 15 years for the tumor to be clinically detectable.

Finally, an historical footnote on computers and deep models is of interest.
Although in the late 1960's we could use the fairly large computers called "mainframes" for
our deep models, getting a program to operate properly was a challenge even
for a brilliant mathematician like Dr. Blumenson. In contrast, my current desktop
computer, an IBM-PC clone, is many times more powerful (and far easier to use)
than the RPMI mainframes of those days. Moreover, there now is standard software
on my hard disk that makes the matrix computations in mathematical modeling
relatively easy. Indeed, the new PC's are so awesome that it may not pay to
do elegant mathematical manipulations of the formulas. Instead of looking for
explicit solutions to equations, the present software can simply compute the
numerical solutions by brute force.

**15. Mathematical models vs. animal models, part II: Breast cancer**

The confrontation between animal model systems and the deep mathematical model
for breast cancer was a direct one. A triumvirate consisting of the new chairman
of the collaborative breast study, a top National Cancer Institute (NCI) bureaucrat,
and an ambitious statistician were determined to take the grant away from RPMI
and to do a new clinical trial of 5-FU as adjuvant chemotherapy for breast
cancer. All three had long association with animal model systems and they "justified" their
actions on the basis of new animal data on 5-FU (Schnitzer and Grunberg, 1958).

In this case, the animal data was being used to directly oppose clinical data
- a scientific absurdity. As the new chairman well knew, we had originally
included 5-FU in the protocols of the previous clinical trial but had been
forced to drop it (with the concurrence of the chairman) because of its excessive
toxicity. Nevertheless, the triumvirate based its grant application on invalid
animal model systems, which contradicted clinical data and the predictions
of the deep mathematical model.

On the basis of clinical facts (and our mathematical model), I strongly opposed
the triumvirate's proposal at what turned into a very stormy meeting. However,
I lost the battle, and the proposed study was carried out. As I had predicted,
it was a total failure. Typically, however, this failure did not deter the
NCI from further use of animal model systems to select drugs for clinical trials.

In my experience, 5-FU is probably the worst chemotherapeutic agent in wide
use. However, the American Cancer Society (ACS) owned 25% of the 5-FU patent
(Moss, 1982), and this may explain why many grant requests included use of
the drug. In the above example, animal model systems were very useful to the
grant winners. The chairman achieved his ambition to take over the collaborative
study, the statistician went on to bigger and better things at Harvard, and
the bureaucrat consolidated his power.

The real losers were the women with breast cancer. A research line that had
started off with a real promise for improving the treatment of breast cancer
was aborted, and an opportunity for control of this disease was lost. Recently,
NCI's own figures show that the age-adjusted breast cancer death rates, which
had been stable for half a century, have suddenly increased (Kolata, 1988).
This human death and suffering is part of the price that the public pays for
the scientific fraud in animal model systems.

**16. Heroic chemotherapy: is toxicity necessary?**

The final example of a deep model represents another effort to understand puzzling
features in the scientific context of a collaborative clinical study. This
study involved what is sometimes called "heroic chemotherapy," the
use of very high doses of toxic chemotherapeutic agents in an attempt to cure
advanced cancer. The name heroic refers to the alleged "courage" of
a physician who was willing to carry his patients to the brink of death in
an effort to cure cancer. Typically, a high dosage of the drug under test would
be used until the white blood count (wbc) was greatly depressed and became
so low that the patient had almost no defense against infection.

A key doctrine of the National Cancer Institute (NCI), which favored heroic
chemotherapy then and now, was that "toxicity is necessary," In other
words, the dangerously low wbc was necessary to insure that the patient got "sufficient
drug for a valid test." A series of studies from the Biostatistics Department
examined the question: Is toxicity really necessary? (Bross et al., 1966a,b,c)
and found that the answer was "no." Chemotherapeutic agents such
as 5-FU were far more effective against the patient's white blood cell system
and other host defense systems than against the cancer, so the high doses were
counterproductive. When our studies using surface models showed this, the NCI
bureaucrat mentioned in the previous section was furious, and NCI did not renew
our contract for the analysis of these studies.

Frankly, I was relieved when they did this, because I had become disenchanted
with heroic chemotherapy. It frequently had side effects that caused severe
suffering, and occasionally it killed the patients. However, in many years
of experience I have seen no evidence that it has cured a single patient with
disseminated solid cancer. I doubt that it is as effective as faith healing.

**17. A deep model for the white blood cell system**

There were a number of very puzzling questions about the relationship between
the dosage of the drug and the white blood count that the surface model had
not answered. For example, one wbc reading would show severe depression but
the next would be far above normal. Often, the subsequent test revealed a normal
level, and clinicians assumed that the unexpected elevated level was a laboratory
error. However, the frequency of these high wbc counts suggested that something
else was going on.

To try to get a better understanding of what was going on in heroic chemotherapy
(and hopefully to find a rational way to use the wbc to control dosage), I
asked Dr. Blumenson to develop a deep model. This time he did the science.
Somehow or other he managed a translation from the medical jargon in the biomedical
literature into mathematical language. He had to describe the extremely complicated
stages of development from a stem cell (which has the potential to become any
one of several types of blood cells) into, for example, a white blood cell.

In essence, the host defense systems are genetically controlled and, while
their biochemical processes are not fully understood, the production is controlled
by feedback loops. Hence, when infection occurs, the invading bacteria trigger
a chemical message to the stem cells, "Make more white cells." When
the infection subsides, the chemical message is, "Go back to normal operations." However,
if there is genetic damage to the cells in the bone marrow, the wrong message
may be sent and therefore the feedback system may not operate properly. Further
details are given in the literature (Blumenson, 1975, Bross and Blumenson,
1979).

This model suggested why heroic chemotherapy had been so counterproductive.
When the white blood count is driven down to near-fatal levels, the feedback
system can go into wild oscillations, even after the drug is stopped. One interesting
by-product of this deep model was a mathematical model of leukemia. A great
excess of white blood cells could be induced in the model by modifying the
feedback parameters to simulate the effects of genetic damage. This model therefore
provides a simple explanation of why genetic damage from mutagens causes myeloid
leukemia in humans.

**18. Mathematical models vs. animal models, part III: Heroic
Chemotherapy**

The deep mathematical model of the human blood system was in a direct confrontation
with results from animal model systems. The main rationale for heroic chemotherapy
(and other heroic cancer therapy) was based on animal models (Skipper et al.,
1965; Schabel, 1977; Hoff et al., 1977; Frei et al., 1980), because there was
little or no valid clinical data to support the dogmas that NCI and ACS promulgate
in the mass media (Cetane et al, 1978).

For example, Dr. Marvin Schniederman, an NCI statistician, showed the paradoxical
result that those patients in whom the drug doses produced an "objective
response" (a detectable reduction in cancer size) actually had a shorter
survival that those patients in whom there was no reduction (1966). An explanation
of this paradox is that the dosages which affected the size of the cancer had
even more effect on the host defense systems that were keeping the patients
alive. We confirmed this theory at RPMI when we addressed the question: Is
toxicity really necessary? The cancer patients who showed toxicity (and supposedly
got "enough" of the drug) did worse than those who did not show toxicity
(and supposedly had not received a sufficient dose).

In summary, the animal studies gave false conclusions that have caused (and
are still causing) unnecessary suffering and even death for thousands of human
cancer patients. Again, despite the many NCI and ACS claims in the mass media,
there is little evidence that any clinically effective chemotherapeutic agent
for human cancer was first found in an animal screening system (Refries, 1987).
On the other hand, some drugs that are now used in the chemotherapy of humans
were rejected by NCI, because they were not effective against mouse leukemia
in the L1210 system that NCI used as the "gold standard" (Bross,
1988). Since experienced clinicians are well aware that the animal model systems
are unreliable for mutagenic diseases such as cancer, it is ironic that the "justification" for
using poisonous drugs that have caused so much unnecessary suffering and death
for human cancer patients has generally come from animal research "in
the name of science."

As an aside, it should be noted that many of the dogmas that were firmly believed
at the NCI cannot be readily documented. They were part of an oral, rather
than written, tradition, as is often the case in medicine. Consequently, it
is difficult to document that cancer research has been dominated by quasi-official
doctrines that had little or no scientific basis. The conviction that "toxicity
is necessary" was like an "article of faith." Similarly, there
was little scientific support for the NCI requirement that drugs could not
be used in patients unless they passed the L-1210 mouse leukemia screen. Because
such policies were difficult to defend on scientific grounds, they were rarely
discussed in print, even though they guided therapeutic and research strategies.
This paper describes widespread attitudes among cancer researchers, which I
recognized during years of work within the cancer establishment. Such beliefs
were expressed in scores of discussions, and they dominated important conferences
in which research strategies were determined. However, the oral discussions
and debates, in which these attitudes were manifested, were not part of the
written record. Thus, written documented proof does not exist for many of the
points of this paper, and I can only provide my personal impressions of the
beliefs of my colleagues.

**19. Why animal models fail for the mutagenic diseases**

Why was animal research so successful in the chemotherapy of the infectious
diseases and such a failure in the chemotherapy of cancer? In infectious diseases,
animals provide a way to grow the microorganism and thereby to test the effects
of chemicals on the microorganism. There is no need for a model of the infectious
disease process in humans. However, cancer is caused by a human cell that differs
from a normal cell only in a chemical lesion in its DNA. A chemotherapeutic
agent affects both the cancer cells and the normal cells and it interacts with
the complex human host defense systems. Hence, for an animal model to be useful
for scientific prediction of the risks and benefits of cancer chemotherapies,
it must be a true model of the human disease.

While there is much to be learned about the genetic control of human body chemistry,
it is obvious that no animal model system can be a true model. No other species,
not even the anthropoid apes, have the same set of genetic instructions, and
these determine the internal chemical environment. Because no other species
has the same internal chemistry as human beings (or even a very similar chemistry),
no other species can be a true model.

Note that all three mathematical models (and their scientific contexts) involve
evaluation of risks or benefits in mutagenic diseases. The deep mathematical
models dealt with risks and benefits of drugs or medical modalities. The intermediate
model dealt with determination of mutagenic risks for low doses of mutagens.
These particular applications happen to constitute the main uses of animal
model systems in biomedical research in general and in cancer research in particular.

In all three examples, the animal model systems were either of little or no
scientific value, or else they were used in ways that were outright fraudulent.
This is typical of the animal research that is funded by U.S. government agencies.
Hence it follows that if the funding of scientific fraud by Executive Branch
health and science agencies were stopped, animal researchers in the United
States would become an endangered species.

**20. Summary and conclusions**

In this overview of mathematical models as an alternative to animal model systems,
the strengths and weaknesses of mathematical models as a biomedical research
instrument have been briefly assessed. There is nothing mysterious or magical
about mathematical models, and, like any other instrument, they have limitations.
Basically, these are the same limitations of natural (or artificial) languages
and of scientific knowledge that apply to any research instrument.

Three kinds of mathematical models can be distinguished on the basis of the
depth of the model - surface, intermediate, and deep. All take advantage of
a strong point of mathematical languages: When properly used, they guarantee
the consistency of the model. While mathematicians have sometimes mistaken
consistency for truth, all that this actually guarantees is freedom from self-contradiction.

The truth of a mathematical model, like the truth of any other scientific theory,
is determined by the prime directive of the empirical sciences: A theory must
fit the facts. One practical advantage of a mathematical model is that "fit" can
be rigorously tested for any actual data. If the fit is good, there is a reasonable
expectation that the model can give accurate predictions

Three specific examples were given of deeper mathematical models that have
been successful. The first example was an intermediate model for the dosage
response curve for mutagens. The second was a deep model of the growth and
spread of breast cancer. Finally there was a deep model of the white blood
cell system.

In all three cases, there was a direct confrontation between mathematical models
and animal models systems, because they gave diametrically opposite results.
In all three cases, the mathematical model gave correct answers while the animal
model gave dangerously wrong answers. Not only were the animal results wrong,
they were used for purposes of scientific fraud and deceit. Moreover, this
fraud has resulted in suffering and death for many thousands of humans. Thus,
animal model systems have not only injured animals, they have been just as
harmful to human beings.

Nevertheless, it would be naive to suppose that mathematical models or any
other alternatives to animal research will be voluntarily phased in (or animal
research phased out) merely because of their scientific advantages. As long
as the business of animal research is so highly profitable to universities
and research institutions, the administrators will insist that animal research
continue, As long as government agencies can take advantage of the main "virtue" of
animal model systems - that by choosing the "right" system, it is
possible to "prove" almost anything that is desired - they will continue
to fund animal research that supports official policies "in the name of
science." As long as the National Research Council's Committee on the
Use of Laboratory Animals in Biomedical and Behavioral Research exists to defend
the vested interests of the animal researchers, they will continue to insist
that the "alternatives will not eliminate the need for animals in the
foreseeable future..." (Holden, 1988).

Hence, in any practical sense, the only way that mathematical models or other
alternatives to animals will replace animal model systems is to stop government
agencies from using public money to fund animal research. There is a legislative
way to stop this funding, because Congress has become increasingly concerned
about scientific fraud in biomedical research. The way to stop such fraud is
to make it a federal crime to use public money to support scientific fraud.
Such a bill does not even have to mention animal research, and it would have
the support of various public interest groups. If this new law stopped government
agencies from funding fraud, it would eliminate most of the current animal
research in the United States.

Eradication of animal research and the other forms of scientific fraud that
are now widely prevalent would also prevent the unnecessary human suffering
and death caused by the failure of Executive Branch agencies to do their job
of protecting the public health and safety. If the billions of dollars each
year that are now wasted on scientific fraud were spent on genuine research,
substantial progress toward control of cancer and other mutagenic diseases
could probably be achieved in the next 20 years.

**References**

Ames, BN (1982) Carcinogens and Anti-Carcinogens, In *Mutagens in Our Environment* (Sorsa
M, Vainio H, eds.) Alan R. Liss, New York.

Blumensan L (1975) "A Comprehensive Modeling Procedure for the Human Granulopoietic
Stystem: Detailed Description and Application to Cancer Chemotherapy", *Mathematical
Biosciences*, 26, 217-239.

Blumenson L and Bross I (1968) "Statistical Testing of a Deep Mathematical
Model for Human Breast Cancer", *Journal of Chronic Diseases*, 21:
493-506.

Blumenson L and Bross I (1969) "A Mathematical Analysis of the Growth
and Spread of Breast Cancer", *Biometrics*, 25(1) 95-109.

Blumenson L and Bross I (1979) "Assessment of Myelotoxic Effects of Chemotherapy
from Early Leukapenic Response: Application of a Mathematical Model for Granulopoiesis", *Journal
of Surgical Oncology*, 11: 171-176.

Bross (1953) *Design for Decision*. The Macmillan Company, New York.

Bross I (1972) "Scientific Strategies in Human Affairs: Use of Deep Mathematical
Models". Transactions of the New York Academy of Sciences, 34(3): 187-199.

Bross I (1975) *Scientific Strategies in Human Affairs: To Tell the Truth*,
Exposition Press, Jericho, New York.

Bross (1981), *Scientific Strategies to Save Your Life*, Marcel Dekker,
New York.

Bross I (1987a) "Why the Dosage Response Curve far Mutagens Has a Maximum",
(abstract) *Environmentrics 87*.

Bross I (1988), *Crimes of Official Science: A Casebook*, Biomedical Metatechnology
Press, Buffalo, New York.

Bross I and Blumensan L (1971) "Predictive Design of Experiments Using
Deep Mathematical Models", *Cancer*, 28(6): 1637-1646.

Bross I, Blumenson L, Slack N, and Priare R (1968) A Two-Disease Model for
Breast Cancer. In *Prognostic Factors in Breast Cancer*, (A.P.M. Forrest
and P. B. Kunkler, eds.) E.& S. L.ivingstone, Edinburgh, Scotland.

Bross I and Bross N (1987), "Do Atomic Veterans Have Excess Cancer? New
Results Correcting for the Healthy Soldier Bias", *American Journal
of Epidemiology*, 126(6), 1042-1051.

Bross I, Rimm A, Slack N, Ausman, and Jones R (1966a) "Is Toxicity Really
Necessary? I. The Question", *Cancer*, 19(12): 1780-1784.

Bross I, Rimm A, Slack N, Ausman, and Jones R (1966b) "Is Toxicity Really
Necessary? II. Source and Analysis of Data", *Cancer*, 19(12): 1785-1795.

Bross I, Rimm A, Slack N, Ausman, and Jones R (1966c) "Is Toxicity Really
Necessary? III. Theoretical Aspects", *Cancer*, 19(12): 1796-1804.

Cetane R, Bono VH, Louie AC, Muggia FM (1978) "High-Dose Methotrexate,
Not a Conventional Treatment", *Cancer Chemotherapy Reports*, 62:
178-180.

Committee on Pathologic Effects of Atomic Radiation, *Long-Term Effects of
Ionizing Radiation from External Sources*, National Academy of Sciences,
National Research Council, Washington, D.C., 1961.

Ehrlich P (1960) "Closing Notes to the Experimental Therapy of Spirillosis",
In *The Collected Papers of Paul Ehrlich Vol III *(P. Himmelwert, at)
New York, Pergamon Press, 1960, pp. 282-309.

Fisher B, Ravdin R, Ausman R, Slack N, Moore 0, Noer R (1968) "Surgical
Adjuvant Chemotherapy in Breast Cancer: Results of a Decade of Cooperative
Investigation", *Annals of Surgery* 168: 337-356.

Frei E, Canelles GP (1980) "Dose: A Critical Factor in Cancer Chemotherapy", *American
Journal of Medicine*, 69: 585-594.

Gross L, Roswit B, Mada E, Dreyfuss Y, Moore LA (1959) "Studies on Radiation-Induced
Leukemia in Mice" *Cancer Research*, 19: 316-320.

Hoff DDV, Penta JS, Helman LI, Slavik M (1977) "Incidence of Drug-Related
Deaths Secondary to High-Dose Methotrexate and Citrovorum Factor Administration", *Cancer
Theatment Reports*, 61: 745-748.

Holden C (1988) "Academy Explores Use of Laboratory Animals", *Science* 242:
185-186.

Kolata G (1988) *New York Times* April 24, p. 72. Medical World News (1970), "Is
the Degree of Mastectomy Academic?", March 13, pp. 23-25.

Moss RW *The Cancer Syndrome*, Grove Press, New York.

Noer RJ (Chairman, Surgical Adjuvant Chemotherapy Breast Group) (1961) "Effectiveness
of Thio-Tepa (Triethylenethiophosphoramide) as an adjuvant to radical mastectomy
for breast cancer", *Annals of Surgery* 154: 629-647.

Reichenbach, H (1949), The Theory of Probability. University of California
Press, Berkeley, California.

Reines, B (1987) *Cancer Research on Animals: Impacts and Alternatives*,
National Anti-vivisection Society, Chicago Illinois.

Rhoads CP (1954-55), "Paul Ehirlich and the Cancer Problem", *New
York Academy of Sciences*, 59: 190-197.

Schabel FM (1977) "Surgical Adjuvant Chemotherapy of Metastatic Murine
Tumors", *Cancer*, 40: 558-568.

Schneiderman MA (1966) "What Shall We Measure in Whom: Why?" *Cancer
Cheomotherapy Reports* 50: 107-112.

Schnitzer RJ, Grunberg E (1958) "Studies on Fluorinated Pyrimidines II.
Effects on Transplant Tumors", *Cancer Research* 18(3): 305-317.

Skipper HE, Schabel FM, Wilcox WS (1965) "Experimental Evaluation of Potential
Anticancer Agents. XIV. Further Study of Certain Basic Concepts Underlying
Chemotherapy of Leukemia", *Cancer Chemotherapy Reports* 45: 5-28.

Slack NH, Blumenson LE, Bross IDJ (1969) "Therapeutic Implications from
a Mathematical Model Characterizing the Course of Breast Cancer" *Cancer*,
24: 960-971.

Subcommittee on Hematologic Effects, Committee on Pathologic Effects of Atomic
Radiation, *Effects of Ionizing Radiation on the Human Hemapoietic System*.
National Academy of Sciences-National Research Council, Washington, D.C., 1961.

Waldren C, Correll I., Sognier M and Puck, T (1986). "Measurement of Low
Levels of X-ray Mutagenesis in Relation to Human Disease", *Proceedings
of the National Academy of Sciences USA*, 83, 4839-4843.

Zubrod CG "Historical Milestones in Curative Chemotherapy" *Seminars
in Oncology* 1979; 6(4):49O-5O5.