Working with Convex Responses: Antifragility from Finance to Oncology

Abstract

We extend techniques and learnings about the stochastic properties of nonlinear responses from finance to medicine, particularly oncology, where it can inform dosing and intervention. We define antifragility. We propose uses of risk analysis for medical problems, through the properties of nonlinear responses (convex or concave). We (1) link the convexity/concavity of the dose-response function to the statistical properties of the results; (2) define “antifragility” as a mathematical property for local beneficial convex responses and the generalization of “fragility” as its opposite, locally concave in the tails of the statistical distribution; (3) propose mathematically tractable relations between dosage, severity of conditions, and iatrogenics. In short, we propose a framework to integrate the necessary consequences of nonlinearities in evidence-based oncology and more general clinical risk management.

1. Introduction: Where the Idea of Antifragility Came From

The notions of fragility and antifragility were inspired by the payoffs and the intricacies of financial derivatives. The concept was introduced in Taleb (2012) [1] and more formalized in Taleb and Douady (2013) [2]. While, in the real world, many phenomena are intuitively known to benefit from an increase in “volatility” (that is, the standard deviation of a random variable, or the variability of a nonrandom one), only quantitative finance had names for such attributes, such as “long gamma” (where the financial “derivative” contract has a positive local second mathematical derivative with respect to the underlying security), “long vega” (where the financial derivative has a positive first derivative with respect to the standard deviation of the underlying security), and similar measures, always associated with some range of variation as these sensitivities are local and have, themselves, higher derivatives. By “local” we refer to the fact that most payoff functions in finance are convex over a certain range, then linear or concave, with the second derivative changing in sign, the so-called “higher Greeks” in [3]) Furthermore, finance links some nonlinear attributes of portfolios to the risk of “blowups”, that is, a loss large enough to be irreversible, such as irrecoverable financial ruin. Such quantitative and qualitative models of ruin can give us a tractable generalizable definition of fragility. However, centrally, derivative risk management, at its core, lies in distinguishing between the properties of a random variable X and a payoff function $f\left(x\right)$, almost always nonlinear.

While Jensen’s inequality (on which more in Appendix B) is concerned with the first moment of the distribution, monotone convex (or concave) functions, and a static expectation operator, financial payoffs are more complicated; the first static moment, while relevant, is not the sole focus as:

• The expectation must be conditioned on absence of “blowup”, that is, the left tail of the distribution must be constrained (see Geman et al., 2015) [4], which involves all higher moments of $f\left(x\right)$.
• The payoff functions are almost never monotone.
• Taking into account higher moments of the distributions is analogous to going beyond second-order effects: third, fourth, etc.

Fragility, as defined in Taleb (2012) and Taleb and Douady (2013) [1], Ref. [2], is related to how a system suffers from the variability of its environment beyond a certain preset threshold (when the threshold is K, it is called K-fragility), see Figure 1, while antifragility refers to when it benefits from this variability—in a similar way to as what we saw quantitative finance calls “vega” for an option or a nonlinear payoff, that is, its sensitivity to volatility or some similar measure of scale of a distribution. (Tail fragility maps to a risk of financial ruin, while local fragility does not necessarily mean ruin). In [2]:

Simply, a coffee cup on a table suffers more from large deviations than from the cumulative effect of some shocks—conditional on being unbroken, it has to suffer more from “tail” events than regular ones around the center of the distribution, the ‘at-the-money’ category. This is the case of elements of nature that have survived: conditional on being in existence, then the class of events around the mean should matter considerably less than tail events, particularly when the probabilities decline faster than the inverse of the harm, which is the case of all used monomodal probability distributions. Further, what has exposure to tail events suffers from uncertainty; typically, when systems—a building, a bridge, a nuclear plant, an airplane, or a bank balance sheet—are made robust to a certain level of variability and stress but may fail or collapse if this level is exceeded, then they are particularly fragile to uncertainty about the distribution of the stressor, hence to model error, as this uncertainty increases the probability of dipping below the robustness level, bringing a higher probability of collapse. In the opposite case, the natural selection of an evolutionary process is particularly antifragile, indeed a more volatile environment increases the relative survival rate of robust species and eliminates those whose superiority over other species is highly dependent on environmental parameters.

The paper above produced theorems linking the second derivative of $f\left(x\right)$ in some ranges of variation to sensitivity to the scale of the distribution of X. Thus, the same sensitivity to the scale of the distribution can also express sensitivity to a stressor (dose increase) in medicine or other fields in its effect on either tail. Thus, one single measure will allow us to express with comfortable precision the exposure to the disorder cluster: (i) uncertainty, (ii) variability, (iii) imperfect, incomplete knowledge, (iv) chance, (v) chaos, (vi) volatility, (vii) disorder, (viii) entropy, (ix) time, (x) the unknown, (xi) randomness, (xii) turmoil, (xiii) stressor, (xiv) error, and (xv) dispersion of outcomes. A positive (negative) sensitivity to one means positive (negative) to all others in the group. Finally—and critically—the paper showed that one does not need to have an exact probability distribution to obtain a useful idea of the exposure, since these metrics are based on acceleration, which washes out up to one order of magnitude the precision errors. Note that for multivariate situations, an additive approach is used without any loss of effectiveness.

Asymmetry of Fragile/Antifragile: The opposite of globally fragile (with respect to a random variable X) is not naively “antifragile”, but is both convex with respect to that variable and has a left tail constraint. In probabilistic representation, $f\left(x\right)$ must have a positively skewed distribution. Furthermore, as with the fragile, antifragility is limited to a specific range of variations, and with respect to a single random variable.

The rest of this article will present medicine and convexity, then apply the notions of fragility–antifragility at two levels: efficient dosing in oncology and an examination of iatrogenics as linked to convexity. Finally, in the appendix, we present an overview of convex responses in medicine.

Note: we use, by convention, the term “convexity” or “convexity effect” in the presence of consequential nonlinearity, which can be concave: if the harm function is defined as positive, it shows as convex; if negative, it shows as concave. Finance uses the expression “convexity bias” for both concave and convex responses (and with the possible additional designation “positive” or “negative” convexity).

2. Medicine and Convexity

Medicine has much simpler payoffs than quantitative finance. Most are generalizations around simple sigmoids, see Figure 2, which were described in the mapping in Taleb and Douady (2013) [2] as belonging to the benign class: the distribution of $f\left(x\right)$ is necessarily thinner-tailed than that of X, owing to the boundedness of the function, dubbed more “binary” than “vanilla”, see [5].

However, in spite of such simplicity, little work in medicine has been conducted about probabilistic effects on convexity—almost always limited to first-order effects and comparative statics. The probabilistic dimension of variability has been made explicitly in some medical domains, for instance, there are a few studies connecting Jensen’s inequality to patient responses with pulmonary ventilators: papers such as Brewster et al. (2005) [6], Amato et al. [7], Funk (2004) [8], Arold et al. (2003) [9], Graham et al. (2005) [10], and Mutch et al. (2007). To summarize the literature, continuous high pressures have been shown to be harmful (leading to increased mortality), but episodic spikes of ventilation pressures can be helpful with the recruitment of collapsed alveoli (natural breathing exhibits some variability, with some breaths deeper than others). However, these papers stop at Jensen’s inequality, and, further, explicit probabilistic formulations are still missing in other domains where the applications of these techniques are most needed, such as intermittent fasting, episodic energy deficit, uneven distribution of sub-groups (say, proteins), vitamin absorption, moderate- and low-intensity training, fractional dosage, the comparative effects of low-intensity and distributed interventions vs. intense and concentrated ones, the chronic vs. the acute, and similar effects. As to the psychology literature, the notion of overcompensation is present Den Hartigh and Hill (2022) [11]. The identification of convexity is still confined to local responses and did not generalize to decision-making under uncertainty and inferences concerning silent risks from the nonlinearity in dose response. For instance, the results did not reach the obvious relation between tumor size and the trade-offs of the intervention, or the extrapolation between the numbers needed to treat (NNT) and the potential severity of the side effects.

The connections we are investigating are necessary and mathematical: they work in both directions. We can illustrate as follows:

• A convex response to energy balance over a fixed time window necessarily implies gains from intermittent fasting in some situations and under some strict conditions (that is, higher variance in the distribution of nutrients) over some range within the limits of that time window;
• The presence of metabolic problems in populations that have a steady supply of food intake, as well as evidence of human fitness to an environment that provides moderate variations in the availability of food, both necessarily imply a concave response to food within a range and time frame.

Finally, a short summary of the above is as follows. Convexity analysis in medicine is in two dimensions; first, working with the nonlinearity of dosing, second, for risk analysis for patients and groups.

Missing second-order effects: One frequent lacuna in the literature is ignoring the second-order effects when making statements derived from from empirical data. One example is dietary recommendations for food group composition rather than frequency. Epidemiological interpretations of the Cretan diet relied solely on composition. A simplified intuition of the second-order effect in nutrition is as follows. Eating once a day vs., say, three times (in an isocaloric way) presents a difference if the response function is nonlinear: an average of response functions is not a response function of an average. However, frequency matters: the Eastern Orthodox Church has, with minor local variations, around two hundred days of vegan fasts per year. This is an episodic protein deprivation; fatty meats are consumed in lumps (on Sundays and holidays), which compensates for such a deprivation (recall the threshold in Figure 4. As shown in the literature review in Appendix A, there is a need for a mathematical bridge between studies of variability, say Martin et al. (2006) [12] and Fontana et al. (2008) [13], on one hand, and the focus on composition—the Longo and Fontana studies, furthermore, narrow the effect of the frequency to a given food type, namely proteins (Lee and Longo (2011) [14] “In the prokaryote E. coli, lack of glucose or nitrogen (comparable to protein restriction in mammals) increase resistance to high levels of $H_2{O}_2$ (15 mm) (Jenkins et al., 1988) [15]”). Further, the computation of the “recommended daily” units may vary markedly if one assumes necessary stochasticity.

Extracting past statistical attributes and frequencies: A central question is if we need a certain dose of stressors, whether in intermittence of nutrition or necessary exercise, might these represent the attributes of an “ideal” environment. Whether evolutionary or not, this is the one to whose stochastic properties we are most adapted. We can, therefore, reverse engineer the stochastic nature of such an “ideal” environment by finding the various conditions that result from a reduction in stressors. We noted that papers such as Kaiser (2003) [16] and Calabrese and Baldwin (2003), [17] do not bridge the results to the point that hormesis may correspond to a “fitness dose”, beyond and below which one departs from such an ideal dispersion of the dose x per time period.

Such a reverse engineering uses the visible dose–response curve to make inferences about the ideal parametrization of the probability distribution of nutritional balance and vice-versa. For example, assessing the benefits of episodic fasting and the length of windows for neoplasms, insulin resistance, and other conditions can lead to understanding some kind of “fitness” to an environment endowed with a certain structure of randomness, either with the $\sigma$ above or some more sophisticated attributes of the probability distributions (such as higher moments, hence different shapes). For example, if insulin resistance can be reduced thanks to occasional deprivation (a certain variance), say one 24-h fast every week, 3 days of fasting per trimester, and a complete week every five years, then we can extract and parametrize a probability distribution of ancestral deprivations. A comprehension of the exact mechanism by which such intermittences work can be helpful but is not needed given the robustness of the mathematical connection between the functional and probabilistic.

Antifragility in Treatment Scheduling

In Figure 5, the input distribution of dose x is subject to the convexity (or concavity or linearity) of the dose-response function, which influences the tail of the outcome distribution. Importantly, an oncologist has “first-mover” advantage [18] and has the benefit of prescribing an “even” treatment protocol with no variance (Figure 4, top row) or an “uneven” treatment protocol with positive variance (Figure 4, bottom row).

In medical practice, treatment protocols are typically fixed with doses administered at regular intervals (e.g., Figure 4). The distribution of dosing is unimodal (purple; continuous dosing), or, at most, bimodal (green; intermittent dosing). Manipulation of dose volatility when designing treatment protocols is under-utilized as a strategy in cancer treatment. In place of a dose x, one can give, say, 120% of x, then 80% of x, with a more favorable outcome if one is in a zone that benefits from unevenness. If antifragile, more unevenness is more beneficial: 140% followed by 60% produces better effects [19].

3. Antifragility in Oncology

Across all treatment modalities, cancer treatment is intended to induce perturbations to environmental conditions within a tumor leading to cell death, altering vasculature, or impacting immune response. However, the most common treatment paradigm is the “maximum tolerable dose” (MTD) dosing protocol, whereby the dose is maximized, and only limited by tolerability, toxicity, and side effects. To re-phrase, oncology research is implicitly focused on maximizing “first-order” treatment effects by increasing the cumulative dose [20] or shortening the time between doses [21]. The “log-kill” law proposes an MTD protocol for cytotoxic chemotherapy agents that decreases the amount of time over which a cumulative dose is delivered as toxicity allows [22]. More recently, metronomic therapy proposes frequent, low doses known to provide an anti-angiogenic effect during chemotherapy, still implicitly optimizing a first-order effect of cumulative dose [23].

Oncology must consider convexity in strategizing treatment protocols. Although convexity was not a consideration of initial clinical design, recent approaches have had success in managing second-order effects through the practice of high/low dosing. Intermittent high dosing of tyrosine kinase inhibitors (TKI) in HER2-driven breast cancers was administered with concentrations of the drugs that would otherwise far exceed toxicity thresholds if dosed continuously [24]. Continuous letrozole in combination with high-dose intermittent ribociclib is currently in clinical trial (NCT02712723; ER-positive breast cancer) [25]. Intermittent high-dose erlotinib delays resistance in an EGFR-mutant non–small cell lung cancer in vivo model [26,27]. Intermittent weekly EGFR-inhibitors reduced tumor load in vivo, compared with daily regimens with identical cumulative doses [28]. Intermittent “pulsatile” high-dose erlotinib once weekly maintains efficacy even after failure of low-dose continuous treatment [29]. Ideal treatment protocols will maximize both first-order effects (cumulative dose) and second-order effects (variance of dose delivered). Studies mentioned previously provide evidence of the tolerability of temporary dose escalation by also employing off-treatment periods to alleviate therapy toxicity.

3.1. Defining (Local) Fragility in Oncology

Local fragility, F, is a measurable quantity, similar to the Jensen gap, defined as the difference in the result of unevenness over evenness (with corresponding unevenness range parameter $\lambda$):

$$F(x,\lambda )=\frac{f(x+\lambda )+f(x-\lambda )}{2}-f\left(x\right)$$

(1)

Which property of cancer cells, $f\left(x\right)$, is important in oncology? Here, we are interested in the advantage of “uneven” high/low schedule over the “even” schedule. More precisely, fragility is the difference between (1) a schedule of two constant doses (termed an “even” dosing strategy), $\overrightarrow{X}=\{x,x\}$ and (2) a schedule of a high dose followed by a low dose (termed an “uneven” dosing strategy), $\overrightarrow{X}=\{x+\lambda ,x-\lambda \}$. We consider the response to two doses over an interval of time T, where the first dose is given at $t=0$ and the second dose is given at $t=T/2$. Given a tumor with an initial population size of $n_0$, the final size of an exponentially growing population is given by:

$$\begin{array}{c}\hfill n_F\left(t\right)=n_{0}exp\left(\gamma \left(x\right)t\right),\end{array}$$

(2)

where $\gamma \left(x\right)$ is the decay rate of the population associated with a dose of x. Fragility can be defined as:

$$F(x,\lambda )=n_{0}exp\left(\gamma (x+\lambda )\frac{T}{2}\right)exp\left(\gamma (x-\lambda )\frac{T}{2}\right)-n_{0}exp\left(\gamma \left(x\right)\frac{T}{2}\right)exp\left(\gamma \left(x\right)\frac{T}{2}\right),$$

(3)

which simplifies to:

$$F(x,\lambda )=n_0\left[exp\left(\left(\gamma (x+\lambda )+\gamma (x-\lambda )\right)\frac{T}{2}\right)-exp\left(\gamma \left(x\right)T\right)\right].$$

(4)

We are interested in the antifragile–fragile boundary, the point at which the population is no longer fragile but antifragile. If $F<0$, the cell population is antifragile by definition (a benefit conferred to uneven dosing for minimizing tumor growth rate), and if $F>0$, then it is fragile. It follows that the previous equation will be negative if:

$$\frac{\gamma (x+\lambda )+\gamma (x-\lambda )}{2}<\gamma \left(x\right)$$

(5)

Thus, the important domain for convexity is the dose-dependent growth rate, $\gamma \left(x\right)$. It is an unfortunate common practice to normalize dose–response curves to obtain fractional survival at the final time point (e.g., $1-n_F/n_0$), which obscures convexity. Drug-induced growth rate inhibition (GR curves) has been introduced as a method to remove the artifactual dependency of IC50 and Emax on the cellular division rate [30]. GR curves preserve convexity, unlike fractional survival.

3.2. Fragility and Taylor Series Approximations

Antifragility is a “second-order” effect. This is shown by first taking a Taylor expansion about x:

$$f(x\pm \lambda )=f\left(x\right)\pm \lambda f^{\prime }\left(x\right)+\frac{{\lambda }^2}{2}{f}^{″}\left(x\right)+O\left({\lambda }^3\right)$$

(6)

where $O\left({\lambda }^3\right)$ represents all third-order or higher terms. Using this expansion, fragility can be written:

$$F(x,\lambda )=\frac{1}{2}\left(2f\left(x\right)-f\left(x\right)-\lambda f^{\prime }\left(x\right)-\frac{{\lambda }^2}{2}{f}^{\prime \prime }\left(x\right)-f\left(x\right)+\lambda f^{\prime }\left(x\right)-\frac{{\lambda }^2}{2}{f}^{\prime \prime }\left(x\right)\right)+O\left({\lambda }^3\right).$$

(7)

The zeroth-order terms and the first-order terms cancel out:

$$F(x,\lambda )=-\frac{{\lambda }^2}{2}{f}^{″}\left(x\right)+O\left({\lambda }^3\right)$$

(8)

For small values of $\lambda$, fragility is proportional to the second derivative, and, thus, known as a second-order effect. Next, we connect the concept to finite difference methods.

3.3. Fragility and Finite Differences

We can approximate the derivative of $f\left(x\right)$ using finite differences. Here, we use the well-known central difference approximation to the second derivative, $f^{″}\left(x\right)$, where h is the width of the interval over which the finite difference is estimated ($h>0$):

$$\begin{array}{c}\hfill f^{″}\left(x\right)=\underset{h\to 0}{lim}\frac{1}{h^2}\left[f(x+h)+f(x-h)-2f\left(x\right)\right].\end{array}$$

(9)

The term on the right-hand side in brackets is related to fragility (Equation (1)), giving us the relationship between $F(x,\lambda =h)$ and $f^{″}\left(x\right)$:

$$\begin{array}{c}\hfill f^{″}\left(x\right)=-\underset{h\to 0}{lim}\frac{2}{h^2}F(x,h),\end{array}$$

(10)

This limit illustrates that $f^{″}\left(x\right)$ approaches F only when h is small. When the value of h is large, the approximation is poor, and, therefore, we employ Equation (1). Figure 6 provides an example of approximation error for the Hill function (see next section).

3.4. Applications of Hill Function

The Hill function is commonly used to describe drug pharmacodynamics [31], where $H\left(x\right)$ is the cell viability in response to a dose x.

$$H\left(x\right)=\frac{E_1-E_0}{1+{\left(\frac{C}{x}\right)}^n}+E_0$$

(11)

where n is the Hill shape parameter, $E_0$ and $E_1$ are the minimal and maximal response (respectively), and C is the half-maximal response (the EC50 value). The second derivative can be written:

$$\frac{d^{2}H}{d{x}^2}=\frac{(E_1-E_0)n{C}^n{x}^{n-2}\left((n-1)C^n-(n+1)x^n\right)}{{(C^n+x^n)}^3}.$$

(12)

Figure 6A shows a sample Hill function and corresponding second derivative. Figure 6B illustrates decreasing error between the numerical (F) and analytical ($\frac{d^{2}H}{d{x}^2}$) as $h\to 0$. The error scales like $h^2$, as predicted in Equation (10). The inflection point, found where $\frac{d^{2}H}{d{x}^2}=0$, defines the boundary between convex and concave regions.

$$x^{*}=C{\left(\frac{n-1}{n+1}\right)}^{1/n}$$

(13)

It can be shown that as n increases, $x^{*}\to C$. Importantly, $x^{*}$ determines the boundary between the antifragile and fragile regions of $f\left(x\right)$. Benefit can, thus, be derived from uneven dosing if $x<x^{*}$. The inverse implies that uneven treatment schedules provide no additional benefit. For a discussion on relaxing the assumption of fixed treatment schedules (e.g., Figure 4) using a probability density function describing dose distribution, see Appendix C, Equations (A4) and (A5).

As shown in Figure 5, the input distribution passing through convex dose response (e.g., on the Hill function below its inflection point) results in a left-tailed outcome distribution, and a concave response function results in a right-tailed distribution.

Thus, we have defined as locally antifragile a situation in which, over a specific interval $[a,b]$, either the expectation rises with the scale parameter of the probability distribution as in Equation (A2), or the dose response is convex (on average) over the same range. The designation in Taleb (2012) [1] meant to accurately describe such situations: anything that gains from an increase in stochasticity or variability (since the scale parameter represents both). Terms such as “resilience”, since they were not mapped mathematically, are vague and even confusing as they meant either resistance or gains from stressors, depending on context. Figure 7 and Figure 8 illustrates the threshold effect of the asymmetric response, and gives the intuition of how they can be described as as antifragile.

3.5. The First-Order Sigmoid Curve

Next, we outline the variety of sigmoids as catalogued by [33]. Define the sigmoid or sigmoidal function as having membership in a class of function $\mathfrak{S}$, $S:\mathbb{R}\to [L,H]$, with additional membership in the ${\mathbb{C}}^2$ class (twice differentiable), monotonic nonincreasing or nondecreasing, that is let $S^{\prime }\left(x\right)$ be the first derivative with respect to x: $S^{\prime }\left(x\right)\ge 0$ for all x or $S^{\prime }\left(x\right)\le 0$. Thus, we have:

$$S\left(x\right)=\left\{\begin{array}{cc}H\hfill & \mathrm{as}x\to +\infty ;\hfill \\ L\hfill & \mathrm{if}x\to -\infty .\hfill \end{array}\right.,$$

which can of course be normalized with $H=1$ and $L=0$ if S is increasing, or vice versa, or alternatively $H=0$ and $L=-1$ if S is increasing. We can define the simple (or first-order) sigmoid curve as having equal convexity in one portion and concavity in another: $\exists k>0\mathrm{s}.\mathrm{t}.\forall x_1<k\mathrm{and}{x}_2>k,$$\mathrm{sgn}\left(S^{″}\left(x_1\right)\right)=-\mathrm{sgn}\left(S^{″}\left(x_2\right)\right)$ if $|S^{″}\left(x_2\right)|\ge 0$.

Now, all functions starting at zero will have three possible properties at inception, as in Figure 9 and Figure 10: concave, linear, and convex. The point of our discussion is the latter becoming sigmoid. Although few medical examples appear, under scrutiny, to belong to the first two cases, one cannot exclude them from analysis. We note that given that the inception of these curves is zero, no linear combination can be initially convex unless the curve is convex, which would not be the case if the start of the reaction is at a level different from zero.

There are many sub-classes of functions producing a sigmoidal effect. Examples include:

• Pure sigmoids with smoothness characteristics expressed in trigonometric or exponential form, $f:\mathbb{R}\to [0,1]$:

  $$f\left(x\right)=\frac{1}{2}tanh\left(\frac{\kappa x}{\pi }\right)+\frac{1}{2}$$

  $$f\left(x\right)=\frac{1}{1-e^{-ax}}$$
• Gompertz functions (a vague classification that includes above curves but can also mean special functions).
• Special functions with support in $\mathbb{R}$ such as the error function $f:\mathbb{R}\to [0,1]$

  $$f\left(x\right)=-\frac{1}{2}\mathrm{erfc}\left(-\frac{x}{\sqrt{2}}\right)$$
• Special functions with support in $[0,1]$, such as $f:[0,1]\to [0,1]$

  $$f\left(x\right)=I_x(a,b),$$

  where $I_{(.)}(.,.)$ is the beta regularized function.
• Special functions with support in $[0,\infty )$

  $$f\left(x\right)=Q\left(a,0,\frac{x}{b}\right)$$

  where $Q\left(.,.,.\right)$ is the gamma regularized function.
• Piecewise sigmoids, such as the CDF of the Student distribution

  $$f\left(x\right)=\left\{\begin{array}{cc}\frac{1}{2}{I}_{\frac{\alpha }{x^2+\alpha }}\left(\frac{\alpha }{2},\frac{1}{2}\right)\hfill & x\le 0\hfill \\ \frac{1}{2}\left(I_{\frac{x^2}{x^2+\alpha }}\left(\frac{1}{2},\frac{\alpha }{2}\right)+1\right)\hfill & x>0\hfill \end{array}\right.$$

We note that the “smoothing” of the step function, or Heaviside theta $\theta (.)$ produces a sigmoid (in a situation of a distribution or convoluted with a test function with compact support), such as $\frac{1}{2}tanh\left(\frac{\kappa x}{\pi }\right)+\frac{1}{2}$, with $\kappa \to \infty$, see Figure 3.

3.6. Some Necessary Relations Leading to a Sigmoid Curve

Let $f_1\left(x\right):{\mathbb{R}}^{+}\to [0,H]$, $H\ge 0$, of class $C^2$ be the first-order dose-response function, satisfying $f_1\left(0\right)=0$, $f_1^{\prime }\left(0\right)|=0$, and ${lim}_{x\to +\infty }{f}_1\left(x\right)=H$, monotonic nondecreasing, that is, $f_1^{\prime }\left(x\right)\ge 0\forall x\in {\mathbb{R}}^{+}$, with a continuous second derivative, and analytic in the vicinity of 0. Then, we conjecture that:

A-There is a zone $[0,b]$ in which $f_1\left(x\right)$ is convex, that is, $f_1^{″}\left(x\right)\ge 0$, with the implication that $\forall a\le b$ a policy of variation of dosage produces beneficial effects:

$$\alpha f_1\left(a\right)+(1-\alpha )f_1\left(b\right)\ge f_1(\alpha a+(1-\alpha )b),0\le \alpha \le 1.$$

(The acute outperforms the chronic).

B-There is a zone $[c,H]$ in which $f_1\left(x\right)$ is concave, that is, $f_1^{″}\left(x\right)\le 0$, with the implication that $\exists d\ge c$ a policy of stability of dosage produces beneficial effects:

$$\alpha f_1\left(c\right)+(1-\alpha )f_1\left(d\right)\le f_1(\alpha c+(1-\alpha )d).$$

(The chronic outperforms the acute).

4. The Generalized Dose–Response Curve

Let $S^N\left(x\right)$: $\mathbb{R}$→ [$k_L$, $k_R$], $S^N\in C^{\infty }$ be a continuous function possessing derivatives ${\left(S^N\right)}^{\left(n\right)}\left(x\right)$ of all orders, expressed as an N-summed and scaled standard sigmoid function:

$$S^N\left(x\right)\triangleq \sum _{i=1}^N\frac{a_k}{1+e^{\left(-b_{k}x+c_k\right)}}$$

(14)

where $a_k,b_k,c_k$ are scaling constants ∈$\mathbb{R}$, satisfying:

• $S^N$(−∞) = $k_L$, and
• $S^N$($+\infty$) = $k_R$, and (equivalently for the first and last of the following conditions)
• $\frac{{\partial }^2{S}^N}{\partial x^2}$≥ 0 for x∈ (−∞, $k_1$), $\frac{{\partial }^2{S}^N}{\partial x^2}$< 0 for x∈ ($k_2$, $k_{>2}$), and $\frac{{\partial }^2{S}^N}{\partial x^2}$≥ 0 for x∈ ($k_{>2}$, ∞), with $k_1>k_2\ge \dots \ge k_N$.

By increasing N, we can approximate a continuous function’s density in a metric space, see Cybenko (1989) [34].

The shapes at different calibrations are shown in Figure 11, in which we combined different values of N = 2 $S^2\left(x;a_1,a_2,b_1,b_2,c_1,c_2\right),$ and the standard sigmoid $S^1\left(x;a_1,b_1,c_1\right)$, with $a_1$ = 1, $b_1$ = 1, and $c_1$ = 0. As we can see, unlike the common sigmoid, the asymptotic response can be lower than the maximum, as our curves are not monotonically increasing. The sigmoid shows benefits increasing rapidly (the convex phase), then increasing at a slower and slower rate until saturation. Our more general case starts by increasing, but the reponse can actually be negative beyond the saturation phase, though in a convex manner. Harm slows down and becomes “flat” when something is totally broken.

Antifragility and Heterogeneity

Tumors are composed of a heterogeneous collection of subpopulations with varied treatment sensitivity. Given N non-interacting populations, the fragility of the total population is given by the sum of each subpopulation i’s fragility, $F_i$, weighted by its frequency within the total population, $w_i$, such that ${\sum }_i{w}_i=1$.

$$\begin{array}{ccc}\hfill F(\overline{x},\sigma )& =& \sum _i^N{w}_i{F}_i(\overline{x},\sigma )\hfill \end{array}$$

(15)

where fragility $F_i$ for a single population is given by Equation (1).

The simplest case of a heterogeneous mixture of two populations, sensitive (with associated dose response $H_1\left(x\right)$) and resistant ($H_2\left(x\right)$), is shown in Figure 12A. In the case that each dose response, $H_i$, is non-increasing, then the mixed dose response will also be non-increasing, but changes in convexity may occur. As seen in Figure 12A, the mixed dose response has an internal plateau (shown for $w=0.5$). Local convexity (fragility) may switch signs multiple times.

5. Nonlinearities and Medical Iatrogenics

Next, we connect nonlinearity to iatrogenics (that is, harm done by the healer) for medicine in general, broadly defined as all manner of net deficit of benefits minus harm from a given intervention.

The Taleb and Douady (2013) [2] theorems state:

• Convexity for a dose-response function increases fragility (from the expansion of the left tail in response to the increase in the scale of the distribution).
• Detection of a nonlinearity allows the prediction of fragility and helps formulate probabilistic decisions without much knowledge of the probability distribution beyond minimum standard attributes.
• The presence of concavity in the tails of the distribution implies a silent risk.

This approach was used in stress testing by the International Monetary Fund (IMF), where the degree of concavity in the tail was used as an indicator of the severity of tail exposure, see Taleb, Canetti et al. [35]. Such a method can transfer to medicine as the convexity of the dose response can be estimated via titration applied to Equation (1).

5.1. Effect Reversal

Radiation might be beneficial in small doses, with reversal later on. In Neumaier et al. (2012) [36] titled “Evidence for formation of DNA repair centers and dose-response nonlinearity in human cells”:

The standard model currently in use applies a linear scale, extrapolating cancer risk from high doses to low doses of ionizing radiation. However, our discovery of DSB clustering over such large distances casts considerable doubts on the general assumption that risk to ionizing radiation is proportional to dose, and instead provides a mechanism that could more accurately address risk dose dependency of ionizing radiation.

Therefore, low-level radiation may cause hormetic overreaction, producing protective effects. Also see Tubiana et al. (2005) [37]. Bharadwaj and Stafford (2010) present similar general sigmoidal effects in hormonal disruptions by chemicals [38].

5.2. Nonlinearity of NNT and the Consequences

Below are applications of convexity analysis in decision-making in dosage, shown in Figure 13 and Figure 14. In short, it is fallacious to translate a policy derived from acute conditions and apply it to milder ones. Mild conditions are different in treatment from an acute ones. Likewise, high risk is qualitatively different from mild risk.

There is active literature on “overdiagnosis”, see Kalager et al. (2012) [39] and Morell et al. (2012) [40]. The point is that treating a tumor that does not kill reduces life expectancy; hence the need to balance iatrogenics and risk of cancer. An application of nonlinearity can shed some light on the approach and clarify the public debate [1].

In a similar spirit of avoiding over-treatment, adaptive therapy in metastatic castrate-resistant prostate cancer (clinical trial NCT02415621) has illustrated the feasibility of irregular treatment protocols based on algorithms that react to tumor response. Adaptive treatment protocols maintain a stable population of sensitive cells in order to suppress the emergence of resistance [41,42]. Resistance, in some cases, is similar to the irreconcilable ruin of a financial “blowup” as patients may develop multi-drug resistance to structurally or functionally different drugs. The irreversibility of such clinical outcomes is similar to that of financial ruin, a tail fragility situation from which the agent cannot exit. Adaptive algorithms decrease the cumulative dose administered to a patient, lessening the selection for resistance [43]. While it is not an explicitly stated goal of adaptive therapy, these schedules increase both intra- and inter-patient dosing variance [44].

Last year (2022) saw the publication of calls from within the FDA to revamp the dose-finding protocols to be suitable for targeted therapies [45]. Traditional dose selection protocols invented for use with cytotoxic chemotherapies may not apply to targeted therapies that have exposure–response curves which plateau at low toxicities to the patient. Differences in convexity between chemotherapies and newly developed targeted therapies lead to differing outcomes in diminished returns of dose escalation and differing curvature of dose–response curves.