Inferring efficacious doses from free plasma concentration, in-vitro efficacy and in-vivo tumour growth inhibition

It has been previously demonstrated that drug response in-vivo can be related to the plasma concentration of the free drug and in-vitro efficacy (as measured by in-vitro IC50 curves)Footnote 4, [3]. Specifically, in an oncology context, a relation between tumor growth inhibition (TGI) and the fraction between the average free plasma concentration and free IC50 has been observed [3, 4]. We therefore aimed to replicate such findings with in house data.

For validation purposes, we utilized data from Boehringer Ingelheim’s internal compound library on MAPK inhibitors (see Fig. (1)). We plotted TGI against the fraction of the average free concentration over free IC50, where each dot represents a xenograft study involving 6 different xenografts and 12 compounds. We then fitted a curve to the logistic function as defined in Eq. (10) from the methods and obtained the following parameters: \(PD_\) = 0.25, \(hill_\) = 0.89, \(TGI_\) = 125, \(TGI_\) = 0.

Fig. 1

A pre-clinical IVIVC relationship between tumor growth inhibition and IC50-exposure coverage using data from Boehringer Ingelheim’s internal projects on receptor kinase inhibitors. A sigmoid fit curve is provided according to Eq. (10)

With this fitting parameters and assuming dose linearity, we were able calculate a stasis dose by Eq. (13) from the methods

$$\begin dose_ = \frac}-TGI_}-1\right) ^}} \, \frac/ \tau }\, , \end$$

(14)

and obtained

$$\begin dose_\approx & 29.8 \, *\,\frac} \end$$

(15)

$$\begin dose_\approx & 1.24 \, *\,\frac} \,, \end$$

(16)

where we also have introduced the dose-normalised average concentration (\(c_\)) and assumed a \(\tau = 24\)h dose interval, hence

$$\begin c_=AUC_/\tau . \end$$

(17)

The relations (15, 16) suggest that the required tumour stasis dose (\(dose_\)) for an "average" compound of this IVIVC relation is proportional to the IC50 of the compound divided by either the dose normalised AUC (\(AUC_\)) or dose-normalised average concentration (\(c_\)).

We therefore have defined the inverse of the (compound and in-vitro model-specific) factors in Eqs. (15) and (16)

$$\begin \frac}\quad \text \quad \frac}\, , \end$$

(18)

as IC50 coverage factors for \(c_\) and \(AUC_\), respectively.

These parameters will play a crucial role in our further investigations to study the required coverage of IC50 by the average concentration or by the time-integrated exposure over a dosing interval.Footnote 5 As evident from Fig. (1) and Eq. (17), both factors exhibit a monotonous trend in relation to tumor growth inhibition (TGI).

Relating empirical efficacious doses to mechanistic TGI Modelling

Above results related the inverse of the compound specific coverage factor (18) to tumour stasis. As this result is only grounded on empirical data, our objective was to ground these findings on a more mechanistic understanding of tumour growth.

Therefore, we asked if a similar relation could be established by using a standard PK/TGI mechanistic model. We therefore have used a model that has been shown to describe the underlying tumour growth mechanism quite well [13, 16, 19] and hence is heavily used in our Boehringer Ingelheim team for in-vivo xenograft study design. For the purpose of the paper, this model is therefore assumed as a fairly accurate coarse-grain description of the underlying tumour growth mechanism, hence being assumed as the “ground truth”.

The model was described in the methods Sect. (2.2). As shown there, we have used a combine model of tumour growth and drug-induced anti-tumour effect

$$\begin dR\,= & \, \left[ g\, -\, d\,\left( \frac} }\right) \,\right] dt \end$$

(19)

$$\begin R\,= & \, R_ + g\tau -\, d\,\int ^\tau _0\, \frac(t)}(t) } \,dt, \end$$

(20)

with the dosing interval \(\tau\) and the initial tumour radius \(R_\). The TGI is then calculated from the volume at time \(t=\tau\) via the relation ( 9), \(Vol=4R^3\,\pi /3\) in the presence and absence of drug \((c_(t))\) concentration, while g and d are in-vivo xenograft specific parameters as described in the methods.

Derivation of model-informed plasma concentrations for tumour stasis

To further the objective of the previous section, we now specifically asked for which IC50 coverage the above tumour model above would predict stasis. We therefore assumed that an equilibrium behaviour for the pharmacokinetics after multiple dosing has been achieved Fig. (2).

For tumour stasis, we then required that the compound exposure at these later cycles (area under the blue curve) is sufficient to halt tumour growth. This leads to the requirement

$$\begin \int \,dR\,0 \, => \, g\,\tau /d \,\, \int ^\tau _0\, \frac(t)}(t) }\, dt\,, \end$$

(21)

where we have used the free plasma concentration \(c_(t)\) over time as the major surrogate for anti-tumour effect (know as free plasma concentration hypothesis).Footnote 6

Fig. 2

A typical pharmacokinetic behaviour for the free plasma concentration over time \(c_(t)\) is given for repeated dosing. Accumulation over repeated dosing cycles leads to a steady state behaviour of the daily exposure. The last dosing interval (in blue) has been assumed to be in steady state and to provide sufficient exposure to put the tumour to stasis (Color figure online)

We next assumed a first order kinetic approximation of the free plasma concentration and again postulate dose linearity. We further assume a fast and hence negligible absorption phase and also assumed an Hill coefficient of 1. We later included some relaxations to these conditions in Sects. 3.5.1 and 3.5.2.

Hence, we have set

$$\begin c_(t) = c_\,e^\qquad k = \frac\,ln\left( \frac}} \right) \end$$

(22)

where we have denoted the maximum (“max”) and minimum (“trough”) free plasma concentration \(c_\) and \(c_\). For brevity and by analogy reasons, we now define the effective AUC, \(AUC_\)

$$\begin AUC_ = \int ^_0\, \frac\,e^}\,e^ + IC50} \,dt \end$$

(23)

and require a threshold for the stasis condition (21)

$$\begin 0 = R-R_0 = g\tau \,-\,d\,AUC_\rightarrow \quad AUC_\, = \,\frac\cdot \tau \, . \end$$

(24)

In the Appendix 7.1, we calculated Eq. (23) as

$$\begin AUC_ = \frac\left( PTR\right) }\text \left[ \frac + IC50} + IC50} \right] \, . \end$$

(25)

We have set here the Peak-trough ratio (\(PTR=\frac}}\)) for convenience as this ratio has manageable variation within one compound class and can be estimated for typical small molecule compounds in oncology projects.

Setting the effective AUC in (B9) into the stasis condition Eq. (21), we were able to relate \(c_\), \(c_\) and \(c_\) with the compounds PTR and the tumour model specific parameters g and d. The detailed derivation is given in Appendix 7.2 and we obtain.

$$\begin c_& = &IC50 \frac-1}}\nonumber \\ c_& = &IC50\,\cdot \, PTR \frac-1}}\nonumber \\ c_& = &IC50\,\cdot \, \frac \frac-1}} . \end$$

(26)

By this, we have established sufficiency condition of above concentrations to achieve tumour stasis when above assumptions are met.Footnote 7

Finally, we used Eq. (26) to scale again the dose such that a stasis dose \(dose_\) is reached. We therefore replace \(c_\) by the total exposure over a dosing interval \(\tau\) according Eq. (17) and assume again dose-linearity Eq. (12),

$$\begin dose_ = \, \frac \frac-1}}\, \frac} \, . \end$$

(27)

For convenience, we define the factor

$$\begin MEF\left( PTR, g/d\right) \,:= & \,\frac \frac-1}} \, \end$$

(28)

$$\begin= & \frac}\,, \end$$

(29)

which is dependent on the compound specific parameter PTR and the in-vivo xenograft specific parameters g and d. We will denote this factor as model efficacy factor which will give us further insights into the driver of this stasis dose and IC50 coverage by \(c_\) in the further analysis.

By this, we can conclude that the stasis dose \(dose_\) can be reached either by optimising exposure over IC50 of the compound (right factor, AUC-drivenness) or by optimising \(MEF\left( PTR, g/d\right) \,\) (non AUC-drivenness).

Calculation of xenograft-model specific IC50 coverage for tumour stasis

Based on the defined coverage laws (26), we now aimed to systematically investigate their dependency on compound (PTR, IC50) and in-vivo xenograft specific parameters (g and d). In line with our internal data set, we used steady-state PTRs with 10-300 and a typical xenograft with a g/d ratio of 0.4 as observed in our pre-clinical studies. We then calculated necessary coverage for \(c_\), \(c_\) and \(c_\) (26) for tumour stasis.

Results are shown in Fig. 3 and indicate that for tumour stasis, IC50 coverage of the average unbound plasma concentration is essential, and less dramatically dependent from min-max variations (PTR). These results suggest “AUC drivenness” in xenografts and compounds with above properties.

Fig. 3

The ratio between the required IC50 coverage for tumour stasis for \(c_\) (dashed line), \(c_\) (solid line) and \(c_\) (dotted line). We see that for typical pre-clinical PTRs of oncology compounds and a reasonable g/d ratio of 0.4, a \(c_\) coverage of about 2 is suggested. The high necessary \(c_\) coverage for larger PTR is to compensate for the less durable target engagement for such low terminal half-life compounds

We then investigated how the coverage factor Eq. (28) changes for the different xenograft models characterised by given g/d ratio with PTR range of 40-150. We therefore plotted the model efficacy factor in Fig. 4. We observed that, when g/d ratios change between 0.3 and 0.7, these efficacy factor (and hence IC50 coverage factors) range from half IC50 coverage to about 10-fold coverage for our MAPK inhibitor compounds.

As expected, low g/d ratios (left edge), consistent with slow growing and relative sensitive xenograft models, required little IC50 coverage (on top of these cell-specific xenografts having potentially also a lower in-vitro IC50). In turn, when the growth rate increases relative to the decay rate \(g/d\rightarrow 1\), high IC50 coverage is necessary for tumour stasis.

Interestingly, these coverage results were again mildly dependent for most assumed pre-clinical PTRs and g/d between 0.3 and 0.7, suggesting that "AUC drivenness" also mostly holds also for mildly sensitive to mildly resistant xenografts and compounds with reasonable max/trough variations.

Overall, these result indicate that the type of xenograft (besides influencing efficacy per se) influences the extent as to how efficacy is dependent on the compound’s PTR. We have thereby also obtained a more mechanistically grounded understanding of our empirical findings in (16), relating the fitting factor in (16) to xenograft- (g/d) and compound-specific (PTR) properties.

Fig. 4

The model efficacy factor MEF given for the depicted Peak-Trough-ratios and different fractions of g/d. This factor indicates necessary IC50 coverage by \(c_\) for stasis. The dashed line indicates MEF = 2. Inlet shows the same plot on a semi-logarithmic axis. PTRs were taken from compounds in Fig. 1. For model used in our research, g/d ratios for xenografts were in the range 0.3\(-\)0.7 (cf. Fig. 8 for our analysis of compounds.)

Expanding the validity of the findings through relaxations of assumptions

For didactic purposed, we have assumed a negligible absorption phase and a Hill coefficient of 1 in our previous derivation. We now make two relaxations, whereby Relaxation 1 leaves the formalism unchanged, while Relaxation will provide further insights into the specific pharmacokinetic drivers of anti-tumor effect, once \(hill>1\) (often denoted as drug co-operativityFootnote 8).

Relaxation 1: application to situations with non-negligible absorption time

Above derivations were performed under the assumption of an exponential decay from \(c_\) to \(c_\) at the entire dosage range. This may be considered as a crude assumption as orally (or other non i.v.) administrated compounds may show a non-negligible absorption phase.

Therefore, in Appendix 8, we demonstrated that above derivations still hold, when the absorption phase is not negligible and maximum concentration \(c_\) is achieved at a certain time \(t_\). We therefore had to postulate that absorption and elimination phase have different temporal dynamics (hence no flip-flop kinetics), which well holds for the investigate MAPK inhibitors reported here.Footnote 9

Relaxation 2: tumour stasis conditions for compounds with \(hill\ne 1\)

An Hill coefficient different to one is often found in the in-vitro function Eq. (7) and is associated with positive or negative co-operative effects of the compound.

Therefore, we have repeated calculations of Sect. 3.3 for situations with \(hill\ne 1\) as described by

$$\begin R = R_ + g\tau -\, d\,\int ^\tau _0\, \frac(t)^} +c_(t)^ } \,dt\, . \end$$

(30)

Using an exponential decay of the free plasma concentration from \(c_\) to \(c_\) we then exploited the fact that the potency of an exponential function gives another factor in the exponential \(\left( e^\right) ^ = e^\).

The derivation followed the line of subsection 3.3 and we finally obtained,

$$\begin dose_= & \, \, MEF\left( PTR, g/d, hill\right) \, \frac/ \tau } \, \end$$

(31)

$$\begin= & \, MEF\left( PTR, g/d, hill\right) \, \frac} \end$$

(32)

with

$$\begin MEF\left( PTR, g/d, hill\right) \,= \,\frac \left[ \frac-1}-PTR^}\right] ^ . \end$$

(33)

Based on equation Eq. (33), we observed that the Hill coefficient appeared as an additional factor in the exponent of the PTR. Indeed, numerical simulation shown in Fig. (5) revealed that Hill coefficients \(hill>1\) made \(MEF\left( PTR, g/d, hill\right)\) of the necessary unbound average concentration (15), or AUC exposure (16), for tumour stasis, highly dependent on PTR.

Fig. 5

The model efficacy factor MEF which describes the necessary IC50 coverage for \(c_\). We observed that the average concentration of the compound that is necessary to cover the in-vitro IC50 for stasis becomes increasingly dependent on the PTR for Hill coefficients \(>1\)

These findings indicate that adjusting the PTR of a compound, while keeping its total exposure or average concentration constant, becomes more crucial in achieving an anti-tumor effect compared to merely changing the average concentration per se.

These observations can be easily understood. When compounds with the same average concentration have higher PTRs, their IC50 coverage in the anti-tumor response is less enduring according to the model in Eq. (30). Of note, this less enduring IC50 coverage can often not be compensated by higher \(c_\) values, once the Hill function Eq. (7) of the pharmacodynamic effect is saturated.Footnote 10

This PTR-dependent response effect is even more pronounced for higher Hill coefficients, which correspond to steeper logistic curves. Consequently, the PTR and the elimination half-time under stasis conditions become increasingly critical for the anti-tumor effect when drug co-operativity is increased (hill > 1).

We finally note that the argument of Sect. 3.5.1 remained valid and, hence, assuming a separate absorption and elimination did not change our results, and both relaxations were combinable (the effect of the absorption phase is accounted for in the PTR).

Understanding variability in the IVIVC by semi-mechanistic modelling

Using the model efficacy factor MEF(PTR, g/d) and the results of Fig. (5), we now can aim to explain a source of variability in the empirical relation between TGI, free IC50, and free average concentration.

Looking specifically into our data set, we observed higher variability at stasis conditions (TGI=100) for different xenografts (upper graph) than for different compounds (lower graph) as seen in Fig. (6). These findings were consistent with our theoretical analysis above that the g/d ratio was more important than the PTR for the necessary IC50 coverage at tumor stasis.

Fig. 6

Variability in the empirical relation between TGI, free IC50 and free average concentration for the data set of Fig. 1 stratified according different xenograft types (having different g/d ratios) and different compounds (different PTRs; NB: compound IC50s are accounted for in the abscissa)

However, since certain compounds were preferably studied in specific xenografts, both parameters were not completely independently studied, and a thorough analysis of the influence of PTR and g/d awaits further experimental studies.

Extension of the method for studying efficacy in clinical populations

We finally sketch how our IVIVC considerations may be applied for studying drug efficacy in human populations, such as informing dosage decisions in clinical trials.Footnote 11

We therefore assumed that each human tumour may be characterised by a certain IC50, and certain growth and decay rates. We first assumed that we can obtain a distributions of free IC50 from an in-vitro tumour cell panel that reflects the sensitivity of individual tumour cells to a specific compound (in a patient population and/or within an individual tumour).

Furthermore, we assumed that g/d rates of individual human tumours can be estimated from clinical data such as using longitudinal PET scans of tumours treated with compounds with comparable mode of action. As example, such longitudinal PET scan data of patient tumours are provided by consortia like the Project Data Sphere Initiative [22, 23].

Postulating that growth and decay rates are in-vivo specific parameters independent from the compound‘s in-vitro pharmacology (as in the pre-clinical case), we hence used above derivations for a human clinical dose estimation.

Replacing now single point estimates for IC50 and g/d with statistical variables \(\left[ \textbf\right]\) and \(\left[ \mathbf\right]\), we obtainedFootnote 12

$$\begin \textbf_} = \tau \,\cdot \, \frac \frac\right] }-1}\right] }} \frac\right] }} \, . \end$$

(34)

With this, we obtained with \(_\textbf}\) a dose distribution to treat a population of individual patients. As can easily been seen from Eq. (34), no tumour regression can be seen for patients with \(g\ge d\) (no positive doses). Notably, from Eq. (34) we inferred that, if the statistical distributions of tumour properties \(\textbf\) and \(\textbf\) of a given population have same medians, the maximum population response would be limited to 50%, independent of the compound’s potency.

We hence suggest that this formalism can describe a tumour idiosyncratic treatment resistance mechanism that is wired in the individual in-vivo tumour growth and decay rates and that cannot be overcome by increased dosing.

DiscussionGeneral learnings

This paper presents a theoretical analysis of the Mayneord-like model, thereby investigating the drug-induced response of tumours and its connection to tumour stasis. Through our analysis, we have confirmed that the model framework is in line with an empirical in-vivo to in-vitro correlation (IVIVC). This correlation relates the free plasma concentration or exposure of a specific compound and its corresponding free in-vitro IC50 to the inhibition of tumour growth (TGI) using a simple formula [3, 4].

Specifically, we have shown that under the assumption of linear pharmacokinetics and the free plasma drug hypothesis, tumour stasis is essentially driven by the IC50 coverage of the unbound plasma drug concentration \(c_ / IC50_\) (or, equivalently, coverage of time integrated exposure \(AUC_/IC50_\)), the compound’s PTR, and the ratio g/d between the xenograft-specific tumour growth and decay rate. Our results will have impact on selecting appropriate xenograft models for proper clinical translation and on the understanding of how co-operativity of drug actions (\(hill>\)1) can determine \(c_\), \(c_\) and \(c_\)-drivenness.

Several further assumptions were made and the following specific conclusions were derived.

The mayneord-like model assuming linear tumour growth

Our analysis is centered around the unique characteristic of the Mayneord-like model, which assumes linear tumour growth over time. This model was derived from in-vivo analysis involving whole tumour resection in Jensen’s rat sarcomas, comparing untreated tissue (control) with tissue treated with X-radiations [16]. The study revealed that tumours grow alongside a rim, characterized by a necrotic core that lacks proliferation. Therefore, mathematical analyses have confirmed the intuitive assumption of sub-exponential growth dynamics, specifically zero-order growth.

It is important to note that certain idealizations were made during the mathematical analysis. For instance, the assumption of an infinitesimally small tumour rim and a tumour that can be mapped on a spherical shape, which may not hold true in experimental settings. Factors such as tumour space limitations, increased tumour vascularization, and changes in immune activity can also influence the growth profile in various ways.

Nonetheless, the Mayneord-like model has proven to be valuable in studying pre-clinical and clinical tumour growth over the past decade [8, 24]. As similar approaches, researchers such as Wang and colleagues, have utilized nonlinear mixed effect models with linear growth (and exponential decay) to describe response data in non-small-cell lung cancer patients [25]. Additionally, other models have been developed based on the assumption of sub-exponential tumour growth over time [26]. However, since exponential tumour growth, specifically when relatively slow (as observed in pre-clinical studies and slower-growing human tumours), can always be approximated using Taylor series expansion of tumour volume or radius, our analysis provides a reasonable approximation. Certainly, further experimental studies in pre-clinical and clinical settings that incorporate imaging and histology would be beneficial for understanding tumour growth dynamics in more detail.

A second specific feature of the Mayneord-like model is the distinction between in-vitro pharmacology (defined by the Hill function with an IC50 value determining the inhibition of cell proliferation) and in-vivo-specific tumour growth and decay rates. These factors establish a connection between the in-vitro pharmacology of a specific compound and in-vivo tumour growth.

While these factors are specific to a particular xenograft model, they are generally assumed to be mostly independent of the compound’s pharmacological profile, at least within a compound class with a similar mode of action. Various such in-vivo characteristics that cannot be captured by in-vitro cell assays include in-vivo tumour growth/aggressiveness, tumour drug exclusion, tumour cell-stroma interaction, and immune modulation in the presence and absence of treatment. Condensing these effects into just two parameters is undoubtedly an oversimplification, albeit one that has proven to be frequently useful in our pre-clinical research.

Pharmacokinetics and posology assumptions

For our investigation, we have made certain assumptions regarding the pharmacokinetics (PK) of the compounds. Specifically, we have assumed dose-linear PK, where the drug concentration in the body increases proportionally with the dose. We have also considered a single daily dose and steady-state kinetics, which involve separate first-order absorption and elimination kinetics. While it is true that many compounds exhibit exposure-limited effects at certain doses, assuming dose-linear PK is not overly restrictive for our analysis purposes. In fact, it is common practice to pre-select compounds during the pre-clinical stage to cover a range of exposures with dose-linearity that can achieve the desired effects in various xenograft models.

In contrast, assumptions about the shape of the pharmacokinetics may be more restrictive. Pre-clinical and clinical pharmacokinetics are complex and often require systematic physiology-based pharmacokinetics modelling (PBPK modelling). This involves integrating several effective compartments (organs) and considering the complex topology of their interconnections, as well as individual metabolic parameters specific to mice or humans [27]. Even if we assume a simple one-compartment model with absorption, obtaining an exact analytical solution for our analysis would require integrating a Bateman function of pharmacokinetics. However, the sum of two terms in the Bateman function would make the Hill function of Eq. (7) too complex for an analytical solution to our problem.

However, in oncology projects, pharmacokinetic (PK) variations often tend to be smaller compared to variations in pharmacodynamic (PD) parameters. This means that the distribution of IC50 values (a measure of drug potency) and the variability of growth and decay rates across tumours (g/d) are potentially more important drivers than the PK variations introduced by our assumption. This smaller dependence of efficacy on PK variations together with the fact that absorption phase (typically 0.5-2 h) and elimination phase (4-8 h) for our compounds were well separated made this assumption of a simpler two-exponential kinetics model reasonable for the class of MAPK inhibitors under investigation and potentially other small molecule compounds.

We finally note that results can be directly applicable to multiple daily doses per day, as long as doses are given in regular intervals. Thereby, the parameter \(\tau\) duration of the dosing interval has to be adapted.

Non-co-operativity in the mayneord-like model leads to AUC-driven effects

Based on the assumptions of dose-linear pharmacokinetics (PK), non-cooperativity in the Mayneord-like model (hill = 1), and a one-compartment PK model with timely separable absorption and elimination phases, our analysis has revealed that the effective dose for tumour stasis is proportional to the fraction of free, dose-normalized area under the curve (AUC) of the compound and the free in-vitro IC50.

An interesting finding was that the necessary IC50 coverage required for tumour stasis shows only mild dependence on the peak-to-trough-ratio ratio (PTR) of the compound and, consequently, on the shape of the dose-response curve. This implies that the relationship between dose and effect is primarily driven by the total free exposure (or average concentration) of the compound, which aligns with the concept of an "AUC-driven" effect. Furthermore, this effect appears to be largely independent of the choice of xenograft model and the specific properties of the compounds, as long as the assumptions of kinetics and non-cooperativity hold true. It is important to note that the xenograft growth-to-decay ratios (g/d values) should fall therefore within the typical range of 0.3 to 0.7.

As shown in Sect. 3.5.2 this relation, however, changes once we assume \(hill\,\ge \,1\). Specifically, higher Hill coefficients lead to an increased influence of the compound’s PTR, and hence its terminal half-life on the IC50 coverage factor for achieving tumour stasis. Our results therefore argue for dose-fractionation studies (with same daily doses, but different posology) to assess potential \(c_/c_\)-drivenness, specifically in such cases.

Analysing non-pharmacology mediated resistance

As stated above, a key feature of the Mayneord-like model is the separation of pharmacology-dependent (Hill function) and xenograft model-dependent (and pharmacology independent) in-vivo properties. Essentially, this effect is captured by the model-dependent factor MEF(g, d, PTR) whose influence of the compound is only given by its PTR and this influence is smaller for broad PTR variations compared to varying g/d ratios when using different xenografts. These results emphasise the importance of choosing the right xenograft model for clinical translation.

Importantly, the model-dependent factor MEF incorporates a limitation where tumour stasis cannot be achieved if the tumour growth parameter (g) is greater than the model-corresponding decay parameter (d). This implies the existence of an in-built in-vivo resistance mechanism, where certain xenograft models would not respond to treatment regardless of the IC50 coverage. A typical mechanism of this solely vivo effect, may be inaccessibility of tumour parts to the compound or resistance through tumour immune editing. This analysis is facilitated by the population formalism in Eq. (34), where we have distinguished between pharmacology resistance (ineffective IC50-coverage) and tumour idiosyncratic resistance (\(g \ge d\)).

This idea of tumour idiosyncratic resistance could be beneficial when transferring these modeling attempts to clinical scenarios. Specifically, we would obtain g/d ratios from longitudinal PET scans of individual patient tumors in clinical trials from dedicated repositories such as Project Data Sphere and integrate them into our model framework. Thereby, the developed model framework would enable mechanistic comprehension of potential resistance mechanisms. This type of clinical back-translation is ongoing in our group.

Applying findings to other therapeutic areas

Although the Mayneord-like model was designed for tumor growth, our results could be applicable to fields beyond oncology, where a pathological effect (here tumor growth) is counteracted by a compound-induced treatment. This generalization is valid as long as the treatment can be represented by a Hill function, possesses in-vitro efficacy parameters, and adheres to the pharmacokinetic assumptions mentioned earlier. Consequently, we can substitute our assumptions of pathological deviation and treatment with an alternative in-vivo efficacy model that includes specific clinical remodeling parameters, in-vitro pharmacological effects, and clinical outcomes tailored to the pathology under study.