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We formulate a dynamic model of vascular tumor growth, in which the interdependence of vascular dynamics with tumor volume is considered. The model describes the angiogenic switch; thus the inhibition of the vascularization process by antiangiogenic drugs may be taken into account explicitly. We validate the model against volume measurement data originating from experiments on mice and analyze the model behavior assuming different inputs corresponding to different therapies. Furthermore, we show that a simple extension of the model is capable of considering cytotoxic and antiangiogenic drugs as inputs simultaneously in qualitatively different ways.

Neovascularization means the formation of new blood vessels. Angiogenesis, an important form of neovascularization, is characterized by hypoxia-driven sprouting of new capillaries from postcapillary venules. This mechanism plays an important role in many physiological (e.g., wound healing [

Lately, much has been revealed about the details of tumor-induced angiogenesis and the underlying biochemical and biomechanical regulatory processes. These studies served as basis for the development of targeted molecular therapies [

Bevacizumab (Avastin) is a pharmacotherapeutic antiangiogenic agent developed to withhold pathological angiogenesis [

In [

On the other hand, based on the new data and paradigms brought to light in biological studies on angiogenesis, computational modeling of tumor-related vasculature development has became popular in the last decades, producing numerous computational models describing tumor growth and tumor-induced angiogenesis under different physiological circumstances (for a review on mathematical modeling of angiogenesis, see [

A large part of the aforementioned models exhibit a quite high level of complexity, which implies that while they may be potentially appropriate for the comparison of different therapeutic approaches under the assumption of a given parameter set, it can be challenging to fit them to single patients. Furthermore, exact therapy optimization may be computationally infeasible if one relies on complex and spatially detailed models as [

Feedback control [

Recently, a simple dynamical model of tumor growth and the effect of the antiangiogenic drug bevacizumab has been published [

First, as it does not include the description of vasculature dynamics, its drawback is that it is unable to interpret advanced measurement data corresponding to tumor and vasculature evolution dynamics, potentially available in the foreseeable future. Recently, several imaging techniques have been described, which allow the reconstruction of vascular microstructures: Doppler optical frequency domain imaging [

Second, minimal models not including vasculature dynamics as [

Third, paper [

Another paper that aimed to formulate a control-oriented dynamical model was also recently published [

According to the above preliminary results and considerations, our aim in this paper is as follows. We formulate a dynamic model, which includes the dynamics of the vasculature volume and describes the interplay between the tumor and vasculature volumes. To achieve this, we also include the dynamics of TAF in the model, which is produced by unsupported tumor and initiates the formation of new blood vessels from existing ones. This way our model will be capable of the interpretation of measurement corresponding to tumor and vasculature evolution dynamics. Furthermore, we aim to formulate a model, the predictions of which are also more acceptable in the therapy-free case.

In the following, we introduce the state-space variables of the model and interpret the state equations with the discussion of the model assumptions. Afterwards, the model parameters, their estimation, and the contextualization of the resulting parameters are described.

The state-space variables of the model are summarized in Table

State variables of the model.

Notation | Variable | Dimension |
---|---|---|

V | Proliferating tumor volume | |

N | Necrotic tumor volume | |

B | Vasculature volume in the tumor | |

T | Concentration of TAF in the tumor | mg/ml |

I | Inhibitor serum level | mg/ml |

We assume a simple spherical tumor. This simplifying assumption (one-dimensional growth in other words) is widely used in the tumor modeling literature (see, e.g., [

The auxiliary variable

The supported ratio of the tumor (

The function

Now let us return our focus to the auxiliary variable

Now, as we have discussed the interpretation of the auxiliary variables and the corresponding assumptions, we may return to the state equations. Equation (

Equation (

Equation (

Equations (

The variable

Some parameters of the model were taken from the literature (see Table

Therapy 1 (protocol-based treatment): five mice (mice C1-C5) were injected with 0.171 mg/ml bevacizumab at day 3 of the treatment (day 0 is considered as the day of the tumor implantation) (see Figure

Therapy 2 (daily, quasi-continuous small amount administration): nine mice (mice E1-E9) received

Nominal model parameters (

Notation | Dimension | Value | Source | |
---|---|---|---|---|

| | 1.4353 | PE | |

| - | | PE | |

| - | | PE | |

| | | PE | |

| | | PE | |

| | | PE | |

| | | PE | |

| | | PE | |

| | | [ | - |

| | | [ | - |

| | | [ | - |

(a) The measured tumor volumes for mice C1-C5 that received therapy 1 and their average. (b) The measured tumor volumes for mice E1-E9 that received therapy 2 and their average.

The nominal parameter set of the model was determined using the average of measurements as reference and minimizing the mean square error of the deviance between the simulated and measured total volumes using the combination of particle swarm global optimization method [

In order to potentially achieve a global optimum with the resulting parameter set, the estimation procedure was started from several initial coordinates in the parameter space. During parameter estimation, the average measurement results of both protocols were used to capture the qualitatively different response of the system to different inputs. Table

In addition, to quantify parameter variance in the context of single trajectories, the model was fitted also for single growth curves as described in the Appendix.

Most estimated parameters of the model are hard to measure individually (in fact some of them are only interpreted inside the framework of this model) and no data are available on them which could serve as basis for comparison.

In this subsection, we compare the model behavior and parameter values to measurement data. Figure

Measured and simulated tumor volumes in the case of therapy 1 (a) and therapy 2 (b).

The possible reason for the better fit in the case of the quasi-continuous therapy may be that the average of the measurements provides a more smooth, exponential-like curve in this case, which was possibly more easy to be achieved by the model.

The final volume of the tumor according to simulation results is 6670

In this subsection, we analyze and compare the dynamic behavior of key model variables in the cases of no therapy, therapy 1, and therapy 2 defined in Section

Figure

(a) Ratio of the well-supported tumor cells (

In general, it can be said that, in the beginning, when the tumor is small, due to the high ratio of periphery cells,

Concentration of the inhibitor (

Figure

(a) Concentration of TAF (

In this figure (Figure

Since the volume trajectories of the model follow an exponential-like fashion in all cases, they are maybe not so informative as, for example, the plot of

Volume trajectories of the model (living volume

In phase I of the experiments described in [

Measured and simulated tumor volumes in the case of no therapy.

The measure of the fit introduced in (

As in control applications, for which the current model is primary proposed, the usual aim is to minimize the final volume of the tumor (under, e.g., constraints corresponding to the total applied drug quantity); it is important to compare the final tumor sizes. As the final day of the experiments was different in the no-therapy case and therapies 1 and 2, Tables

Simulated and measured average total volumes on day 21 (

Therapy | | |
---|---|---|

No therapy | 1.03 | No data |

Therapy 1 | 6.67 | 6.60 |

Therapy 2 | 3.99 | 3.26 |

Simulated and measured average total volumes on day 19 (

Therapy | | |
---|---|---|

No therapy | 4.68 | 6.15 |

Therapy 1 | 3.10 | 4.44 |

Therapy 2 | 2.13 | 2.03 |

For the sake of comparison to previous literature results, let us note that the simulated volume on day 19 assuming no therapy is

In the following two subsections, we present some results related to the parameter sensitivity and structural identifiability of the proposed model.

In this subsection, we analyze the parameter sensitivity of the model for the estimated parameters. The sensitivity analysis is an important tool to characterize how the model parameters affect the simulation output. The presence of very large differences in the sensitivities of parameters may point to identifiability problems.

In order to formalize this analysis, we define the sensitivity measure detailed in the following equation:

In (

Sensitivities (

-20% | -10% | -5% | +5% | +10% | +20% | |
---|---|---|---|---|---|---|

| 0.1321 | 0.0240 | 0.0052 | 0.0040 | 0.0143 | 0.0462 |

| 0.4715 | 0.1601 | 0.0467 | 0.0640 | 0.2996 | 1.6456 |

| 0.0653 | 0.0173 | 0.0044 | 0.0047 | 0.0193 | 0.0821 |

| 0.0072 | 0.0017 | 0.0004 | 0.0004 | 0.0016 | 0.0061 |

| 0.2287 | 0.0769 | 0.0224 | 0.0309 | 0.1454 | 0.8147 |

| 0.2287 | 0.0769 | 0.0224 | 0.0309 | 0.1454 | 0.8148 |

| 0.5561 | 0.0868 | 0.0176 | 0.0120 | 0.0405 | 0.1185 |

| 0 | 0 | 0 | 0 | 0 | 0 |

First, it is conspicuous that the sensitivity to the parameter

Second, we can see in Table

Third, the model is the most sensitive to the parameter

Sensitivities (

-20% | -10% | -5% | +5% | +10% | +20% | |
---|---|---|---|---|---|---|

| 0.2926 | 0.0525 | 0.0114 | 0.0086 | 0.0305 | 0.0975 |

| 0.9383 | 0.3259 | 0.0962 | 0.1353 | 0.6422 | 3.6265 |

| 0.1375 | 0.0368 | 0.0095 | 0.0102 | 0.0420 | 0.1807 |

| 0.0181 | 0.0043 | 0.0011 | 0.0010 | 0.0039 | 0.0149 |

| 0.4498 | 0.1518 | 0.0442 | 0.0611 | 0.2889 | 1.6242 |

| 0.4594 | 0.1559 | 0.0456 | 0.0634 | 0.3011 | 1.7086 |

| 0.9364 | 0.1493 | 0.0305 | 0.0213 | 0.0720 | 0.2128 |

| 0.0100 | 0.0023 | 0.0005 | 0.0005 | 0.0019 | 0.0069 |

Table

Sensitivities (

-20% | -10% | -5% | +5% | +10% | +20% | |
---|---|---|---|---|---|---|

| 0.1304 | 0.0230 | 0.0050 | 0.0038 | 0.0131 | 0.0415 |

| 0.4012 | 0.1422 | 0.0427 | 0.0620 | 0.3015 | 1.7891 |

| 0.0582 | 0.0157 | 0.0041 | 0.0044 | 0.0185 | 0.0798 |

| 0.0090 | 0.0021 | 0.0005 | 0.0005 | 0.0019 | 0.0072 |

| 0.1232 | 0.0402 | 0.0116 | 0.0155 | 0.0722 | 0.3971 |

| 0.1332 | 0.0448 | 0.0131 | 0.0179 | 0.0859 | 0.4902 |

| 0.1690 | 0.0288 | 0.0061 | 0.0045 | 0.0154 | 0.0473 |

| 0.0135 | 0.0030 | 0.0007 | 0.0006 | 0.0023 | 0.0081 |

Apart from this, the results are similar to the case of therapy 1.

Altogether, based on the results of the sensitivity analysis, it can be said that further experiments focusing solely on pharmacokinetics of the applied drugs are desirable to estimate the parameter

Structural identifiability properties of a system describe whether there is a theoretical possibility for the unique determination of system parameters from appropriate input-output measurements or not. It is important to emphasize that identifiability is a property of the model structure. Basic early references for studying identifiability of dynamical systems are [

First, let us note that as our model uses the variable

Second, let us consider the factors that make the structural identifiability analysis challenging in our case. Structural identifiability methods usually rely on iterative computation of (Lie-) derivatives of the output (see, e.g., [

For identifiability analysis, let us consider a reduced version of the proposed model, which assumes no input (no antiangiogenic drug is present). The simplified form of the model is described by (

In this case, if only the total volume may be measured (as in the case of our measurements used for the parameter estimation),

Based on the above considerations, for the structural identifiability analysis, we use the freely available GenSSI [

According to the results of this software, the parameter ^{th}-order Lie-derivatives, which has been proven to be the computational limit in our case.

Nevertheless, let us discuss this topic a bit further from the point of view of possible future measurements with regard to the proposed model. In the recent years, multiple imaging techniques have been developed, which allow the 3D reconstruction of vascular microstructures: Doppler optical frequency domain imaging [

If these methods will be applicable in the case of animals used in the experiments, total tumor volume ^{th} order.

The complete identifiability tableaus of the reduced model are depicted in Figure

Complete identifiability tableaus of the reduced model in the case when the output is

Based on the above, it may be suspected that the model will have beneficial properties shall it be fitted for measurements planned to be carried out in the foreseeable future.

In the clinical practice, antiangiogenic drugs are often used together with conventional cytotoxic substances. In this setup, while the cytotoxic agent enhances the degeneration/necrosis of tumor cells, the antiangiogenic drugs are resposible for cutting the tumor from metabolic support via the inhibition of angiogenesis. Several results have been published recently corresponding to these combined therapies [

Models with predictive power regarding the efficiency of combined therapies and model-based optimization of such treatments are not prevalent in literature. Some initial results on the optimization of combined therapies are described in [

As the proposed model is taking into account vasculature and tumor cell dynamics in a differentiated way, it is able to distinguish between qualitatively different inputs related to different therapeutic agents. As a consequence, the proposed model may be easily extended to consider not only angiogenic drugs but also cytotoxic drugs. Let us consider the following modified state-space model described in the following equations:

First, the new equation (

The effect of the cytotoxic drug is modeled in this case as an enzymatic reaction, in which the cytotoxic drug acts as an enzyme, turning living cells to necrotic cells. This mechanism is described by the term

This way the effects of the two drugs are considered in qualitatively different ways in the model. While the antiangiogenic drug acts explicitly on the formation of new blood vessels by binding to TAF and thus inhibiting angiogenesis, the cytotoxic drug acts as an enzyme, driving living tumor cells to necrosis, independent of the actual vascular state of the tumor.

In this article, we formulated a dynamic model of vascular tumor growth, which accounts for the vasculature and TAF concentration development of the tumor and thus is able to reproduce the phenomenon of the angiogenic switch. We validated the model against volume measurement data originating from experiments on mice and found that the model provides a good fit for tumor volume data in both cases of the two analyzed therapies. The extension of the model described in Section

When comparing the proposed model to literature results, we may state the following. Regarding the model in [

Comparing the model described in the current article to [

Regarding future work, in the framework of the project Tamed Cancer (ERC grant agreement number 679681), animal experiments (mice) aiming to characterize the vasculature development during tumor growth are planned in the near future. These experiments will provide reference data for both vasculature volumes and tumor volumes, so we will be able to fit the model in either dimension against experimental data. This will allow further validation, refinement, or recalibration of the model.

Experiments regarding the efficiency of various combined therapies are also expected in the future, which will serve as reference scenarios regarding the identification of the extended model described in Section

Once the model is identified and validated from multiple aspects, studies on therapy optimization in open-loop and closed-loop setup will take place.

In this appendix, we detail the fitting of the model to the individual trajectories corresponding to single mice in the case of either therapy 1 or therapy 2.

Fitting the model to individual trajectories in the case of therapy 1: simulated output (with the parameters obtained from fitting the model to the specific trajectory (

Fitting the model to individual trajectories in the case of therapy 2: simulated output (with the parameters obtained from fitting the model to the specific trajectory (

In every case, the simulated output assuming the nominal parameters detailed in Table

Model parameter values resulting from fitting to individual trajectories in the case of therapy 1 (mice

Parameter | | | | | | Scale |
---|---|---|---|---|---|---|

| 0.0336 | 0.0127 | 0.0289 | 0.0374 | 0.0246 | |

| 1.4336 | 1.4454 | 1.55 | 1.435 | 1.4154 | |

| 1.4403 | 1.9642 | 1.4604 | 2.2084 | 1.2167 | |

| 0.0838 | 0.1684 | 0.0737 | 0.0739 | 0.0791 | |

| 4.4337 | 5.5455 | 10.2252 | 4.4032 | 4.4909 | |

| 1.8021 | 0.5612 | 0.7111 | 1.9291 | 1.4795 | |

| 0.2762 | 0.3917 | 0.2517 | 0.1173 | 0.3162 | |

| 0.1995 | 0.1637 | 0.1701 | 0.2114 | 0.1369 |

Model parameter values resulting from fitting to individual trajectories in the case of therapy 2 (mice

Par. | | | | | | | | | | Scale |
---|---|---|---|---|---|---|---|---|---|---|

| 0.0322 | 0.0393 | 0.0331 | 0.0184 | 0.039 | 0.0179 | 0.0422 | 0.0223 | 0.0331 | |

| 1.622 | 0.9568 | 1.436 | 1.433 | 1.408 | 1.442 | 1.634 | 1.609 | 1.432 | |

| 1.463 | 2.01 | 1.373 | 2.026 | 1.195 | 2.004 | 1.173 | 0.9805 | 0.6332 | |

| 0.0933 | 0.0552 | 0.1056 | 0.0567 | 0.0618 | 0.1236 | 0.0933 | 0.0425 | 0.1056 | |

| 17.34 | 18.67 | 18.08 | 18.66 | 18.66 | 18.08 | 18.08 | 18.65 | 18.08 | |

| 0.444 | 0.6039 | 0.4612 | 0.4512 | 0.923 | 0.1138 | 0.6959 | 0.335 | 0.9693 | |

| 0.2834 | 0.2657 | 0.4071 | 0.3201 | 0.3063 | 0.4105 | 0.3123 | 0.3311 | 0.4066 | |

| 0.2449 | 0.1011 | 0.2605 | 0.2546 | 0.259 | 0.2496 | 0.1749 | 0.2643 | 0.2603 |

Table

Standard deviation (STD) of the estimated parameters, regarding fitting to individual trajectories compared to their nominal value in %.

Parameter | STD (%) |
---|---|

| 10.96 |

| 23.40 |

| 39.60 |

| 33.33 |

| 52.64 |

| 47.20 |

| 31.40 |

| 32.57 |

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement no. 679681).

simulate_MMAGS.m is the file that performs the simulation of the model in MATLAB; it uses the following files: inj_fnc_prot_1_discrete.m, description of the injection function in the case of therapy 1; inj_fnc_prot_2_discrete.m, description of the injection function in the case of therapy 2; f_gamma.m, implementation of the function ∖gamma detailed in the article (see (