Drug delivery: Experiments, mathematical modelling and machine learning

Abstract

We address the problem of determining from laboratory experiments the data necessary for a proper modeling of drug delivery and efficacy in anticancer therapy. There is an inherent difficulty in extracting the necessary parameters, because the experiments often yield an insufficient quantity of information. To overcome this difficulty, we propose to combine real experiments, numerical simulation, and Machine Learning (ML) based on Artificial Neural Networks (ANN), aiming at a reliable identification of the physical model factors, e.g. the killing action of the drug. To this purpose, we exploit the employed mathematical-numerical model for tumor growth and drug delivery, together with the ANN - ML procedure, to integrate the results of the experimental tests and feed back the model itself, thus obtaining a reliable predictive tool. The procedure represents a hybrid data-driven, physics-informed approach to machine learning. The physical and mathematical model employed for the numerical simulations is without extracellular matrix (ECM) and healthy cells because of the experimental conditions we reproduce.

1. INTRODUCTION

Delivery and evaluation of the efficacy of drugs is a crucial step in anticancer therapy. Help in this context can be obtained from numerical simulations, possibly patient-specific. For this purpose, a computer model for tumor growth and a bio distribution model for the delivery of drugs to the tumor microenvironment are necessary. Of great help in developing such models is the framework of transport oncophysics [1], which states that physical properties of biological barriers control cell, particle, and molecule transport across tissues and this transport and its dysregulation play an overarching role in cancer physics. It is the physics of mass transport within body compartments and across biological barriers which differentiates cancers from healthy tissues, and cancer is seen as a proliferative disease of mass transport dysregulation. Transport aspects also control delivery of therapeutic agents e.g. chemotherapeutics or molecularly targeted therapeutics such as T cells, antibodies for immunotherapy, particles, multistage platforms. In facts probes that are used to investigate the mass transport properties of tissues acting as imaging contrast enhancers, can be used as directed vectors for the localized, preferential release of therapeutics into tumors. Particles and multistage platforms must pass through different and heterogeneous tumor and healthy compartments (e.g. vascular, stroma) with their respective barriers and distinct physical properties [2–6]. Drug delivery has been addressed previously in [7] with a theoretical model, and with numerical models in [8–13].

Our tumor growth model is of the multiphysics type and includes aspects of Thermodynamically Constrained Averaging Theory (TCAT) [14], where the model derivation proceeds systematically from known microscale relations to mathematically and physically consistent larger scale relations. The most recent versions, including also angiogenesis, are found in [15, 16]. Other growth models are of the multi-parameter type [17–21]. Multi-parameter models are based on mixture theory where the relevant balance equations are written directly at the level of interest and the thermodynamic consistency is satisfied at the same level. The biodistribution models are based on works in [22–26]. While for the growth [15–16, 27–28] and the bio-distribution model [22–26] the necessary parameters have already been obtained and are listed in the above cited papers, those for the drug action are still pending. The most important parameters are those related to the mass exchange due to drug uptake and the death rate of living tumor cells due to the action of the drug (killing mechanism). These parameters have to be obtained from experiments which are rather cumbersome. Further, they have to be iteratively adjusted (e.g. with least squares, linear regressions) to get plausible and realistic simulation results.

In this work, to address this issue, we combine in vitro experiments, numerical simulations and machine learning based on Artificial Neural Networks (ANNs). To exploit the mathematical model for the simulation of the drug action, we need to integrate it with a machine learning approach, which can give give some of the constitutive parameters characterizing the process.

The size of the data needed for ANN machine learning depends upon the features and the conditions of the problem at hand. In our case, the in vitro experiments generate a data set too small for an effective net training and a reliable identification of the physical model parameters. In these situations, the use of ANNs is almost a challenge [29]. In the following, we show that this obstacle can be overcome by a hybrid data-driven physics-informed approach, i.e. a combination of experimental data and simulations with a validated computational model. In detail, we limit ourselves to the tumor microenvironment where in our case the extracellular matrix ECM and healthy cells are absent. The reason is that spheroids in a culture medium grow initially without ECM, unless a decellularized ECM is present, see e.g. [30]. Only at a later stage there is deposition of ECM [31]. Healthy cells are missing because the spheroids have been grown in a culture medium without co-culture of a healthy tissue. The general model from which the application is extracted is however a four-phase model: one solid and the other three liquids where the solid phase would be the here missing ECM. We do not have interfaces because the model is written in the framework of porous media mechanics, exploiting Thermodynamically Constrained Averaging Theory TCAT to pass to the macroscale, where everything is smeared. We focus here on the death rate of living tumor cells due to the action of the drug.

The paper is outlined as follows: in Section 2 the mathematical model is presented, pointing out the enhancement due to the introduction of the drug as a new species, and in Section 3 the laboratory experiments are described. Successively, in Section 4 the experimental and numerical results are commented, and in Section 5 the machine learning approach, based on the Artificial Neural Networks, is outlined. Conclusions follow in Section 6 and the numerical details are given in the Appendix A at the end of the paper.

2. THE MATHEMATICAL MODEL

2.1. The multiphase tumor growth model

The model for the simulation of tumor growth is a continuum model based on porous media mechanics. The tumor is modelled as a four-phase system: one solid and three liquid. The extracellular matrix (ECM) is the only solid phase and constitutes the solid skeleton; the tumor cells, the healthy cells and the interstitial fluid are described as three different liquid phases. The porous medium is fully saturated. The scale of the tumor representation is the macroscale, sufficient to describe the tumor development while filtering out high frequency spatial variability. The relations at the macroscale are obtained by means of the Thermodynamically Constrained Averaging Theory (TCAT). TCAT consistently transforms microscale conservation and thermodynamic equations to the macroscale and converts averages of microscale derivatives into derivatives of macroscale average quantities. The final model is made of a set of equations for average quantities, where solid-fluid interfaces are not explicitly considered. A recent version of the governing equations has been derived in [15, 16] and here only the final form at macroscopic level is given. This model is enhanced with the introduction of the drug as a new species, to numerically simulate the effect of a therapeutic agent (chemotherapy, Nano Particles, etc.) within a growing tumor. The therapeutic agent is transported by the interstitial fluid, together with nutrients and other substances. The tumor cells grow by consuming oxygen and nutrients and reduce their growth by drug effect. Further, the tumor cells may become necrotic upon exposure to low nutrient concentrations or excessive mechanical pressure or drug action. The necrotic cells may then become part to the interstitial fluid by lysis process.

2.2. The governing equations

We present here the general model where in the following mass and momentum balance equations, α denotes a generic phase, t the tumor cells (TCs), h the healthy cells (HCs), s the solid phase (ECM), and l the interstitial fluid (IF). In the numerical implementation of this model it is possible to neglect the volume fractions of some phases, such as e. g. ECM and HCs, without compromising the results [31].

The voids of the porous scaffold (ECM) are filled by the fluid phases, IF, TC and HC. With a porosity of the scaffold ε, the volume fraction of the solid phase is ε^s =1- ε. The rest of the volume is occupied by the other phases: tumor cells (ε^t), healthy cells (ε^h) and interstitial fluid (ε^l). Because of the application the model is intended for, it is limited here to avascular tumor growth. The constraint on the volume fractions is:

$${\epsilon }^s+{\epsilon }^h+{\epsilon }^t+{\epsilon }^l=1$$ (1)

The mass balance equation for the ECM phase at macroscale is:

$$\frac{\partial \epsilon }{\partial t}=\nabla \cdot v^{\overline{s}}+\frac{(1-\epsilon )}{{\rho }^s}\frac{\partial {\rho }^s}{\partial t}-\nabla \cdot \left(\epsilon v^{\overline{s}}\right)-\frac{\underset{ECM}{\overset{l\to s}{M}}}{{\rho }^s}$$ (2)

where $v^{\overline{s}}$ and ρ^s are the velocity and the density of the solid phase, $\begin{array}{c}l\to s\\ M\\ ECM\end{array}$ is the mass exchange term between IF phase and ECM phase to take into account the possible deposition of ECM by tumor cells.

The mass balance equations of TCs and IF read:

$$\frac{\partial \left({\epsilon }^t{\rho }^t\right)}{\partial t}+\nabla \cdot \left({\epsilon }^t{\rho }^t{v}^t\right)=\begin{array}{c}l\to t\\ M\\ growth\end{array}-\begin{array}{c}t\to l\\ M\\ lysis\end{array}$$ (3)

$$\frac{\partial \left({\epsilon }^l{\rho }^l\right)}{\partial t}+\nabla \cdot \left({\epsilon }^l{\rho }^l{v}^l\right)=\begin{array}{c}t\to l\\ M\\ lysis\end{array}-\begin{array}{c}l\to t\\ M\\ growth\end{array}-\begin{array}{c}l\to s\\ M\\ ECM\end{array}$$ (4)

where v^t and ρ^t are the velocity and the density of the tumor cells phase, v^l and ρ^l are the velocity and the density of the IF phase. $\begin{array}{c}l\to t\\ M\\ growth\end{array}$ is the mass exchange term between IF and TCs phases and takes into account the consumption of oxygen to the tumor cells growth; $\begin{array}{c}t\to l\\ M\\ lysis\end{array}$ is the mass exchange term between TC and IF to represent the necrotic part of TC that became liquid because of lysis.

The mass balance equation of HCs is:

$$\frac{\partial \left({\epsilon }^h{\rho }^h\right)}{\partial t}+\nabla \cdot \left({\epsilon }^h{\rho }^h{v}^h\right)=0$$ (5)

We assume no exchange terms for the HCs, hence they are considered in homeostasis. The TCs are divided in two species, the living tumor cells (LTCs) and necrotic tumor cells (NTCs). Their mass balance equations are respectively:

$$\frac{\partial \left[{\epsilon }^t{\rho }^t\left(1-{\omega }^{N\overline{t}}\right)\right]}{\partial t}+\nabla \cdot \left[{\epsilon }^t{\rho }^t\left(1-{\omega }^{N\overline{t}}\right)v^{\overline{t}}\right]=\underset{growth}{\stackrel{l\to t}{M}}-{\epsilon }^t{r}^{Nt}$$ (6)

$$\frac{\partial \left({\epsilon }^t{\rho }^t{\omega }^{N\overline{t}}\right)}{\partial t}+\nabla \cdot \left({\epsilon }^t{\rho }^t{\omega }^{N\overline{t}}{v}^{\overline{t}}\right)={\epsilon }^t{r}^{Nt}-\underset{lysis}{\stackrel{l\to t}{M}}$$ (7)

where ${\omega }^{N\overline{t}}$ is the mass fraction of the necrotic tumor cells and $\left(1-{\omega }^{N\overline{t}}\right)$ is the mass fraction of living tumor cells; the reaction term ε^tr^Nt is an intra-phase exchange term and represents the death rate of tumor cells, i.e. the rate of generation of necrotic cells. It contains also the killing term which will be given below. The sum of the previous two equations (6) and (7) gives the mass balance equation for tumor cells (eq. (3) above). Since the experiments considered in this paper concern the growth of multicellular spheroids in vitro, we consider two transported species in the IF phase, the nutrient (oxygen) and the drug. The mass balance equation for the nutrient is:

$$\frac{\partial \left({\epsilon }^l{\rho }^l{\omega }^{nl}\right)}{\partial t}+\nabla \cdot \left({\epsilon }^l{\rho }^l{\omega }^{nl}{v}^l\right)-\nabla \cdot \left({\epsilon }^l{\rho }^l{D}_{\text{eff}}^{nl}\nabla {\omega }^{nl}\right)=-\stackrel{nl\to t}{M}$$ (8)

where $D_{eff}^{\overline{nl}}$ is the effective diffusivity of the nutrient species in the extracellular space, ω^nl the mass fraction of nutrient species n and $\stackrel{nl\to t}{M}$ is the mass of nutrient consumed by tumor cells via metabolism and growth.

For the drug we have:

$$\frac{\partial \left({\epsilon }^l{\rho }^l{\omega }^{drug}\right)}{\partial t}+\nabla \cdot \left({\epsilon }^l{\rho }^l{\omega }^{drug}{v}^l\right)-\nabla \cdot \left({\epsilon }^l{\rho }^l{D}_{\text{eff}}^{drqg}\nabla {\omega }^{drug}\right)=-\begin{array}{c}l\to t\\ M\\ drug\end{array}$$ (9)

in which ω^drug is the mass fraction of the drug, $D_{\text{eff}}^{drug}$ is the diffusion coefficient of the drug and $\begin{array}{c}l\to t\\ M\\ drug\end{array}$ is the mass exchange term between the drug and the tumor phase that accounts for the uptake of the drug by the tumor cells.

The last governing equation of the model is the linear momentum balance of the solid phase expressed in rate form as:

$$\nabla \cdot \left(\frac{\partial t_{eff}^{\overline{\overline{s}}}}{\partial t}-\overline{\alpha }\frac{\partial p^s}{\partial t}1\right)=0$$ (10)

where the interaction between the solid and the three fluid phases is accounted for through the effective stress $t_{eff}^{\overline{\overline{s}}}$ in the sense of porous media mechanics:

$$t_{eff}^{\overline{\overline{s}}}=t^{\overline{\overline{s}}}+\overline{\alpha }{p}^{s}1$$ (11)

where 1 is the unit tensor, $t^{\overline{\overline{s}}}$ is the total stress tensor in the solid phase, $\overline{\alpha }$ is the Biot’s coefficient $\overline{\alpha }=1-K/K_s$, with K the compressibility of the ECM material. In the modeled problem, K/K_s tends to be zero hence we can assume a Biot’s coefficient equal to 1. The solid pressure p^s is given as

$$p^s=S^h{p}^h+S^t{p}^t+S^l{p}^l=p^l+\left(1-S^l\right)p^{hl}+S^t{p}^{th}$$ (12)

where the solid surface fraction of each fluid phase in contact with the solid phase (Bishop parameter) has been taken equal to its own degree of saturation. In the equation (11) the normal stresses as well as the fluid pressures are positive for compression, as usually done in fluid mechanics.

The input data for the full model are: the domain geometry and the computational mesh; the initial and boundary conditions; the values of the physical governing parameters, such as: ECM permeability and stiffness, dynamic viscosity of the fluids, interfacial tensions, differential pressure-saturation relationship, oxygen consumption, cell growth and death rate, drug concentration and others. The outputs of the model are the distribution of the volume fraction for each phase such as the tumor cells; the mass fraction distribution for each species, such as the oxygen and drug concentration and the necrotic cells distribution; the pressure of TCs, IF and HCs; the displacement field, the stress and strains of the solid phase (ECM).

2.3. The constitutive equations.

Constitutive equations have been extensively dealt with in [15] and will not be repeated here. We concentrate our attention on the new constitutive equations needed for drug delivery.

The mass exchange term between the drug and the tumor phase in eq. (9) accounting for the uptake of the drug by the tumor cells has been chosen according to [32, 33] where the simplest kinetics for drug uptake (i.e. linear) was assumed:

$$\begin{array}{c}t\to l\\ M\\ drug\end{array}={\gamma }^{drug}{\omega }^{drug}\left(1-{\omega }^{Nt}\right){\epsilon }^t$$ (13)

with γ^drug a coefficient accounting for the drug uptake rate by living TCs. Another form of the mass exchange term may be obtained by assuming a linear relationship between the chemical potentials of the two involved phases l and t and a linear equilibrium isotherm, with ${\widehat{K}}_{lt}^{drug}$ and ${\widehat{B}}_{lt}^{drug}$ the respective proportionality coefficients:

$$\begin{array}{c}l\to t\\ M\\ drug\end{array}={\widehat{K}}_{lt}^{drug}\left({\omega }^{drug,\overline{l}}-{\widehat{B}}_{lt}^{drug}{\omega }^{drug,\overline{t}}\right)$$ (14)

with ${\omega }^{drug,\overline{l}}$ mass fraction of drug in the IF, ${\omega }^{drug,\overline{t}}$ mass fraction of drug in the tumor. However, this form would require an augmented number of mass balance equations. In the following simulations we have considered eq. (13) with a fixed value of γ^drug. The quantity of interest here is the killing action of the drug. It appears in the reaction term of the mass balance equations (6) and (7) of the tumor cell species expressing the death rate of living TCs and is given by the last term of the following equation:

$${\epsilon }^t{r}^{N^t}=-\left[{\gamma }_{necrosis}^t{\langle \frac{{\omega }^{\overline{nl}}-{\omega }_{crit}^{\overline{nl}}}{{\omega }_{env}^{\overline{nl}}-{\omega }_{crit}^{\overline{nl}}}\rangle }_{-}-{\delta }_a^{t}H\left(p^t-p_{necr}^t\right)\right]\left(1-{\omega }^{N\overline{t}}\right)\epsilon S^t-{\lambda }_{drug}{\omega }^{drug}\left(1-{\omega }^{N\overline{t}}\right){\epsilon }^{t}f$$ (15)

where the Macaulay brackets ⟨•⟩_- indicate the negative value of their argument, λ_drug is the rate of drug induced cell death, f is the activation function that assumes value 0 if the drug is not yet active, thus canceling the term for cell death induced by the therapeutic agent, and 1 after the activation of the drug. This activation function f allows to consider the drug activation time, i.e. the time when the drug does not affect the growth yet. The remaining terms have already been dealt with in [34].

2.4. Boundary and Initial Conditions.

Figure 1 shows the geometrical domain and the initial and boundary conditions reflecting the experimental setting described in the next section. Given the geometry of the physical problem addressed here, axisymmetric boundary conditions are imposed along the radial axes B₁, that is to say zero flux is imposed for all phases, nutrients and drug due to the radial symmetry (Neumann’s boundary condition). Additional boundary conditions are imposed on the outer contour B₂, where all input variables are fixed (Dirichlet’s boundary condition). The atmospheric pressure is the reference pressure. On B₂ and throughout the computational domain at t = 0, the pressure of the interstitial fluid is fixed to zero, the oxygen mass fraction is fixed to the value of ${\omega }_{env}^{nl}$ and the drug mass fraction is fixed to the value ${\omega }_{env}^{drug}$. The initial tumor volume fraction is 0.02 in the red area and equals zero in the blue area. We recall that we are modeling the growth of a tumor spheroid in vitro, hence in the whole domain the healthy cells and the ECM are absent and possible deposition of ECM by TCs is neglected; just the interstitial fluid is present in the culture medium with ε_l =1. The values of the governing parameters are summarized in Table 1.

Figure 1.

Geometry and boundary/initial conditions of the model

Table 1:

values of the model parameters considered for the numerical simulations

Parameter	Symbol	Value
Growth coefficient of tumor cells	${\gamma }_{growth}^t$	1.8·10−2
Necrosis coefficient	${\gamma }_{necrosis}^t$	0.7·10−2
Normal mass fraction of oxygen in tissue	${\omega }_{env}^{\overline{n}l}$	4.9·10−6
Diffusion coefficient for the drug	$D^{drug}$	9.375·10−14
Drug uptake rate by tumor cells	${\gamma }_{drug}^t$	1.157·10−2
Drug-induced cell death rate	${\lambda }_{drug}$	Variable
Mass fraction of drug	${\omega }^{drug}$	Variable

3. THE EXPERIMENTAL TESTS: TUMOR MULTI-CELLULAR SPHEROIDS OF GLIOBLASTOMA MULTIFORME

3.1. Description of the method

To elucidate more clearly the effect of drug transport within a solid tumor mass, multicellular spheroids were generated and used to mimic the 3D structure of actual malignancies. Indeed, the use of tumor spheroids over more conventional 2D cell monolayers significantly strengthens the clinical implication of the presented results. In tumor spheroids, drugs would need to first diffuse deep into the tumor mass, then enter cells and eventually deploy their cytotoxic activity. This scenario represents more closely the transport of drugs into the tumor tissue than 2D cell monolayers. The evaluation of drugs’ therapeutic efficacy was performed by measuring the growth of spheroids over time, analyzing the difference between treated and untreated spheroids.

3.2. Cell source and Materials

U-87 MG cells were purchased from American Type Culture Collection (ATCC) as a Glioblastoma Multiforme brain tumor model. Cells were cultured, according to manufacturer’s instructions, in Eagle’s Minimum Essential Medium (EMEM), supplemented with 10% of fetal bovine serum, both purchased from Gibco (Italy). Polydimethylsiloxane (PDMS) (Sylgard 184) was obtained from Dow Corning (USA). DTXL was purchased from Alfa Aesar (Germany).

3.3. Performed tests and results

1×10³ U-87 MG cells were seeded on a 96 well plate (GravityTRAP™ ULA Plate, in-sphero, Switzerland), treated to avoid cell adhesion. Then, cells were centrifuged at 200 x g for 2 minutes to facilitate their gathering at the center of the well and a formation of a single structure. After 24 hours, individual spheroids were transferred into poly(dimethylsiloxane) (PDMS)-coated 24 well-plate. This coating was used because PDMS does not provide a viable substrate for cell adhesion. Moreover, PDMS coating limits also the diffusion of liquid within its matrix, thus avoiding drug loss over time. Spheroids were treated with a chemotherapeutic agent, docetaxel (DTXL), at different concentrations, namely 1, 10, 50, 100, and 1000 nM, based on previous studies [35–37], documenting that the IC₅₀ of DTXL on a U-87 MG cell monolayer is in the order of few tens of nM. Notice that the presentation of drug molecules to cells in tumor spheroids is reduced as compared to that of a cell monolayer. Indeed, in the latter case, all cells in the culture are readily exposed to the drug molecules whereas in the case of cancer spheroids malignant cells deep into the core are exposed to limited amounts of therapeutic agents. This would justify the broad range of drug concentrations used here. The proper amount of DTXL needed to reach the desired concentration was added to the cell medium and then added to the wells containing the spheroids. The medium remained unchanged in the wells until the end of the experiment. As the IC₅₀ for a cell monolayer is conventionally quantified at 24, 48 and 72 hours post treatment initiation, the response of the spheroid was monitored for even longer times, all the way up to 8 days. At different time points (0, 1, 2, 3, 6, 7 and 8 days), each spheroid was analyzed by an inverted epifluorescent microscope (Leica). The analysis of microscopy images through ImageJ software determined the spheroids radius, considered as the more reliable measure to evaluate tumor spheroids growth over time (Table 2). Untreated spheroids (CTR in Table 2) were used as a comparison to establish drug efficacy.

Table 2:

Measured individual spheroids radius at all the examined time points (green background: data used for ANN training)

Spheroids Radius (μm)
Conditions	Day 0	Day 1	Day 2	Day 3	Day 6	Day 7	Day 8
CTR	126.122	136.6078	159.004	197.8372	322.1303	359.9451	389.9716
CTR	123.8085	134.4084	161.7739	184.1797	312.6216	349.0405	390.7251
CTR	120.6301	133.3662	158.9127	186.4775	310.7503	338.4063	385.0138
CTR	118.2563	136.5874	158.2087	186.5703	309.6708	352.167	386.4737
CTR	114.1155	131.9399	159.4771	183.4175	306.3293	343.4811	372.7906
CTR	118.826	137.203	155.2458	187.8463	316.8757	355.4906	389.7511
CTR	117.1562	130.2283	158.3286	191.1013	309.9987	344.0623	377.319
1 nM	123.9211	131.5714	146.4967	174.1615	254.6313	283.1415	308.4014
1 nM	120.5699	138.8126	155.4973	183.8369	266.2529	294.6937	329.3085
1 nM	118.1666	134.2962	151.917	185.6706	270.2188	301.3829	334.2042
1 nM	119.8782	131.495	157.1858	183.5939	263.6936	293.9237	317.4619
10 nM	127.3989	133.7841	150.6293	148.8401	153.1209	148.7427	145.4722
10 nM	124.4514	136.0264	142.9526	156.0598	160.0082	157.8875	149.0351
10 nM	128.5503	136.0921	143.708	154.0476	152.5951	149.8455	149.8046
10 nM	127.8406	137.3047	145.1764	152.1079	156.5026	156.3422	142.1265
50 nM	129.7732	140.4992	136.3011	136.6773	122.4395	127.5566	122.1155
50 nM	125.4698	139.9658	137.6336	130.4509	114.0078	121.841	110.6389
50 nM	120.9305	135.6525	134.6076	130.0524	122.79	117.4323	112.1471
50 nM	126.2547	136.2437	133.8508	137.7876	119.0888	115.9159	115.699
100 nM	122.8809	138.6235	133.5168	132.0245	124.7158	120.4403	115.8966
100 nM	126.3166	138.3656	136.6732	133.031	120.0689	113.8364	108.9973
100 nM	117.9065	130.9889	124.1731	130.7117	117.8734	116.7985	115.9592
100 nM	123.0851	132.6992	129.6957	127.9977	116.9417	113.905	108.5201
1 μM	127.6702	138.881	139.914	140.5508	129.6053	130.7929	131.0912
1 μM	116.9989	131.8299	133.6004	138.0991	126.8985	125.5277	123.0353
1 μM	118.5533	128.7195	133.8508	133.3286	130.4338	125.8962	122.517
1 μM	121.9462	140.1213	134.2838	138.6557	128.0979	127.1139	122.4532

4. MODEL RESULTS

The simulations presented in the following have a double aim: the validation of the model enhanced with the introduction of the chemotherapeutic agent, by the comparison of the results with the experimental data, as well as the model prediction of the drug influence on tumor growth, by comparison between the results of different drug concentration applications.

In the laboratory experimental data shown in Figure 2, we can see the growth of the untreated tumor spheroid (black line): the radius increases in time with an exponential initial phase and a following linear trend. The other curves show the effects of different concentrations of the chemotherapeutic agent on the tumor proliferation. The inhibition of tumor growth is not linear with drug concentration: the growth is significantly reduced till 50 nM of drug, for higher concentrations the efficacy decreases more and more, to have the same result for concentrations above 100 nM evidencing an effect of saturation. Further, till after the first day the various curves overlap and start to differentiate only around the second day of treatment, showing that the therapy needs an activation time to become effective.

Figure 2.

Experimental data for tumor growth with different concentrations of drug. Continuous lines represent the best fit.

Figure 3 shows the comparison between the experimental data and the simulations results. In detail, Figure 3a shows the untreated case and Figure 3b represents the case for the drug concentration of 1 nM and the drug activation after 1 day. Both simulations show a very good agreement with the real tests, the R² of the untreated and 1 nM concentration are 0.9969 and 0.9965 respectively. The validation of the model with and without drug is confirmed by the very good agreement between experimental and numerical results shown in Figures 3a, 3b. Figures 3c, 3d, 3e and 3f illustrate the cases for drug concentration respectively 10 nM and drug activation after 2 days, 50 nM and drug activation after 1 day, 100 nM and drug activation after 1 day, 1000nM and drug activation after 1 or 1.25 or 1.5 days. The different line colors indicate different values of λ_drug as indicated in the images. These Figures will be discussed in more detail in the next section.

Figure 3.

Experimental data and results of simulations carried out with the full numerical model. (a) shows the untreated case and (b) the case for the drug activation after 1 day and drug concentration of 1nM. The validation of the model is confirmed with these figures. (c) represents the case for drug concentration 10 nM and activation after 2 days, (d) drug concentration 50 nM and activation after 1 day, (e) concentration of 100 nM and activation after 1 day, (f) concentration 1000nM and activation after 1 or 1.25 or 1.5 days. Figures (c) to (f) evidence the difficulties encountered when trying to identify the parameters with the full mathematical model.

5. MACHINE LEARNING

The physics of tumor growth is extremely multifaceted so that the set of partial differential equations is necessarily complex to be able to catch the fundamental processes correctly. Furthermore, investigating the drug delivery involves additional features to be considered. To this point, as already mentioned, besides being able to formulate the proper equations for the drug action, it is necessary to obtain the involved parameters. In the following we focus on the drug killing action, expressed by the last term of equation (15):

$${\left({\epsilon }^t{r}^{Nt}\right)}_{kill}={\lambda }_{drug}{\omega }^{dnug}\left(1-{\omega }^{Nt}\right){\epsilon }^{t}f$$ (16)

we recall that λ_drug, ω^drug and f are the drug-induced cell death rate, the mass fraction of drug and the activation function, respectively. It is easy to understand that λ_drug is very difficult to obtain experimentally, or even not possible at all. However, in order to have good predictive performances, the numerical model must be supplemented by the constitutive parameters as close as possible to reality. Figure 3a illustrates how, in the absence of drug, the model is able to predict the growth curve of the spheroid. Figure 3b shows the case of the spheroid subjected to a low concentration of drug (1nM): by using standard interpolation techniques to determine the parameter λ_drug (e.g. least squares, linear regression), the model is able to reconstruct the entire growth curve and to fit very well the experimental data. In contrast, figures 3c-d-e-f show that, for higher drug concentrations, although using the same numerical techniques to identify λ_drug and iteratively trying to vary its value, it is not possible to fit the experimental data well. It is not accidental: for higher drug concentrations new phenomena take place, for example the activation time (introduced in the model by means of the function f), which makes it impossible to predict the λ_drug parameter with standard numerical techniques. In this framework, the use of machine learning is extremely helpful, because it is able to catch the more complex physics and therefore to identify the necessary parameters. In other words, although it is impossible to know how to define explicitly the relation between the involved quantities, by having at hand the set of corresponding input - output data, the mapping can be learned by machine learning methods. In the following, ANN-based machine learning will be used to obtain the suitable value of the drug-induced cell death rate parameter, as a function of the drug concentration ω^drug, time and tumor size. Concerning the drug uptake, in this work the drug concentration is kept constant and the activation term f is a stepwise function (its value is 0 or 1). It can be easily seen from Figure 2 that the drug uptake does not depend on the drug concentration: all curves overlap during the first day of observation. A more sophisticated modeling would consider the gradient of the drug concentration across the tumor, by dropping the activation function and using a variable ω^drug. This is the object of an on-going work.

Artificial Neural Networks are a mathematical - computational model that is inspired by the structure and functional aspects of biological neural networks. They are composed of a collection of nodes – also called neurons – gathered into layers. The nodes in the first layer contain the input data of the problem, the nodes in the last layer contain the desired output. Evidently, the number of nodes in those two layers depends upon the physics of the problem at hand. Between the input and output layers, nodes are organized in the so called “hidden layers”: the number of hidden layers and the number of nodes per each hidden layer is decided by the user. A sketch of the topology of an Artificial Neural Network applied to this problem is depicted in Figure 4.

Figure 4.

Topology of an Artificial Neural Network applied to this problem

5.1. Training of the Artificial Neural Networks

The first step of an ANN-based machine learning model consists in the neural network training. To this purpose a set of corresponding input-output data is needed, i.e. ω^drug, tumor radius R, growth time t and corresponding (known) λ_drug to approximate the functional dependence:

$${ℝ}^3\to ℝ:\left\{{\omega }^{drug},R,t\right\}\to {\lambda }_{drug}$$ (17)

As detailed in the following, we tried to use only the raw experimental data (Table 2), but it turned out that they were not sufficient to have a good prediction. Therefore, we needed to enrich the data set and it was possible by means of the mathematical model. For each in vitro experiment, we calculated more points than those actually recorded. Furthermore, we run in-silico experiments with other combinations of λ_drug and ω^drug. It is important to underline that in this phase a very accurate pre-analysis of the data is needed to compile a reliable training catalogue.

In detail, six in vitro tests were available, but the experiment performed with a concentration of the therapeutic agent of 1000nM showed an inhibition capability worse than the 50nM and 100nM concentration, revealing a saturation mechanism. As a consequence, it was excluded from the neural network learning set. Since the target was the identification of the drug induced death rate, also the control case (no drug) wasn’t considered. The spheroid radius was recorded at day 0, 1, 2, 3, 6, 7, 8 and the experimental tests were repeated four times for each drug concentration (Table 1). Therefore, from in vitro records, 4×7×4 items in total (see Table 1, green background) were available and they proved to be insufficient to train a network. By reproducing the in vitro tests with the computational model, we could obtain the tumor growth as a continuous function of time, so that we could consider more frequent observation, i.e. every six hours. In this way we collected a set of data made of 33 points for each of the four drug concentrations (ω=1, 10, 50 and 100 nM) considered in vitro (Figure 5). By training the network with these 132 items, the convergence was achieved, however the network did not show generalizing capabilities: it failed to predict λ_drug in recall mode. To this aim, different networks with several topologies were tested, but none of them succeeded in predicting the needed parameter. Consequently, we enriched the training set with purely in-silico tests, namely simulating the growth of the spheroid for other drug concentrations, i.e. ω= 3, 6, 20 nM. By adding these entirely numerical tests to the previous set, we could rely on a total of 231 input-output points. At this moment, we excluded the concentration of 1 nM to use it in recall mode and trained an Artificial Neural Network with 198 sample points, corresponding to 33 points for six considered drug concentration (ω= 3, 6, 10, 20, 50 and 100 nM). Among these, 20 patterns (called test set) randomly selected were set aside to test for network overtraining and to check the integrity of the model. With this database (training set: 178 patterns, test set: 20 patterns), the learning phase was successful, and the trained net performed very well. The root mean square error (RMSE) of both the training set and the test set was monitored during learning. The evolution of the training set RMSE indicates the velocity of the network learning, the frequency of instabilities and the general state of convergence (or divergence), while the evolution of the test set RMSE shows the possible overtraining status and how well the network responds to cases not contained in the training set. To avoid overfitting, the training phase was stopped just before the root mean square error of the test set started to increase. In Figure 6a it is visible that a local minimum has occurred in the test set error shortly after the beginning of the training, and it would have been an error to finish at that point because after some iterations the RMSE of the test set decreased stably.

Figure 5.

Data set used for ANN training. First, the in vitro points were used (single black marks, series indicated by “exp” in the legend). Second, numerical results for w=1, 10, 50 and 100 nM (marks within lines) were considered. Finally, ω=3, 6, 10, 20, 50, 100 nM (marks within lines) proved to be a suitable learning set, leaving 1nM drug concentration for the recall mode.

Figure 6:

Quality of the net training. (a) shows the evolution of the root mean square error and (b) of the correlation degree as a function of the number of learning iterations.

Concerning the neural network architecture, several combinations of topologies were also tested, depending upon the learning set used. For the cited example, a network with three hidden layers with 3, 2, 2 nodes respectively (ANN 3_3_2_2_1) gave the best results, reaching a root mean square error of 0.000256 for the training set and 0.000172 for the test set. The correlation degree (CD) was 0.999999 for both sets. The evolution of RMSE and CD as a function of the number of learning iterations are shown in Figure 6a and 6b respectively.

In summary, as presented above, the database size had a strong influence on the ANN learning and prediction. By using a small set, made only of the in vitro tests available (Table 2), convergence was not achieved because there were too few points. By using a rather bigger set of data (132 patterns), it was possible to train a net, but it failed in recall mode. It means that, for a net trainability, it was necessary to have more than 7 records for each concentration (33 showed to be a suitable set) and, for a net generalization capability, more than 4 different concentrations of drug were needed. Finally, with a reasonable size of the data set (198 learning patterns), very good results were obtained, as shown in the following section.

5.2. Recall mode

In Figure 7 the training target is compared with the response of ANN for learning patterns. The values of the drug-induced cell death rate are reported as a function of the input patterns. The stepwise aspect of the curve is due to the fact that for each experiment the value of λ_drug is constant. The dotted red line shows the target (i.e. the known output values, obtained by means of the experimental test and the numerical computations), while the single marks show the values predicted by the network at the end of the training: the green crosses refer to the training set output, the blue squares to the test set output. A very good agreement is evident, the error between the ANN predictions and the target is very small.

Figure 7:

Comparison between the target values (red dotted line) and the response of the network for the training set (green crosses) and the test set (blue squares).

Once the network is successfully trained, it can be used in recall mode to process new input patterns and obtain the parameters to be identified. To show the validity of the approach, Figure 8 (a) shows the ANN predictions for the case of drug concentration of 1 nM, which was not used for the net training. An accurate prediction was obtained: the expected value was λ_drug =5 10^6, the Artificial Neural Network prediction was 5.18 10^6. The comparison between the experimental data and the numerical results of the model obtained by using the drug-induced cell death rate identified by the ANN is shown in Figure 8 (b). A very good match can be noticed.

Figure 8.

(a) shows the comparison between the target values (blue line) and the predictions of the network (orange line). (b) shows the comparison between the experimental data (red marks) and the numerical results obtained by the full numerical model by using the value of the rate of drug induced cell death λ_drug identified by means of the neural network.

6. CONCLUSIONS

Efficient mathematical-numerical models for evaluating the efficacy of cancer drugs such as those developed within the framework of transport oncophysics, require reliable parameters which have to be extracted from laboratory tests. While determination of growth parameters has received much attention in the past, data needed for simulating drug delivery and efficacy have received much less attention. This is due to the fact that drug delivery models are much less frequent than growth models and because there is an inherent difficulty in extracting the necessary data from experiments. Such experiments are often tedious and above all yield only a limited set of data. To overcome this difficulty, we have combined real experiments, numerical simulations and ANN machine learning to obtain a reliable identification of the physical model parameters, in this case for the killing action of the drug. The physical and mathematical model for the numerical simulation is without ECM and healthy cells because of the experimental conditions we have reproduced. The results are encouraging and have shown that, by using a hybrid data-driven, physics-informed approach, based on tumor growth model computing and recurrent neural network running, we were able to mimic the complex processes of delivery and efficacy of drug.

APPENDIX A

Numerical solution and computational procedure.

The weak form of the governing equations is obtained by means of the standard Galerkin procedure and is then discretized in space by means of the finite element method [38]. Integration in the time domain is carried out by the Finite Difference Method adopting a quasi-Crank–Nicolson scheme (θ-Wilson method, with θ = 0.52). Within each time step the equations are linearized by the Newton-Raphson method.

For the numerical solution of the resulting system of equations, a staggered scheme is chosen, with iterations within each time step to preserve the coupled nature of the system. Five computational units are used in the staggered scheme: the first is for the nutrient mass fraction ${\overline{\omega }}^{\overline{nl}}$, the second for the drug mass fraction ${\overline{\omega }}^{\overline{drug}}$, the third to compute ${\overline{p}}^{\text{th}}$, ${\overline{p}}^{hl}$, ${\overline{p}}^l$, and the fourth is used to obtain the displacement vector ${\overline{u}}^s$.

The final system of equations can be expressed in matrix form

$$C_{ij}(x)\frac{\partial x}{\partial t}+K_{ij}(x)x=f_i(x)$$ (A.1)

where some of the coupling terms have been placed in the source terms and are updated at each iteration to preserve the coupled nature of the problem. The nonlinear coefficient matrices C_ij (x), K_ij (x) and f_i (x) are defined as

$$\begin{gathered} C_{ij}=\left(\begin{array}{cccccc}{C}_{nn}& 0& 0& 0& 0& 0\\ 0& C_{dd}& 0& 0& 0& 0\\ 0& 0& C_{tt}& C_{th}& C_{tl}& 0\\ 0& 0& C_{ht}& C_{hh}& C_{tl}& 0\\ 0& 0& C_{lt}& C_{lh}& C_{ll}& 0\\ 0& 0& 0& 0& 0& C_{uu}\end{array}\right) \\ {K}_{ij}=\left(\begin{array}{cccccc}{K}_{nn}& 0& 0& 0& 0& 0\\ 0& K_{dd}& 0& 0& 0& 0\\ 0& 0& K_{tt}& K_{th}& K_{tl}& 0\\ 0& 0& K_{ht}& K_{hh}& K_{hl}& 0\\ 0& 0& K_{lt}& K_{lh}& K_{ll}& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right),f_i=\left(\begin{array}{c}{f}_n\\ f_d\\ f_t\\ f_h\\ f_l\\ f_u\end{array}\right) \end{gathered}$$ (A.2)

and $x^{\text{T}}=\left\{{\overline{\omega }}^{\overline{nl}},{\overline{\omega }}^{\overline{drug}},{\overline{p}}^{th},{\overline{p}}^{hl},{\overline{p}}^l,{\overline{u}}^{\text{s}}\right\}$. These matrices C_ij (x), K_ij (x) and f_i (x) are given in [34]. The modular computational structure allows for more than one chemical species to be taken into account, simply by adding a computational unit (equivalent to the first one used for the nutrient) for each of the additional chemical species considered. In this paper a new computational unit for the drug has been introduced; the terms marked with the index dd in the matrices C_ij (x) and K_ij (x) and d in the matrix f_i (x) refer to the drug. They are defined as:

$$C_{dd}=\underset{\text{Ω}}{\int }{N}_d^T\left(\epsilon S^l{N}_d\right)d\text{Ω}$$ (A.3)

$$K_{dd}=\underset{\text{Ω}}{\int }{\left(\nabla N_d\right)}^{\text{T}}\left(\epsilon S^l{D}_{eff}^{\overline{drug}}\nabla N_d\right)d\text{Ω}$$ (A.4)

(A.4)

$$f_d={\int }_{\text{Ω}}{N}_d^T\left(\frac{1}{{\rho }^l}\left({\omega }^{drug}\left(\left(\begin{array}{c}l\to t\\ M\\ growth\end{array}-\begin{array}{c}t\to l\\ M\\ lysis\end{array}\right)+\begin{array}{c}l\to s\\ M\\ ECM\end{array}\right)-\stackrel{drug\to t}{M}\right)-\epsilon S^l{v}^{\overline{l}}\cdot \nabla {\omega }^{\overline{drug}}\right)d\text{Ω}$$ (A.5)