Introduction

Cancer continues to pose a significant challenge to global health, with its incidence increasing annually and affecting millions across the globe. This rising prevalence highlights the critical demand for innovative and effective treatment approaches. Throughout the years, considerable research has been dedicated to creating targeted therapies, with treatment options tailored to factors like tumor location, disease stage, and the overall health of the patient1. Common treatment options encompass surgery, solid tumor removal, radiotherapy, thermal necrosis, and chemotherapy2,3. Among these, chemotherapy has gained significant recognition due to its promising results in clinical settings4. This non-invasive treatment approach prioritizes patient comfort and can be utilized across a wide range of age groups5,6. Given the wide array of tumor types and their inherent diversity, there remains an ongoing effort to develop diverse chemotherapy drugs. However, the effective delivery of these drugs presents a significant challenge7. Intravenous injection stands as one of the primary and most widely employed drug delivery methods8. However, chemotherapy drugs often possess a high clearance rate, leading to poor blood circulation and limited bioavailability within the tumor9. Consequently, frequent injections are required, which increases the likelihood of side effects10. For instance, repeated administration of doxorubicin has been reported to cause serious damage to the cardiovascular system11. To address these concerns, encapsulation of chemotherapy drugs has emerged as a developed method aimed at increasing their half-life in blood circulation and reducing associated side effects12. However, some reports suggest that nanoparticles, commonly employed for encapsulation, may not effectively extravasate into tumor tissue13,14. This limitation hampers the achievement of an effective therapeutic concentration within the tumor, thereby limiting treatment efficacy. Consequently, attention has shifted towards exploring the potential of intratumoral delivery as an alternative approach.

Local chemotherapy, also known as the local delivery of chemotherapeutic agents, involves administering drugs directly into the tumor or adjacent areas15,16,17. This targeted approach has shown promising results in the field of cancer treatment and management18,19,20. By delivering chemotherapy drugs through direct injection at precise locations and carefully adjusting the release profile, local chemotherapy offers several advantages. Firstly, local administration allows for the attainment of an effective drug concentration within the tumor, thereby increasing the likelihood of cell death and tumor regression21,22,23. This targeted approach maximizes the therapeutic effect while minimizing systemic exposure, reducing the risk of adverse side effects commonly associated with intravenous injection. It is important to note that the success of intratumoral treatment relies on the availability of tumor’s primary lesions or metastases for direct injection24. Various types of cancer, including breast, colon, pancreas, liver, ovary, head and neck cancers, can benefit from this approach. The ability to directly access the tumor site enables precise drug delivery and enhances the chances of achieving desired treatment outcomes25.

The efficacy of intratumoral injection as a treatment modality is linked to the characteristics of the tumor’s tissue structure. Effective diffusion throughout the tissue is crucial for the agents to target and suppress a substantial proportion of the tumor cells, leading to potential tumor eradication26. Tumor tissue is characterized by an unfunctional lymphatic drainage system, which inadvertently aids treatment by preventing the rapid efflux of therapeutic agents27. However, this advantage is mitigated by the compromised blood microvessels leakage to tumor tissues, which contributes to fluid accumulation and elevated interstitial pressure28. These factors, compounded by high cellular density, pose significant barriers to the uniform distribution of therapeutic agents within the tumor matrix. Furthermore, it has been observed that the interstitial pressure within tumors is comparable to the pressure within microvessels29. As a consequence, therapeutic agents may diffuse into the bloodstream, undermining their localized therapeutic activity. This phenomenon, along with the increased density of tumor microvessels that accelerates the removal of therapeutic compounds from the extracellular space, poses a significant challenge to the success of intratumoral chemotherapy30. To address this and improve the bioavailability of therapeutic agents, the combination of chemotherapy and anti-angiogenic agents has shown promise in preclinical models for inhibiting tumor growth31,32,33,34,35.

In this study, we examined the impediments that arise during the direct intratumoral injection of therapeutics, particularly due to the high functional microvascular density found in primary and nascent tumors36,37,38,39,40,41. High microvascular density often leads to premature entry of therapeutic agents into the circulatory system, reducing their efficacy. To explore this issue, we employed a mathematical modeling platform to examine the direct delivery and distribution of therapeutic agents within the interstitium of a well-vascularized tumor. The initial phase of our modeling focuses on the distribution of free drugs post-injection, considering the diverse properties of anti-cancer drugs—particularly their varying rates of diffusion and blood microvascular drainage. Subsequently, to mitigate the challenges of washout of therapeutic agents and their entry into the bloodstream, an angiogenesis inhibitor is introduced. This agent aims to impair blood microvessels, thereby reducing microvascular density and diminishing the blood microvascular drainage effect. Following the administration of the anti-angiogenic agent, a chemotherapy drug is injected. The subsequent analysis focuses on determining the distribution and depth of penetration of the drugs within the tumor tissue. In the last phase of the investigation, nanoparticles with negative surface charge are introduced to increase the depth of penetration. These nanoparticles are selected for their high diffusion coefficient, facilitating their extensive dispersion within the tumor. To minimize uptake by cancer cells which hinder nanoparticles movement in the extracellular space, the surface charge of the nanoparticles is aligned with that of the cancer cells, reducing interactive forces and thereby further enhancing penetration depth within the interstitium. The study culminates with a synthesis discussing the key factors influencing the efficacy of direct intratumoral injection therapy. Illustrative of the research, Fig. 1 provides a detailed schematic of the methodology used in the current study.

Fig. 1: Scenarios for intratumoral delivery of therapeutic agents.
figure 1

In the first scenario, free drugs are injected into a well-vascularized tumor. The high microvascular density and blood microvascular drainage facilitate the entry of chemotherapeutic drugs into the bloodstream from the extracellular space. In the second scenario, an anti-angiogenic drug is administered prior to the chemotherapeutic drug. This anti-angiogenic drug effectively reduces microvascular density, enhancing the subsequent penetration of the chemotherapy drug by reducing drainage. The third scenario involves the injection of drug-loaded nanoparticles. The surface charge of these nanoparticles is crucial for their penetration into the extracellular space. By matching the surface charge of the nanoparticles with that of the cells, physical interactions between the cells and nanoparticles are minimized, enabling the nanoparticles to penetrate deeper into the extracellular space and release their cargo in the deeper layers of the tumor. The figure was created by the authors using Adobe Photoshop and 3ds Max.

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Results

In Section “Free Drug Injection” the focus lies on the injection of free drug, where various parameters such as diffusion coefficient and blood microvessels drainage rate are examined to determine the concentration, bioavailability, and penetration depth of the free drugs. This section provides insights into how the distribution of therapeutic agents can be affected by the high microvascular density in a well-vascularized tumor, and how the behavior of therapeutic agents varies with specific characteristics including diffusion coefficient and elimination rate. Section “Anti-Angiogenesis Injection Followed by Free Drug Injection” delves into the injection of an anti-angiogenic drug to disrupt microvascular density. The aim is to understand how the introduction of an anti-angiogenic drug, with a specific diffusion coefficient for microvascular suppression, can enhance the distribution and bioavailability of doxorubicin in the extracellular space. In Section “Injection of Drug-Loaded Nanoparticles”, nanoparticles are discussed as a means to maintain drug concentration levels in the extracellular space. Specifically, nanoparticles with a negative surface charge are utilized to improve the distribution of the drug.

Free drug injection

In this section, the administration of free drug via needle injection at the tissue center of a well-vascularized tumor is considered. The injection is carried out continuously for 72 h with a concentration of \(500{\mu g}/{ml}\) and flow rate \(3.0{\mu L}/\min\). According to Eq. (4), the concentration of free drugs in the tissue depends on various parameters including the drainage rate and diffusion coefficient. Figure 2A illustrates the impact of these two crucial parameters on the average concentration of free drugs in the extracellular space. It is clear that therapeutic agents with higher diffusion coefficients result in higher average concentration levels in the tumor’s extracellular space. This is because a higher diffusion coefficient allows the drug to spread more extensively throughout the tissue, exposing a larger area to a higher drug concentration. In contrast, drugs with lower diffusion coefficients are limited in their ability to disperse within the tissue, resulting in high concentration levels only in the vicinity of the injection site. Therefore, chemotherapy drugs like Fluorouracil, which possess better diffusion coefficients, can achieve higher concentration levels in the same injection compared to drugs like Cisplatin. Additionally, the drainage rate also significantly impacts the drug concentration level. Drugs with lower drainage rates can maintain higher concentration levels in the extracellular space. This is due to the slower drainage rate, which allows the drug to remain in the extracellular space for a longer duration, thereby increasing its overall amount during the injection. These two key factors, diffusion coefficient and drainage rate, play a crucial role in the distribution of therapeutic agents within the tissue and also determine the bioavailability of chemotherapy drugs for effective treatment. In Fig. 2B, the area under the curve (AUC), known as bioavailability42,43,44, is presented. It is observed that reducing the drainage rate significantly increases the AUC by helping to maintain a high concentration of the drug in the tissue. Similarly, the improvement in the diffusion coefficient, leading to a higher concentration spread across a wider area of the tissue, also contributes to a significant enhancement in the AUC.

Fig. 2: Injection of free drug for 72 h.
figure 2

A Temporal distribution of the average concentration of free drug in the extracellular space. B The amount of AUC, considering the effects of diffusion coefficient and drainage rate. \(D\): Diffusion coefficient of free drug, \({k}_{b}\): Blood microvascular drainage rate of free drug.

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Figure 3 presents the spatial distribution and penetration depth of a chemotherapy drug with consideration of different characteristics of diffusion coefficient and drainage rate in the axial, radial, and needle wall directions. Evaluating the concentration in the radial and needle wall directions underscores the significance of the diffusion mechanism and the diffusion coefficient. Furthermore, assessing the concentration in the axial direction, in addition to the aforementioned aspects, demonstrates the critical role of convection-enhanced delivery. Generally, the depth of penetration is greater in the axial direction due to the path of the injection flow. In Fig. 3A, it is evident that the maximum concentration of the drug is observed at the tip of the needle. As the distance from the needle surface increases, the concentration decreases according to the specific trend dictated by the diffusion coefficient and drainage rate. When the diffusion coefficient is lower and the drainage rate is higher, the drug concentration decreases sharply in all three specified directions. Conversely, higher diffusion coefficients and reduced drainage rates result in a milder slope of concentration decrease, thereby maintaining a higher concentration level at a distance from the needle. Figure 3B illustrates the maximum penetration depth. The results demonstrate that improving the diffusion coefficient and reducing the drainage coefficient significantly increase the penetration depth in all three specified directions.

Fig. 3: Spatial distribution and maximum penetration depth of free drug over 72 h.
figure 3

A Spatial distribution of free drug, and B Maximum penetration depth considering the diffusion coefficient and different drainage rates. In the figures, the small gray cylinder symbolizes the needle, while the red line indicates the path along which the concentration of therapeutic agents is measured. \(D\): Diffusion coefficient of free drug, \({k}_{b}\): Blood microvascular drainage rate of free drug.

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Anti-angiogenesis injection followed by free drug injection

This section emphasizes the importance of injecting anti-angiogenic drugs to decrease microvascular density and subsequently reduce the drainage rate of chemotherapeutic drugs. As mentioned earlier, four anti-angiogenic drugs with different diffusion coefficients (designated as \({10}^{-13}\,(\frac{{m}^{2}}{s})\), \({10}^{-12}\,(\frac{{m}^{2}}{s})\), \({10}^{-11}\,(\frac{{m}^{2}}{s})\), and \({10}^{-10}\,(\frac{{m}^{2}}{s})\)) were taken into consideration. Figure 4A demonstrates that the diffusion coefficient has a significant impact on the average concentration of anti-angiogenic drugs in the extracellular space. Specifically, anti-angiogenic drugs with diffusion coefficients of \({10}^{-13}\,(\frac{{m}^{2}}{s})\) and \({10}^{-12}\,(\frac{{m}^{2}}{s})\) exhibit a notably low average concentration. On the other hand, enhancing the diffusion coefficient to \({10}^{-11}\,(\frac{{m}^{2}}{s})\) leads to an increase in the average concentration of the anti-angiogenic drug. Moreover, increasing the diffusion coefficient to \({10}^{-10}\,(\frac{{m}^{2}}{s})\) results in a considerably higher increase in the average concentration compared to other cases in the extracellular space. This noteworthy increase of the anti-angiogenic drug with a diffusion coefficient of \({10}^{-10}\,(\frac{{m}^{2}}{s})\) can be attributed to its extensive distribution throughout the tumor extracellular space, as depicted in Fig. 4B. Conversely, larger anti-angiogenic drugs with weaker diffusion coefficients tend to accumulate near the injection site. The spatial distribution and penetration depth of the anti-angiogenic drug in the axial, radial, and needle wall directions, as shown in Fig. 4C, further demonstrate that a higher diffusion coefficient exposes the tumor tissue to a greater concentration of the anti-angiogenic drug in all three directions. Conversely, reducing the diffusion coefficient limits the penetration depth of the anti-angiogenic drug. For instance, if the diffusion coefficient of the anti-angiogenic drug is \({10}^{-13}\,(\frac{{m}^{2}}{s})\), the maximum distance from the needle tip in all three directions is less than 1 mm.

Fig. 4: Injection of anti-angiogenic drugs with different diffusion coefficients.
figure 4

A Temporal distribution of the average concentration in the extracellular space, B Spatial distribution contour of anti-angiogenic drugs at the end of injection (72 h) in the extracellular space, C Spatial distribution of the drug concentration in axial, radial, and needle wall directions at the end of injection (72 h). In the figures, the small gray cylinder symbolizes the needle, while the red line indicates the path along which the concentration of therapeutic agents is measured. \(D\): Diffusion coefficient of anti-angiogenic drug.

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The presence of anti-angiogenic drugs in the extracellular space plays a crucial role in reducing microvascular density. As depicted in Fig. 4A, the anti-angiogenic drug with a diffusion coefficient of \({10}^{-10}\,(\frac{{m}^{2}}{s})\) exhibits a substantial concentration, and Fig. 5A confirms a significant decrease in microvascular density. Conversely, other anti-angiogenic drugs have minimal impact on microvascular density due to their low diffusion coefficients. As illustrated in Fig. 5B, suggests that microvascular density is expected to decrease based on the anti-angiogenic drug distribution. Figure 5C demonstrates a reduction in microvascular density in all three axial, radial, and needle wall directions. Furthermore, the penetration of the anti-angiogenic drug with a high concentration and a diffusion coefficient of \({10}^{-10}\,(\frac{{m}^{2}}{s})\) leads to a wider area near the injection site where microvascular density decreases to zero, and the microvascular density towards the tumor border is significantly diminished.

Fig. 5: Variations in microvascular density resulting from the injection of anti-angiogenic drugs with different diffusion coefficients.
figure 5

A Average microvascular density of tumor tissue, B Spatial distribution contour of microvasculature in the tumor tissue at the end of anti-angiogenic drug injection (72 h), C Spatial distance of microvascular density in axial and radial, and needle wall directions at the end of anti-angiogenic drug injection (72 h). In the figures, the small gray cylinder symbolizes the needle, while the red line indicates the path along which the concentration of therapeutic agents is measured. \(D\): Diffusion coefficient of anti-angiogenic drug.

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Following the completion of the 72-h injection of anti-angiogenic drugs, the chemotherapy drug doxorubicin is administered with a concentration of \(500{\mu g}/{ml}\) and flow rate \(3.0{\mu L}/\min\), which is known to have a high drainage rate. The reduction in microvascular density resulting from the anti-angiogenic drug injection significantly affects the average concentration of doxorubicin in the extracellular space (Supplementary Fig. 1A). Specifically, the injection of the anti-angiogenic drug with a diffusion coefficient of \({10}^{-10}\,(\frac{{m}^{2}}{s})\), which leads to a substantial decrease in microvascular density, yields a higher average concentration of doxorubicin in the extracellular space compared to the injection of other anti-angiogenic drugs. AUC represents the bioavailability of doxorubicin in the extracellular space which indicates that the decrease in microvascular density, resulting from the prevention of drainage, contributes to an increase in the AUC (Supplementary Fig. 1B). Notably, the improvement in the diffusion coefficient of the anti-angiogenic drug from \({10}^{-13}\,(\frac{{m}^{2}}{s})\) to \({10}^{-10}\,(\frac{{m}^{2}}{s})\), due to the enhanced reduction in microvascular density, causes the AUC to increase by more than sevenfold.

Based on Fig. 6A, it becomes evident that the lower microvascular density resulting from treatment with the anti-angiogenic drug possessing a diffusion coefficient of \({10}^{-10}\,(\frac{{m}^{2}}{s})\) enables doxorubicin to maintain a high concentration level at a greater distance from the needle tip. As the distance from the needle tip increases, along with the rise in microvascular density and the rapid drainage rate of doxorubicin, the concentration of the drug experiences a significant decrease. This phenomenon holds true for all three axial, radial, and needle wall directions. The effective penetration depth of doxorubicin is illustrated in Fig. 6B. It is well-established that anti-angiogenic drugs with higher diffusion coefficients are capable of achieving a greater effective penetration depth, primarily due to their effective suppression of microvascular density. Additionally, the injection path influences the penetration depth, with a higher depth observed in the axial direction. Figure 6C presents the contour of distribution of doxorubicin in the extracellular space, considering the impact of the injected anti-angiogenic drugs. Reduced microvascular density results in a decreased rate of doxorubicin drainage. Additionally, the depth of doxorubicin penetration is influenced by the drug’s low diffusion coefficient and other factors, including inactivation through enzymatic or non-enzymatic reactions, as well as consumption by cancer cells (Eq. S4). The results indicate that the injection of an anti-angiogenic drug with a lower diffusion coefficient, which fails to sufficiently reduce microvascular density, leads to the accumulation of doxorubicin solely near the injection site due to its high drainage rate. Conversely, the reduction of extensive microvascular density widens the distribution of doxorubicin by preventing its drainage.

Fig. 6: The effect of injecting anti-angiogenic drugs with different diffusion coefficients on the spatial distribution of free doxorubicin.
figure 6

A Spatial effective distance of free doxorubicin in the axial, radial, and needle wall directions at the end of anti-angiogenic drug injection (72 h), B The effective penetration depth of free doxorubicin at the end of anti-angiogenic drug injection (72 h), C free doxorubicin spatial distribution contour in the extracellular space at the end of anti-angiogenic drug injection (72 h). In the figures, the small gray cylinder symbolizes the needle, while the red line indicates the path along which the concentration of therapeutic agents is measured. \(D\): Diffusion coefficient of anti-angiogenic drug.

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Injection of drug-loaded nanoparticles

This section explores the function of injecting 60 nm nanoparticles with different surface charges (\({Z}_{P}=0{mV}\), \(-5{mV}\) and \(-15{mV}\)) in the distribution of drug-loaded nanoparticles and the release of the doxorubicin. The nanoparticles are injected for a duration of 72 h, with a concentration of \(500{\mu g}/{ml}\) and flow rate \(3.0{\mu L}/\min\). The nanoparticles release their cargo at three different rates: \({10}^{-3}\,(\frac{1}{s})\), \({10}^{-4}\,(\frac{1}{s})\), and \({10}^{-5}\,(\frac{1}{s})\). Figure 7 presents the temporal distribution of the average concentration of nanoparticles with varying surface charges and release rates in the extracellular space. Changing the surface charge from neutral to negative significantly increases the diffusion coefficient, facilitating the spread of nanoparticles throughout the tumor. Consequently, the average concentration of negatively charged nanoparticles is much higher than that of neutral nanoparticles. Moreover, negatively charged nanoparticles reach their maximum value in the extracellular space more quickly. The release rate also plays a crucial role in determining the concentration of drug-loaded nanoparticles. Based on the release term in Eq. (2), a higher release rate (e.g., \({10}^{-3}\,(\frac{1}{s})\)) leads to an explosive release of cargo, resulting in a decrease in the concentration of drug-loaded nanoparticles. On the other hand, a sustained release rate of \({10}^{-5}\,(\frac{1}{s})\), has a lesser impact on reducing the average concentration of nanoparticles. Thus, a weaker release rate is more effective in maintaining a higher concentration of drug-loaded nanoparticles in the extracellular space.

Fig. 7: Temporal distribution of average concentration of nanoparticles with various zeta potential values.
figure 7

A \(0{mV}\) - Equilibrium concentration values:\({C}_{{k}_{r}={10}^{-5}\left({s}^{-1}\right)}=0.237\,\left(\mu g/{ml}\right),\,{C}_{{k}_{r}={10}^{-4}\left({s}^{-1}\right)}=0.05\,\left(\mu g/{ml}\right)\), \({C}_{{k}_{r}={10}^{-3}\left({s}^{-1}\right)}=0.01\,\left(\mu g/{ml}\right)\), B \(-5{mV}\) - Equilibrium concentration values:\({C}_{{k}_{r}={10}^{-5}\left({s}^{-1}\right)}=325\,\left(\mu g/{ml}\right),\,{C}_{{k}_{r}={10}^{-4}\left({s}^{-1}\right)}=210\,\left(\mu g/{ml}\right)\), \({C}_{{k}_{r}={10}^{-3}\left({s}^{-1}\right)}=57\,\left(\mu g/{ml}\right)\), and C \(-15{mV}\) - Equilibrium concentration values:\({C}_{{k}_{r}={10}^{-5}\left({s}^{-1}\right)}=360\,\left(\mu g/{ml}\right),{C}_{{k}_{r}={10}^{-4}\left({s}^{-1}\right)}=330\left(\mu g/{ml}\right)\), \({C}_{{k}_{r}={10}^{-3}\left({s}^{-1}\right)}=220\,\left(\mu g/{ml}\right)\); considering different drug release rates. \({Z}_{P}\): Zeta potential, \({k}_{r}\): Drug release rate.

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Figure 8 illustrates the spatial distribution of nanoparticles in the axial, radial, and needle wall directions. Decreasing the release rate leads to a higher concentration of nanoparticles being maintained. According to the diffusion mechanism and the concentration gradient, nanoparticles with a lower release rate can maintain a higher concentration level and achieve a greater penetration depth from the needle surface. When using nanoparticles with a neutral surface charge, the penetration depth in all three directions is less than \(2{mm}\). However, when utilizing nanoparticles with a negative surface charge, the concentration level remains high up to the tumor border. Essentially, the entire tumor is exposed to a high concentration of nanoparticles (Supplementary Fig. 2). Notably, nanoparticles with a lower release rate can generate a wider area with a high concentration near the injection site. In summary, employing a lower release rate and utilizing nanoparticles with a negative surface charge ensures complete exposure of the tumor tissue to a higher concentration of nanoparticles. Additionally, it is worth mentioning that the limited drainage rate of \(60{nm}\) nanoparticles contributes to a more effective accumulation of nanoparticles in the extracellular space.

Fig. 8: Spatial effective distance of nanoparticles with different zeta potential values.
figure 8

A \(0{mV}\), B \(-5{mV}\), and C \(-15{mV}\)) in axial, radial, and needle wall directions considering different drug release rates at the end of injection (72 h). In the figures, the small gray cylinder symbolizes the needle, while the red line indicates the path along which the concentration of therapeutic agents is measured. \({Z}_{P}\): Zeta potential of nanoparticles, \({k}_{r}\): Drug release rate.

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The concentration of drug-loaded nanoparticles and the drug release rate, as described by Eq. (4), have a significant impact on the average concentration of doxorubicin in the extracellular space. In Fig. 9A, the average concentration of doxorubicin is depicted, and unlike nanoparticles, a high release rate leads to a higher concentration of the drug. This is because a high release rate results in an explosive release of the drug, causing it to accumulate in the tissue at a rate that surpasses doxorubicin drainage. Conversely, the slow release of doxorubicin from nanoparticles fails to overcome its drainage rate, resulting in minimal drug accumulation. This is because the released drug is quickly washed away by the blood microvessels. Furthermore, nanoparticles with a negative surface charge, which provide a higher average concentration in the extracellular space, significantly increase the average concentration of doxorubicin. The rate of drug release from drug-loaded nanoparticles and the average concentration of nanoparticles in the extracellular space directly affect the bioavailability of the drug. Figure 9B demonstrates that nanoparticles with a neutral surface charge exhibit a much lower AUC compared to nanoparticles with a negative surface charge. Additionally, a release rate of \({10}^{-3}\,(\frac{1}{s})\) achieves a significantly higher AUC due to the creation of a higher average concentration level. In summary, the concentration of drug-loaded nanoparticles, the drug release rate, and the surface charge of nanoparticles play crucial roles in determining the average concentration of doxorubicin in the extracellular space and its bioavailability.

Fig. 9: Temporal distribution and AUC of doxorubicin with various zeta potential values and drug release rates.
figure 9

A Temporal distribution of average concentration of doxorubicin, and B AUC, considering injection of nanoparticles with different zeta potential values (\(0{mV}\), \(-5{mV}\), and \(-15{mV}\)) and different drug release rates. \({Z}_{P}\): Zeta potential of nanoparticles, \({k}_{r}\): Drug release rate.

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Figure 10A presents the penetration depth and spatial distribution of free doxorubicin. When using nanoparticles with a neutral surface charge, the penetration depth of doxorubicin in the axial, radial, and needle wall directions is very limited, measuring less than \(1{mm}\). However, the use of nanoparticles with a negative surface charge significantly increases the depth of drug penetration, thanks to their enhanced ability to penetrate tissue. Furthermore, the release rate also influences the extent of free drug penetration in the extracellular space. A release rate of \({10}^{-3}\,(\frac{1}{s})\) results in greater penetration depth due to the higher drug concentration and concentration gradient it generates. Additionally, the use of nanoparticles with a negative surface charge exposes the tissue to a higher concentration of the drug. In Fig. 10B, which represents the depth of effective penetration, it is evident that a release rate of \({10}^{-5}\,(\frac{1}{s})\) does not achieve effective penetration for cancer cell eradication. Conversely, release rates of \({10}^{-3}\,(\frac{1}{s})\) and \({10}^{-4}\,(\frac{1}{s})\) exhibit much better performance in terms of effective penetration. Notably, the combination of nanoparticles with a negative surface charge and a release rate of \({10}^{-3}\,(\frac{1}{s})\) demonstrates the highest penetration depth, capable of covering the entire tissue. Consequently, this combination holds great potential for successful tumor eradication. The free doxorubicin distribution in the extracellular space shows that neutral nanoparticles and low release rates, unlike nanoparticles with a negative surface charge and higher release rates, have not been successful in providing a high concentration of free drug for therapeutic purposes (Supplementary Fig. 3).

Fig. 10: Spatial distribution and effective penetration depth of free doxorubicin.
figure 10

A Spatial distribution of free doxorubicin in in axial, radial, and needle wall directions, and B Effective penetration depth, considering injection of nanoparticles with zeta potential \(0{mV}\), \(-5{mV}\), and \(-15{mV}\), as well as drug release rate. In the figures, the small gray cylinder symbolizes the needle, while the red line indicates the path along which the concentration of therapeutic agents is measured. \({Z}_{P}\): Zeta potential, \({k}_{r}\): Release rate.

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Validation of computational results

Mathematical models face a significant challenge in verifying their accuracy through in vivo observations due to the difficulty in measuring crucial interactions under these conditions. Despite this limitation, the current mathematical framework has been validated through multiple research investigations, reinforcing its reliability. The main purpose of mathematical models is to explore the effectiveness of novel concepts, such as the one being investigated here, and to optimize them to reduce the number of animal experiments and expedite clinical translation of these new anti-cancer approaches. Due to the complexity of tumor microenvironment, it is important to acknowledge that comparable empirical evidence may not always be readily available to authenticate all aspects of the investigation.

This section focuses on validating the outcomes impacting the dispersion of therapeutic agents within the tissue, considering both the tumor microenvironment’s characteristics and the diffusion coefficient of the therapeutic agents. One important characteristic of the tumor microenvironment is the interstitial fluid pressure (IFP), where the elevated IFP is identified as one of the main barriers to efficient drug delivery into solid tumors. The mean IFP value of the tumor was calculated to be 11.3 mmHg, consistent with previous experimental investigations by Boucher and Jain45,46, which reported tumor pressures ranging from 3.67 to 30.88 mmHg. In their work, they simultaneously measured IFP and microvascular pressure MVP in 13 tissue-isolated R3230AC mammary adenocarcinomas transplanted in rats. Comparing the IFP between different tumor tissues reveals significantly elevated pressure within the tumor tissue, closely resembling the intravascular pressure (Fig. 11A). Consequently, this high pressure facilitates the removal of therapeutic agents from the tissue, allowing them to enter the bloodstream through a diffusion mechanism.

Fig. 11: Validation of our modeling approach.
figure 11

A Comparing the predicted IFP (resulting from a microvascular pressure of 15.3 mmHg) with the IFP and microvascular pressure obtained from in vivo studies. IFP was estimated using Darcy’s law (Eq. (11)). B Comparing the temporal volume infused distribution results of Evans Blue and Albumin under local infusion in the present study with an experimental model47. To calculate - Volume distributions: volume infused - calculated volume distribution based on the average concentration of the agent throughout the tumor per the concentration of agent infused over time.

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To validate the intratumoral delivery simulation, which primarily relies on the diffusion mechanism and diffusion coefficients, the updated diffusion coefficients have been incorporated, and the results are compared with the experimental data presented in Fig. 11B. These results, obtained under the same operating conditions as those used by Neeves et al.47, demonstrate a strong agreement between the current modeling approach and the experimental models. Neeves et al. employed microfluidic probes to study convection-enhanced delivery (CED), infusing Evans Blue and albumin into hydrogel brain phantoms. For this study, the diffusion coefficient for Evans Blue and Albumin is assumed to be \(2\times {10}^{-6}\left({{cm}}^{2}/s\right)\) and \(8.3\times {10}^{-7}\left({{cm}}^{2}/s\right)\), respectively.

Discussion

The presence of tumors can limit invasive treatments, such as surgery and tumor resection, due to various factors including the patient’s condition and the number of primary and metastatic tumors. As a result, local chemotherapy has emerged as a promising and minimally invasive method. This approach involves directly injecting the chemotherapy drug into the tissue. It is important to carefully consider the duration and rate of the injection, considering the patient’s condition and the characteristics of the tissue. The effectiveness of local chemotherapy primarily depends on the tumor characteristics and the specific drug being used. Tumor tissue has the potential to impact the therapeutic response in various ways, with one of the key factors being the facilitation of therapeutic agent accumulation and bioavailability within the tissue. As previously discussed, the high pressure within the tumor tissue, which is similar to the intravascular pressure, increases the likelihood of therapeutic agents entering the bloodstream through concentration gradients and diffusion mechanisms in the extracellular space. Consequently, the presence of high microvascular density can pose a challenge in suppressing the therapeutic response in local chemotherapy48,49,50,51,52,53. Therefore, this study specifically focuses on the distribution of therapeutic agents through intratumoral injection in a well-vascularized tumor. By examining this aspect, it aims to gain insights into optimizing the delivery of therapeutic agents and improving the overall efficacy of local chemotherapy.

The analysis of free drug injection in Section “Free Drug Injection” revealed that drugs with a high drainage rate are prone to being washed away from the extracellular space by blood vessels, resulting in ineffective therapeutic outcomes. Therefore, in the direct injection and local chemotherapy of well-vascularized tumors, it is advisable to utilize chemotherapeutic drugs with a low drainage rate that can offer optimal bioavailability. Additionally, for a wider distribution and to induce damage throughout the tissue, a high diffusion coefficient is necessary. This is because therapeutic agents with a low diffusion coefficient tend to accumulate only near the injection site. However, reducing the drainage rate not only helps maintain a high concentration but also has an impact on the diffusion mechanism and concentration gradient. Based on the results, chemotherapy drugs with a diffusion coefficient of \({10}^{-9}\,(\frac{{m}^{2}}{s})\) and an elimination rate of \({10}^{-2}\,(\frac{1}{s})\) offer 14 and 17 times more AUC value and penetration depth compared to chemotherapy drugs with a diffusion coefficient of \({10}^{-11}\,(\frac{{m}^{2}}{s})\) and an elimination rate of \({10}^{-2}\,(\frac{1}{s})\). In summary, the achievement of therapeutic response in direct injection of free chemotherapy drugs in well-vascularized tumors is dependent on the utilization of drugs with lower drainage rates and higher diffusion coefficients.

Doxorubicin is a widely utilized chemotherapy drug for various tumor types. However, despite its moderate diffusion coefficient, its high drainage rate hampers its efficacy in intratumoral injection and local chemotherapy, often resulting in suboptimal therapeutic responses. Thus, preventing drug drainage becomes imperative to achieve desirable treatment outcomes. Section “Anti-Angiogenesis Injection Followed by Free Drug Injection” suggests employing anti-angiogenic drug injections to mitigate this issue by reducing microvascular density, thereby decreasing Doxorubicin drainage. It is important to note that not all anti-angiogenic drugs are suitable for intratumoral injection, particularly those with low diffusion coefficients that only impact microvascular density near the injection site. To ensure broad distribution of chemotherapy drugs, anti-angiogenic agents with high diffusion coefficients are necessary. These drugs can effectively reduce microvascular density across a significant portion of the tumor tissue and, by preventing drug drainage, enhance the overall chemotherapy response. The results suggest that anti-angiogenic drugs with a diffusion coefficient of \({10}^{-10}\,(\frac{{m}^{2}}{s})\) can decrease the average density of microvessels by 50% when compared to anti-angiogenic drugs with a lower diffusion coefficient. This reduction in microvessel density can enhance the penetration depth by four times by lowering the elimination rate of doxorubicin. In clinical settings, anti-angiogenic agents are typically administered intravenously. To optimize therapeutic outcomes, it is crucial to coordinate the delivery of chemotherapy, either prior to or concurrently with the anti-angiogenic agents, to prevent any potential interference with the accumulation of chemotherapy in the extracellular space. In contrast, this study employs a direct injection approach. In this context, the initial administration of anti-angiogenic agents is key, as it reduces microvascular density and modulates drainage rates through the blood microvessels, ultimately enhancing the effectiveness of localized chemotherapy.

The tumor microenvironment, characterized by high cell density and enhanced uptake of therapeutic agents by cancer cells, posses challenges to the distribution of these agents to distant areas from the injection site. However, the negative surface charge of cancer cells suggests the potential utility of nanoparticles with a negative surface charge to minimize interactions with cancer cells. Consequently, such nanoparticles which are not readily taken up by cancer cells, exhibit a higher diffusion coefficient in the tissue and can travel greater distances from the source. It is noteworthy that factors like collagen, which possesses a partial positive charge, may affect the diffusion of nanoparticles. Nonetheless, nanoparticles with a negative charge generally demonstrate a high diffusion coefficient in tissue. In Section “Injection of Drug-Loaded Nanoparticles”, the utilization of nanoparticles with negative surface charge showcased their effectiveness in covering the entire tumor and releasing their cargo in response to tumor-specific characteristics, such as extracellular acidity. While the drainage of doxorubicin was not entirely prevented, the use of nanoparticles facilitated their widespread distribution throughout the tissue. The rate of drug release from nanoparticles emerges as a critical parameter influencing therapeutic response. A burst release rate leads to a high accumulation of drug in the extracellular space, achieving an effective concentration for therapeutic efficacy. Moreover, the substantial volume of released drug overcomes the impact of drainage, ensuring an effective concentration of the drug throughout the tissue. Conversely, a low release rate leads to limited availability of drugs in the extracellular matrix, making them susceptible to being washed away by the significant impact of microvascular blood drainage compared to the rate of drug supply. In conclusion, the direct injection of chemotherapy drug-loaded nanoparticles with a negative surface charge and a high release rate holds significant promise for localized chemotherapy.

In summary, results suggest that the direct intratumoral injection of nanoparticles with negative surface charge holds promise as a strategy for localized chemotherapy and tumor eradication. However, to ensure its effectiveness, it is necessary to evaluate its performance in in vitro cell culture studies and eventually in vivo animal models. The current study utilized a mathematical platform to evaluate a combo therapy including anti-angiogenesis and nanomedicine Utilizing such mathematical models enables the identification of multiple optimal strategies, allowing only those with the highest potential to advance to cell culture and animal experiments. This approach accelerates the translation of promising therapies to clinical applications. It is important to acknowledge that the current model has limitations, including the absence of considerations for the functioning of certain cells, such as immune cells, as well as enzymatic and non-enzymatic reactions on nanoparticles and anti-angiogenic drugs. Also, the impact of anti-angiogenic drugs on the upregulation of angiogenic factors and their influence on cancer cells is not addressed in the current mathematical model due to the complexity involved in integrating these factors. Furthermore, the study does not directly evaluate the elimination of cancer cells, and the absence of experimental data specific to this model remains a limitation that requires further investigation. Despite the limitations, the study effectively addresses critical factors such as the diffusion of therapeutic agents and their drainage rate, which significantly influence therapeutic response. Therefore, it can be considered a reliable and valuable resource for advancing future research in this field.

Methods

This section delineates the foundational governing equations, the parameters influencing the solution, as well as the strategic approach adopted for solving this problem.

Mathematical model

Biological tissues are widely regarded as porous media, attributed to their inherent physical characteristics. Consequently, to characterize fluid flow phenomena within such tissues, Darcy’s law is employed. This law enables a description of fluid flow within the interstitial spaces, considering source and sink terms originating from the activity of blood microvessels and the lymphatic drainage system integrated within the tissue matrix. This can be formulated as follows54,55:

$$\begin{array}{c}{v}_{i}=-\kappa \nabla {P}_{i}\\ \nabla {v}_{i}=\mathop{\underbrace{\frac{{L}_{P}S}{V}\left({P}_{v}-{P}_{i}-{\sigma }_{s}\left({\pi }_{B}-{\pi }_{i}\right)\right)}}\limits_{{\rm{sourceterm}}}-\mathop{\underbrace{{L}_{{PL}}{\left(\frac{S}{V}\right)}_{L}\left({P}_{i}-{P}_{L}\right)}}\limits_{{\rm{sinkterm}}}\end{array}$$
(1)

In the context of vascular dynamics, \({L}_{P}\) characterizes the hydraulic conductivity of the microvessel wall, while \(\frac{S}{V}\) quantifies the surface area of blood vessels per unit volume of tissue. \({P}_{v}\) represents the vascular pressures, and \({\sigma }_{s}\) signifies the average osmotic reflection coefficient. The osmotic pressures of the plasma and interstitial fluid are denoted as \({\pi }_{B}\) and \({\pi }_{i}\), respectively. The lymphatic drainage rate is influenced by the pressure difference between the interstitial fluid and the lymphatic system. Notably, the lymphatic system is not taken into account when examining tumor tissue. Specifically, \({L}_{{PL}}\), \({\left(\frac{S}{V}\right)}_{L}\), and \({P}_{L}\) correspond to the hydraulic conductivity of the lymphatic wall, the surface area of lymphatic vessels per unit volume of tissue, and the intra-lymphatic pressure, respectively. Reports indicate that the drainage system in tumor tissue is notably unfunctional. Considering this, the study assumes the sink term for the tumor tissue to be negligible, effectively considering it as zero.

Both the tumor and its surrounding normal tissue are structurally divided into three prominent compartments: the extracellular space (ECS), the cellular membrane (CM), and the intracellular space (ICS). In Fig. 12, a schematic visualization of governing multicompartment of the drug transport dynamics that occur during the administration of therapeutic agents employing both drug-loaded nanoparticles and free drugs. The formulated equation that governs the concentration dynamics of drugs encapsulated within nanoparticles (\({C}_{n}\)) is presented as follows40:

$$\mathop{\underbrace{\frac{\partial {C}_{n}}{\partial t}}}\limits_{\begin{array}{c}{\rm{Nanoparticles}}\\ {Concentration}\end{array}}=\mathop{\underbrace{{D}_{n,{ECS}}{\nabla }^{2}{C}_{n}}}\limits_{\begin{array}{c}{Diffusion}\\ {Mechanism}\end{array}}-\mathop{\underbrace{{v}_{i}\nabla {C}_{n}}}\limits_{\begin{array}{c}{Convection}\\ {Mechanism}\end{array}}-\mathop{\underbrace{{k}_{{rel}}{C}_{n}}}\limits_{\begin{array}{c}{Drug}\\ {Release}\end{array}}-\mathop{\underbrace{{P}_{n}\frac{S}{V}{C}_{n}}}\limits_{\begin{array}{c}{BloodMicrovessels}\\ {Drainage}\end{array}}$$
(2)
Fig. 12: Multicompartment model of the drug transport dynamics.
figure 12

Therapeutic agents (i.e., free drugs, anti-angiogenic drugs, and drug-loaded nanoparticles) are directly injected into the extracellular space within the tumor. Each agent interacts differently within the biophysical tumor microenvironments due to its unique nature and therapeutic purpose. AA Anti-Angiogenic Drug, NP Drug-Loaded Nanoparticle, FD Free Drug, BD Bound Drug. The figure was created by the authors using Adobe Photoshop and 3ds Max.

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In the equation, \({D}_{n,{ECS}}\) represents the diffusivity of nanoparticles within the interstitial fluid. The variable \({k}_{{rel}}\) is employed to characterize the drug release rate, while \({P}_{n}\) signifies the vascular permeability of nanoparticles.

The amounts of total free drugs (unbound) and protein-bound drugs present in the tissue are quantified as \({C}_{F,{total}}=\alpha {C}_{F,{ECS}}+\beta {C}_{F,{ICS}}+(1-\alpha -\beta ){C}_{F,{CM}}\) and \({C}_{B,{total}}=\alpha {C}_{B,{ECS}}+\beta {C}_{B,{ICS}}+(1-\alpha -\beta ){C}_{B,{CM}}\), respectively, where \(\alpha\) denotes the volume fraction of ECS, and \(\beta\) signifies the volume fraction of ICS. The concentration of free drugs within tissue can be ascertained by accounting for a range of influencing factors. These include the release of drugs from nanoparticles, their diffusion and convection within the interstitial space, removal by blood microvascular drainage, interactions with proteins, and both enzymatic and non-enzymatic degradation reactions. Collectively, these elements dictate the accumulation of free drugs in ECS. Consequently, the calculation of free drug availability is defined the following equation56:

$$\mathop{\underbrace{\frac{\partial {C}_{F}}{\partial t}}}\limits_{\begin{array}{c}{\rm{Free\; Drug}}\,\\ {Concentration}\end{array}}=\mathop{\underbrace{{\alpha D}_{F,{ECS}}{\nabla }^{2}{C}_{F,{ECS}}}}\limits_{\begin{array}{c}{Diffusion}\\ {Mechanism}\end{array}}-\mathop{\underbrace{{\alpha v}_{i}\nabla {C}_{F,{ECS}}}}\limits_{\begin{array}{c}{Convection}\\ {Mechanism}\end{array}}-\mathop{\underbrace{\alpha {P}_{F}\frac{S}{V}{C}_{F,{ECS}}}}\limits_{\begin{array}{c}{Blood\; Microvessels}\\ {Drainage}\end{array}}-\mathop{\underbrace{{\alpha k}_{{Fe}}{C}_{F,{ECS}}}}\limits_{\begin{array}{c}\frac{{Enzymatic}}{{non}-{enzymatic}}\,\\ {Reactions}\end{array}}-\mathop{\underbrace{\beta {k}_{{Fe}}{C}_{F,{ICS}}}}\limits_{\begin{array}{c}\frac{{Enzymatic}}{{non}-{enzymatic}}\\ {Reactions}\end{array}}-\mathop{\underbrace{\frac{\partial {C}_{B}}{\partial t}}}\limits_{\begin{array}{c}{\rm{Bound\; Drug}}\,\\ {Concentration}\end{array}}+\mathop{\underbrace{{k}_{{rel}}{C}_{n}}}\limits_{\begin{array}{c}{Drug}\\ {Release}\end{array}}$$
(3)

\({D}_{F,{ECS}}\) denotes the diffusivity of free drugs in the ECS. \({P}_{F}\) represents the microvascular permeability of free drugs, and \({k}_{{Fe}}\) is the rate at which free drugs are eliminated due to enzymatic and non-enzymatic reactions. Under the presupposition of certain conditions, we can streamline the equation as follows: firstly, assuming that the concentrations of both free drugs and protein-bound drugs are linearly related within ECS and ICS, as denoted by (\({K}_{{ECS}}={C}_{B,{ECS}}/{C}_{F,{ECS}}\); \({K}_{{ICS}}={C}_{B,{ICS}}/{C}_{F,{ICS}}\)). Secondly, it is assumed that the distribution of free drugs attains equilibrium across the three tissue compartments, which is represented by (\({P}_{{ICS}-{ECS}}={C}_{F,{ICS}}/{C}_{F,{ECS}}\); \({P}_{{CM}-{ECS}}={C}_{F,{CM}}/{C}_{F,{ECS}}\)). With these assumptions in place, the equation can be modified as:

$$\frac{\partial {C}_{F,{ECS}}}{\partial t}\mathop{\underbrace{=(\frac{\alpha }{\varpi }){D}_{F,{ECS}}{\nabla }^{2}{C}_{F,{ECS}}}}\limits_{\begin{array}{c}{Diffusion}\\ {Mechanism}\end{array}}-\mathop{\underbrace{(\frac{\alpha }{\varpi }){\nu }_{i}\nabla {C}_{F,{ECS}}}}\limits_{\begin{array}{c}{Convection}\\ {Mechanism}\end{array}}-\mathop{\underbrace{\frac{[\alpha {P}_{F}\frac{S}{V}+(\alpha +\beta ){k}_{{Fe}}]}{\varpi }{C}_{F,{ECS}}}}\limits_{{Elimination}}+\mathop{\underbrace{\frac{{k}_{{rel}}}{\varpi }{C}_{n}}}\limits_{\begin{array}{c}{Drug}\\ {Release}\end{array}}$$
(4)

where \(\varpi\) is given by,

$$\varpi =\alpha (1+{K}_{{ECS}})+\beta {P}_{{ICS}-{ECS}}(1+{K}_{{ICS}})+(1-\alpha -\beta ){P}_{{CM}-{ECS}}$$
(5)

The concentration level of antiangiogenic drugs within ECS are determined by the mechanisms of convective and diffusive transport, alongside the process of elimination, as detailed in the following equation57,58.

$$\mathop{\underbrace{\frac{\partial {C}_{{Aa}}}{\partial t}}}\limits_{\begin{array}{c}{\rm{Antiangiogenic\; Drug}}\,\\ {Concentration}\end{array}}=\mathop{\underbrace{{D}_{{Aa},{ECS}}{\nabla }^{2}{C}_{{Aa}}}}\limits_{\begin{array}{c}{Diffusion}\\ {Mechanism}\end{array}}-\mathop{\underbrace{{v}_{i}\nabla {C}_{{Aa}}}}\limits_{\begin{array}{c}{Convection}\\ {Mechanism}\end{array}}-\mathop{\underbrace{{k}_{{AA}}{C}_{{Aa}}}}\limits_{\begin{array}{c}{Drug}\\ {Reaction}\end{array}}$$
(6)

In this context, \({D}_{{Aa},{ECS}}\) represents the diffusivity of antiangiogenic drugs within the tissue ECS, while \({k}_{{AA}}\) signifies its elimination rate.

The inhibitory effect on angiogenesis and microvascular density can be characterized by the following equation57,58:

$$\mathop{\underbrace{\frac{\partial \varphi }{\partial t}}}\limits_{\begin{array}{c}{Scaling\; Factor\; for}\\ {MicrovasculatureDensity}\end{array}}=\mathop{\underbrace{\varphi \left(\varepsilon +\delta \varphi +\gamma {\varphi }^{2}\right)}}\limits_{\begin{array}{c}{Term\; of}\\ {Natural\; Angiogenesis}\end{array}}-\mathop{\underbrace{{k}_{a}\varphi {C}_{{Aa}}}}\limits_{\begin{array}{c}{\rm{Antiangiogenic\; Drug}}\\ {Effect}\end{array}}$$
(7)

\(\varphi\) serves as the scaling factor for microvascular density, while \({k}_{a}\) represents the antiangiogenic rate. Additionally, \(\varepsilon\), \(\delta\), and \(\gamma\) are utilized to account for the natural angiogenesis process within the tissue.

Model parameters

In this modeling analysis, we assumed the study parameters governing drug transport and the geometrical characteristics remain unchanged throughout the investigation period. This simplification is justifiable due to the relatively short temporal scope of the study, which is significantly shorter than the rate of tumor progression. Tables 1 and 2 enumerate the pertinent parameter values, delineating the properties of both the tumor and the adjacent normal tissue, as well as the chemotherapeutic and antiangiogenic drugs. In the context of sensitivity analysis, a wide array of parameters, including those determining the distribution of nanoparticles, are considered as variables. Further elaborations on this subject are set for the subsequent subsection.

Table 1 The model parameters associated with the tumor and normal tissue
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Table 2 The model parameters associated with drugs
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Solution strategy and assumptions

The present study concentrates on examining a solid tumor with an 8 mm radius, surrounded by normal tissue three times its size. At the center of the tumor, an infusion catheter with a 1 mm diameter is precisely positioned for the direct administration of therapeutic agents through its tip. The simulation utilizes a two-dimensional axisymmetric model, and further details on the boundary conditions can be found in Table 3.

Table 3 The boundary conditions for this study
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To computationally resolve this scenario, the methodology is divided into two key steps: the establishing a steady state phase and conducting a time-dependent analysis. Initially, Darcy’s law is applied to ascertain the IFP and interstitial fluid velocity (IFV) in the steady-state phase, laying the groundwork for solving time-dependent solute transport equations. The coupled nonlinear set of governing equations, along with the boundary conditions, are evaluated using the finite element method. The simulation is conducted using the commercial finite element software COMSOL Multiphysics 6.1 (COMSOL, Inc., Burlington, MA, USA). For numerical solution, a segregated approach is utilized with a time-step of 1 s and a relative tolerance of 0.001. Solution field meshing is considered in the study of triangular elements with an extra finer level. The drug delivery analysis spans a time period of 96 h.

The results of this investigation are contingent upon several pivotal assumptions that have been carefully considered. These key assumptions are outlined below.

  • Considering that clinical trials documented the intratumoral infusion of drugs at a rate of \(0.5 \sim 10.0{\mu L}/\min\) over a period ranging from 2 to 5 days, the current study implements an intratumoral infusion at a modulated rate of \(3.0{\mu L}/\min\) with a duration of 3 days59,60. According to previous studies where anti-angiogenesis agents were infused at concentrations of \(7.26\,\times \,{10}^{-5}\) M61, the present simulations adhere to these established parameters. Moreover, both free drugs and nanoparticles in the current study are infused at concentrations of \(500{\mu g}/{ml}\). These parameters ensure the relevance and applicability of the study’s findings within the context of existing clinical and experimental data.

  • The ratio of microvascular surface area to tissue volume, termed microvascular density, is indicative of functional microvessels. Based on previous studies, a ratio of \(\frac{S}{V}=\mathrm{20,000}\,({m}^{-1})\) is recognized as the highest functional microvascular density of a tumor. This study posits that a tumor exhibiting \(\frac{S}{V}=\mathrm{20,000}\,({m}^{-1})\) is characteristic of a well-vascularized tumor62,63,64,65.

  • The aim of this research is centered on well-vascularized tumors. This implied that the ratio of microvascular surface area to tissue volume is assumed to be uniformly \(\mathrm{20,000}\,({m}^{-1})\)62\(.\) The anti-angiogenesis effect on microvessel density is represented as \(\varphi\). To incorporate this effect into the equations governing the distribution of therapeutic agents, the term ‘Blood Microvessels Drainage’ is mathematically adjusted to ‘\({P}_{{Therapeutic\; agent}}\bullet \left(\varphi \bullet \frac{S}{V}\right){C}_{{Therapeutic\; agent}}\)’.

  • In this study, the primary focus is on the anticancer drug doxorubicin, which has a default blood microvessels drainage rate of \(6\times {10}^{-2}\left(\frac{1}{s}\right)\)66 and a diffusion coefficient of \(3.4\times {10}^{-10}\,(\frac{{m}^{2}}{s})\)66\(.\) However, it is important to note that there are other chemotherapy drugs available, such as Fluorouracil and Cisplatin, which have diffusion coefficients around \({10}^{-9}\,(\frac{{m}^{2}}{s})\) and \({10}^{-11}\,(\frac{{m}^{2}}{s})\)67,68\(,\) respectively. Additionally, the blood microvessels drainage rates can vary for different drugs, for example, Carmustine has an elimination rate of \({10}^{-2}\left(\frac{1}{s}\right)\)69\(,\) while Paclitaxel’s elimination rate is much lower, around \({10}^{-4}\left(\frac{1}{s}\right)\)70\(.\) To comprehensively analyze how the distribution of infused free drugs is affected, the study considers a broader range of diffusion coefficients, spanning from \({10}^{-11}\,{to}\,{10}^{-9}\,(\frac{{m}^{2}}{s})\), and drainage rates ranging from \({10}^{-4}\,{to}\,{10}^{-2}\,(\frac{1}{s})\) for well-vascularized tumors, as will be discussed in Section “Free drug injection”. This approach enables a thorough examination of the sensitivity of drug distribution in various scenarios, taking into account the diverse pharmacokinetic properties of different chemotherapy agents.

  • The size of agents in biological tissues is closely linked to their diffusion coefficient. Antiangiogenic drugs exhibit a range of sizes, from large ones like Bevacizumab to smaller ones like Thalidomide and Sunitinib71. Previous studies have identified the diffusion coefficient as the most significant characteristic influencing the distribution efficacy of antiangiogenic drugs57. For this analysis, the diffusion coefficient has been considered within the range of \({10}^{-13}\,{to}\,{10}^{-10}\,(\frac{{m}^{2}}{s})\), based on the findings of the previous studies57,72. The results of this sensitivity analysis can be found in Section “Anti-angiogenesis injection followed by free drug injection”.

  • Equations (6) and (7) highlight the dynamic nature of angiogenesis, where a reduction in the concentration of the antiangiogenic agent results in an increase in microvascular density. It can be posited that the reduction of microvascular density is reached promptly after the antiangiogenic process concludes. Administering the chemotherapy agent at this juncture is crucial for optimizing tumor penetration efficiency.

  • In this study, 60 nm nanoparticles as drug carrier were utilized to carry and release the drug within the extracellular space. Compared to smaller nanoparticles, these 60 nm nanoparticles have a higher loading capacity and a lower clearance rate in blood microvessels. In addition, they have a higher diffusion coefficient and can penetrate deeper into tissues compared to larger nanoparticles. These 60 nm nanoparticles with a neutral surface charge have a diffusion coefficient of \(2\times {10}^{-12}\,(\frac{{m}^{2}}{s})\) and a blood microvessels drainage rate of \(5.6\times {10}^{-7}\,(\frac{1}{s})\)73,74\(.\) To enhance the penetration depth and distribution of agents in the tissue, nanoparticles with a negative surface charge were chosen to minimize their uptake by cancer cells, which also possess a negative surface charge75. Therefore, nanoparticles with zeta potentials of \(-5{mV}\) and \(-15{mV}\) were considered, and their diffusion coefficients are \(2\times {10}^{-8}\,(\frac{{m}^{2}}{s})\) and \(2.5\times {10}^{-7}\,\left(\frac{{m}^{2}}{s}\right)\), respectively76. The blood microvessels drainage rate of negatively charged nanoparticles is assumed to be the same as that of the neutral ones. The results of this sensitivity analysis can be found in Section “Injection of drug-loaded nanoparticles”.

  • The drug release rate from drug-loaded nanoparticles in the extracellular space significantly affects both the concentration of drug-loaded nanoparticles and the concentration of free drug in the extracellular space. The drug release rate also influences the penetration depth of the drug into internal layers of tumor due to the diffusion mechanism and concentration gradient3. In this study, three different release rates have been examined: \({10}^{-5}\,(\frac{1}{s})\), \({10}^{-4}\,(\frac{1}{s})\), and \({10}^{-3}\,(\frac{1}{s})\)77\(.\) A very slow release rate is commonly referred to as sustained release, while a high release rate is known as burst release. The drug release from nanoparticles can occur through diffusion or degradation mechanisms in response to specific characteristics of the tumor microenvironment, such as pH. The results of sensitivity analysis for different drug release rates can be found in the result section.

  • The effective penetration depth of doxorubicin is defined as the depth at which the concentration reaches a level of approximately 1 μg/ml, which is sufficient to eradicate 90% of cancer cells.