The blood flow rate flowing in the volume is Q i n from the arterial side, and the blood flow rate leaving the domain is Q out , respectively. Since mass is conserved, Q i n is equal to Q out as shown in Equation 1. In Figure 1, the arteries (red) and veins (blue) are considered to be the segmented vasculature for which the length and diameter can be calculated from imaging data. Q = Q i n = Q out (1)

The blood entering the tissue domain is considered to be uniformly distributed (Schreiner, 1993; Schreiner et al., 1995; Xing et al., 2022; Correa-Alfonso et al., 2022). Thus, the flow rates leaving the arterial terminal nodes 3 are considered to be equal. With the equal distribution assumption, the blood flow rates in the terminal vessels are calculated using Equation 2, where n represents the number of bifurcation generations. In the illustrated example, n is equal to 2. Q term = Q 2 n (2)

Using the dimensions of the vasculature, the equivalent flow resistance offered by each arterial and venous vascular tree can be calculated using parallel and series resistance methods. The equivalent resistance of arterial and venous trees are represented as R A and R V , respectively, and shown in Figure 2. Figure 2 represents the simplification of the domain from Figure 1 by calculating equivalent flow resistances of segmented arteries and veins.

Using the simplified blood vessel model, the blood flow equations in the arterial and venous trees can be written as Equation 3 and Equation 4, respectively. P A , t e r m represents the pressure at node 3 where the segmented arteries end, and P V , t e r m represents the pressure at node 4 where the segmented veins end. P A , t e r m = P i n − Q R A (3) P V , t e r m = P out + Q R V (4)

The total pressure drop between terminal arteries and terminal veins represents the pressure drop across unsegmented vasculature. This pressure drop can be calculated using Equation 5, where Δ P A , t e r m → t and Δ P t → V , t e r m represent the pressure drop between the terminal artery and arterial compartment and venous compartment and terminal vein, respectively. Δ P t represents the pressure drop across arterial and venous compartments within the tissue and is calculated using the perfusion coefficient, α , and total volume, V , of the tissue domain as shown in Equation 6. P A , t e r m − P V , t e r m = Δ P term = Δ P A , t e r m → t + Δ P t + Δ P t → V , t e r m (5) Δ P t = Q α V (6)

A correlation between the flow resistance of arterial and venous trees and the pressure drop between terminal vessels and tissue is proposed in Equation 7. The correlation states that the ratio of pressure drop between unsegmented arteries to capillary bed and unsegmented veins to capillary bed is equal to the ratio of overall flow resistance offered by segmented arteries and veins. The constand ψ is used for derivation only and it represents the ratio of effective arterial resistance to effective venous resistance. Δ P A , t e r m → t Δ P t → V , t e r m = R A R V = ψ (7) Δ P A , t e r m → t = ψ Δ P t → V , t e r m (8)

Using Equation 8 and substituting in Equation 5, equations for Δ P A , t e r m → t and Δ P t → V , t e r m are derived, shown in Equation 9 and Equation 10, respectively. Δ P A , t e r m → t = ψ Δ P term − Δ P t ψ + 1 (9) Δ P t → V , t e r m = Δ P term − Δ P t ψ + 1 (10)

The resistance equation for flow in virtual vessels can be written as Equations 11, 12 where the resistance offered by virtual arterial and venous networks is represented using blood viscosity μ and pressure drop parameter γ β . Δ P A , t e r m → t = Q A , t e r m μ γ a (11) Δ P t → V , t e r m = Q V , t e r m μ γ v (12)

Using Equations 9–12, the equations to determine pressure drop parameter for the arterial and venous tree is derived as shown in Equation 13 and Equation 14, respectively. γ a = μ Q 2 n R A R V + 1 R A R V Δ P − Q R A + R V + α V − 1 (13) γ v = μ Q 2 n R A R V + 1 Δ P − Q R A + R V + α V − 1 (14)

Equation 13 and Equation 14 are the final forms of equations to calculate the pressure drop parameter for arterial and venous trees. It can be seen from these equations that the pressure drop parameters can be calculated using total blood flow rate in the simulation domain, pressure drop across the simulation domain, and the equivalent resistance of arterial and venous trees calculated from the segmented vasculature. The demonstration of how to use the pressure drop equations for a given biological domain is shown in Figure 3.

To verify the applicability of the Equation 13 and Equation 14 on a 3D domain, a test domain shown in Figure 4 was generated using Rhinoceros (McNeel et al., 2010). The domain has 32 terminals for arterial and venous trees and the cuboidal tissue size encasing the vasculature was 20 c m × 20 c m × 20 c m . The voxel dimensions are 2.5 m m × 2.5 m m × 2.5 m m . The 32 terminal domain was considered as a reference solution (most segmented data available) for comparison and will be referred to as Case 1. Similar to previous work (Amare et al., 2023) on a 2D domain, the number of terminals was reduced gradually to create different simulation domains representing lack of segmented data as shown in Figure 4. The resultant domains are termed Case 2, Case 3, Case 4, and Case 5 for 16, 8, 4, and 2 terminals, respectively (Figure 4). The dimensions of the arterial and venous trees were considered same for this simulation for simplicity and are given in Table 1.

Since this was a 3D modeled domain, a reference volumetric blood flow rate had to be determined. The VoM-PhyS framework (Amare et al., 2022; Hodneland et al., 2019) was used with the flow parameters given in Table 2 to calculate the overall volumetric blood flow rate in Case 1. The pressure drop parameter for this simulation was considered as 1 m 3 . This ensured that only the flow resistance of segmented vessels in Case 1 determine the overall blood flow rate.

The parameters used in this 3D blood flow and heat transfer simulation model are based on established physiological values and commonly accepted approximations in biomedical engineering. The blood viscosity ( μ ) and thermal conductivity ( k t ) are consistent with values reported in the IT’IS database for biological tissues (Hasgall et al., 2018). The arterial and venous permeability values ( K a , K v ) are supported by previous computational models of tissue perfusion (Amare et al., 2022; Hodneland et al., 2016; Hodneland et al., 2019). The perfusion rate ( α ) falls within the typical range for various tissue types (Jeong et al., 2023). The specific heat ( c p t ) and density ( ρ t ) of the tissue are approximated to those of water, a common practice in biological heat transfer models (Duck, 2013). The ambient temperature ( T ∞ ) represents a standard room temperature, while the inlet blood temperature ( T i n ) is slightly below core body temperature, accounting for cooler peripheral blood. The metabolic heat generation rate is within the range observed in various tissues, albeit on the higher end (Wahyudi et al., 2022). The convective heat transfer coefficients for ambient and blood ( h amb , h b ) are typical values used in biological heat transfer simulations. While the inlet and outlet pressures are lower than physiological arterial pressures, they are representative for the specific modeling conditions and simulate a pressure drop of arround 1000 P a (Timothy, 2016; Li et al., 2012).

Using Equations 13, 14, the pressure drop parameters for all five cases were calculated. The blood flow was simulated with the respective pressure drop parameters and the pressure maps were compared with the reference Case 1.

The pressure drop parameters calculated using Equation 13 and Equation 14 for the 3D reference domain (Figure 4) are shown in Table 3. Nt represents the number of vascular terminals for respective cases. To understand the importance of pressure drop parameter and its effect on pressure solution, Figure 5 is shown. Here the percentage error in pressure at arterial nodes 6 , 5 , and 4 of Case 2 is plotted for ± 1%, ± 2%, and ± 5% change in γ a . Nodes 5 represent the last segemnted node in Case 2, Nodes 4 represent the nodes connected to Nodes 5, and Nodes 3 represent the pressure nodes connected to Nodes 4. There are many biological factors like non-newtonian behavior of blood and vasomotion which are not considered in Equations 13, 14. Thus, these equations can only provide an accurate flow resistance behavior of unsegmented vessels. A 0.6% change in pressure for 5% in pressudre drop parameter can be seen from Figure 5. This shows that an accurate pressure drop parameter will be sufficient for preliminary analysis and simulation of blood flow.

The flow equations were solved using the pressure drop parameters of respective cases and pramaters shown in Table 2. The resultant pressure values are given in Tables 4–6, for the arterial tree, venous tree, and tissue compartments, respectively. In Table 6, the average, maximum, and minimum pressures in the arterial and venous compartment of all the cases is shown. A contour map for pressure error between Cases 2 to 5 and Case 1, is shown in Figure 6, at z = 80 the location of z in the domain shown in Figure 7. The pressure difference between Case 5 and Case 1 at each tissue voxel is within ± 0.75 P a . Case 5 represents the worst case possible with the least segmented vasculature available.

To analyse the effect of lack of segmentation data on bioheat transfer, Case 1 temperature profile was considered as reference solution, and all other Cases were compared to it. The temperature errors of Case 2, Case 3, Case 4, and Case 5 at z = 80 are shown in Figure 8. The location of z = 80 was selected as it had the maximum temperature error in the entire domain. The dimensionless temperature error was calculated using Equation 15. T amb represents the ambient temperature used for simulation, T i n represents the inlet blood temperature, T ref,i represents the temperature of ith voxel in Case 1, and T c , i represents the temperature of ith voxel in Case c where c ∈ ( 2,3,4,5 ) . θ i = T c , i − T ref,i T i n − T amb (15)

Between two tissue voxels, heat transfer takes place via advection and conduction. As the blood flows from one porous tissue voxel to another, it carries the heat with it resulting in advection. As the pressure map was consistent with maximum error within ± 0.75 P a , there was negligible change in the flow rate among tissue voxels. However, as fewer blood vessels are segmented, they are modeled as part of tissue porosity. The porous tissue voxels now consist of larger blood vessel than the reference domain (Case 1). Though there was no change in heat transfer due to advection, thermal conduction among tissue voxels can change due to larger unsegmented vessels. In literature (Chen and Holmes, 1980; Keller and Seiler, 1971; Weinbaum et al., 1984; Weinbaum and Jiji, 1985; Roetzel and Xuan, 1997; Nakayama and Kuwahara, 2008), effective thermal conductivity is used to consider the effect of unsegmented blood vessels and counter-current heat exchange. The same concept of effective thermal conductivity was used to compensate for the effect of cross-flow heat exchange between unsegmented blood vessels. The thermal conductivity of the tissue voxels was varied between 0.5 W m − 1 ◦ C − 1 ◦ to 2.0 W m − 1 ◦ C − 1 ◦ . The resultant temperature errors between Case 5 and Case 1 for different values of tissue thermal conductivity are shown in Table 7 with the temperature contours plots at z = 80 shown in Figure 9. From Figure 9 and Table 7 it can be seen that as the tissue thermal conductivity increases, the temperature error begins to reduce till it reaches a threshold value, beyond which the temperature error seems to increase.

Sphere of Influence (SoI) is a parameter introduced in previous work (Hodneland et al., 2016; Hodneland et al., 2019) and is critical in the VoM-PhyS framework (Amare et al., 2022) for coupling 1D flow with 3D flow. The SoI is a volume of sphere with origin center at a terminal vessel. The tissue voxels that fall within a given SoI are considered to exchange blood with the respective terminal vessel. The radius, ϵ , of the SoI is an empirical quantity. The effect of ϵ on blood flow and pressure drop was studied (Amare et al., 2023) and ϵ did not demonstrate any effect on pressure drop. However, ϵ has been shown to affect thermal maps in tissue (Amare et al., 2022). Thus, different values of ϵ were studied to understand its effect on reduction of temperature error due to unsegmented vessels. The temperature error values for different ϵ and different values of tissue thermal conductivity, k t , are given in Tables 8, 9. The temperature difference contours at z = 80 for Case 5 with k t as 0.5 W m − 1 ◦ C − 1 is shown for different values of ϵ . From Table 8 and Figure 10, it can be seen that as the ϵ increases, the temperature error begins to decrease and reaches a threshold value beyond which there is no change in the temperature error. This behavior is different than the effect of k t , where beyond the threshold, the temperature error begins to increase again.

To understand the effect of these two methods on temperature error reduction, root mean square error (RMSE) and the sum of temperature error (STE) in the entire domain were calculated. The values of RMSE and STE are shown in Tables 8, 9, and a graphical plot of RMSE and ∑ Δ T t for the different values of k t and ϵ is shown in Figures 11, 12, respectively. In Figure 11, it can be seen that as k t increases, the RMSE decreases for ϵ to a threshold value. Beyond the threshold, the RMSE increases with an increase in k t . Using polynomial regression, six polynomial functions of the fourth order were fitted for RMSE for each value of k t , respectively. Similarly, six polynomial functions of fourth order were fitted for STE for each value of k t . Using the RMSE function, the threshold where minimum RMSE occurs was calculated, along with the corresponding STE. These values are given in Table 10. It is noteworthy that the STE is positive for all k t at the threshold ϵ of minimum RMSE. This denotes that the VoM-PhyS framework would result in higher temperatures than the reference for a less segmented vascular domain. Similarly, the ϵ where STE equals zero was calculated along with the corresponding RMSE. These values are given in Table 11. The values of ϵ where STE is zero are not the same where minimum RMSE occurs, as can be seen. A further detailed statistical analysis of these values could lead to greater insight into the performance of the proposed energy error reduction methods. A further analysis is needed to determine what parameters affect the value of ϵ for any given vascular data. Another noteworthy observation is that the minimum temperature for larger SoI is for k t = 0.5 W m − 1 ◦ C − 1 , and this error increases for the same SoI when tissue thermal conductivity is varied. A SoI larger than the minimum required for 100% coverage ensures the tissue domain lies closer to the source than the periphery of the SoI. This provides more blood flow to the entire tissue domain than the simulation case when SoI is restricted to minimum ϵ for 100% coverage. As the SoI is increased, it is expected to achieve equal distribution in the entire tissue domain, and is expected when the SoI is considerably larger than the domain dimensions. This behavior ensures that we reach a plateau beyond which the temperature error cannot be decreased even if the SoI is increased.

The use of the equations to calculate correct pressure drop parameter for a given vasculature opens avenues for furthering this research field. Availability of segmentable vascular model remains a challenge, but the use of the pressure drop parameter equation provides a novel way to overcome this challenge and obtain a detailed accurate blood flow simulation. This study clearly shows the effect of unsegmented vasculature is dominant on heat transfer simulation. Further research is required to better understand methods to quantify this error and be able to provide rectification methods from available data. Similar to the equations proposed in this paper for calculating pressure drop parameter, if equations to rectify the temperature error could be derived, that would provide a major revolution in the field. Such equations could help simulate large scale domains which otherwise could not be easily modeled due to their computational memory requirement.

These findings also highlight that correcting flow resistance alone is insufficient for accurate heat transfer modeling. This insight is critical for researchers developing bioheat transfer models, as it underscores the need for additional considerations beyond flow resistance.

One of the major limitation of this work is tied to the SoI. The SoI is an empirical value and no data is available to determine its accuracy. The SoI radius is expected to vary based on the biological domain under consideration, tissue properties, vasomotion, and various other biological features. A detail study is required to better understand this parameter.

To derive the equations of pressure drop parameter, a simulation domain where segmented vasculature could be gradually removed was required. Obtaining such a simulation domain from medical imaging data still remains a challenge as discussed in the Introduction. The resolution of image required to get the desired segmentable vasculature is very high, increasing the computational overhead for simulation. The pressure drop parameter itself is the solution to model flow resistance of unsegmented blood vessels and hence an artificial 3D domain with blood vessels was developed for derivation of mathematical equations. Future work will include demonstrating the use of these equations on biological organs obtained from medical imaging scans.

While our proposed methods significantly improve bioheat transfer simulations with limited vascular data, several limitations should be noted. The pressure drop parameter equations assume steady-state flow and do not account for pulsatile effects or non-Newtonian blood behavior. The effectiveness of the error reduction methods may vary depending on the specific organ or tissue being modeled. Additionally, validation against in vivo measurements remains challenging due to the complexity of obtaining high-resolution temperature data in living tissues.

The broader implications of this work extend to various fields. In medical applications, more accurate bioheat transfer models could improve the planning and execution of thermal therapies, such as hyperthermia treatments for cancer. Our current research focuses on this and will be published in future work. In physiological research, these methods could enhance our understanding of thermoregulation in different organs. For computational biology, our approach offers a pathway to simulate large-scale domains that were previously computationally prohibitive, potentially enabling more comprehensive whole-organ or even full-body simulations.

These findings provide novel equations to calculate the pressure drop parameter ( γ ) for unsegmented vasculature using only the net flow rate and flow resistance of segmented vessels. This advancement allows for accurate pressure mapping in simulations with limited vascular network data, addressing a significant challenge in the field. The study provides a detailed analysis of how the lack of segmented vascular data affects temperature profiles in bioheat transfer simulations and proposes two approaches to mitigate these errors: implementing an effective thermal conductivity approach and assuming equal spatial distribution from terminal vessels in the tissue domain. These methods show significant improvements in simulation accuracy, with the effective thermal conductivity approach reducing maximum absolute temperature errors from 2.74 ◦ C to 1.45 ◦ C in the test case.

To contextualize our temperature error results, we compared them with literature benchmarks. Weinbaum and Jiji (Weinbaum and Jiji, 1985) reported temperature variations of up to 2 - 3 ◦ C due to vascular effects in their bioheat transfer model. Our maximum absolute temperature error of 2.74 ◦ C in the worst-case scenario (Case 5) is consistent with this range. However, our proposed error reduction methods significantly improve upon this, with the effective thermal conductivity approach reducing the maximum error to 1.45 ◦ C . This improvement is particularly significant in the context of hyperthermia treatments, where temperature accuracy within 1- 2 ◦ C is crucial for treatment efficacy and safety (Kok et al., 2015).

In summary, this research contributes to the ongoing effort to develop more accurate and computationally efficient bioheat transfer models. It bridges the gap between high-resolution vascular modeling and practical limitations in medical imaging, offering a more accessible yet accurate method for simulating physiological processes. The novel approaches presented in this study have the potential to significantly impact various fields, from medical research to thermal regulation studies, paving the way for more sophisticated and realistic modeling of heat transfer in living tissues.